Microscopic Spatio-Temporal Dynamics of
Semiconductor Quantum Well Lasers and Amplifiers
Von der Fakulta¨t Mathematik und Physik der Universita¨t Stuttgart
zur Erlangung der Wu¨rde eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigte Abhandlung
Vorgelegt von
Klaus Bo¨hringer
aus Waiblingen
Hauptberichter: Prof. Dr. O. Hess
Mitberichter: Prof. Dr. G. Mahler
Tag der mu¨ndlichen Pru¨fung: 4. April 2007
Institut fu¨r Technische Physik
Deutsches Zentrum fu¨r Luft- und Raumfahrt (DLR)
Pfaffenwaldring 38–40
D-70569 Stuttgart
2007
Summary
This work discusses light-matter interaction and optical nonlinearities in semiconductor
nanostructures and presents a detailed numerical analysis of the spatio-temporal dynam-
ics in novel high-power diode lasers. We derive a microscopic, spatially resolved model
that combines a density matrix approach to the optoelectronic properties of quantum
well gain media with the macroscopic Maxwell equations for the electromagnetic field
dynamics. We present Maxwell semiconductor Bloch equations in full time-domain that
cover many-particle interactions, a diversity of time scales and gain saturation mech-
anisms, and inclose the fast-oscillating carrier wave and a sub-wavelength spatial res-
olution. Microscopically calculated scattering rates are incorporated into our spatially
resolved model.
Our work focuses on ultrafast carrier effects, a quantitative understanding of optical
nonlinearities, the engineering of the mode structure in microcavities, and their im-
pact on the laser emission characteristics. Optical dephasing and carrier and energy
redistribution due to the screened Coulomb interaction and scattering with phonons are
explored in detail.
We study the technologically important structure of a broad area edge-emitting laser
within the framework of the paraxial wave approximation. The excitation of multiple
transverse modes and the occurrence of unstable optical filaments are quantitatively
analysed. We show how transverse instabilities originate from spatial hole burning,
gain- and index-guiding and from self-focussing. We investigate the dependence of emis-
sion dynamics on characteristics of the gain material (e.g. the amplitude-phase coupling
factor), the stripe width, pumping and carrier diffusion. Depending on the width of
the laser, several dynamic emission regimes can be distinguished. We also project the
spatio-temporal dynamics onto the laser modes.
We analyse VCSEL devices with a periodically structured defect as an example of a
photonic band edge band gap laser. In particular, we explore the utilisation of photonic
crystal structures: gain enhancement for band edge modes due to the more efficient
interaction of photons with the gain medium and increased localisation over the active
layers, and the reduction of optical losses. We numerically confirm that photonic crystal
effects can be obtained for finite crystal structures, and demonstrate that they lead to a
significant improvement in laser performance, e.g. reduced lasing thresholds.
Optically pumped VECSEL are a device concept designed to increase the power out-
put of surface-emitters in combination with near-diffraction-limited beam quality. We
explore the complex interplay between the intracavity optical fields and the quantum
well gain material in VECSEL structures. Our simulations reveal the dynamical balance
between carrier generation due to pumping into high energy states, momentum relax-
ation of carriers, and stimulated recombination from states near the band edge. We
show that the longitudinal multi-mode behaviour is composed of several external cavity
modes.
We also consider the interaction of high-intensity femtosecond and picosecond pulses
with semiconductor structures. We identify the microscopic origin of the fast nonlin-
earities, and consider the physical effects behind the various saturation mechanisms.
We also obtain the nonlinear gain coefficients and recovery rates. It is demonstrated
that group velocity dispersion, dynamical gain saturation and fast self-phase modula-
tion are the main causes for changes and asymmetries in the amplified pulse shape and
spectrum. We show that the time constants of the intraband scattering processes are
critical to gain recovery. Our results are essential for the interpretation and the quan-
titative understanding of nonlinear pulse reshaping in semiconductor optical amplifiers
and absorbers.
The accurate and spectrally broad modelling of semiconductor gain and complex struc-
tured laser cavities that is presented in this work extends the scientific discussion of
semiconductor laser systems. Built upon efficient numerical algorithms and the increased
availability of inexpensive high-performance computing resources, our microscopic time-
domain approach is also well suitable for the engineering and design optimisation of
modern nanostructured high-power diode lasers.
CONTENTS v
Contents
Summary iii
Contents v
1 Introduction, Overview and Outlook 1
2 Microscopic Description of the Gain Dynamics 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Parabolic Band Structure Approximation and Confinement Functions . . 13
2.3 Density Matrix Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Coherent Coupling to the Optical Field . . . . . . . . . . . . . . . . . . . 20
2.5 Coupling to a Full Time-Domain Scheme . . . . . . . . . . . . . . . . . . 23
2.6 Phenomenological Terms and Additional Many-Body Hamiltonians . . . 26
2.7 Many-Body Interactions—Hartree-Fock Terms . . . . . . . . . . . . . . . 30
2.8 Many-Body Interactions—Correlation Terms . . . . . . . . . . . . . . . . 32
2.9 Quantum Dot Lasers and Multi-Level Bloch Equations . . . . . . . . . . 40
2.10 Fitting the Dielectric Function . . . . . . . . . . . . . . . . . . . . . . . . 42
2.11 Spectral Summation and Coupling to the Maxwell Equations . . . . . . . 44
2.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Light Field Dynamics in Laser Cavities 55
3.1 Introduction—Macroscopic Maxwell Equations . . . . . . . . . . . . . . . 55
3.2 Paraxial Approximation—Transverse Wave Equation . . . . . . . . . . . 58
3.3 Full Time-Domain Maxwell Equations . . . . . . . . . . . . . . . . . . . 64
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Semiconductor Laser Fundamentals 69
4.1 Calculation of Scattering Rates . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.1 Microscopic Scattering Rates in Semiconductor Quantum Wells . 69
4.1.2 Extension to the Multi-Subband Case . . . . . . . . . . . . . . . . 74
4.2 Calculations of Quantum Well Laser Gain . . . . . . . . . . . . . . . . . 78
4.3 Relaxation Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Quantisation of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.5 Thermodynamics of Semiconductor Lasers . . . . . . . . . . . . . . . . . 97
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
vi CONTENTS
5 Transverse Multi-Mode Laser Dynamics 101
5.1 Introduction—Effects of Spatial Degrees of Freedom . . . . . . . . . . . . 101
5.2 Transverse Instabilities in Broad Area Lasers . . . . . . . . . . . . . . . . 103
5.3 Simulations—Different Dynamic Emission Regimes . . . . . . . . . . . . 105
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Longitudinal Multi-Mode Laser Dynamics 117
6.1 VCSEL with Resonant Periodic Gain and Refractive Index Structures . . 117
6.2 Optically Pumped VECSEL . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.3 Small Signal Gain Calculations . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 Nonlinear Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.5 Ultrafast Gain Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.6 Chirped Pulse Amplification . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.7 Ultrashort Coherent Optical Pulse Interactions . . . . . . . . . . . . . . . 150
6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
A Discretisation Schemes and Numerical Implementations 159
A.1 Discretisation Scheme of the Transverse Wave Equation Model . . . . . . 160
A.2 Discretisation Scheme of the Longitudinal Full Time-Domain Model . . . 162
A.3 Numerical Analysis of the Scattering Integrals . . . . . . . . . . . . . . . 167
A.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
B Dynamical Treatment of the Scattering Contributions 173
C Zusammenfassung 177
C.1 Kurzzusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
C.2 Einleitung und Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 178
C.3 U¨berblick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
C.4 Ausblick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Bibliography 189
1 Introduction, Overview and Outlook
Introduction and Motivation
The concept of light amplification by the process of stimulated emission of radiation
(i.e. LASER operation [1]) in semiconductor gain materials was first demonstrated in the
year 1962. Following a theoretical suggestion by Basov, three laboratories independently
reported lasing in direct band gap compound semiconductor structures, with pulsed op-
eration at cryogenic temperatures [2–4]. Stimulated emission of radiation was realised in
these semiconductor diode lasers by the radiative recombination of electron-hole pairs in-
jected across a pn-junction. The population inversion necessary for lasing was created in
the depletion region of a GaAs(P) homojunction. The introduction of heterostructures in
1970 was a pioneering contribution to the development of efficient optoelectronic devices
and a breakthrough towards industrial applications. Heterostructures are composed of
multiple layers of compound semiconductors with different compositions. In particular,
double heterostructures consisting of an intrinsic active GaAs layer, a thin film for light
generation and amplification, sandwiched between two doped AlGaAs cladding layers
enabled continuous wave operation at room temperature [3]. The improved performance
characteristics (e.g. high efficiency, low heat dissipation) and reduced lasing thresholds
in these structures were achieved by enhanced carrier and optical confinement [2–4].
Sophisticated crystal growth techniques (e.g. molecular beam epitaxy, metal-organic
chemical vapor deposition, liquid-phase epitaxy, self-organised growth methods) [5], and
processing and etching technologies, allow the manufacture of high quality, complex semi-
conductor nanostructures (such as graded index layers, superlattices, quantum wells and
quantum dots, or photonic crystals). These technologies permit better control over the
electronic properties of the gain material by employing the concepts of size quantisa-
tion and reduced dimensionality, and engineering of the optical mode structure and the
density of states in functional photonic materials. At the same time, theory and simula-
tion of semiconductor laser dynamics and structures has developed into a topic of more
than only an academic interest. Due to its predictive character, computational mod-
elling constitutes an invaluable tool for the engineering of novel active structures and
improved device designs. Driven by the increasing demands for higher power output, dif-
ferent operating wavelengths, and high-speed modulation performance, the development
of novel laser and amplifier structures and the introduction of new gain material systems
and lasing concepts have been pushed forward. This is accompanied with the succes-
sive miniaturisation of coherent light sources and amplifiers (with the aim of on-chip
realisation). This intense research in the world of semiconductor lasers stimulated the
transition from a research structure to a mass product with applications in our everyday
2 INTRODUCTION, OVERVIEW AND OUTLOOK
life. Semiconductor diode lasers feature some key advantages compared to other coher-
ent light sources: compact size, reliability, very high conversion efficiency of pump energy
into coherent light, tunability and tailoring of the optoelectronic properties, direct signal
modulation up to tens of GHz, and inexpensive production costs. This has created a
huge market for diode lasers, with worldwide sales revenue for semiconductor lasers es-
timated at $ 3.2 billion in 2005. The main industrial applications of diode lasers include
optical storage (CD- and DVD-based technologies), optical fibre communications, high-
power applications (such as solid-state laser pumping) and medical therapeutics. Besides
their technological relevance, the complexity of physical interactions in diode lasers, the
nonlinear coupling of various subsystems, and the diversity of time and length scales also
make semiconductor lasers ideal laboratories for investigating nonlinear [6,7] and quan-
tum optical phenomena [8], as well as ultrafast interaction processes in semiconductors,
and concepts of nonlinear dynamics and synergetics [9, 10].
New and exciting developments are aimed to control and manipulate light-matter
interaction, light emission and propagation, and to engineer the optoelectronic properties
of semiconductor gain media. In the following we give an overview on recent research
activities that are related to this work.
Laser and amplifier structures: Vertical cavity surface-emitting lasers (VCSEL) [11,
12] are designed as highly reflective Bragg mirrors with an enclosed defect cavity and
additional transverse confinement (using oxide apertures). In particular, VCSEL show
strong coupling between active gain material and the light field. Microcavities [13, 14]
promise high quality factors, small mode volumes and low lasing thresholds. They are
fundamental tools to study and tailor the emission characteristics of a light source,
and allow the modification of light-matter interactions, e.g. spontaneous emission, as
the local optical density of states is defined by the dielectric structure. Semiconductor
optical amplifiers (SOA) are attractive as repeaters and functional devices in all-optical
networks. Resonant periodic gain structures [11,15] offer an enhancement of the effective
modal gain. Vertical external cavity surface-emitters (VECSEL) [16, 17] are a design
scheme for increasing laser output powers with high quality output beams, and give
easy access to nonlinear optical effects, such as dynamical gain saturation and pulse
shaping or frequency conversion. Photonic crystals and photonic band gap materials
facilitate light propagation with very low group velocities [18,19]. They can function as
frequency-selective mirrors or optical microcavities, and offer a guiding mechanism based
on multiple Bragg interference and not on total internal reflection [20]. This guiding in
the low index core (air) is used in photonic crystal fibres.
Semiconductor gain material systems: New semiconductor compound materials (bi-
nary, ternary and quaternary alloys) can be designed for a wide range of optical emission
wavelengths spanning from the infrared to the ultraviolet. Most recent research focuses
on II-VI compounds, wide band gap group-III nitride-based semiconductors [21–23],
and dilute nitrides, e.g. GaInNAs [24], for advanced telecommunications device applica-
tions. Quantum confined active nanostructures [25] with reduced dimensionality, such
as quantum wells [26], nanowires or quantum dashes [27, 28], and quantum dots [29],
fundamentally change the electronic band structure and optoelectronic properties by
3applying the concept of size quantisation, resulting in more efficient carrier inversions.
The application of tensile or compressive strain allows further band and gain engineer-
ing. In addition, organic semiconductor lasers [30] may become viable candidates for
visible solid-state semiconductor lasers and provide green light in display and illumi-
nation technologies. The scope for simple fabrication and shaping of these conjugated
polymer films is attractive [31]. Most lasers of this type utilise a corrugated structure
which acts as a resonator based on the concept of distributed feedback.
Laser and gain concepts: Optical pumping schemes provide an alternative to electrical
pumping, and offer the possibility of defining the spatial distribution of the population
inversion over large areas. The generation of ultrashort optical pulses by passive mode-
locking using saturable absorber elements [32,33] is based upon the systematic utilisation
of nonlinear effects and the complex dynamic interplay between gain and light field. The
proposed gain material concept and structure of quantum cascade lasers depends on in-
tersubband transitions (i.e. intraband polarisations). This intersubband nature (involv-
ing only a single type of carriers) gives rise to several key advantages compared to devices
based on stimulated electron-hole recombinations [34]. For example, tunability on basis
of the size quantisation concept in multiple quantum well heterostructures (i.e. no re-
striction given by the energy band gaps), and higher efficiency. Achieving optical gain
for recombination processes in indirect band gap semiconductor materials is challenging,
which motivates the investigation of other physical interactions and gain mechanisms.
Recently, light amplification and lasing in silicon has been demonstrated [35], where
amplification is achieved by stimulated Raman scattering.
The numerical simulation of laser dynamics plays an important role in the develop-
ment of novel structures, concepts, and improved designs of active semiconductor laser
devices. It is also valuable in analysing the underlying physical limitations and optical
and electronic properties of the various subsystems. Because of the complex nature of
the problem and the nonlinear character of the governing coupled equations, an analyt-
ical treatment is difficult. With only little restriction on the geometrical and physical
setup, time-domain methods offer a flexible and expandable tool appropriate to tackle
the coupled dynamics of intracavity optical fields and active gain material. The only
restriction in the implementation of such methods is set by the required computing re-
sources. The basis for semiconductor laser models are the fundamental laser equations,
derived by Haken and Lamb, Jr. [36]: The basic set of dynamical variables is given by
the optical field, the induced polarisation of the active material, and the carrier inver-
sion. An overview of commonly applied time-domain models to simulate semiconductor
lasers within the semiclassical framework (i.e. the combination of classical electromag-
netic fields with the quantum electronic properties of the active material) is presented
in [3, 37]. We summarise the various approaches to modelling semiconductor gain dy-
namics in Figure 1.1, and the dynamics of the optical fields in Figure 1.2. Therein,
we also define the scope and merits of our novel models compared to other methods.
In distinction to other laser structures, semiconductor diode lasers are characterised by
some unique features that have to be reflected in realistic computer models [2,4,37]: Due
to the high carrier density, many-body interactions are important, particularly in gain
4 INTRODUCTION, OVERVIEW AND OUTLOOK
q
ua
ntum
 kinetic
 theo
ry
ba
nd
-resolved
 se
m
ico
nd
ucto
r
 Bloch
 eq
uatio
ns
 (SBE)
 with
H
a
rtree
-F
ock
a
nd
 co
rrelatio
n
 co
ntrib
utio
ns
 fro
m
 m
a
ny
-body
inte
ractio
ns
 
(m
ulti
-subba
nd)
 SBE
 in
 rela
xatio
n
 rate
 app
ro
xim
atio
n
 (
 
 
 
 
 
 
 
 )
plus
 additio
nal
 phe
no
m
e
nological
 te
rm
s;
 three
 fo
rm
ulatio
ns:
a
.
 
 
 
 
-pola
risatio
n
 as
 d
e
rived
 fro
m
 de
nsity
 m
atrix
 theo
ry
,
real
-valued
 pola
risatio
ns:
 
 b
.
 (2
.37)+(2
.38)
 
 c
.
 (2
.41)
-(2
.44)
 
ba
nd
-resolved
 m
odels
effective
 SBE
:
 fitting
 the
 dielectric
 fu
nctio
n
 (susceptibility)
,
the
 (sm
all
 sig
nal)
 g
ain
 a
nd
 ind
uced
 refractive
 inde
x
 spectra
m
ulti
-le
vel
 Bloch
 eq
uatio
ns
(Lo
re
ntzia
n
lineshap
e)
 plus
D
ebye
 a
nd
 D
rud
e
DE
m
acroscopic
 SBE
,
adiabatic
 elim
inatio
n
 of
 P
 
effective
 SBE
 m
odels
tw
o
-le
vel
 Bloch
 eq
uatio
ns
phe
no
m
e
nological
 g
ain
descriptio
n
ato
m
ic
-like
 m
ate
rial
 m
od
els
T
aylo
r
 e
xp
a
nsio
n
 of
 the
 W
ig
ne
r
 rep
rese
ntatio
n
 of
 m
icroscopic
va
riables,
 sep
a
ratio
n
 of
 scale
s:
 p
a
ra
m
etric sp
ace
 dep
e
nde
nce
sim
plified
 treatm
e
nt
 of
 co
rrelatio
n
 co
ntrib
utio
n
s/scatte
ring
no
 sp
ectral
 hole
 b
u
rning
,
 n
o
 n
o
nlin
ea
r
 g
ain
 satu
ratio
n
,
 loss
 of
physical
 m
otivatio
n
 of
 co
upling
 b
etw
ee
n
 the
 diffe
re
nt
 state
s
(ho
m
og
e
neo
u
s
a
nd
 inho
m
og
e
neo
us b
roade
ning);
b
ut:
m
uch
 red
uced
 n
u
m
e
rical
 co
m
ple
xity
 a
nd
 co
m
p
utatio
n
al
req
uire
m
e
nts
sep
a
ratio
n
 of
 (decay)
 tim
e
 scales
u
n
realistic co
upling
 of
 a
m
plitude
 a
nd
 ph
ase
 dyna
m
ics
u
n
realistic m
odelling
 of
 g
ain
 sp
ectra
 a
nd
 d
ephasing
m
acroscopic
 SBE
 with
dyna
m
ical
 p
ola
risatio
n
full
 tim
e
-d
o
m
ain
 
 
 
 freq
u
e
n
cy
-/tim
e
-d
o
m
ain
Figure 1.1: Overview on commonly applied approaches for describing the semicon-
ductor quantum well gain dynamics as the source term of an optical wave equation.
Symbols and applied approximations are discussed in Chapter 2. Our simulations em-
ploy the band-resolved semiconductor Bloch equations in relaxation rate approximation
and effective SBE models (as highlighted in the drawing).
5PDE
O
DE
full
-vecto
rial(m
acroscopic)
 M
a
xw
ell
 curl
 eq
uatio
ns
q
u
a
ntu
m
 m
e
ch
a
nical
 fo
rm
ulatio
n
:
 op
e
rato
r
 dyna
m
ics
,
 Q
ED
se
m
i
-vecto
rialo
r
 scala
r
 w
a
ve
 eq
uatio
n
,
 tw
o
-dim
.
 o
r
o
ne
-dim
e
nsio
nal
 M
a
xw
ell
 eq
uatio
ns
pa
ra
xial
 w
a
ve
 eq
uatio
n
tra
velling
 w
a
ve
 eq
uatio
n
m
ea
n
-field
 app
ro
xim
atio
n
rate
 eq
uatio
n
 fo
r
 electric
 field
rate
 eq
uatio
n
 fo
r
 photo
n
 de
nsity
full
 tim
e
-do
m
ain
freq
ue
ncy
-/tim
e
-do
m
ain
H
elm
holtz
 eq
uatio
n
loss of
 phase
 info
rm
atio
n/dyna
m
ics of
 optical
 fields
no
 sp
atial
 resolutio
n
,
 sp
atially a
ve
rag
ed
 va
riables
no
 tra
nsve
rse
 m
ode
 dyn
a
m
ics,
 pla
ne
 w
a
ve
 app
ro
xim
atio
n
no
 resolutio
n
 of
 lo
ngitudin
al
 reso
nato
r
 structu
re
p
a
rtial
 tra
nsfo
rm
atio
ns
 to
 F
o
u
rie
r
 (freq
ue
ncy
 a
nd
 m
o
m
e
ntu
m)
sp
ace
,
 m
o
noch
ro
m
aticity(single
 lo
ngitudinal
 m
ode)
,
 RW
A
,
SVAA
,
 sep
a
ratio
n
 of
 scales
,
 effective
 inde
x app
ro
xim
atio
n
tra
nsfo
rm
atio
n
 to
 freq
ue
ncy
 sp
ace
no
 ab
initio
 d
escriptio
n
 of
 q
ua
ntu
m
 noise
,
 sp
o
nta
ne
o
us e
m
issio
n
infinite
 w
a
veg
uide
,
 do
m
ina
nt
 w
a
ve
vecto
r
co
m
p
o
ne
nt
(tra
nsve
rsality)
,
 p
a
rticula
r
 p
ola
risatio
n
 a
ssu
m
ed
classical
 fo
rm
ulatio
n
Figure 1.2: Overview on mathematical descriptions of the optical field dynamics in
diode laser models. The laser fields are coupled to the semiconductor material response
by the induced polarisation. Symbols and assumptions are described in Chapter
3. The optical models implemented in this work, namely the scalar one-dimensional
Maxwell curl equations and the paraxial transverse wave equation, are highlighted.
6 INTRODUCTION, OVERVIEW AND OUTLOOK
structures with quantum confinement, and there is a strong dephasing of the induced
polarisation. Semiconductor gain materials are characterised by broad gain spectra, and
a strong coupling of the amplitude and phase dynamics (qualified by a high α factor).
Diode lasers include a multitude of relevant time scales ranging from a few femtoseconds
(for intraband Coulomb scattering) to several nanoseconds (for macroscopic transport
processes such as carrier diffusion). Furthermore, strong spatial and spectral hole burn-
ing, nonlinearities and saturation effects are all important. Modern semiconductor diode
lasers are composed of complex structured cavities. The lasing modes may strongly dif-
fer from cold-cavity modes because the lasing action and carrier dynamics (e.g. hole
burning and thermal effects) change the refractive index structure. The output per-
formance and characteristics of a diode laser are limited by the nonlinear properties of
the gain material, important aspects comprise the quantum well carrier dynamics (e.g.
spectral hole burning and scattering processes), spatial effects (e.g. spatial hole burning,
self-focussing) and thermally induced changes of the gain medium. In conclusion, the
theory of semiconductor lasers has to be continuously revised and successively adapted
to the novel structures, concepts, and gain materials, which have been described in the
previous sections.
The main objective of this work is developing a theoretical description of the spatio-
temporal dynamics in novel high-power semiconductor lasing structures. We also aim to
obtain a quantitative understanding of the nonlinear interplay between the semiconduc-
tor gain dynamics and the intracavity light field dynamics. To this end, we revise and
extend existing theoretical approaches to semiconductor laser dynamics to account for
the challenging requirements imposed by the improved gain and cavity designs discussed
above. Special attention is paid with regard to quantum confined gain structures, com-
plex dielectric microcavities, and photonic structures. Taking into account the above
specified requirements for a realistic model of semiconductor laser dynamics, our time-
domain approach within the semiclassical framework combines the spatially resolved
Maxwell equations, or an approximated paraxial wave equation, with an energy-resolved
description of the material response, as given by the semiconductor Bloch equations. Two
classes of models are developed: 1) a transverse model for the analysis of transverse multi-
mode dynamics in broad area lasers, and 2) a longitudinal model to target ultrashort
pulse propagation in semiconductor optical amplifiers. We also employ the latter model
to study complex structured active laser devices, e.g. vertical cavity surface-emitting
lasers with a periodically structured defect, and optically pumped vertical external cav-
ity surface-emitters. To solve the model equations for the optical field dynamics (partial
differential equations) simultaneously with the polarisation and carrier dynamics of the
active semiconductor structures, we convert them to finite-difference equations which
are numerically integrated on a uniform grid.
Overview of Thesis
This work aims to improve the quantitative understanding of lasing systems with tech-
nological or fundamental relevance. On basis of a full microscopic gain model with
7many-body interactions we focus on the investigation of femtosecond and picosecond
phenomena and analyse spatial optical near-field patterns. We analyse the microscopic
origin of optical nonlinearities and the impact of the carrier dynamics on the output
characteristics of diode lasers. We concentrate on high-power structures such as broad
area lasers and semiconductor optical amplifiers. We numerically validate novel concepts
to increase the output powers of surface-emitting lasers, e.g. by the engineering of the
optical mode structure. More specifically, we investigate photonic band edge band gap
lasers and optically pumped VECSEL. The thesis is organised in the following way:
In Chapter 2 we derive the multi-subband semiconductor Bloch equations, which
consist of a quantum mechanical description of the ultrafast gain dynamics in quantum
wells based on the density matrix formalism. Our approach, formulated in momentum
space, covers the relevant interactions in semiconductor-based optically active gain ma-
terials, and a diversity of time scales. Light interaction with semiconductor gain media
is modelled, within the semiclassical framework, using the electric dipole approximation.
Many-body interactions, namely the screened Coulomb interaction and the scattering
with phonons, effect the renormalisation of transition energies, the Coulomb enhance-
ment of the Rabi frequency, and relaxation and dephasing processes. We perform full
microscopic many-body calculations. The equations are also derived in full time-domain.
To the best of our knowledge, this formulation of the full time-domain semiconductor
Bloch equations has not been reported before. Our approach does not imply the usual
rotating wave approximation, and thus it represents an accurate and spectrally broad
modelling of the gain medium. Our model allows the simulation of ultrafast nonlinear
pulse interactions in semiconductor lasers and amplifiers. The response of the amplifier
medium is qualified by the induced macroscopic polarisation P(r, t), the carrier density,
and the material dispersion nbackground(r;ω). In addition, we propose the fitting and
parametrisation of the complex dielectric susceptibility using only a few oscillators as
an effective way to reproduce the optoelectronic properties of a complex semiconductor
gain medium.
In Chapter 3 we consider the description of the light field dynamics in laser cav-
ities. The propagation, diffraction, reflection and guiding of electromagnetic fields in
refractive index structures and optical cavities of laser resonators is modelled using the
macroscopic Maxwell curl equations, or a derived approximated wave equations. The
spectrally resolved induced macroscopic polarisation, that we calculate in Chapter 2,
acts as a source term for the optical laser fields. We develop two different models to ac-
count for the passive problem: 1) The full time-domain Maxwell equations are targeted
towards problems that involve a very broad range of relevant frequencies or spatially
strong localised cavity modes. This approach, however, is limited to micrometer-sized
active devices. We investigate microcavity lasers and nonlinear pulse interactions in
semiconductor amplifiers. We combine the Maxwell curl equations with a band-resolved
description of the semiconductor gain dynamics, within the finite-difference time-domain
framework. Our numerically challenging approach is generic and has not been imple-
mented before. 2) Depending on the resonator characteristics the paraxial, slowly varying
amplitude and rotating wave approximations are applied to simplify the model equations.
8 INTRODUCTION, OVERVIEW AND OUTLOOK
A transverse scalar wave equation model is developed for the analysis of high-power large
edge-emitting semiconductor lasers and amplifier structures, and for the study of trans-
verse multi-mode laser dynamics.
In Chapter 4 we consider the local properties of the active laser gain medium using
models which are not spatially resolved. Instead, the resonator structure is accounted
for through effective parameters. Microscopic scattering and dephasing rates for the
Coulomb and Fro¨hlich interaction in GaAs quantum well systems are calculated for
various carrier sheet densities and temperatures. We employ these microscopic param-
eters in our simulations of the spatio-temporal laser dynamics, which is demonstrated
to improve the model quantitatively. We show that the relaxation rates are a crucial
factor for the accurate modelling of the gain properties, nonlinear pulse propagation,
and the dynamical saturation behaviour of gain and absorber elements. A dynami-
cal treatment of scattering is presented in Appendix B. The important optical gain
properties of semiconductor amplifier media, the spontaneous emission spectra and the
radiative recombination rates are calculated using a microscopic approach, derived from
first principles. Basic concepts of semiconductor lasers, such as density pinning and gain
clamping, gain saturation, and thermal rollover, are discussed within the framework of
our models. The transient dynamic response when a laser is switched-on or perturbed
during operation, i.e. laser relaxation oscillations, is investigated. We obtain a realis-
tic set of input parameters for the analysis of spatio-temporal laser dynamics, which
improve our approach quantitatively. The close agreement between numerical results
and experimental findings, as well as theoretical predictions from more simple models,
verifies our approach.
In Chapter 5 we study broad area lasers by applying the paraxial optical model
introduced in Chapter 3. We show that on increasing the width of the active zone or
the pump power, more transverse modes are excited, spatial and temporal fluctuations
in the near-field occur, and unstable optical filaments are formed. We link the transverse
multi-mode and filamentation behaviour with the carrier and gain dynamics, and to the
selective depletion of the carrier density by the various transverse laser modes. We show
that the coexistence of multiple modes is supported by the interaction of the various
modes with different domains of the gain material, spatially due to the transverse degree
of freedom and spectrally as a result of inhomogeneous broadening. The impact of optical
nonlinearities due to spatial hole burning and of the physical effects of carrier diffusion,
gain- and index-guiding, self-focussing and diffraction, are quantitatively analysed. We
present a linear stability analysis of the transverse instability. This allows us to identify
the main control parameters: the stripe width, and the linewidth enhancement factor.
We numerically show that, depending on the width of the structure, different degrees
of spatio-temporal complexity and dynamic emission regimes can be distinguished. We
employ a complex mode analysis of the numerically calculated optical field amplitudes
to find out the lasing frequencies, and project the spatio-temporal near-fields onto the
laser modes (in the quasi-periodic regime).
In Chapter 6 we apply the full time-domain model, i.e. the Maxwell equations cou-
pled to the semiconductor Bloch equations, to investigate the longitudinal multi-mode
dynamics of novel laser structures and the interaction of femtosecond and picosecond
9pulses with an optically active medium. VCSEL with embedded periodic gain and
refractive index structures are considered. The calculation of the laser modes, lasing fre-
quencies and thresholds, numerically shows that photonic crystal effects can be obtained
for finite crystal structures, and lead to a significant improvement of the laser perfor-
mance. We show how photonic band edge lasers capitalise on the special properties of
the singularities in the optical band structure diagrams, e.g. the flat dispersion at band
edges. Our results demonstrate gain enhancement by an increased localisation of the
modes over the active quantum confined structures and by the more efficient interac-
tion of photons with the gain medium. The latter is due to the reduced group velocity.
We analyse the suppression of optical losses of the inner finite photonic structure by
introducing a surrounding photonic band gap region. Realistic optically pumped exter-
nal cavity surface-emitting laser structures are studied. We show that the longitudinal
multi-mode behaviour in such structures is composed of several external cavity modes.
A microscopic analysis reveals the dynamical balance between carrier generation from
pumping into high energy states, relaxation of carriers towards the Fermi-Dirac distri-
bution, and lasing from states near the band edge. We also consider the propagation
of femtosecond and picosecond pulses in active semiconductor structures. We identify
the microscopic origin of the fast nonlinearities, and discuss the physical effects behind
the various saturation mechanisms, e.g. the depletion of available resonant carriers for
stimulated emission. We also calculate the nonlinear gain coefficients and the different
recovery rates. Group velocity dispersion, the dynamical saturation and fast self-phase
modulation, are the main causes of the asymmetries observed in pulse shape and spec-
trum. The most critical parameters of gain recovery are given by the time constants of
intraband scattering processes. We also investigate nonlinear coherent pulse propagation
phenomena in active gain media, in particular the pulse area theorem and self-induced
transparency. Our numerical full time-domain simulations are employed with realistic
parameters for the dephasing processes and the homogeneous broadenings. The results
show that coherent ultrafast nonlinear propagation effects become less distinctive in
semiconductor quantum well gain materials at room temperature.
Appendix A provides details about the numerical implementation of the theoreti-
cal time-domain models, such as the applied discretisations of the fields and differential
operators on regular grids, and the integration schemes used. We also discuss the numer-
ical complexity, accuracy and the stability of our algorithms. For the paraxial transverse
wave equation model we propose the Hopscotch method as an efficient, stable and reason-
able accurate solver. We employ a numerical algorithm which involves the partitioning
of the grid into two groups of grid points, and an integration scheme with alternating
explicit and implicit discretisations. The merit of this method is that it does not require
solving a matrix-valued problem. We discuss the finite-difference time-domain (FDTD)
method which solves the first order Maxwell curl equations by arranging the electric
and magnetic field quantities on staggered grids in time and space according to the
Yee scheme. A fully explicit numerical implementation of our full time-domain model
of the active nonlinear material response and the passive refractive index structure is
developed.
10 INTRODUCTION, OVERVIEW AND OUTLOOK
Outlook
In summary, in this work we develop and present time-domain methods, which are
demonstrated to be valuable tools in the analysis of the coupled intracavity optical field
dynamics and band-resolved semiconductor material response. Our novel approaches are
successfully applied to the investigation of one-dimensional photonic band edge band gap
lasers and optically pumped VECSEL. We also consider high-power structures, such as
broad area lasers and semiconductor optical amplifiers, where optical nonlinearities and
ultrafast processes are significant. The theoretical framework presented in this work is
rather general, and thus can be easily extended to more complex gain systems and laser
structures. Their implementation, however, is limited by the available supercomputing
resources.
Possible extensions of the model and topics for further research include microcavity
and photonic crystal lasers, which would require to consider the three-dimensional full-
vectorial Maxwell equations [14, 38]. The study of σ-type electric dipole transitions,
i.e. circular polarisation, can be implemented with complex-valued electromagnetic and
polarisation fields [39]. Femtosecond nonlinear pulse interactions in semiconductors can
be analysed. To this end, a microscopic dynamical treatment of the carrier scattering
and energy redistribution [40], and the consideration of gain and material dispersion [41],
are very important. We note that with the miniaturisation of laser structures (e.g.
microcavities), the study of spontaneous emission becomes a significant topic as the
radiative lifetimes can be controlled by the dielectric structure and cavity design, and
thresholdless lasing is possible [20,42–44]. We thus suggest our model could be extended
by including a description of quantum noise and spontaneous emission, as well as cavity
quantum electrodynamics, into the time-domain approach [45–47].
2 Microscopic Description of the Gain
Dynamics
2.1 Introduction
The static and dynamical performance characteristics of a diode laser or semiconduc-
tor optical amplifier device are determined by both, the electronic and optoelectronic
properties of the active gain medium and the refractive index structure and geometry
of the optical resonator, modelled by the use of the macroscopic Maxwell equations or
some derived approximated wave equations (Chapter 3) [2]. In this chapter, the re-
sponse of the semiconductor material to applied electromagnetic fields is calculated. The
response of the material system includes the nonlinear and noninstantaneous induced
electric polarisation P(r, t) of the active gain medium and the dispersion of the back-
ground material nbackground(r;ω), which is accounting for far off-resonant excitations or
resonances of the host medium.
There are a couple of different model equations to qualify the physics of gain media
and to describe the coherent processes of stimulated emission and absorption [3, 10]. In
solid-state or gas lasers one can regard the active medium effectively as an ensemble of
absorption or amplification centres (like e.g. atoms, molecules) with only two electronic
energy levels which couple to the resonant optical field mode [1]. Other electronic states
are used to excite or pump the system. Depending on the occupations, amplification
or absorption of the light fields takes place. The optical Bloch equations [2, 10], which
are a set of equations for the population inversion and the induced electric polarisation,
form the very basis of the semiclassical two-level model. The two-level system with
homogeneous broadening describes a driven damped harmonic oscillator with a sym-
metric Lorentzian gain profile. This, however, does not constitute a sufficiently precise
model for the gain profile of a semiconductor-based active medium [3,24,48–51]. The big
advantages of this approach are low complexity of the description and the fact that it
requires only marginal numerical efforts. Besides its shortcomings, such a simple model
enables the possibility to perform extensive parameter scans, for instance in a bifurcation
analysis of nonlinear laser dynamics [9, 10, 52]. In other situations, e.g. when concen-
trating on the optical properties of the cavity structure of novel nanostructured devices
or microcavity lasers [13], a three-dimensional modelling is necessary. As the numerical
effort may already be high, a two-level gain model can come into consideration [14]. In
reality, the gain spectrum of a semiconductor medium is highly asymmetric and shows
a typical profile in frequency, i.e. a sharp structure at the direct band edge and absorp-
tion for high frequencies [4, 51]. As starting point, one could interpret a semiconductor
12 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
medium as the sum of two-level systems with different transition frequencies as deter-
mined by the electronic band structure and with separated carrier inversions, and then
make a superposition of the contributions of the various transitions. This would result
in an inhomogeneously broadened system. However, due to the high carrier densities in
a semiconductor laser and operation at room temperature, the diverse emission centres
are strongly coupled and many-body effects like the carrier-carrier Coulomb interac-
tion or the interaction of the carrier subsystem with quantised excitations of the lattice
(phonons) are prominent features [4,48,53]. Besides that, the coupling between the phase
and amplitude dynamics in semiconductor-based active structures is typically large [2].
Other effects, such as the renormalisation of the transition energies, the Coulomb en-
hancement of the Rabi frequency, various scattering and dephasing processes, spectral
hole burning and multi-mode operation, the diversity of optical nonlinearities, poor beam
quality or enhanced laser linewidths are covered by the semiconductor Bloch model. As a
result, the semiconductor laser is both, an inhomogeneously and homogeneously broad-
ened system. The most important laser properties, such as the transparency density,
threshold current, differential gain factor, nonlinear saturation behaviour or the density
dependent linewidth enhancement factor [4] are calculated in a straightforward and ap-
propriate way. To summarise, we could say that, the more detailed the description of
the active semiconductor material, the less need for phenomenological factors, which are
typical for more generic laser models [54].
In this chapter we will introduce a detailed quantum mechanical description based on
the density matrix formalism. The gain dynamics includes many-body effects. Multi-
subband semiconductor Bloch equations are derived in both, a time-domain formulation
with a split-off frequency and in a full time-domain formulation. All the active material
properties and microscopic interactions determine the induced macroscopic polarisation
P[E, n◦k, N, Tpl, Tlat], which is a dynamical material response functional. This source
term of the optical wave equation is connected with the microscopic gain dynamics by a
spectral summation, and in low-dimensional nanostructures by an additional necessary
transformation to a volume-based density quantity. An adiabatic elimination of the
polarisation is an approximation which is motivated by the separation of decay time
scales of the different macroscopic variables. This enables us to derive an effective
microscopic model from the microscopic set of equations [36,55]. A further simplification
is to neglect the phase dynamics of the optical fields. This leads to standard rate
equation models [9, 52]. For complicated active gain media, e.g. polymer or organic
materials [30,31], carbon nanotubes [56], or other nanostructures [27,57], it is considered
impractical to microscopically calculate the optoelectronic material response properties.
An approach to model the interaction of the active material with electromagnetic fields
is to fit the susceptibility coefficients with as few as possible excitations of Lorentzian,
Debye or Drude type [58, 59] (a multi-level system). Figure 1.1 gives an overview on
commonly applied models which can be used to simulate for the semiconductor gain
dynamics or generic optically active gain media. The various levels of description are
developed and discussed in more detail in the following sections.
2.2 Parabolic Band Structure Approximation and Confinement Functions 13
2.2 Parabolic Band Structure Approximation and
Confinement Functions
In this section we review the basic concepts, terminology and properties of semicon-
ductor crystals [5]. The simulations in this work are done for the Indium Gallium Ar-
senide/Aluminium Gallium Arsenide InxGa1−xAs/AlyGa1−yAs material system, which
is a typical representative of the III-V semiconductors [60]. III-V semiconductors are
composed of atoms of the group of metals (In,Ga,Al) and atoms of the nitrogen group
(group-V of the periodic table of elements). They often crystallise with a cubic crystal
symmetry, the zincblende structure. The chemical bonding in these materials is medi-
ated through the exchange interaction between the eight outer electrons (per unit cell),
and sp3 hybrid orbitals are formed. To describe the formation of a crystal from first prin-
ciples one would have to solve the many-body Schro¨dinger equation for a macroscopic
system with about 1023 atoms, which is apparently not possible. However, in order
to understand the optoelectronic properties of bulk semiconductor crystals around the
band edge, a single particle approximation can be applied, where the time-independent
Schro¨dinger equation for a representative electron is solved. The interaction with the
nuclei and the inner strongly bound electrons and other free electrons is summarised by
an effective periodic potential Veff. The spin degree of freedom and the resulting spin-
orbit coupling are accounted for by an additional term Hspin-orbit. The single particle
Hamiltonian can thus be written as
H0 =
p2
2m0
+ Veff(x) +Hspin-orbit, Veff(x) = Veff(x−X). (2.1)
Due to the translation symmetry of the lattice, the eigensolutions obey the Bloch theorem
and can be labelled with a wavevector k. Equation (2.1) is the starting point for band
structure calculations using various numerical methods, such as tight-binding theory
(or linear combination of atomic orbitals, abbr. LCAO), pseudo-potential methods or
semi-empirical, perturbatively k · p theory [5, 61]. The eigenstates are delocalised over
the crystal and form rather complicated energy band structure diagrams with several
conduction and valence bands. The for lasing crucial process of coherent interaction
between light fields and the two-component carrier plasma involves only a small portion
of the band structure. Transitions are designed to occur with energies around the band
gap and around the high-symmetry Γ point (i.e. k0 = 0) of the direct band gap gain
material. Photon momentum transfer is negligible, and optical transitions primarily
take place at the band edge. A simplification of the energy level structure for a bulk
semiconductor laser is the effective mass approximation which introduced a parabolic
dispersion of the conduction band (cb) and the heavy-hole valence band (vb). Then, by
replacing the free electron mass with (in the general case tensor-valued) effective masses,
we can write
Eek =
~
2k2
2me
, Eh−k =
~
2k2
2mh
(2.2)
14 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
for the energies of free electrons and holes. For k0 = 0 some similarities to atomic physics
may be drawn, and the different energy bands can be characterised by atomic quantum
numbers for the total and projected orbital and spin angular momenta |l s;mlms〉 or,
if we assume the spin-orbit coupling in the Russell-Saunders scheme, by total angu-
lar momentum numbers |l s; j mj〉. The conduction band states are s-like and show a
two-fold degeneracy due to the spin degree of freedom |0 1/2; 1/2 ± 1/2〉, whereas the
valence band states are p-like and six-fold degenerated. The consideration of the spin-
orbit coupling partially lifts this energetic degeneracy and separates the split-off band
|1 1/2; 1/2 ± 1/2〉 from the heavy-hole, light-hole energy band states |1 1/2; 3/2 ± 3/2〉
and |1 1/2; 3/2 ± 1/2〉 [4].
Sophisticated crystal growth techniques, namely molecular beam epitaxy (MBE),
liquid-phase epitaxy (LPE), metal-organic chemical vapor deposition (MOCVD) [5],
and modern self-organised growth methods [29] are used to manufacture high quality
semiconductor structures (e.g. layers, superlattices or quantum dots). The precision of
some of these techniques allows to create or grow quantum confined structures, and to
engineer the band gap and energy levels. In a quantum well heterostructure the carriers
are quantum confined in one direction, namely in the growth direction of the structure.
The motion in the two perpendicular directions is free. We choose the z-direction, in the
transverse wave equation model it would be the lateral y-direction. Quantum confine-
ment occurs if one dimension of the structure is comparable to the characteristic length
scale of elementary excitations. In summary, we have a quasi two-dimensional system
with a wavevector k = (kx, ky). As the result of the lower symmetries and dimensions
properties are modified. The envelope function approach incorporates the quantum well
confinement into the k · p method. The wave functions are approximated in plane wave
mode expansions [62]
ψcbi,k(z, r) =
1√
A
eik·r
∑
n(bands)
ucbn,k0(z, r)φn,i,k(z), (2.3)
ψvbj,k(z, r) =
1√
A
eik·r
∑
n(bands)
uvbn,k0(z, r)φn,j,k(z), (2.4)
in which ucbn,k0(z, r), u
vb
n,k0
(z, r) denote the lattice-periodic Bloch functions at the band
minima, and A is a normalisation area. φi(z), φj(z) are the electronic confinement or
envelope functions for the electrons and holes, respectively. The Bloch functions vary on
the atomic length scale in contrast to the envelope (confinement) functions which vary
on the nanoscale. With the above given ansatz we solve the matrix-valued eigenvalue
problem
∑
m
Hnm
(
p2
2m0
, Veff(x), Hspin-orbit, Vconf(z),E, e, . . .
)
φm,i,k = Ei,kφn,i,k, (2.5)
where i represents a combination of quantum numbers. The diagonalisation of this ma-
trix equation returns the energy eigenvalues and the envelope functions. The Hamilton
operator covers the effective crystal symmetry and the relativistic spin-orbit coupling,
2.2 Parabolic Band Structure Approximation and Confinement Functions 15
Figure 2.1: For direct optical transitions relevant section of the energy band struc-
ture diagram around the high-symmetry point Γ. Eigensolutions of the one-electron
Schro¨dinger equation for the (In)GaAs gain material system in the case of bulk (left)
and quantum well (right graph) structures [4,5]: The energy bands are classified by the
total and projected angular momentum numbers. The split-off energy ∆so is often com-
parably large (not for phosphide and nitride compounds). The energy level structure
is modified as a result of lowered dimension and symmetries (for a quantum well, right
diagram). Because of the effects of mass reversal and level-anticrossing the energy dis-
persion relations are more complicated, the hole energy bands become nonparabolic. In
our microscopic time-domain simulations the energy structures of active gain elements
are approximated by effective masses me/m0(GaAs) = 0.067, mh/m0(GaAs) = 0.33
and me/m0(InGaAs) = 0.06, mh/m0(InGaAs) = 0.35 (proportional to the inverse
curvatures of the energy dispersions), and by the band gap and quantum confinement
energy. Completely filled (inner) bands and resonances of lattice and strong bound elec-
trons do not contribute directly to the dynamical material response P(t), that means
to resonant optical transitions and excitations. However, they are included into the
linear, static response specified by the background refractive index nbackground(λ).
16 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
usually quantified by the experimentally determined Kohn-Luttinger parameters [60],
the carrier quantum confinement potential Vconf(z) which is formed by the band edge
discontinuities, and the impact of tensile or compressive elastic strain e (in not perfectly
lattice matched heterostructures). In compounds of group-III nitrides, which are im-
portant for wide band gap materials emitting in the green to ultraviolet [21, 22], the
calculations are even more complicated. These compounds crystallise in the (asymmet-
ric) hexagonal wurtzite structure, and strain induced piezoelectric fields have to be taken
into account by solving the Poisson equation together with the Schro¨dinger equation.
We would like to give a brief sketch of the different levels of that problem [62] and than
introduce the set of single particle energy eigenstates used in the rest of this work. The
simplest representative of k ·p models applies a 4×4 matrix Hamiltonian, which already
features effects like mass reversal, heavy-hole and light-hole band mixing as typical for
quantum confined structures [5]. In active laser compounds based on phosphides or ni-
trides with comparatively small split-off energies the additional coupling of the split-off
band enlarges the problem to a 6× 6 matrix eigenvalue problem [4]. More complicated
Hamiltonians exist to model the mixture of conduction and valence bands and leading
to nonparabolic conduction subbands [61,62].
Based on the fact that we concentrate on (In)GaAs quantum wells, the basic op-
toelectronic properties can be well approximated by restricting the description to the
conduction and the heavy-hole band. We apply the effective mass approximation, and
assume an isotropic (averaging over all possible directions of k) and parabolic band
structure. As first step we solve the following one-dimensional problems: The quan-
tum confinement of the carriers as imposed by the grown layer heterostructure1 and
possible external potentials give rise to the following Schro¨dinger-like energy eigenvalue
problems2[
−∂z
(
~
2
2me,z(z)
∂z
)
+
~
2k2
2me,⊥
+ Vconf(z)
]
φi(z) =
(
Eei +
~
2k2
2me,⊥
)
φi(z), (2.6)[
−∂z
(
~
2
2mh,z(z)
∂z
)
+
~
2k2
2mh,⊥
+ Vconf(z)
]
φj(z) =
(
Ehj +
~
2k2
2mh,⊥
)
φj(z). (2.7)
In addition to this electronic confinement of the eigensolutions φi(z), φj(z), the grown
heterostructure also represents a refractive index waveguiding structure n(z) which fo-
cuses the optical field profiles over the active quantum wells. Due to the very different
length scales, configurations which separate these two confinements have been proposed3.
1The band edge discontinuities in heterostructures form a potential Vconf(z). For an infinitely deep
potential well the wave functions have to vanish at the layer interfaces, and we can calculate the
confinement energies analytically as En = ~
2/(2mz)π
2/d2QW · n2.
2The demand of continuity of the quantum mechanical wave function and of the probability current
at band gap discontinuities (at positions zint) give rise to the so-called Bastard boundary conditions
[5, 26] 1/mz(z
−
int)∂zφn(z
−
int) = 1/mz(z
+
int)∂zφn(z
+
int).
3The basic objective of these structures is to enhance the modal gainGmod,m, the effective amplification
of the spatial field profile or mode m. The dynamics is given by the multi-mode rate equation
1/vgr∂t|E˜m|2 = (Gmod,m − 2γresonator,m)|E˜m|2 [63]. The modal gain can be enhanced by increasing
the semiconductor material gain g(ω = ωm) (e.g. by an electronic quantum confinement) and/or by
optimising the optical confinement Γm, this is because of the relation Gmod,m = 2κ˜Γmg(ωm).
2.2 Parabolic Band Structure Approximation and Confinement Functions 17
Figure 2.2: Overview on epitaxially grown heterostructures with electronic and ad-
ditional optical confinement: (Top) The lowest confined eigenstates in a type-I het-
erostructure (e.g. GaAs/AlGaAs quantum wells with a one-dimensional quantum con-
finement potential Vconf(z), and an inverted profile n(z) for the refractive index) are
displayed. This gives rise to the existence of subbands. The energy dispersion rela-
tions are composed by the energy eigenvalue plus by a free motion in-plane. (Middle)
Layer structures which separate the optical from the electronic confinement. (Bottom)
Calculated electronic confinement in a type-II heterostructure (e.g. GaAsSb/InGaAs
quantum wells with a spatially indirect band gap). The radiative transition rates are
mainly determined by the overlap integrals of the envelope functions
∫
dzφ∗i (z)φj(z).
18 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
The above eigenvalues Eei , E
h
j of bound states have to be added to the dispersion relation
(2.2) of a free electron, hole
Eei,k = E
e
i +
~
2k2
2me
, Ehj,−k = E
h
j +
~
2k2
2mh
. (2.8)
As a result, a finite number of subbands are formed. We neglect the unconfined states of
the quantum well. The energy eigenstates are fully determined by the two-dimensional
wavevector k and the number of the subband i, j. To account for recombination and loss
processes, to be flexible concerning the number of electrons and holes, and for symmetries
associated with fermionic particles, the description of the semiconductor gain medium
by the wave function of the system is extended to second quantisation. The following
plane wave mode expansion is made [64]
ψ(z, r) =
1√
A
∑
i
∑
k
φi(z)ci,ke
ik·r +
1√
A
∑
j
∑
k
φj(z)d
†
j,−ke
ik·r. (2.9)
The creation, annihilation operators for electrons in the states of the conduction subband
i and with wavevector k are introduced as c†i,k, ci,k. d
†
j,−k, dj,−k are the corresponding
quantities for the holes.
2.3 Density Matrix Formalism
A quantum theoretical treatment of a macroscopic system of many identical, indis-
tinguishable and interacting particles (e.g. 1023) by calculating the many-body wave
function from Schro¨dinger equation is far too complicated. This is due to interaction
processes such as recombination, collision, energy and momentum relaxation, the inter-
action with classical radiation fields, and the Coulomb interaction. In addition, open
systems such as a laser are characterised by a coupling to its external environment.
Therefore, a statistical description and the concepts of ensembles and baths are applied.
The quantum mechanical equivalent to the classical phase space distribution is the sta-
tistical or density operator ρ, a Hermitian, positive semidefinite and basis-independent
operator [65]. Expectation or average values of quantum mechanical operators O (ob-
servables) can be derived from a trace operation over their product with the statistical
operator
〈O〉 = tr[Oρ] = tr[ρO] =
∑
n,n′
ρn,n′On′,n, (2.10)
and include both the quantum theoretical uncertainty On′,n and an incoherent summa-
tion, which expresses statistical ensemble averaging. The dynamical evolution of an
expectation value could be either computed by calculating the dynamics of the density
operator using the non-dissipative Liouville-von Neumann equation together with cou-
pling terms to reservoirs and baths or by calculating the dynamics of O in the Heisenberg
picture. In the latter case, we use ∂tO = i/~ [H,O] and a static statistical operator ρ(t0).
2.3 Density Matrix Formalism 19
The statistical operator can be represented in the basis of a complete set of orthonormal
vectors as a sum of projection operators (of ket and bra vectors)
ρ =
∑
n,n′
ρn,n′ |n〉 〈n′| , tr[ρ] =
∑
n
ρn(,n) = 1. (2.11)
ρn,n′ is called the density matrix. The diagonal elements represent probabilities of finding
the system in the states |n〉, whereas nondiagonal elements give information about rela-
tive phases, about coherences between the different states |n〉 and |n′〉 [64]. We will work
in the basis of the energy eigenfunctions of the single particle HamiltonianH =
∑
kHk =∑
kEkNk, which is the basis of eigenfunctions of the number operators Nk at the same
time. The collision or interaction processes and the coupling to reservoirs drive the di-
agonal part of the expectation value of the statistical operator to a thermal equilibrium
function fk = tr[ρNk] =
(
1 + exp(β(Ek − µ))
)−1
, β = 1/(kBT ). This is the Fermi-Dirac
function which can be calculated for a (quantum) grand canonical equilibrium ensem-
ble represented by the density operator ρ = 1/Z
∏
k ρk = 1/Zexp(−β
∑
k(Hk − µNk)),
and a partition function Z = tr[
∏
k exp(−β(Hk − µNk))]. Coherences described by
nondiagonal elements are frequently lost in statistical averaging operations.
In the following we concentrate on single particle density operators. The electronic
density operator ρ→ n(r, r′) as an example of a matrix in real space representation can
be written as
〈n(r, r′)〉 = 〈ψ†(r)ψ(r′)〉 . (2.12)
By inserting the mode expansion ansatz ψ(r) = 1/
√
V
∑
k ckφk(r) with plane wave
expansion functions φk, a representation in momentum space nk,k′ =
〈
c†kck′
〉
can be
created. The mathematical concept of the Wigner distribution function [66], a fractional
Fourier transformation with respect to a relative coordinate, gives a representation in
mixed real and momentum space
〈nk(r)〉 =
∫
d3r′
〈
n
(
r− r
′
2
, r+
r′
2
)〉
e−ikr
′
=
∑
k′
nk−k′/2,k+k′/2e
ik′r. (2.13)
This allows for the modelling of spatially inhomogeneous structures [67] and is one basis
of quantum kinetic transport theory. A systematic Taylor expansion of (2.13) in real
and momentum space [36] would lead in lowest order to terms in which the position has
only parametric character. This means, that locally one has the same dynamics as in
the homogeneous case [66] which can be explained by the fact that the density matrix
elements vary in space and momentum with very different scales4.
4In the following we will limit our calculations therefore on measures diagonal in Fourier space and
account for spatial dependencies and carrier or energy transport not on the microscopic but on the
macroscopic level. This means that the basic dynamical variables are reduced to the expectation
values nk =
〈
c†kck
〉
. As the macroscopic fields define the equilibrium microscopic functions fk(r)
spatial inhomogeneities enter the microscopic description.
20 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
2.4 Coherent Coupling to the Optical Field
In this section we motivate the equations which describe the coherent interaction of
the electronic subsystem with an externally applied classical (electromagnetic) radiation
field. The basic single particle carrier energy and interaction Hamiltonian H = Hcarrier+
Hcarrier-light is given by
H =
1
2m◦
(
p− eA)2 + Veff =
(
− ~
2
2m◦
∆+ Veff
)
− e
m◦
A · p+ e
2
2m◦
A2, ◦ = e, h.
(2.14)
The electromagnetic vector potential A(r, t)5 specifies the terms of the minimal coupling
and the canonical momenta6. The effective potential Veff seen by a representative car-
rier, and additional terms accounting for spin-orbit coupling, strain, piezoelectric fields,
quantum confinement potentials etc., define the single particle energies Eei,k, E
h
j,−k [62].
Based on the fact that the wavelength of the electromagnetic field is far larger than a
typical atomic dimension Ratomic/λ ≪ 1, we may neglect the spatial variation of the
vector potential on the atomic length scale (long-wavelength approximation [26]). A
unitary time-dependent transformation T (t) is introduced to transform the interaction
Hamiltonian in minimal coupling ∝ A · p to the direct coupling er · E →Meh · E [68].
For this, an electric dipole moment of the charge distribution with respect to the
origin Meh = e/V(unit cell)
∫
d3Rucbk0(R)Ru
vb
k0
(R) and the unitary translation operator
T (t) = exp
[−i/~Meh ·A(R = 0, r, t] are defined [68]. In the new representation, the
electric dipole approximation, the Hamiltonian is given by
H ′(t) = TH(t)T † + i~ (∂tT )T
† = Hcarrier +Hcarrier-light = Hcarrier −Meh · E(r, t).
(2.15)
The light-carrier interaction couples the dipole moment to the external electric field. A
more intuitive motivation could be given by calculating the energy of a dipole moment
(volume) density in an external field −VP(r) · E(r) [69,70].
In second quantisation the single particle Hamiltonian (2.15) of the electron-hole
plasma interacting with a classical external light field is defined by
Hcarrier +Hcarrier-light =
∫
d3rψ†(r)
(
Hcarrier −Meh · E
)
ψ(r). (2.16)
We utilise the orthonormality relation of the expansion basis and analyse the above
expression after inserting the mode expansion ansatz (2.9). The dipole interaction term
couples only bilinear terms (two states) which are diagonal in k. These terms describe
5The subsequent derivation is done in the Coulomb gauge. The scalar electromagnetic potential is
assumed to vanish, and we demand the transversality condition divA = 0.
6The quadratic term in the vector potential in equation (2.14) may be neglected compared to the linear
terms due to its smallness as two-photon processes (products of annihilation and creation operators)
are much less probable than the one-photon processes described by the linear terms, except in the
case of extremely high intensities.
2.4 Coherent Coupling to the Optical Field 21
transitions between different subbands (c†i1,kci2,k or d
†
j1,−k
dj2,−k) and interband transitions
(polarisations) with one electron and one hole operator. When modelling laser devices
or the pulse propagation in active semiconductor materials, we assume that the optical
frequencies are in the order of magnitude of the band edge. Therefore, intersubband
transitions which are far off-resonant may be neglected7. The strength of the coupling
between a transition and the optical field can be approximated as the product of Meh
with an overlap integral of the envelope functions
Mehij =M
eh
∫
dzφ∗i (z)φj(z). (2.17)
With these assumptions the multi-subband Bloch Hamiltonian can be written as
Hcarrier +Hcarrier-light =
∑
k
Hk =
∑
i
∑
k
Eei,kc
†
i,kci,k +
∑
j
∑
k
Ehj,−kd
†
j,−kdj,−k
−
∑
i,j
∑
k
[
Mehij c
†
i,kd
†
j,−k +M
eh∗
ij dj,−kci,k
]
· E. (2.18)
The basic dynamical variables in the microscopic equations of motion are defined using
a single particle density matrix
nei1i2,k =
〈
c†i1,kci2,k
〉
, nhj1j2,−k =
〈
d†j1,−kdj2,−k
〉
. (2.19)
The diagonal elements of these two intraband expectation values are distribution func-
tions while the nondiagonal parts describe coherences between different subband states
(intersubband transitions), namely the intraband polarisations. In addition we define
the interband polarisations, the material response to the applied electric field, as
pji,k =
〈
dj,−kci,k
〉
, p∗ji,k =
〈
c†i,kd
†
j,−k
〉
. (2.20)
We can link the nonlinear induced macroscopic polarisation8, as the source term in the
later derived wave equations (3.37) and (3.46), to the microscopic dynamical variables
by
P =
∑
i,j
1
A
∑
k
(
Mehij p
∗
ji,k +M
eh∗
ij pji,k
)
. (2.21)
7The proposed novel gain material concept and structure of quantum cascade lasers is based on inter-
subband transitions and thus on intraband polarisations. This intersubband nature (only a single
type of carriers, e.g. electrons needed) gives rise to several key advantages compared to devices based
on stimulated electron-hole recombinations [34]. Tunability by using the concept of size quantisa-
tion in multi quantum well heterostructures (i.e. no restriction given by the energy band gaps) and
efficiency are two key features.
8The dipole matrix elements, which in the general case are complex-valued vectors, represent the con-
nection between the optical excitation E and the material response functional P. After an adiabatic
elimination of the microscopic polarisations, one can write P =
∑
i,j 1/A
∑
k Ck
(
Mehij ⊗Mehij
)
E. If
we assume that the light field is linearly polarised and the dipole vectors are randomly oriented the
ensemble average is given by
(
Mehij
)2
= 1/3
( ∣∣Mehij,x∣∣2 + ∣∣Mehij,y∣∣2 + ∣∣Mehij,z∣∣2 ). In the following sim-
ulations we use the material parameters Meh(GaAs) = 0.3 enm = 4.8 · 10−29 Cm, Meh(InGaAs) =
8 · 10−29 Cm.
22 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
The derivation of equations of motion in the Heisenberg picture for both, the particle
number and polarisation operators involves the computation of commutator operations
with the bilinear operators of the basic Hamiltonian (2.18). The result is a closed set of
equations9, the multi-subband semiconductor (optical) Bloch equations10 [64]
∂tn
e
i1i2,k
= − i
~
(
Eei2,k − Eei1,k
)
nei1i2,k +
i
~
∑
j
(
Mehi2jp
∗
ji1,k
−Meh∗i1j pji2,k
) · E, (2.22)
∂tn
h
j1j2,−k
= − i
~
(
Ehj2,−k − Ehj1,−k
)
nhj1j2,−k +
i
~
∑
i
(
Mehij2p
∗
j1i,k
−Meh∗ij1 pj2i,k
) · E,
(2.23)
∂tpj1i1,k = −
i
~
(
Eei1,k + E
h
j1,−k
+ Egap
)
pj1i1,k
− i
~
(∑
i
Mehij1n
e
ii1,k
+
∑
j
Mehi1jn
h
jj1,−k
−Mehi1j1
)
· E (2.24)
for the basic dynamical variables. In order to gain a formulation in frequency-/time-
domain, we decompose the electric field and the induced (microscopic and macroscopic)
polarisation terms into a slowly varying amplitude and in a fast, with a frequency ω
oscillating phase factor. In other words, we transform the equations into a frame which
rotates with the frequency ω [37]. We note that we will apply the same approach in the
derivation of the paraxial optical wave equation (in Chapter 3). In the rotating wave
approximation we now only take into account slowly varying terms and neglect terms
that are ≈ ω out of phase. Applied to the Bloch equations, this results in
∂tn
e
k = ∂tn
h
−k = −g˜k =
i
2~
(
Mehp˜∗k · E˜−Meh∗p˜k · E˜∗
)
, (2.25)
∂tp˜k = − i
~
(
Eek + E
h
−k + Egap − ~ω
)
p˜k − i
2~
Meh · E˜ (nek + nh−k − 1) , (2.26)
where we consider the special case of only one electron and one hole subband. The above
set of equations describe a system equivalent to an ensemble of uncoupled harmonic
oscillators with different resonance frequencies which are driven by the product of an
electric field with the carrier inversion (nek + n
h
−k − 1). This corresponds to a purely
inhomogeneously broadened system. The factor g˜k is called the spectrally resolved carrier
generation rate and qualifies the impact of the laser field on the gain medium. Both, the
spatial and spectral hole burning and the gain saturation are related to g˜k. The effect
of hole burning is considered to be a limiting factor in high-speed applications and of
great importance in laser amplifier structures and pulse propagation [2, 73].
9The following decomposition (to anti-commutators) may be helpful to evaluate the commutators in
the Heisenberg picture and shows that the resulting term is a sum of bilinear operators [AB,CD] =
A[B,C]+D − C[A,D]+B − [A,C]+BD + CA[B,D]+.
10The optical Bloch equations do conserve the quantities d2k =
(
nek + n
h
−k − 1
)2
+4 |pk|2 , ∂tdk = 0. The
time evolution of N-level quantum systems may be characterised by the evolution of a number of
independent coherence vectors. For each of these vectors and for the generalised pseudo-spin vector
a nonlinear constant of motion (associated with the length of the vector) can be derived [71,72].
2.5 Coupling to a Full Time-Domain Scheme 23
2.5 Coupling to a Full Time-Domain Scheme
The (semiclassical) semiconductor Bloch equations describe the coherent interaction be-
tween electromagnetic fields and the carrier subsystem. As they were derived by apply-
ing the quantum mechanical density matrix formalism, they are formulated in complex
space. In order to incorporate the dynamics of active semiconductor gain media into a
full time-domain description of the optical fields as specified by the macroscopic Maxwell
equations11 it is advantageous to transform the Bloch equations to real space [14, 74].
This is in the following done for a two-band gain model
∂tpk = − (iΩk + γpk) pk − i
M ehE
~
(
nek + n
h
−k − 1
)
, ~Ωk = Egap + E
e
k + E
h
−k,
(2.27)
∂tp
∗
k = (∂tpk)
∗ = − (−iΩk + γpk) p∗k + i
M ehE
~
(
nek + n
h
−k − 1
)
, (2.28)
∂tnk = i
M ehE
~
(
p∗k − pk
)
=
2M ehE
~
Im (pk) . (2.29)
Note that here, the dipole matrix element M eh (in general a rank-one tensor) and the
electric field E are defined as scalar measures and are assumed to be real-valued12. In
addition, we have added a dephasing term γpk for the nonlinear induced polarisation.
Equations (2.27)+(2.28) for the dynamical polarisation pk(t),R→ C, and the conjugate
complex variable are first order in time. It is well known that this oscillator equations
are equivalent to an equation second order in time for a real quantity pk(t)
∂2t pk + 2γ
p
k∂tpk + ω
2
kpk = −Ck
M ehE
~
(
nek + n
h
−k − 1
)
, ωk =
√
Ω2k + (γ
p
k)
2 (2.30)
→ ∂tPk = AkPk + Fk. (2.31)
The vectors Pk(t),R → R2, and Fk and the real-valued quadratic matrix Ak are given
by
Pk =
(
pk
∂tpk
)
, Fk =
(
0
−CkM ehE/~
(
nek + n
h
−k − 1
) ) , Ak =
(
0 1
−ω2k −2γpk
)
.
(2.32)
With the objective to derive a real space equation from the complex counterpart we
apply the mathematical techniques of diagonalisation and principal axis transformation.
11The Maxwell equations are formulated for real quantities as the computational costs of operating on
complex variables would be doubled and all physical, measurable quantities have to be real-valued.
12This assumption is not a restriction in the case of linear polarised light, as via the multiplication with
an appropriate phase factor (and with it a rotation in complex space) Meh may be chosen to be
real. For systems with elliptically polarised light a more general ansatz for the (dipole) carrier-light
interaction given by the substitution Meh ·E(z, t)→∑i=x,y (Re(Mehi )Re(Ei)+Im(Mehi )Im(Ei)) =∑
i=x,y 1/2
(
Meh∗i Ei +M
eh
i E
∗
i
)
is advantageous [39]. This term represents the energy of a dipole
(vector) in an external electromagnetic field, a measurable quantity that must be real-valued.
24 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
Therefore, one determines the eigenvalues of Ak as λ±,k = −
(
γpk ± iΩk
)
and λ−,k =
λ∗+,k, and constructs a transformation matrix Uk from the two linearly independent
eigenvectors Uk =
(
U+,k U
∗
+,k
)
with UT+,k =
(
1 λ+,k
)
. With the help of the inverse
matrix U−1k , the quadratic matrix Ak can be diagonalised
∂t
(U−1k Pk) = (U−1k AkUk) (U−1k Pk)+ (U−1k Fk) = Dk (U−1k Pk)+ (U−1k Fk) , (2.33)
Uk =
(
1 1
λ+,k λ
∗
+,k
)
, U−1k =
1
i2Ωk
(
λ∗+,k −1
−λ+,k 1
)
, Dk =
(
λ+,k 0
0 λ∗+,k
)
.
(2.34)
The transformation to the new vector P˜k = U−1k Pk results in a set of two conjugate
complex equations similar to (2.27)+(2.28)
∂t
(
pk + i
(
γpk
Ωk
pk +
1
Ωk
∂tpk
))
= λ+,k
(
pk + i
(
γpk
Ωk
pk +
1
Ωk
∂tpk
))
− iCk
Ωk
M ehE
~
(
nek + n
h
−k − 1
)
. (2.35)
To create a mapping of the equations and to connect the two formulations pk ∈ C ↔
pk, ∂tpk ∈ R, we identify
Re (pk) = pk → P = 2
A
∑
k
M ehpk, Im (pk) =
1
Ωk
∂tpk +
γpk
Ωk
pk, Ck = Ωk. (2.36)
Thus, the equivalent microscopic dynamical equations of the active semiconductor ma-
terial response in R for our simulations within full time-domain are defined as
∂2t pk + 2γ
p
k∂tpk + ω
2
kpk = −Ωk
M ehE
~
(
nek + n
h
−k − 1
)
, (2.37)
∂tnk = 2
M ehE
~
1
Ωk
(
∂tpk + γ
p
kpk
)
. (2.38)
A more intuitive derivation of the coherent carrier generation term 1/Ωk∂tpk can be given
from Poynting’s energy density theorem (in the case of ∂tpk ≫ γpkpk → ωk ≈ Ωk). In the
next sections we will discuss the strong many-body interactions in semiconductors which
distinguish these materials from other laser gain media. As a result changes of transition
energies ~Ωk and Rabi frequencies M
ehE/~ have to be incorporated, which would make
the equivalence transformation more complicated13. An extension to the multi-subband
semiconductor Bloch equations, however, is straightforward as each independent (in-
terband or intraband) polarisation term is linked with an harmonic damped oscillator
equation (2.37) and describes a homogeneously broadened Lorentz medium [58], driven
13As a consequence the quadratic matrix Ak is now implicitly time-dependent, and the same is true
for the basis of the diagonalisation transformation λ±,k,U±,k,Uk,U−1k (t). The transformation to a
new vector (2.33) may be written as ∂tP˜k =
(Dk − U−1k (∂tUk))P˜k + F˜k.
2.5 Coupling to a Full Time-Domain Scheme 25
by the electric field times the respective carrier inversion. Lasing action in a four-level
atomic model system, implemented in a similar way, is reported in [38,75].
A fairly different approach to incorporate the Bloch equations for a simple two-level
system into the full time-domain method was suggested in [76]. In close analogy to
atomic or nuclear spin systems (interacting with a static and an oscillating magnetic
field) the carrier state is represented by a pseudo-spin vector. We generalise this approach
to a more realistic band-resolved description by introducing the following vectors with
three components for each transition identified by the wavevector k
ρ1,k = 2Re (pk) , ρ2,k = 2Im (pk) , (2.39)
ρ3,k = n
e
k + n
h
−k − 1. (2.40)
The dynamics of the pseudo-vector, which completely quantifies the occupation proba-
bilities and coherences of the carrier system, is determined by
∂tρ1,k = −γpkρ1,k + Ωkρ2,k, (2.41)
∂tρ2,k = −Ωkρ1,k − γpkρ2,k − 2
M ehE
~
ρ3,k, (2.42)
∂tρ3,k = 2
M ehE
~
ρ2,k, (2.43)
or written in a matrix form for the vectors ρk(t),R→ R3,
∂tρk = Rkρk − Tk (ρk − ρeqk ) = Rk × ρk − Tk (ρk − ρeqk ) , (2.44)
Rk =

 0 Ωk 0−Ωk 0 −2ΩRabi
0 2ΩRabi 0

 , Rk =

 2ΩRabi0
−Ωk

 , ΩRabi = M ehE
~
, (2.45)
Tk =

 −γpk 0 00 −γpk 0
0 0 −γk

 , ρeqk =

 00
ρtherm3,k

 . (2.46)
We have added a relaxation term −γk
(
ρ3,k− ρtherm3,k
)
towards the quasi-equilibrium state
to equation (2.43). ρtherm3,k = f
e
k+f
h
−k−1 represents the thermal occupation (or inversion)
for each k. We conclude this section, and note that there are three homomorphic formu-
lations of the semiconductor Bloch equations, which are coupled to the full time-domain
Maxwell curl equations model:
The formulation (2.27)–(2.29) as derived from quantum mechanical density matrix
theory operates on complex-valued microscopic quantities what doubles the numerical
effort and memory requirements. This set of equations provides a straightforward start-
ing point for dynamical models in frequency-/time-domain wherein a mode expansion
ansatz (for E and pk) and a partial transformation to Fourier space are carried out. To
our knowledge this full time-domain formulation has only been implemented once [77],
namely for the simulation of the dynamics of a two-level atom in a photonic crystal cav-
ity, and the investigation of normal-mode coupling or Rabi splitting in the transmission
26 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
spectra. This is a nonclassical phenomenon with the coherent exchange of energy in (ex-
citonic) polariton systems [78,79], in which the strong coupling exceeds the cavity decay
rate and the decoherence linewidth. (2.27)–(2.29) are a preferred choice for simulating
active systems with circularly or elliptically polarised light as the electromagnetic fields
are then in general complex-valued anyway.
The pseudo-spin approach (2.41)–(2.43) is strongly adapted to resonant atomic-like
systems, i.e. systems with only two relevant electronic energy states. It helps to obtain a
deeper insight into nonlinear coherent resonant optical phenomena, such as photon echos,
π or 2π pulses, pulse area theorem [80], self-induced transparency or transmission [81,82],
soliton solutions [76], or carrier wave Rabi flopping [83] (see Section 6.7). All these
effects are easy to visualise using the ∂tρ = R× ρ equation with the pseudo-spin Bloch
vector ρ and the torque vector R. They are based on the oscillation of the electronic
inversion (in systems in which the Rabi frequency fulfils certain restrictive conditions)
and on the resonance of the transition frequency with the frequency of the optical wave.
However, we note that in active semiconductor structures at room temperature and
for high carrier densities (even in novel rather atomic-like quantum dots) these coherent
resonant occurrences are highly unlikely [82,84,85], due to strong dephasing mechanisms,
renormalisation, the presence of homogeneous and inhomogeneous broadenings, and a
statistical spread of material parameters (e.g. dipole matrix elements) in ensembles of
active structures.
For our full time-domain simulations we have opted for the approach (2.37)–(2.38) as
we consider its numerical implementation to be more efficient compared to the two other
formulations. With an appropriate discretisation and iteration scheme it is possible to
time-step the material gain dynamics in synchronism with Maxwell curl equations and
to avoid computationally costly Runge-Kutta or predictor-corrector integration schemes
(see Section A.2). This is significant as we are not focussed on simple two-level systems
but rather on band-resolved gain models. It enables us to simulate semiconductor laser
and amplifier structures (typical multi-scale systems) on time scales of up to a few
nanoseconds which is necessary due to the rather slow macroscopic carrier dynamics
and recombination or relaxation processes.
2.6 Phenomenological Terms and Additional
Many-Body Hamiltonians
The Bloch equations derived in the previous sections describe only the coherent interac-
tion between the two-component carrier plasma and an external electric field. To gain a
more realistic description of a semiconductor laser or amplifier device other important,
mostly incoherent processes have to be incorporated into the laser model. For example
diverse recombination processes in semiconductor quantum wells are known to drasti-
cally reduce the efficiency of light-emitting structures [86]. The main recombination
processes and carrier loss channels discussed in literature [2,53] are nonradiative recom-
bination, spontaneous emission, Auger recombination, carrier leakage and thermionic
2.6 Phenomenological Terms and Additional Many-Body Hamiltonians 27
emission out of the optically active states. In imperfect semiconductors impurities, crys-
tal defects and surfaces are responsible for states with energies deep within the band
gap. These traps allow for nonradiative recombination. The most common mechanism
is called Shockley-Read-Hall recombination. The process of spontaneous emission also
removes carriers which are then no longer available for coherent stimulated emission into
the laser mode. The radiative recombination rate in direct band gap semiconductors is
proportional to the product of γspij,k with n
e
ii,kn
h
jj,−k, and as two particles are involved
it is often referred to as bimolecular recombination. However, for high densities this
process is not proportional to N2 because of the degenerate character of the distribution
functions (for more details see Section 4.4). For high densities and longer wavelength
material systems the mechanism of Auger recombination becomes more and more im-
portant. One distinguishes between direct and phonon-assisted processes and classifies
the interactions wherein four carrier states are involved [87] by the incorporated bands
of the initial and final states [2]. In our simulations we include the different carrier loss
channels by
∂t n
e
ii,k
∣∣
loss
= −γnrneii,k −
∑
j
γspij,kn
e
ii,kn
h
jj,−k − γe,Augerii,k , (2.47)
γspij,k =
2n
3ǫ0~π
|Mehij |2
1
c3~3
(
Eei,k + E
h
j,−k + Egap
)3
. (2.48)
γspij,k is evaluated using the Weißkopf-Wigner theory. The Auger recombination is nor-
mally implemented on the macroscopic level
∂tN = −ΓAugerN = −ΓAugerN3. (2.49)
The different recombination processes (and loss channels) and the total carrier lifetime,
which is typically assumed as τ = (A+BN +CN2)−1, ∂tN = −1/τN , strongly depend
on the actual used semiconductor material system, the grown heterostructure, and the
applied operating conditions, such as temperature or carrier density (pumping). High
pressure techniques are a powerful diagnostic tool as they allow for a change of the band
gap and the laser frequency [88]. This technique can be used to investigate and optimise
semiconductor-based laser devices.
To compensate for the various loss channels and to build up inversion and reach the
threshold density, carriers and energy have to be injected into the active regions. This
is essential for lasing, i.e. for the establishment of coherent stimulated emission. An
applied electric contact (over dact) results in the macroscopic carrier pump rate
Λ =
ηJ
e
, (2.50)
where dact describes the thickness of the active area and η the efficiency of the carrier
injection. One assumes that the carriers reach the quantum wells in a thermalised state,
but because of the Pauli exclusion principle for fermionic particles there has to be an
unoccupied final state available. The effective carrier pump includes this pump blocking
28 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
and is described by
∂t n
e
ii,k
∣∣
pump
= Λeii,k = Λ
f eii,k
(
1− neii,k
)
∑
i 1/A
∑
k f
e
ii,k
(
1− neii,k
) . (2.51)
Besides the electrical pumping over current contacts, in novel laser structures like ver-
tical extended (external) cavity surface-emitters [16, 17] optical pumping schemes are
discussed. The central frequency of the pumping light has to be chosen carefully: The
pumping wavelength has to be smaller than the lasing wavelength. The coupling to the
active medium is stronger for states higher in the energy band as the electronic density
of states is larger. However, this also results in increasing thermal energy flows. The
optical pumping term may be written as14
∂t n
e
ii,k
∣∣
pump
= −
∑
j
∣∣Mehij · E˜pump∣∣2
2~2
γpij,k(
γpij,k
)2
+ (Ωij,k − ωpump)2
(
neii,k + n
h
jj,−k − 1
)
.
(2.52)
The last phenomenological terms we want to introduce are relaxation rates for the
distribution functions and (interband) polarisations which are measures how fast the
system relaxes towards the quasi-equilibrium state. We will see that these processes are
due to scattering interactions and will discuss these correlation contributions in more
detail in a later section. At this point we introduce the relaxation time approximation
by imposing simple scattering rates
∂t n
e
ii,k
∣∣
relax
= −γe (neii,k − f eii,k) , (2.53)
∂t pji,k|relax = −γppji,k, (2.54)
which account for relaxation to quasi-equilibrium.
One of the most characteristic and distinguishing properties of semiconductor gain
media is the importance of many-body interactions [3,48]. A general two-particle Hamil-
tonian in second quantisation is given by [89]
H =
1
2
∫
d3xd3x′ψ†(x)ψ†(x′)W (x,x′)ψ(x′)ψ(x), (2.55)
wherein W (x,x′) is the matrix element in real space representation. An important case
is the Coulomb interaction between charged particles
V (r− r′, z − z′) = e
2
4πǫ0ǫ
1√
(r− r′)2 + (z − z′)2 . (2.56)
14From a physical point of view the optical pumping as stimulated absorption process should be mod-
elled exactly in the same way the coherent process of stimulated emission is treated, i.e. by the optical
Bloch equations with an associated dynamical polarisation response at the pumping frequency. This
is not possible in commonly applied frequency-/time-domain models, but with our band-resolved
full time-domain approach.
2.6 Phenomenological Terms and Additional Many-Body Hamiltonians 29
Using the mode expansion (2.9) we can transform the above interaction Hamiltonian
to momentum space. The orthonormality of the plane waves (the expansion functions)
secures the total momentum conservation of the interaction, and the integrations over the
envelope functions modifies the interaction strength of the quasi two-dimensional system.
The expansion would give rise to 16 different combinations but terms with an odd number
of electron and hole operators (describing processes like Auger recombination or impact
ionisation), terms changing the number of particles in the conduction and hole bands, and
interband exchange interaction terms (arising from mixed charge densities) are neglected.
One ends up with a carrier-carrier Hamiltonian which has the form
Hcarrier-carrier =
1
2
∑
k,k′,q
∑
i1,i2,i3,i4
Vi1i2i3i4(q)c
†
i1,k+q
c†i2,k′−qci3,k′ci4,k
+
1
2
∑
k,k′,q
∑
j1,j2,j3,j4
Vj4j3j2j1(q)d
†
j1,k+q
d†j2,k′−qdj3,k′dj4,k
−
∑
k,k′,q
∑
i1,j1,j2,i2
Vi1j2j1i2(q)c
†
i1,k+q
d†j1,k′−qdj2,k′ci2,k. (2.57)
The Coulomb matrix elements in momentum space are given by [64]
Vn1n2n3n4(q) =
e2
2ǫ0ǫ
1
A
1
q
Fn1n2n3n4(q). (2.58)
The above introduced form factors are defined as [90]
Fn1n2n3n4(q) =
∫ ∞
−∞
dzdz′φ∗n1(z)φ
∗
n2
(z′)e−q|z−z
′|φn3(z
′)φn4(z). (2.59)
In this work we also consider the energy exchange mechanism between carriers and the
crystalline lattice (carrier cooling) as the interactions of the quantised lattice oscillation
modes of the polar semiconductor medium with the electronic subsystem15. The most
important scattering processes are related to longitudinal optical phonons due to the
polar Fro¨hlich interaction. The interaction Hamiltonian is given by
Hcarrier-phonon =
∑
k,q,qz
∑
i1,i2
[
γi1i2(q, qz)c
†
i1,k+q
bq,qzci2,k + γ
∗
i1i2
(q, qz)c
†
i2,k
b†q,qzci1,k+q
]
+
∑
k,q,qz
∑
j1,j2
[
γj1j2(q, qz)d
†
j1,k+q
bq,qzdj2,k + γ
∗
j1j2
(q, qz)d
†
j2,k
b†q,qzdi1,k+q
]
,
(2.60)
15Details of the lattice unit cell define the vibrational properties of a semiconductor, the dynamics can be
composed of different (independent) normal modes, leading to distinctive phonon dispersion branches
(b†q,qzbq,qz ). These branches are classified into acoustic and optical, longitudinal and transverse
phonons, the possible interaction processes can be divided into polar and that over deformation
potentials [5, 66]. In this work the effect of the confinement structure on the phonons is neglected
[64]. As only longitudinal phonons couple effectively to electrons and holes and since in the long-
wavelength limit around the Γ point the acoustic branches (with ωq,qz ≈ cl−aq) are vanishing, we
do only consider longitudinal optical phonons.
30 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
where we are assuming bulk phonons with a constant dispersion relation ωq,qz = ωl-o. An
interaction process of the carriers with phonons is possible under absorption and under
emission of a longitudinal optical phonon. The strength of the interaction is given by
the matrix elements
γi1i2(q, qz) = −
i
ǫ0
√
e2~
2γωl-oALref
1√
q2 + q2z
∫ ∞
−∞
dzφ∗i1(z)φi2(z)e
iqzz, (2.61)
γ =
1
ω2l-oǫ0
(
1
ǫ∞
− 1
ǫs
)−1
, (2.62)
γj1j2(q, qz) =
i
ǫ0
√
e2~
2γωl-oALref
1√
q2 + q2z
∫ ∞
−∞
dzφj1(z)φ
∗
j2
(z)eiqzz. (2.63)
2.7 Many-Body Interactions—Hartree-Fock Terms
In this section we will show the consequences that arise when one takes the step from
a free non-interacting electron-hole plasma, whose dynamics is described by a number
of uncoupled Bloch equations, to a more realistic model for the dynamics of the gain
medium. The many-body interactions strongly affect the optical and electronic proper-
ties of the active medium. The name comes from the fact that more than one particle
or coupled electron-hole quasi-particle is involved. The Coulomb interaction appears as
interband attraction between electrons and holes and as intraband repulsion, whereas
the interaction with phonons couples the subsystem carriers with the lattice. In the
derivation of the equations of motion for the single particle expectation values, namely
distribution functions and polarisations, the Hamiltonian for the Coulomb interaction
couples the dynamics of these variables with higher order correlations, with products
of four fermionic operators. The differential equations for these new variables would
be governed by products of six operators, and so on. An infinite hierarchy of coupled
differential equations is the result. There are different levels of approximation to close
the set of equations and to truncate the hierarchy [4]. We will start with the lowest
contributions of the Coulomb interactions, the so-called mean-field approximation or
Hartree-Fock terms. As starting point we evaluate the Heisenberg equation, calculate
the commutators of the one particle operators with the carrier-carrier Hamiltonian and
compute the expectation values
∂t n
e
i1i2,k
∣∣
carrier-carrier
=− i
~
∑
k′,q
∑
i5,i6,i7,i8
Vi5i6i7i8 (q)
[
δi2,i5
〈
c†i1,kc
†
i6,k′
ci7,k′+qci8,k−q
〉
−δi1,i8
〈
c†i5,k−qc
†
i6,k′+q
ci7,k′ci2,k
〉]
+
i
~
∑
k′,q
∑
i5,j6,j7,i8
Vi5j7j6i8(q)
[
δi2,i5
〈
c†i1,kd
†
j6,−k′−q
dj7,−k′ci8,k−q
〉
−δi1,i8
〈
c†i5,k−qd
†
j6,−k′
dj7,−k′−qci2,k
〉]
. (2.64)
2.7 Many-Body Interactions—Hartree-Fock Terms 31
We realise that on the right-hand side of the equation new quantities, namely expectation
values of products of four operators emerge. If one does not want to introduce new
variables one may truncate the hierarchy by factorising these new measures in all possible
combinations of the expectation values of products of two operators (the basic dynamical
variables). As an example we could factorise the expression16
〈
c†i1,kc
†
i6,k′
ci7,k′+qci8,k−q
〉
≈ δq,0nei1i8,knei6i7,k′ − δk′,k−qnei1i7,knei6i8,k−q. (2.65)
This level of truncation is called Hartree-Fock approximation17. The additional terms
to the optical Bloch equations can be written down in a very concise way by introducing
the self-energy matrices arising from electron-electron or hole-hole interaction terms, and
internal fields resulting from the electron-hole Coulomb interaction [49,50]
δEei1i2,k = −
∑
k′
∑
i3,i4
Vi1i3i2i4(k− k′)nei3i4,k′ , (2.66)
δEhj1j2,−k = −
∑
k′
∑
j3,j4
Vj4j2j3j1(k− k′)nhj3j4,−k′ , (2.67)
∆i1j1,k = −
∑
k′
∑
i2,j2
Vi1j2j1i2(k− k′)pj2i2,k′ . (2.68)
One result is that different k states are now strongly coupled and the equations become
nonlinear. Thereby, single particle energies are renormalised
Eei1i2,k = E
e
i1,k
δi1,i2 + δE
e
i1i2,k
, (2.69)
Ehj1j2,−k = E
h
j1,−k
δj1,j2 + δE
h
j1j2,−k
. (2.70)
This means that with increasing carrier density the repulsive interaction leads to lower
energies and reduced transition frequencies. Furthermore, the optical Rabi frequency is
shifted due to the field renormalisation
ΩRabii1j1,k =
Ui1j1,k
~
=
1
~
(
Mehi1j1 · E−∆i1j1,k
)
. (2.71)
The attractive electrostatic interaction between electrons and holes effects in a higher
Rabi frequency, the so-called the Coulomb enhancement. At low temperature and low
density conditions this term should not be neglected as the resulting excitonic resonances
are then dominant effects. In summary, the dynamics of an interacting electron-hole
plasma in an optical field (without higher correlations) can be described by the multi-
16Below direct Hartree terms with δq,0 will be neglected.
17For an expectation value of a product of four operators (on the right-hand side) we can specify the
general approach ∂t 〈AB〉 = 〈CDEF 〉H-F + (〈CDEF 〉 − 〈CDEF 〉H-F) = 〈CDEF 〉H-F + δ 〈CDEF 〉.
The contributions beyond Hartree-Fock are called higher correlations, the Heisenberg equation would
link the dynamical evolution of 〈CDEF 〉 with expectation values of products of six operators.
32 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
subband semiconductor optical Bloch equations18
∂tn
e
i1i2,k
= − i
~
∑
i3,i4
(
Eei2i4,kδi1,i3 − Eei3i1,kδi2,i4
)
nei3i4,k
+
i
~
∑
j
(
Ui2j,kp
∗
ji1,k
− U∗i1j,kpji2,k
)
, (2.72)
∂tn
h
j1j2,−k
= − i
~
∑
j3,j4
(
Ehj2j4,−kδj1,j3 − Ehj3j1,−kδj2,j4
)
nhj3j4,−k
+
i
~
∑
i
(
Uij2,kp
∗
j1i,k
− U∗ij1,kpj2i,k
)
, (2.73)
∂tpj1i1,k = −
i
~
∑
i2,j2
(
Eei1i2,kδj1,j2 + E
h
j1j2,−k
δi1,i2 + Egapδi1,i2δj1,j2
)
pj2i2,k
− i
~
(∑
i
Uij1,kn
e
ii1,k
+
∑
j
Ui1j,kn
h
jj1,−k
− Ui1j1,k
)
, (2.74)
which describe an ensemble of driven and coupled oscillators.
2.8 Many-Body Interactions—Correlation Terms
Apart from the discussed modifications of the optical Bloch equations by adding the
coherent processes of energy and field renormalisation other important consequences of
the Coulomb interaction arise. Carrier-carrier scattering drives (on a sub-picosecond
time scale) the dynamical distribution functions of electron and hole states towards the
quasi-equilibrium Fermi-Dirac distributions. These processes also dominantly contribute
to the fast optical dephasing or loss of coherence due to a decay of the various polarisa-
tions. Another effect of the Coulomb interaction is the plasma screening, this means that
due to the presence of all other charged carriers an individual carrier experiences a weak-
ened interaction. The spectra of the optical gain and induced refractive index change
are altered by diagonal and nondiagonal terms arising from the many-body interactions.
Finally, the carrier-phonon interaction couples the hot carrier system with the colder
crystalline lattice which in turn is coupled to the bath. Similar to the Coulomb inter-
action, these scattering processes redistribute the carriers and destroy coherence within
the carrier system. The specified processes can be reproduced by the many-body ap-
proach, but we will outline an approximative ansatz where some of these correlations
are only treated in a phenomenological way. This approach is necessary because of the
complexity of the investigated systems, namely by reason that in the simulations the
energy-resolved dynamics has to be calculated on a huge spatial grid for a long time
18When we disregard intraband (intersubband) polarisations the renormalised equations have a more
simple structure δEeii,k = −
∑
k′
∑
i3
Vii3ii3(k−k′)nei3i3,k′ , ∂tneii,k = i/~
∑
j
(
Uij,kp
∗
ji,k−U∗ij,kpji,k
)
,
∂tpji,k = −i/~
(
Eeii,k + E
h
jj,−k + Egap
)
pji,k − i/~Uij,k
(
neii,k + n
h
jj,−k − 1
)
.
2.8 Many-Body Interactions—Correlation Terms 33
window. To give a rough idea about these higher correlation contributions and the used
assumptions, we calculate the dynamical equations for the product of four operators in
(2.65). In doing so we only evaluate the commutators with the single particle energy
Hamiltonian and the repulsive part of the Coulomb interaction
∂t
〈
c†i1,kc
†
i2,k′
ci3,k′+qci4,k−q
〉
= − i
~
(−Eei1,k − Eei2,k′ + Eei3,k′+q + Eei4,k−q) 〈c†i1,kc†i2,k′ci3,k′+qci4,k−q〉
− i
~
∑
k′′,q′
∑
i5,i6,i7,i8
{
Vi5i6i7i8(q
′)
[
δi4,i5
(
δi3,i6δk′+q,k′′−q′
〈
c†i1,kc
†
i2,k′
ci7,k′+q+q′ci8,k−q−q′
〉
−
〈
c†i1,kc
†
i2,k′
c†i6,k′′ci3,k′+qci7,k′′+q′ci8,k−q−q′
〉)
+ δi3,i5
〈
c†i1,kc
†
i2,k′
c†i6,k′′ci4,k−qci7,k′′+q′ci8,k′+q−q′
〉]
+ Vi8i7i6i5(q
′)
[
δi1,i5
(
−δi2,i6δk′,k′′
〈
c†i8,k−q′c
†
i7,k′+q′
ci3,k′+qci4,k−q
〉
+
〈
c†i8,k−q′c
†
i7,k′′+q′
c†i2,k′ci6,k′′ci3,k′+qci4,k−q
〉)
− δi2,i5
〈
c†i8,k′−q′c
†
i7,k′′+q′
c†i1,kci6,k′′ci3,k′+qci4,k−q
〉]}
. (2.75)
Other terms arising from −∑k,k′,q∑i5,j6,j7,i8 Vi5j7j6i8(q)c†i5,k+qd†j6,k′−qdj7,k′ci8,k have a
similar structure. We abbreviate the deviations of the full correlation terms from the
factorised Hartree-Fock approximation terms with δ 〈〉 (t). An example of this deviation
and its time derivative ∂tδ 〈〉 (t) is
δ
〈
c†i1,kc
†
i2,k′
ci3,k′+qci4,k−q
〉
=
〈
c†i1,kc
†
i2,k′
ci3,k′+qci4,k−q
〉
+ δk′,k−qn
e
i1i3,k
nei2i4,k−q,
∂tδ
〈
c†i1,kc
†
i2,k′
ci3,k′+qci4,k−q
〉
= ∂t
〈
c†i1,kc
†
i2,k′
ci3,k′+qci4,k−q
〉
(2.76)
+ δk′,k−q
(
∂t
(
nei1i3,k
)
nei2i4,k−q + n
e
i1i3,k
∂t
(
nei2i4,k−q
))
.
This can be written in the general form
∂tδ 〈〉4 = −iΩδ 〈〉4 −
i
~
I [〈〉2] , Ω = Ωei1i2i3i4,kk′q, (2.77)
where the subscript indicates the number of operators. Equation (2.77) has to be solved
along with the equation of motion of the basic dynamical variables, that is
∂t 〈〉2 = 〈〉2,optical Bloch terms + 〈〉4,H-F + δ 〈〉4 . (2.78)
To truncate the hierarchy of coupled differential equations involving expectation values,
the inhomogeneity I(t) is factorised into products of distribution functions, intraband
polarisations and interband polarisations. Self-energies for correlation contributions and
a self-consistent treatment of plasma screening are disregarded. Indeed, the mechanism
of screening is included in the above Coulomb interaction terms, but for computational
34 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
efficiency we model this mechanism merely by replacing the bar potential by an effective
screened potential. Furthermore, we neglect terms with q = 0,q′ = 0,k = k′ in the
factorisation and disregard exchange terms with V (k−k′−q) [64]. We see from equation
(2.77) that quantum mechanical scattering processes are not instantaneous as assumed
in the usual Boltzmann ansatz. Energy-time uncertainty, transitions not allowed by
classical selection rules and Coulombic memory effects play a role. The formal integration
of (2.77) reads
δ 〈〉 (t) = δ 〈〉 (t0)e−iΩ(t−t0) − i
~
∫ t
t0
dt′e−iΩ(t−t
′)I(t′). (2.79)
It is therefore possible to transform the microscopic equations of motion into non-
Markovian integro-differential equations with memory effects for the single particle den-
sity matrices. In active laser systems which require rather high carrier densities neglect-
ing Coulombic memory effects and applying the Markov approximation is considered a
reasonable approach until we deal with extremely short time scales. We assume an adi-
abatic activation of the interaction by introducing a phenomenological decay constant,
Ω→ Ω− iγ, γ → 0. As a result we obtain
δ 〈〉 (t) = lim
γ→0
(
− i
~
1
iΩ + γ
I(t)
)
= −i
(
πδ(~Ω)− i P
~Ω
)
I(t), (2.80)
wherein P describes the Cauchy integral principal value, and δ(~Ω) secures the strict
energy conservation of the Coulomb interaction on long time scales (the generalised δ
or Heitler-Zeta function). The correlation contributions can be expressed by scattering
matrices Γ and internal fields ∆ [64]. In summary, the scattering terms for the electron
expectation values (and similar for the hole matrices) and the interband polarisations
can be written as
∂t n
e
i1i2,k
∣∣
relax
=
∑
i5
[−Γee,outi2i5,knei1i5,k − Γee,out∗i1i5,k ne∗i2i5,k
+ Γee,ini2i5,k
(
δi1,i5 − nei1i5,k
)
+ Γee,in∗i1i5,k
(
δi2,i5 − ne∗i2i5,k
)]
− i
~
∑
j5
[
∆ehi2j5,kp
∗
j5i1,k
−∆eh∗i1j5,kpj5i2,k
]
, (2.81)
∂t pj1i1,k|relax = −
∑
i5
(
Γee,outi1i5,k + Γ
ee,in
i1i5,k
)
pj1i5,k −
∑
j5
(
Γhh,outj1j5,k + Γ
hh,in
j1j5,k
)
pj5i1,k
+
∑
i5
[
Γhe,outj1i5,kn
e
i5i1,k
+ Γhe,inj1i5,k
(
δi5,i1 − nei5i1,k
)]
+
∑
j5
[
Γeh,outi1j5,kn
h
j5j1,−k
+ Γeh,ini1j5,k
(
δj5,j1 − nhj5j1,−k
)]
. (2.82)
A numerical simulation of the above equations is still complicated. However, in semi-
conductor laser systems operated at optical frequencies, it is reasonable to disregard
2.8 Many-Body Interactions—Correlation Terms 35
scattering processes comprising intraband polarisations, and to neglect nonlinear terms
in the interband polarisations. On that level the scattering is described by the quantum
Boltzmann collision terms in second Born approximation
∂t n
e
ii,k
∣∣
relax
= −2Γee,outii,k neii,k + 2Γee,inii,k
(
1− neii,k
)
= −2
(
Γee,outii,k + Γ
ee,in
ii,k
)
neii,k + 2Γ
ee,in
ii,k . (2.83)
The scattering matrices for the out-scattering of an electron in a state characterised by
the momentum k and the subband index i1 are given by
k→ k− q, k′ → k′ + q :
Γee,outi1i1,k =
π
~
∑
k′,q
∑
i2,i3,i4
δ
(−Eei1,k − Eei2,k′ + Eei3,k′+q + Eei4,k−q) |Vi1i2i3i4(q)|2
× nei2i2,k′
(
1− nei3i3,k′+q
) (
1− nei4i4,k−q
)
+
π
~
∑
k′,q
∑
j2,j3,i4
δ
(−Eei1,k − Ehj2,−k′ + Ehj3,−k′−q + Eei4,k−q) |Vi1j3j2i4(q)|2
× nhj2j2,−k′
(
1− nhj3j3,−k′−q
) (
1− nei4i4,k−q
)
. (2.84)
In a similar way we can determine the matrices of in-scattering contributions
k− q→ k, k′ + q→ k′ :
Γee,ini1i1,k =
π
~
∑
k′,q
∑
i2,i3,i4
δ
(−Eei1,k − Eei2,k′ + Eei3,k′+q + Eei4,k−q) |Vi1i2i3i4(q)|2
× (1− nei2i2,k′)nei3i3,k′+qnei4i4,k−q
+
π
~
∑
k′,q
∑
j2,j3,i4
δ
(−Eei1,k − Ehj2,−k′ + Ehj3,−k′−q + Eei4,k−q) |Vi1j3j2i4(q)|2
× (1− nhj2j2,−k′)nhj3j3,−k′−qnei4i4,k−q. (2.85)
Scattering processes and associated matrices mediated through the Coulomb interaction
have a typical Boltzmann structure (see Figure 2.3): The δ function in the product
guarantees for the strict energy conservation (on long time scales), the strength of the
interaction is quantified by |V (q)|2. The single scattering process is proportional to the
occupation of the initial states and the available unoccupied final states (Pauli principle),
and finally we have to sum over all possibilities. The collision terms in the equation for
the interband polarisation [51] are given by
∂t pji,k|relax = −
∑
k′
Λij,kk′pji,k′ −
∑
k′
∑
i5 6=i,j5 6=j
Λ˜iji5j5,kk′pj5i5,k′ . (2.86)
The diagonal relaxation elements are determined by the above given matrices (2.84)–
(2.85) as Λij,kk = Γ
ee,out
ii,k + Γ
ee,in
ii,k + Γ
hh,out
jj,k + Γ
hh,in
jj,k . The off-diagonal elements Λji,kk′
19
19These nondiagonal elements determine in parts the shape of the gain spectra [4]: Calculating the
gain spectra with solely diagonal relaxation terms leads to noticeable amplification or gain below
36 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
initial
states
final
states
out-scattering
initial
states
final
states
in-scattering
nek 1− nek−q
n◦k′ 1− n◦k′+q
nek−q 1− nek
n◦k′+q 1− n◦k′
|V sc(q)|2|V sc(q)|2
k→ k− q k− q→ k
Figure 2.3: In order to visualise and describe the correlation contributions from
quantum theoretical many-body interactions the schematic Feynman diagrams for out-
scattering and in-scattering processes, respectively, are given. This illustrates the char-
acteristic structure of the scattering matrices: The initial states (before the interaction)
are plotted on the left, the final states (on the right) have to be available (as per Pauli
exclusion principle for fermionic particles). The strength of the interaction is given by
|V sc(q)|2, and we do sum over all possible scattering partners ◦ = e, h and∑k′,q (and in
the case of quantum confined structures over all thinkable combinations of the discrete
subband quantum numbers).
have a similar structure and partially compensate the impacts of the diagonal terms [4].
Apart from the conservation of the total momentum, the carrier-carrier scattering does
neither change the particle numbers (the carrier density) nor the total kinetic energy. If
a hot carrier non-equilibrium distribution is generated, for example by high-power pulse
excitation or pumping at high energy states, the kinetic energy density (as the second
moment of the distribution) is characterised by an effective plasma temperature which
may be well above the lattice temperature (by some ten Kelvin) [91].
The process of thermal relaxation is characterised by the coupling of the carrier to
the lattice subsystem and is mediated by the Fro¨hlich interaction. There is no coherent
the band edge. That is clear from the structure of the polarisation differential equation as a sum of
(independent) damped harmonic oscillators which can also be driven (because of the finite decoher-
ence time) at frequencies unequal the respective resonance frequency. The more general ansatz for
the decoherence terms removes this problem.
2.8 Many-Body Interactions—Correlation Terms 37
(Hartree-Fock) contribution of this interaction. Using similar approximations as above
the scattering matrices in (2.81) and (2.82) can be written as
k→ k− q : Γee,outi1i1,k =
π
~
∑
q,qz
∑
i2
∑
±
δ
(−Eei1,k + Eei2,k−q ± ~ωl-o) |γi1i2(q, qz)|2
× (1− nei2i2,k−q)
(
nph,q,qz +
1
2
± 1
2
)
, (2.87)
k− q→ k : Γee,ini1i1,k =
π
~
∑
q,qz
∑
i2
∑
±
δ
(−Eei1,k + Eei2,k−q ∓ ~ωl-o) |γi1i2(q, qz)|2
× nei2i2,k−q
(
nph,q,qz +
1
2
± 1
2
)
. (2.88)
The Boltzmann collision integrals contain factors guaranteeing for the energy conser-
vation, describing the strength of the phonon-carrier interaction, and the probability
factors for initial and final states. A scattering is possible under emission (upper sign)
and under absorption (lower sign) of a longitudinal optical phonon. The transfer of
kinetic energy to the phonon bath results in a cooling of the hot carrier system. To
simplify the problem, we assume the phonons to be in an equilibrium state qualified by
the Bose-Einstein distribution
nph,q,qz =
1
exp (β~ωl-o)− 1 , β =
1
kBTl-o
. (2.89)
In typical semiconductor lasers with high carrier densities, equation (2.83) may be
further approximated [37]. One usually can assume that carrier-carrier scattering drives
the distribution on a very fast sub-picosecond time scale towards the quasi-equilibrium
distribution. For these distributions the correlation contributions vanish
2Γee,outii,k {f ◦k}f eii,k = 2Γee,inii,k {f ◦k}
(
1− f eii,k
)
. (2.90)
The above relation describes the principle of detailed balance, for which the scattering
into a state is exactly balanced by the out-scattering term. If the deviations of the
distribution from the quasi-equilibrium function are relatively small (and if we look at
time scales of some times the characteristic momentum scattering time), one can linearise
the Boltzmann equation in this deviation and introduce relaxation rates
γeii,k = 2
(
Γee,outii,k {f ◦k}+ Γee,inii,k {f ◦k}
)
, (2.91)
∂t n
e
ii,k
∣∣
relax
= −γeii,k
(
neii,k − f eii,k
)
. (2.92)
This ansatz with k-dependent scattering times (rates) fails to preserve the carrier den-
sity, and there is another criticism on this widely used approach [4]. Relaxation times
descriptions are a good approximation in the case of analysing a system interacting with
an infinitely big bath. However, in a semiconductor laser there is no distinction be-
tween a system and a bath, and the reservoir carriers are part of the system [4]. Recent
38 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
numerical investigations [92] (and Appendix B) show through a comparison with a dy-
namical treatment of the scattering interactions that the relaxation rate approximation
is a surprisingly well-justified approach. We conclude that (2.92) is a reasonable accurate
description except for the cases of the propagation of few femtosecond or very strong
optical pulses.
In a last step, we would like to include the screening effect into our model equations.
We will do this using a phenomenological approach by replacing the bare Coulomb
potential by the screened potential
V sc(q, ω) =
V (q)
ǫ(q, ω)
. (2.93)
For simplicity, we neglect for a moment the dependence on four subband quantum num-
bers. The spatial and spectral dispersion in the density dependent (longitudinal) dielec-
tric function of the two-component plasma are given by the Lindhard formula, which is
derived within the random phase approximation (RPA), with ω → ω + iδ [25]
ǫ(q, ω) = 1− V (q)
∑
k
[∑
i
neii,k+q − neii,k
Eei,k+q − Eei,k − ~ω
+
∑
j
nhjj,−k−q − nhjj,−k
Ehj,−k−q − Ehj,−k − ~ω
]
.
(2.94)
In this work the long-wavelength (|q| → 0) and the static (ω → 0) limits of the Lindhard
formula are calculated. Two approximations are discussed for the screened Coulomb
potential [36]. The (total) inverse screening length is calculated by
κ =
e2
2ǫ0ǫπ~2
[
me
∑
i
f eii,k=0 +mh
∑
j
fhjj,−k=0
]
. (2.95)
We can write the free-carrier dispersion as
ǫ(q→ 0, ω → 0) = 1 + κ
q
, (2.96)
which would result in the simple Yukawa potential. In the single effective plasmon
pole approximation one replaces the continuum of resonances, the multiple poles in the
Lindhard formula, by an effective plasmon frequency and get an expression similar to the
classical Drude dielectric function with a numerical constant C which is usually taken
between 1 and 4 [4, 48]
V sc(q) = V (q)
(
1− ω
2
pl
ω2q
)
, (2.97)
ω2pl =
e2N
2ǫ0ǫmr
q, ω2q = ω
2
pl
(
1 +
q
κ
)
+
C
4
(
~q2
2mr
)2
. (2.98)
This phenomenologically defined screened potential modifies all our above calculations,
for example the band gap renormalisation (shrinkage), which in the screened Hartree-
Fock approximation is given by δESXij,k = δE
e,SX
ii,k + δE
h,SX
jj,−k, or the renormalisation of
2.8 Many-Body Interactions—Correlation Terms 39
Figure 2.4: Microscopic dephasing rates (as integrals over all possible individual scat-
tering processes) of the interband polarisation γp11,k versus the carrier density and the
momentum due to carrier-carrier scattering (upper graph) and due to the interaction of
carriers with longitudinal optical phonons (below). These momentum relaxation rates
γk(N, T ) are incorporated into the spatially resolved time-domain simulations using
lookup tables. The sharp structures in γ
p(ph)
k are caused by the restrictive threshold con-
dition for out-scattering processes which involve the emission of an optical phonon (and
for in-scattering processes with the absorption of a phonon). For realistic laser operat-
ing conditions (that imply states near the band edge and a density of the active GaAs
quantum well structure of around the threshold carrier sheet density ≈ 2 · 1012 cm−2)
typical dephasing times for the screened Coulomb interaction are 50 fs, and for the
Fro¨hlich interaction 350 fs. For more details see Section 4.1. In the calculations of the
scattering matrices and rates due to the carrier-carrier interaction we have neglected ex-
change contributions. At high densities these terms may weaken the overall interaction
strengths and reduce the relaxation rates.
40 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
the Rabi frequency. In addition to the exchange shift of the single particle energies, a
further term arises due to the presence of the two-component plasma: The Debye shift20
or Coulomb hole self-energy
δECH =
∑
q6=0
[V sc(q)− V (q)] (2.99)
contributes to a change in transition energies ~Ωij,k(N) = E
e
i,k+E
h
j,−k+ δE
SX
ij,k+ δE
CH+
Egap.
2.9 Quantum Dot Lasers and Multi-Level Bloch
Equations
In recent years the application of the size quantisation concept has been extended to
two dimensions (quantum wires [28, 57]) and to three dimensions, namely to quantum
dot systems [29]. This process has been motivated by the success of optically active
semiconductor multi-layer heterostructures, such as quantum well-based lasers and am-
plifiers. The electronic and optoelectronic properties of quantum dots (fabricated by
self-organised growth of islands on the nanometer scale) are rather comparable to that
of single atoms than to the characteristics known from solid state semiconductor systems.
Accompanied with the electronic quantum confinement is a strong modification of the
electronic density of states, states at high energies are relatively suppressed, and states
around the band edge (where the lasing process takes place) are pronounced [25]. For this
reason quantum dot lasers should outperform semiconductor bulk or quantum well laser
structures with respect to parameters like the transparency or threshold current, the
material gain, the linewidth enhancement factor or the beam quality [29,93]. Due to the
discrete energy level structure one would expect only very few (intradot) relaxation and
decoherence channels as the scattering processes are not allowed by reason of strict en-
ergy conservation (in Markov approximation). Recent experimental investigations [94],
however, show ultrafast carrier relaxation and recovery of the gain dynamics and fast
dephasing times, both of which makes quantum dot structures promising materials for
high-speed applications such as amplifiers or saturable absorbers.
Starting from a paraxial semiconductor quantum dot Maxwell Bloch model [95] we will
specify equations of the dynamics of the carriers within a quantum dot interacting with a
classical electromagnetic field. The atomic-like gain system may be modelled by multi-
level Bloch equations, including additional phenomenological terms. The individual
two-level systems (i.e. dipole transitions) are characterised by the energy levels i, j for
electrons and holes, respectively, with energetic degeneracies dei , d
h
j , which account for
20For quantum well structures with a quasi two-dimensional (two-component) plasma this shift is
approximately given by δECH = −e2κ/(4πǫ0ǫ) ln
(
1+
√
8e2Nmr/(Cǫ0ǫ~2κ3)
)
[48], but the exchange
contributions to the shifts of the transition energies and the field renormalisation (the Coulomb
enhancement) have to be calculated numerically applying for the screened potential with (2.98)
V sc(q) = e2/(2ǫ0ǫAq)F (q)
(
q/κ+ q3Cǫ0ǫ~
2/(8e2Nmr)
) (
1 + q/κ+ q3Cǫ0ǫ~
2/(8e2Nmr)
)−1
.
2.9 Quantum Dot Lasers and Multi-Level Bloch Equations 41
the maximum occupations nei ∈ [0, dei ], nhj ∈ [0, dhj ], by the transition frequencies Ωij,
the homogeneous broadenings γpij, and the dipole matrix elements M
eh
ij (measures of the
strength of the interaction). The dynamics is given by the differential equations of driven
damped harmonic oscillators, which here are written in full time-domain
∂2t pji + 2γ
p
ij∂tpji + ω
2
ijpji = −Ωij
M ehij E
~
(
nein
h
j − (dei − nei )(dhj − nhj )
)
= −Ωij
M ehij E
~
(
dhjn
e
i + d
e
in
h
j − deidhj
)
. (2.100)
The electron or hole interlevel polarisations have been neglected on the basis of their typ-
ically strong energetic non-resonant nature with the optical fields. The carrier dynamics
of the electron and hole populations is presented by
∂tn
e
i =
∑
j
2
M ehij E
~
1
Ωij
(
∂tpji + γ
p
ijpji
)
+ Λ
f ei (d
e
i − nei )∑
i f
e
i (d
e
i − nei )
− γnri nei −
∑
j
γspij n
e
in
h
j − γAugeri + ∂tnei |relax , (2.101)
∂tn
h
j =
∑
i
2
M ehij E
~
1
Ωij
(
∂tpji + γ
p
ijpji
)
+ ∂tn
h
j
∣∣
ph
+ ∂tn
h
j
∣∣
relax
. (2.102)
The electrical pumping term contains the Pauli blocking effect for fermionic parti-
cles, and quasi-equilibrium populations have been introduced. But the various (in-
terlevel) scattering, thermalisation and relaxation processes, many-body interactions
and renormalisation effects, the coupling to the wetting layer states by carrier cap-
ture (via the interaction processes phonons, Coulomb, Auger and impact ionisation)
and thermionic emission have to be addressed in more detail [95]. Furthermore, a
many-particle treatment would allow for a quantum statistical description of energy
level occupations and a more precise definition of blocking factors. A spectral summa-
tion results in N e = 2
∑
i n
e
i , N
h = 2
∑
j n
h
j , P = 2 · 2
∑
i,j M
eh
ij pji. The inclusion of
these active quantum dot measures into macroscopic field equations like the Maxwell
wave equation requires to treat ensembles of quantum dots, or define macroscopic ef-
fective quantum dots, and a conversion into carrier and dipole polarisation volume den-
sities D = ǫ0ǫE + ΓNQD/VrefP . The transition from the electron and hole representa-
tion to a more general electronic multi-level picture can be done by the replacement
nhj → dej − nej , dej = dhj , where the induced polarisation pji is driven by the appropriate
occupation inversion dejn
e
i − deinej .
A set of few uncoupled and independent two-level systems, each modelled by a ho-
mogeneously broadened harmonic oscillator in time-domain is equivalent to a sum of
Lorentzians in frequency-domain. The superpositions of Lorentzians can be used to cre-
ate an efficient realisation of the asymmetric semiconductor gain profile. The effective
semiconductor Bloch equations [12,55,96] are an approximative fit to the dielectric dis-
persion and absorption (or amplification), given by the many-body calculations of the
42 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
semiconductor permittivity ǫ(ω,N, T ) or electric susceptibility χ(ω,N, T ). For more de-
tails see the section on spectral summation 2.11. By treating the macroscopic fields such
as the carrier densities or occupations as dynamical variables, nonlinear behaviour and
the effect of gain saturation are included.
2.10 Fitting the Dielectric Function
We now consider more complicated active gain media, and try to understand the nature
of electromagnetic pulse interactions with these novel materials, which feature linear and
nonlinear dispersion and nonlinearities, over a broad frequency range. Possible systems
include optically active carbon nanotubes [56,97], organic [30] or polymer [31] materials
for lasers, metallic nanoparticles and one-dimensional nanostructures [27], or biological
materials subject to impulsive excitation [98]. These dispersive media will be modelled
within the finite-difference time-domain framework [99] by the Maxwell curl equations
curlE+ ∂tB = 0,
1
µ0
curlB− ǫ0ǫ∂tE = J+ ∂tP, (2.103)
and constitutive relations between electromagnetic excitations and the dielectric response
D(r, t) = ǫ0ǫ(r)E+P(r, t), P = ǫ0ǫχE, (2.104)
E˜(ω)e−iωt, P˜ (ω)e−iωt, J˜(ω)e−iωt, ∂t ↔ −iω; χ(ω) = P˜ (ω)
ǫ0ǫE˜(ω)
, ǫ(ω) = ǫ (1 + χ(ω)) .
(2.105)
The complex-valued dielectric function ǫ(ω) is crucial to understand the optical prop-
erties of metallic, dielectric systems. Its characteristics are associated with intraband
free-electron excitations and an additive complex contribution from interband transi-
tions. Theoretically this can be explained by the Drude model, and multiple-pole Debye
and Lorentz dispersions. A positive imaginary part of ǫ is associated with absorption, a
negative real part with reflection. The dispersion in frequency-domain is translated to a
temporally nonlocal response in the time-domain. Convolution relations or differential
equations connect the effective dielectric response with the excitation. The Kramers-
Kronig relations connect the real and imaginary part of a any complex-valued response
function of a linear and causal physical process. From Cauchy theorem for an analytic
and bound function in the upper half plane (from causality principle and analyticity
domain) we can give the reformulations for positive values of the frequency [69]
Re(ǫ(ω)) = ǫ+
2
π
P
∫ ∞
0
ω′Im(ǫ(ω′))
ω′2 − ω2 dω
′, Im(ǫ(ω)) = −2ω
π
P
∫ ∞
0
Re(ǫ(ω′))− ǫ
ω′2 − ω2 dω
′,
(2.106)
with ǫ = ǫbackground = ǫ∞. The empirical knowledge of the imaginary part of ǫ(ω)
from absorption studies and measurements enables the calculation of the real part. It
2.10 Fitting the Dielectric Function 43
turns out that Re(ǫ) is monotonically increasing between the absorption bands, but
shows anomalous dispersion within resonances. A fit over a given frequency range is
possible by a superposition of a finite number of Lorentz, Debye and Drude dispersions,
which will inherently satisfy causality requirements. Two efficient, rather complementary
approaches are known to model nonmagnetic, dispersive (and hence lossy, absorptive)
dielectric materials [58, 59, 72], such as insulators, semiconductors or metals21. We will
follow the auxiliary differential equations approach (ADE) which introduce polarisation
and induced polarisation current terms P,J ∈ R into the Maxwell equations (Ampe`re’s
law) which are the time-domain equivalent of the complex-valued dielectric constant
ǫ(ω) ∈ C in frequency-domain [58]. The ADE approach operates on real-valued fields.
The basic model to describe the material response of bound charge carriers to an applied
electromagnetic field is the Lorentz model. For a more general material response we
assume that l labels a discrete resonance with frequency ωl, with the damping factor
γl, and ∆ǫl as the relative permittivity increment or strength of the coupling. The
time-domain description is given by a damped driven harmonic oscillator
(
∂2t + 2γl∂t + ω
2
l
)
Pl = ǫ0∆ǫlω
2
l E. (2.107)
Transformed from the differential equations representation we attain the electric suscep-
tibility in frequency-domain
χ(ω) =
1
ǫ
∑
l
∆ǫlω
2
l ·
(
ω2l − ω2
(ω2l − ω2)2 + 4γ2l ω2
+ i
2γlω
(ω2l − ω2)2 + 4γ2l ω2
)
, (2.108)
χ(ω) ≈ 1
ǫ
∑
l
∆ǫlω
2
l ·
(
1
2ω
ωl − ω
(ωl − ω)2 + γ2l
+ i
1
2ω
γl
(ωl − ω)2 + γ2l
)
, (2.109)
where the second line is an approximation near the resonance frequency ω2l − ω2 ≈
2ω(ωl−ω). Other models discussed in literature can be derived from the Lorentz model.
In the slow rise time signal regime limit [72], that is ∂2t (ω
2)≪ ω2l , 2γl/ω2l → τd, one ob-
tains the Debye rotational relaxation model with real-valued single-poles. This material
model is widely applied as a result of the growing interest in electromagnetic pulse in-
teractions with biological (i.e. water-based) substances and accounts for the excitations
21The piecewise-linear recursive convolution (PLRC) method [58, 59] is based on a systematic evalu-
ation of ǫ0ǫ[χ(t) ∗ E(t)]. It incorporates the discretised form of the convolution integral into the
finite-difference time-domain paradigm Dn = ǫ0ǫE
n + ǫ0ǫ
∫ t
t0
dt′χ(t′)E(t − t′), t = t0 + n∆t. For
greater accuracy of second order, E(t− t′) is approximated by piecewise-linear line segments in the
integration of the time-domain convolution integrals. An efficient on-the-fly approach is realised by
the introduction of a new, additional time-dependent variable Ψn ∈ C [59], representing the recursive
accumulator (which may be complex-valued). An updating of E(t) is performed without explicitly
evaluating the convolutional sum (thereby it is not required to store the complete time history). It
can be reduced to a cumulative sum (to add the most recent contributions) that is updated at each
time step by Ψn(En, En−1,Ψn−1), En+1(En,Hn+1/2,Ψn).
44 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
of permanent dipole moments (as established in water)
(τd∂t + 1)Pd = ǫ0∆ǫdE, (2.110)
χ(ω) =
1
ǫ
∑
d
∆ǫd ·
(
1
1 + τ 2dω
2
+ i
τdω
1 + τ 2dω
2
)
, ǫ(ω) = ǫ+
∑
d
∆ǫd
1− iωτd . (2.111)
In the fast time response limit of the Lorentz model [72], ∂2t (ω
2) ≫ ω2l ,
∑
l∆ǫlω
2
l →
ω2m,pl, 2γl → 1/τm, the standard Drude model emerges as applicable for metals, plasmas
of free charge carriers or intraband transitions
J
.
= ∂tP →
(
∂t +
1
τm
)
Jm = ǫ0ω
2
m,plE, (2.112)
χf (ω) =
∑
m
(
− ω
2
m,plτ
2
m
1 + ω2τ 2m
+ i
ω2m,plτm
ω(1 + ω2τ 2m)
)
, ǫf (ω) = 1 + χf (ω)→ ǫ(ω) + ǫf (ω).
(2.113)
ωm,pl is the bulk plasmon frequency and τm a collision time. There is a considerable
interest to study light interacting with metallic nanoparticles [100], in particular, to in-
vestigate surface plasmon polaritons as resonance interactions of light with the electronic
charge density near the metal surface. Arrays of such particles could be used to confine
and guide light at the sub-diffraction dimension, as coupled plasmons are generated [101].
The above approaches to incorporate dynamic material responses into the Maxwell equa-
tions can be extended to model nonlinear dispersive media (D = ǫ0ǫE+P
linear+P nonlinear)
which are characterised by nonlinear susceptibility kernels χ(n)(t− t1, . . . , t− tn)22 [6].
2.11 Spectral Summation and Coupling to the Maxwell
Equations
In this section we establish the connection between our microscopic description of the
active gain medium in momentum space and the macroscopic real space description
22The third order nonlinear polarisations χ(3)(t− t1, t− t2, t− t3) (Kerr effect or inelastic Raman scat-
tering by phonons) are of particular interest [7], specifically convolution integrals (as the constitutive
relations) that depend on the product of electric field squared and χ˜(3) [58, 102]. Using the Born
Oppenheimer approximation [58] we may decompose the susceptibility function into a Kerr-type in-
stantaneous nonlinearity plus a Raman-type dispersive nonlinearity P nonlinear,(3) = PKerr+PRaman,
more precisely P nonlinear,(3) = ǫ0ǫE(t)
∑
p χ
(3)
0,p
∫ t
t0
dt′[E(t′)]2(αpδ(t− t′) + (1− αp)χ˜(3)p (t− t′)) (χ(3)0,p
is the strength of the third order nonlinearity, αp determines the ratio of the two types). The
ADE method allows for the modelling of arbitrary nonlinear dispersive media and is very effi-
cient (because in contrast to the PLRC ansatz finding the roots of nonlinear polynomial equa-
tions for the updated field values at each time step is not necessary). In close analogy to the
linear case, see for example (2.107), additional (polarisation) variables, the convolutions Rp(t) =
ǫ0χ
(3)
0,p(1−αp)[χ˜(3)p (t) ∗E2(t)] ∈ R are time stepped in synchronism with the Maxwell curl equations
by (∂2t + 2γp∂t + ω
2
p)Rp = ǫ0χ
(3)
0,p(1 − αp)E2(t). The Bloch equations models represent a different
character of nonlinear material description, nonlinear behaviour arises from the implementation of
additional variables, the carrier density and distributions, and the effects of hole burning.
2.11 Spectral Summation and Coupling to the Maxwell Equations 45
including the dynamics of optical, thermal and carrier fields. Macroscopic quantities
can be derived from microscopic variables by a spectral summation over the portion of
interest of the band structure. The macroscopic fields, on the other hand, specify the
quasi-equilibrium distribution functions with a parametric spatial dependency. The ther-
mal quasi-equilibrium is understood as a local quasi-stationary state that a subsystem,
e.g. electrons, tries to establish due to fast scattering processes. The various subsystems
and variables and their coupling and connections are summarised in Figure 2.5. The
carrier sheet density, for example, is connected with the microscopic carrier distribution
functions by
N e =
∑
i
1
A
∑
k
neii,k, N
h =
∑
j
1
A
∑
k
nhjj,−k. (2.114)
The summation over the momentum vector k is frequently replaced by an continuous
integral∑
kx
→
∫
dkx
Lx
2π
,
∑
ky
→
∫
dky
Ly
2π
, (2.115)
where we assume periodic boundary conditions in the semiconductor. In semiconductor
quantum well gain materials with an isotropic band structure diagram, i.e. d2k = 2πkdk,
one can write∑
i
1
A
∑
k
→
∑
i
1
A
A
(2π)2
· 2
∫
d2k =
∑
i
1
π
∫ kcut-off
0
dkk, ρ2Dk =
k
π
. (2.116)
The electronic density of states in momentum space ρ2Dk includes a factor of two be-
cause of the summation over the two degenerated spin states. This translates to an
energy-based density of states which is is given by ρ2DE =
∑
imi/(π~
2)Θ(E − Ei) (for
a parabolic band structure)23. In a similar way, the microscopic interband polarisation
can be summed up to the induced macroscopic polarisation
P =
∑
i,j
1
A
∑
k
(
Mehij p
∗
ji,k +M
eh∗
ij pji,k
)
, (2.117)
which is a sheet density related measure. In order to couple this quantity to the Maxwell
equation, the density of quantum wells has to be accounted for by applying the factor
NQW/Lref. For more details see Figure 2.6. In the hydrodynamic approach [103] the
23The equivalent calculations performed for a bulk medium give ρ3Dk = k
2/π2, ρ3DE = m/(π
2
~
3)
√
2mE
[3], novel (quasi) one-dimensional quantum wire nanostructures [28, 57] are characterised by ρ1Dk =
2/π, ρ1DE =
∑
i di2mi/(π~)/
√
2mi(E − Ei)Θ(E − Ei), and atomic-like quantum dots [29] by ρ0DE =
2
∑
i diδ(E − Ei). In structures where more than one direction is quantum confined there may
be besides the degeneracy of two because of the spin degree of freedom an additional energetic
degeneracy (different combinations of quantum numbers leading to the same energy) di. These
considerations show that states at high energies are relatively suppressed, states near the band
edge (here the lasing takes place) are pronounced. The quantum confinement (the size quantisation
concept) in novel systems with reduced dimensionality thus increases the efficiency of the inversion.
46 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
m
ac
ro
s
copic
 clas
sical
 d
e
sc
riptio
n
 (in
 re
al
 sp
ace)
m
ic
ro
sc
opic
 q
u
a
ntu
m
 th
e
o
retical
 d
e
sc
riptio
n
 (in
 m
o
m
e
ntu
m
 sp
ace)
spectral summation
inte
ractio
n
s
-
additio
nal
 phe
no
m
e
nological
 te
rm
s
-
spatial
 tra
nspo
rt:
-
ca
rrie
r
 diffusio
n
-he
at
 co
nd
uctio
n
-
co
upling
 of
 
 
 
 
 
 
 to
 rese
rvoir
 heatsink
-
va
rio
us
 ca
rrie
r
 loss
 cha
n
nels
-
electrical
 ca
rrie
r
 p
um
ping
-
ca
rrie
r
-light
 inte
ractio
n
,
 optical
 p
um
ping
-
m
a
ny
-b
ody
 inte
ractio
ns:
 re
no
rm
alisatio
ns
,
C
o
ulo
m
b
 scatte
ring
-pho
no
n
 dyna
m
ics:
 pho
no
n
 scatte
ring
 
 
 
 
 
 
,
 
no
n
-eq
uilib
rium
 pho
no
n
 distrib
utio
n
 d
ecays
into
 aco
ustic
 pho
no
ns
(m
ulti
-subba
nd)
 se
m
ico
nd
ucto
r
 Bloch
 eq
uatio
ns
,
ba
nd
-resolved
 g
ain
 dyna
m
ics
M
a
xw
ell
 w
a
ve
eq
uatio
n
m
easu
re
m
e
nt
co
upling
 of
 th
e
 diffe
re
nt
s
ub
sy
ste
m
s
 a
nd
 v
a
riable
s
lase
r
 pa
ra
m
ete
rs
e
.g
.
- specification of quasi-equilibrium
distribution functions with parametric    
spatial dependency
- optical excitation
Figure 2.5: Coupling of the various subsystems (levels of description) and variables
of a semiconductor laser structure, and overview on interactions and phenomenological
terms.
2.11 Spectral Summation and Coupling to the Maxwell Equations 47
Figure 2.6: Coupling of the induced polarisation of the active gain medium with the
electric field. The microscopic variables are dimensionless quantities. In the spectral
summation the various spectral components are weighted by the respective electronic
density of states. This leads to terms which are dependent on the dimension of the
quantum confined structure, e.g. P [C/m] is a dipole sheet densities for quantum wells.
The physical polarisation quantity which couples to the electric field has to be given
by an electric dipole volume density, that may qualify for a volume conversion factor
E ← VconvΓ ·P/ǫ0. (Left) In the case of an edge-emitting structure the coupling term in
the paraxial wave equation is given by the product of a confinement factor of the slab
optical waveguide (Γy)dact with the density of quantum wells (NQWdQW/dact) · 1/dQW,
that is E(x, z, t)← (Γy)dact NQW/dact ·P (x, z, t)/ǫ0. The strength is thus independent of
the width of the quantum wells but on the density of active sheets. (Right) If resolving
the spatial direction of quantum confinement, the reference length is given by Lref = ∆z,
and the correct coupling is expressed by E(z, t)← Γx,yNQW/∆z · P (z, t)/ǫ0.
carrier state is characterised by the first moments of the distributions: density, momen-
tum density and energy density. Non-equilibrium distribution functions can be calcu-
lated from the microscopic semiconductor Bloch equations including the discussed Boltz-
mann transport or scattering terms (Section 2.8). A quantity of particular interest is
the second moment, the plasma energy density
U e =
∑
i
1
A
∑
k
(
Eei,k + δE
e,SX
ii,k
)
neii,k, U
h =
∑
j
1
A
∑
k
(
Ehj,−k + δE
h,SX
jj,−k
)
nhjj,−k.
(2.118)
As discussed before, an important result of the Coulomb many-body interaction is the
establishment of a quasi-equilibrium distribution function. The Fermi-Dirac distribution
of the electron plasma is determined by the carrier density and energy density (2.118).
By replacing the extensive variables N,U e by the intensive measures temperature T epl
and chemical potential µe, the Fermi-Dirac distribution for the electrons can be written
48 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
as
f eii,k(N
e, U e;µe, T
e
pl) =
1
1 + exp
(
β
(
Ee,SXii,k − µe
)) , (2.119)
βµe = ln
[
exp
(
βπ~2N e
me
)
− 1
]
, β =
1
kBT epl
. (2.120)
The chemical potential µ can be calculated analytically as in (2.120) in the case of one
subband. For the multi-subband case this has to be done numerically.
In summary, the microscopic carrier dynamics in the active quantum well regions
∂tn
e
ii,k = −
∑
j
g˜ij,k − γnrneii,k −
∑
j
γspij,kn
e
ii,kn
h
jj,−k − γe,Augerii,k
+ Λ
f eii,k(Tlat)
(
1− neii,k(T epl)
)
∑
i 1/A
∑
k f
e
ii,k
(
1− neii,k
) − γe(ph)ii,k (neii,k(T epl)− f eii,k(Tlat))
− γee(cc)ii,k
(
neii,k(T
e
pl)− f eii,k(T epl)
)− γeh(cc)ii,k (neii,k(T epl)− f eii,k(T hpl)) (2.121)
comprises the processes of coherent coupling to an electromagnetic field, various carrier
recombination and loss channels, and an electrical pumping term. Here, the scattering
contributions due to many-body interactions are specified in the relaxation rate approxi-
mation. A macroscopic equation in real space for the sheet carrier density can be derived
by applying a spectral summation over (2.121)∑
i
1
A
∑
k
·(2.121)→ ∂tN = −G˜N − ΓnrN − ΓspN − ΓAugerN + Λ+DN∆‖N. (2.122)
On this level of description, a phenomenological ambipolar diffusion term [48,53] that re-
duces density inhomogeneities has been added. A systematic derivation of this term from
the hydrodynamic approach starting from the microscopic equations including Boltz-
mann transport terms is presented in [36,103]. We try to motivate this balancing process
in the following: A complete treatment of electron and hole carrier dynamics includes
classical transport in bulk regions, the connectivity to quantum wells or more general
nanostructures by quantum carrier capture and thermionic emission, self-heating or self-
cooling effects (hot electrons and hot phonons). Besides a formulation in energy space by
(2.121) electronic equations in real space [63] are necessary, e.g. to describe the dynamic
impact of the electrostatic potential which is arising from the Poisson equation [63,104]
∇ · ǫ∇Φ = − ρ
ǫ0
=
e
ǫ0
(
N e −Nh +N−A −N+D
)
. (2.123)
Here, the divergence of the dielectric displacement is given by the space charge density
due to a deviation from local charge neutrality. N−A , N
+
D are ionised acceptor and donor
concentrations, respectively. In addition, the real space electronic problem includes
generalised continuity equations for electrons (and for holes) [63,104]
∇ · jNe
3D
+ ∂tN
e
3D + Γ
e,net loss
Ne
3D
+ Ce,net capNe
3D
,Ne = 0, (2.124)
∇ · jNe + ∂tN e + Γe,net lossNe + G˜N − Ce,net capNe
3D
,Ne = 0. (2.125)
2.11 Spectral Summation and Coupling to the Maxwell Equations 49
The above equations balance the divergence of the carrier fluxes by recombination and
loss processes and by the macroscopic carrier generation rate in the active areas. Abrupt
heterojunctions that define the quantum wells or embedded nanostructures (the quantum
regime) are treated as scattering centres characterised by the net carrier capture rates
from the wetting layers. Carrier capture represents a source term for bound electrons
and a drain for continuum carriers. In bulk material or in the plane of the quantum
well the Shockley drift-diffusion theory can be applied. It relates the carrier transport,
which is a flow of charged electrons, to a drift component and a diffusive process
jNe
3D
= µeN e3D∇Φ−De∇N e3D, (2.126)
qualified by the particle mobility µe and the diffusion constant De. While the diffu-
sion is caused by density gradients, the drift originates from the electrostatic force. In
semiconductor diode lasers, the carrier densities are typically high, and the ultrafast
interband coupling due to Coulomb electron-hole scattering dominates over the carrier-
phonon interaction. This has two implications. Firstly, a common plasma temperature
T epl = T
h
pl = Tpl can be assumed in good approximation. Secondly, the charge imbalances
are usually very small, which justifies to set N e = Nh = N . This local charge neutrality
allows to use the so-called ambipolar diffusion approximation [103]
jN = −DN∇‖N, DN = µ
eDh + µhDe
µe + µh
. (2.127)
In microlasers (such as VCSEL or microcavity lasers) the heat dissipation from the active
quantum regime to the ambient is seriously hindered. In general, thermal effects are more
problematic for smaller and more complicated structured lasers as they significantly
contribute to the saturation of the gain and a reduction of output power [105–108]. The
effects of carrier and lattice self-heating have been included in self-consistent electro-
thermo-optical simulators [63,109]. Whereas, the concept of temperatures as dynamical
variables Tpl, Tlat demands for formulations that balance the various sinks and sources
of the electron-hole plasma energy density [12,108]
∂tU = ∂t
(
U e + Uh
)
=
∑
i
1
A
∑
k
Ee,SXii,k · (2.121) +
∑
j
1
A
∑
k
Eh,SXjj,−k
(
∂tn
h
jj,−k
)→
∂tU = −G˜U − ΓnrU − ΓspU −
(
Γe,AugerU + Γ
h,Auger
U
)
+ ΛU
−
∑
i
1
A
∑
k
γ
e(ph)
ii,k E
e,SX
ii,k
(
neii,k(Tpl)− f eii,k(Tlat)
)
−
∑
j
1
A
∑
k
γ
h(ph)
jj,−kE
h,SX
jj,−k
(
nhjj,−k(Tpl)− fhjj,−k(Tlat)
)
+DU∆‖U. (2.128)
A phenomenological balance process to equalise existing inhomogeneities has been added.
Of particular importance is the pumping with Pauli blocking. For numerical efficiency,
it is of advantage to calculate the total differential of the plasma temperature by the
50 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
changes of the related extensive measures N and U
∂tTpl = JN [N, Tpl]∂tN − JU [N, Tpl]∂tU, (2.129)
where the coefficients JN and JU are nonlinear functionals of the carrier density and
the plasma temperature [91, 106]. The dynamics of the phonon system is not treated
on a microscopic level, but by a macroscopic phenomenological equation for the lattice
temperature
∂tTlat = Γ
nr (U +NEgap)
NQW
Lref
1
cq,V
+
(
δU e(ph) + δUh(ph)
) NQW
Lref
1
cq,V
− σthTlat − Tamb
dambdact
1
cq,V
+ ρJ2
1
cq,V
. (2.130)
The hot carrier system is coupled to the lattice subsystem by means of carrier-phonon
scattering, by the terms δU e(ph), δUh(ph). In addition, the active region is coupled to
the reservoir heatsink by thermal conduction (cooling due to phonon diffusion). This
relaxation to the ambient temperature is strongly dependent on the device design. σth
[J/(Ksm)] represents the heat conductivity, cq,V [J/(Km
3)] is the specific heat per vol-
ume. An important process is the Joule heating determined by the specific resistivity ρ
[Vsm/C] and the injection current density. One should keep in mind, that the approach is
based on the assumption that the electron and phonon subsystems are in different quasi-
equilibrium states with individual temperatures, although they are coupled. Thus, the
dynamics is described only correct if the coupling is weak, and when looking at compar-
atively long time scales of a few characteristic energy relaxation or thermalisation time
constants (that is some ten picoseconds or longer). Then, a quasi-stationary situation
can be established within the subsystems.
In the following we establish an effective semiconductor Bloch equations model [36,
55], based on self-consistency and the spectral summation over microscopic (quasi-
equilibrium) variables: The full microscopic model is known to give the correct nonlinear
dependence of the gain and refractive index on the carrier density and temperature, and
realistic dispersions of the susceptibility. The basic idea now is to reproduce this complex
behaviour by an effective and numerically less demanding model on the basis of a macro-
scopic, effectively two-level [36] or a multi-level [12,41,55] system; for each macroscopic
state the parameterised susceptibility spectrum is approximated as the superposition of
few Lorentzians. Because of the very different decay time scales of the various dynam-
ical variables, more specifically γmicr, γpolarisation, γenergy relaxation ≫ γelectric field ≫ γdensity,
one can assume that the polarisation follows the variations of E, N and Tpl, Tlat quasi-
instantaneously. For an overview see Figure 2.7. In this case, the polarisation can be
adiabatically eliminated [10]. The idea behind the effective microscopic model is: We
first calculate the Fermi-Dirac distributions and the renormalised individual transition
energies for a given carrier density and given temperatures. The band gap shrinkage
due to the lattice temperature is thereby modelled by a phenomenological fit with the
2.11 Spectral Summation and Coupling to the Maxwell Equations 51
macroscopic variables
(in real space)
microscopic variables
(in momentum space)
photon lifetime in laser cavity, resonator loss
(measured by the Q-factor): some picoseconds
loss of coherence, dephasing: few ten femtosecondscarrier-carrier and carrier-phonon
momentum scattering, relaxation
towards quasi-equilibrium state:
few ten femtoseconds
recombination processes, carrier loss channels,
diffusion: some nanoseconds
coupling to (reservoir) heatsink by spatial
transport process  
decay time scales of the dynamical variables of diode laser systems:
main processes, interactions, effects & typical time scales
recombination processes: some nanoseconds
carrier-phonon scattering, coupling to lattice:
around a picosecond
SBE
dy
n
am
ic
al
 
te
m
pe
ra
tu
re
s
Figure 2.7: Overview on the various dynamical variables and belonging decay time
scales. By treating the plasma and the lattice temperatures as parameters we are left
with three macroscopic variables, namely the electric field E(r, t), the carrier density
N and the induced polarisation P. The number of relevant dynamical variables char-
acterises the system of coupled differential equations and thus the possible solutions of
that nonlinear dynamical system [9]. Other quantities with very fast decay time scales
adiabatically follow the slowly changing variables. Semiconductor-based single-mode
lasers are known to be a prototype of a class-B laser with the dynamical variables
|E(t)|2 and N(t) [52]. Typical solutions of this set of coupled nonlinear differential
equations are represented by damped relaxation oscillations, a fixed point attractor.
A class-A laser system can be derived by adiabatically eliminating the carrier inver-
sion, mathematically specified by a partial differential equation known as the complex
Ginzburg-Landau or complex Swift-Hohenberg equation. However, typical time scales
in semiconductor-based active systems do not justify this approach.
52 MICROSCOPIC DESCRIPTION OF THE GAIN DYNAMICS
Varshni band gap parameters [60],
~Ωij,k(N, Tpl, Tlat) = E
e
i,k + E
h
j,−k + δE
SX
ij,k(N, Tpl) + δE
CH(N, Tpl) + Egap(Tlat),
(2.131)
Egap(Tlat) = Egap(T
0
lat)−
(
α(Tlat)
2
β + Tlat
− α(T
0
lat)
2
β + T 0lat
)
. (2.132)
The Coulomb enhancement in Pade´ approximation [4, 48] is an approximative solution
of nonlinear convolution integral equations for the polarisations pji,k(t). It is quantified
by a complex, dimensionless factor
qij,k = − i
~
∑
k′ 6=k
V scijji(k− k′)
f eii,k′ + f
h
jj,−k′ − 1
i (Ωij,k′ − ω) + γpij,k′
, Qij,k(ω,N, Tpl) =
1
1− qij,k .
(2.133)
Subsequently, we perform an adiabatic elimination of the polarisation variables in the
semiconductor Bloch equations. After spectral summation, we can calculate the linear
susceptibility kernel in frequency-domain
χ(ω,N, Tpl, Tlat) =
P˜ (ω,N, Tpl, Tlat; t)
ǫ0ǫE˜(ω; t)
=
1
ǫ0ǫ
NQW
Lref
×
−
∑
i,j
1
A
∑
k
(Ωij,k − ω) + iγpij,k
(Ωij,k − ω)2 +
(
γpij,k
)2 1
~
∣∣M ehij ∣∣2 (f eii,k + fhjj,−k − 1) ·Qij,k, (2.134)
where the electric field amplitude is assumed to be constant over 1/γp. In a last step,
χ(ω,N, Tpl, Tlat) is inserted into the macroscopic field equations. With this approach
active material properties and microscopic interactions and processes are effectively in-
cluded into a macroscopic model. The advantage compared to common phenomenologi-
cal models is that realistic device simulations can be performed. E.g., this computation-
ally efficient model has been applied to study the transverse mode dynamics of (broad
area) high-power semiconductor lasers [12, 36]. The validity of the model is restricted
due to the applied approximations: Strong electric fields which lead to pronounced non-
linear behaviour, such as spectral carrier depletion, will violate the quasi-equilibrium
assumption. The same holds for ultrashort pulse excitations which are shorter than the
characteristic time scales associated with relaxation to quasi-equilibrium.
2.12 Conclusion
In this chapter we have derived the multi-subband semiconductor Bloch equations, a
quantum mechanical description of the gain dynamics in low-dimensional systems, more
precisely in quantum wells, based on the density matrix formalism. We have discussed
the possibilities of band structure engineering of the optoelectronic properties in quantum
2.12 Conclusion 53
confined nanostructures, and introduced the basic concepts and physics of semiconduc-
tor heterostructures. The presented approach is formulated in momentum space and
covers the relevant interactions in semiconductor-based optically active gain materials.
It is associated with a diversity of time scales and nonlinearities. The interaction of
light with semiconductor gain media within the semiclassical framework is modelled in
the electric dipole approximation. The important many-body interactions, namely the
screened Coulomb interaction and the scattering with phonons, involve the renormalisa-
tion of transition energies and the Coulomb enhancement of the Rabi frequency (screened
Hartree-Fock terms), and the relaxation and dephasing processes. These many-body in-
teractions have been presented together with other significant carrier loss channels in
the context of a microscopic, band-resolved approach. The equations are derived both in
a frequency-/time-domain formulation with split-off central frequency as well as in full
time-domain. This novel full time-domain approach (without the usual rotating wave ap-
proximation) represents an accurate and spectrally broad modelling of the gain medium
and offers the possibility to study sub-picosecond pulse interactions in semiconductor
lasers. To our knowledge, the full time-domain semiconductor Bloch equations have not
been reported before. We have shown how to couple the microscopic quantum theoretical
description in momentum space with the macroscopic classical description in real space
and have motivated further phenomenological terms that balance inhomogeneities. The
dynamics of the gain medium is measured by the induced macroscopic polarisation, the
carrier density, and the material dispersion. The models presented in this chapter will
be applied to complex and realistic systems. However, the size of these laser systems
may require an effective microscopic approach. Therefore, we have proposed the fitting
of the complex dielectric susceptibility by only a few oscillators as an efficient method
to reproduce the optoelectronic properties of the semiconductor gain medium.
3 Light Field Dynamics in Laser
Cavities
3.1 Introduction—Macroscopic Maxwell Equations
The realistic modelling of a diode laser requires a spatially resolved consideration of the
optical cavity, refractive index structure and laser fields, and a theoretical description of
the optoelectronic properties of the semiconductor amplifier medium. Within a classical
framework, the dynamics of the optical fields are governed by the macroscopic Maxwell
equations (here written in a differential form and in the SI unit system) [69,110]
curlE+ ∂tB = 0 Faraday’s law, (3.1)
divB = 0 absence of free magnetic poles, (3.2)
divD = ρ Coulomb’s law, (3.3)
curlH− ∂tD = J Ampe`re’s law. (3.4)
The macroscopic electric and magnetic field quantities E(r, t),R3 × R → R3 (electric
field) and B (magnetic induction) are defined as spatial averages over numerous atomic
dimensions of the microscopic fields, which are governed by the vacuum Maxwell equa-
tions. Derived fields include the influence of the microscopic bound sources, and ρ(r, t)
and J describe the free macroscopic charge density and charge current density, respec-
tively. The derived fields dielectric displacement D and magnetic field H can have
complex multi-linear functional dependencies [69,70]
D[E,B] = ǫ0ǫE+P, (3.5)
H[E,B] =
1
µ0
B−M. (3.6)
It is very challenging in theory and simulation of active devices to develop a compre-
hensive picture of the material properties, namely of the induced macroscopic electric
P and magnetic M polarisations as dynamical material responses to an applied elec-
tromagnetic field. In general, the nonlinear functionals may be expanded in a Volterra
series, as product of convolution integrals [5, 6]
Pi[E] =
∞∑
n=1
∫ t
t0
. . .
∫ t
t0
dt1 . . . dtnǫ0ǫχ
(n)
i;j1...jn
(r, t− t1, . . . , t− tn)Ej1(r, t1) . . . Ejn(r, tn).
(3.7)
56 LIGHT FIELD DYNAMICS IN LASER CAVITIES
This approach assumes causality, invariance under translations in time and spatial
locality. We note that the Fourier transforms of the tensorial susceptibility kernels
χ
(n)
i;j1...jn
(r, t − t1, . . . , t − tn) to frequency domain lead to the measurable nonlinear sus-
ceptibility coefficients χ(n)(r,−ω, ω1, . . . , ωn) [7]. In the general case, the permittivity or
susceptibility are dispersive, absorbing, inhomogeneous, anisotropic and nonlinear ma-
terial properties. The active material response is noninstantaneous. Maxwell equations
(3.1)–(3.4) and the material response (3.5) form a closed set of equations. A solution
is possible for arbitrary refractive index structures and cavity geometries, but because
of the finite analysis windows correct boundary conditions of Dirichlet, von Neumann,
combined type, or periodic boundary conditions have to be specified.
For describing the optical properties of semiconductor lasers it is a common ansatz to
apply some reasonable simplifications and approximations to the macroscopic Maxwell
equations. One typically assumes the absence of free field sources (ρ = 0 and J = 0)
and regards nonmagnetic materials M = 0. Therefore only two quantities measure
the passive and active material response in equation (3.5): ǫ(r) ∈ R, which may have
a fairly complex spatial structure in novel monolithic semiconductor devices, denotes
the permittivity of the host background medium. P(r, t) is the induced polarisation of
the active gain material. For a semiconductor lasing medium the passive background
refractive index nbackground(λ) =
√
ǫ takes into account resonant excitations of the lattice
with the strong bound inner electrons. It quantifies the linear, static, instantaneous
material response1. A typical value for a InGaAlAs-based device is ≈ 3.5. In general,
the permittivity is a rank-two tensor, a 3 × 3 matrix, but often a scalar dependence is
assumed. The induced polarisation P describes the gain and induced refractive index
change. It contains the dynamical nonlinear interaction of the laser fields (with photon
energies of about the band gap, few eV) with the charge carriers in the conduction
and valence bands. In this work, one of the main foci is the detailed modelling of
the induced material properties. One way to classify the various semiconductor laser
models discussed in literature is the level of description of the induced polarisation in
the active zone [37]: from the microscopic semiconductor Bloch equations to very simple
phenomenological two-level gain models. An overview is given in Figure 1.1. From
the above set of partial differential equations (3.1)–(3.4), and (3.5), one can deduce a
full-vectorial wave equation for the electric field with the induced polarisation as source
term
grad (divE)−∇2E+ ǫ
c2
∂2tE = −
1
ǫ0c2
∂2tP. (3.8)
The auxiliary condition of transversality, that is divD = 0, leads to
ǫ0ǫdivE+ ǫ0gradǫ · E+ divP = 0, P = 0→ divE = −1
ǫ
gradǫ · E. (3.9)
1Refractive index variations of the host or background material as a result of nonlinear optical effects
(such as the third-order optical Kerr nonlinearity) will be disregarded. Thus, nbackground specifies
the (passive) static and linear response to applied electromagnetic fields.
3.1 Introduction—Macroscopic Maxwell Equations 57
The Maxwell curl equations are a set of two coupled first order partial differential equa-
tions for the electric and magnetic field quantities. This is equivalent to a decoupled
formulation second order for the electric or magnetic field, respectively.
In addition to the time-domain approach, diode laser systems can be characterised
in the mode picture, i.e. in frequency domain: We study the passive optical eigenvalue
problem, that means P = 0, assume a time-harmonic behaviour E ∝ exp(−iωt) and
transform the problem to Fourier space
1
ǫ
∇× (∇× E) = −1
ǫ
(
grad
(
1
ǫ
E · gradǫ
)
+∇2E
)
− ω
2
c2
E = 0, (3.10)
∇×
(
1
ǫ
∇×H
)
=
1
ǫ
(
grad(divH)−∇2H)− 1
ǫ2
(gradǫ)× (curlH)− ω
2
c2
H = 0.
(3.11)
The above linear partial differential equations together with the geometry, ǫ(r) and cor-
rect boundary conditions determine the eigenfrequencies and optical field distributions
within the resonator cavity, the so-called stationary cold-cavity modes. In many appli-
cations vector A(r, t) and scalar Φ electromagnetic potentials are introduced to simplify
the solution of the two homogeneous macroscopic Maxwell equations
B = curlA, E = −gradΦ− ∂tA. (3.12)
The arbitrariness in the choice of the potentials connected with the gauge invariance of
the physical, that means measurable fields under gauge transformations can be restricted
by demanding the transverse Coulomb gauge [69]
divA = 0, Φ = 0. (3.13)
We attain the following Helmholtz equation for the vector potential(
1
ǫ
∇2 + ω
2
c2
)
A = 0. (3.14)
The full-vectorial wave equation or corresponding Helmholtz eigenvalue problem can be
solved by various numerical methods [111]. In the case of metallic boundary conditions
or an infinite analysis region (with periodic boundary conditions) we find a complete
set of eigenfunctions with real eigenvalues, as we have a linear Hermitian differential
operator. The cold-cavity modes represent standing waves or travelling plane waves2,
respectively. For general dielectric cavities and open boundary conditions the solutions
have complex-valued frequencies and therefore lose energy across the boundary of the
problem space. The completeness and orthogonality of the solutions cannot be proven.
Different laser models vary in and can be classified on the basis of how the optical
fields are described and the level of approximations applied [111], ranging from the full-
vectorial wave equation (without any approximations) to simple, spatially not resolved
2More precisely, the stationary solutions in periodic systems can be written as the product of a phase
factor (plane wave) with a unit cell-periodic function exp(ik · r)uωk(r) (Bloch theorem [5]).
58 LIGHT FIELD DYNAMICS IN LASER CAVITIES
rate equation approaches (see Figure 1.2). In the following sections we will introduce
two models: The transverse scalar wave equation model obtained by a paraxial approxi-
mation is suitable to the numerical simulation of edge-emitting semiconductor laser and
amplifier structures. The full time-domain Maxwell equations, on the other hand, allow
us to investigate problems which imply a broad range of relevant frequencies or complex
dielectric structures.
3.2 Paraxial Approximation—Transverse Wave Equation
In this section an optical model for a broad area edge-emitting laser or amplifier struc-
ture with a simple Fabry-Pe´rot resonator geometry and a multi-layered dielectric slab
waveguide structure is developed. Typical dimensions of such high-power semiconductor
devices are: the stripe width of the current contact in the transverse x-direction (around
100µm), the length of the resonator in the propagation z-direction (some 100µm to
several mm), and the height of the zone with embedded active materials in lateral y-
direction, that is in the direction of the grown layer structure (around 100µm). The
actual waveguide structure is typically a few hundred nm thick [2]. For more details
see Figure 3.1. The aim of the subsequent considerations is to derive an approximated
optical wave equation with source term for such a type of laser configuration and device
size. Due to the fact that a simulation of the full-vectorial wave equation is only feasible
for micrometer-sized active structures, such as microdisk or microgear lasers [13, 14],
we will derive a scalar transverse wave equation applying the paraxial, effective index,
rotating wave and slowly varying amplitude approximations.
In symmetric slab optical waveguides with embedded active quantum wells in the core
layer and different differential gain coefficients for TE and TM linear polarised light [4],
the transverse electric polarisation is normally predominant. We can assume Ey = 0.
With this approximation, the wave equation (3.8) and transversality condition (3.9) read
∂x∂zEz − ∂2yEx − ∂2zEx +
ǫ
c2
∂2tEx +
1
ǫ0c2
∂2t Px = 0, (3.15)
∂z∂xEx − ∂2yEz − ∂2xEz +
ǫ
c2
∂2tEz +
1
ǫ0c2
∂2t Pz = 0, (3.16)
ǫ0ǫ (∂xEx + ∂zEz) + ǫ0 (∂xǫ)Ex + ǫ0 (∂zǫ)Ez + ∂xPx + ∂zPz = 0. (3.17)
In order to gain simple solutions, which serve as basis of an expansion ansatz for solv-
ing the equations (3.15)–(3.17), we do not take into account the nonlinear dynamical
polarisation and neglect any transverse or longitudinal dielectric structure. Then we get
∂x∂zEz − ∂2zEx − ∂2yEx +
ǫ(y)
c2
∂2tEx = 0, (3.18)
∂z∂xEx − ∂2xEz − ∂2yEz +
ǫ(y)
c2
∂2tEz = 0, (3.19)
∂xEx + ∂zEz = 0. (3.20)
3.2 Paraxial Approximation—Transverse Wave Equation 59
Figure 3.1: Semiconductor Fabry-Pe´rot edge-emitting laser structure: We recognise
the optical resonator, a slab waveguiding heterostructure, the active gain medium (in
our case semiconductor multi quantum wells), and the electrical carrier pumping mech-
anism. More complicated contact profiles, e.g. multi-stripe lasers [37, 112] or tapered
semiconductor optical amplifiers [106], have been proposed. Novel designs like dis-
tributed feedback (DFB) lasers with longitudinal single-mode operation or multi-section
lasers [113] have been suggested. These rather large and consequently high-power
output devices suffer from poor beam quality, transverse multi-mode behaviour and
spatio-temporal instabilities [3, 36]. An expansion of the electric fields in forward and
backward travelling plane waves and the subsequent application of the paraxial, slowly
varying amplitude and rotating wave approximations are a well justified approach [37].
We investigate the transverse multi-mode dynamics and optical filamentation in broad
area edge-emitting lasers in Chapter 5, by applying the transverse scalar optical wave
equation (3.37).
The solutions of the above system of partial differential equations with periodic boundary
conditions are given by transverse electric (TE) plane waves which propagate forward
and backward in the longitudinal direction Ex ∝ exp(±iωt) exp(±ik0neffz), Ez = 0,
ω = ck0. The impact of the lateral optical waveguide structure is accounted for by the
introduction of an effective index in the phase velocity term. In the above equations we
perform the replacement
(−c2∂2y + ǫ(y)∂2t )Ex|z → n2eff∂2tEx|z. neff is the refractive index
a travelling plane wave experiences in the layer structure. This approach is called the
effective index approximation. It reduces the dynamical problem to a two-dimensional
one. For the full problem (3.15)–(3.17) we choose an ansatz that decomposes the elec-
tromagnetic fields into slowly varying field amplitudes, and fast-oscillating phase factors
which are represented by the above complete set of orthogonal eigenfunctions (related to
the eigenfrequencies of the resonator). Furthermore, we assume longitudinal single-mode
60 LIGHT FIELD DYNAMICS IN LASER CAVITIES
operation3, monochromaticity with a positive frequency and obtain
Ex|z(r, t;ω)→ E˜x|z(x, z, t)φ(y)e−iωtζ(z), R3 × R→ C, (3.21)
Px|z(r, t;ω)→ Px|z
[
E˜x|z
]
(x, z, t)Θ(y)φ(y)e−iωtζ(z). (3.22)
The expansion modes in the above product ansatz are specified by spatially delocalised
plane waves
ζ(z) = eik0neffz, k0 =
ω
c
. (3.23)
Effectively, a partial Fourier transformation to frequency and to momentum space has
been carried out. In the ansatz (3.22) we have implicitly made a further approximation,
the rotating wave approximation (RWA). The total electric field at frequency ω is a
superposition of plane waves travelling to the left and to the right with amplitudes
E˜±x|z(x, z, t) ∈ C
Ex|z(r, t;ω) = φ(y) · 1
2
[
e−iωt
(
E˜+x|zζ(z) + E˜
−
x|zζ
∗(z)
)
+ conjugate complex
]
. (3.24)
To evaluate the material functional P[E], counter-rotating parts and terms with phase
factors which oscillate with multiples of the phase factor of the electric field amplitude
are neglected. Mathematically speaking
P[E]→ P˜+x|z(x, z, t)Θ(y)φ(y)e−iωtζ(z), P˜+x|z(x, z, t) ≈ Px|z
[
E˜+x|z
]
(x, z, t). (3.25)
The function φ(y) records the stationary lateral field profile, which is determined by the
slab optical waveguide and refractive index structure in y-direction. Θ(y) filters out the
active layers. That means, the quantity is one at the position of the quantum wells, the
areas where the optical field is amplified by stimulated emission, and zero for all other
layers.
Additionally, a smallness parameter f ≪ 1 is introduced as a measure to quantify
the deviations from a plane wave solution, which is an eigenfunction of the passive
problem in an infinite slab optical waveguide. The consistent expansion of the electric
and polarisation field amplitudes in this parameter reads
E˜x|z(x, z, t) = f
0E
(0)
x|z + f
1E
(1)
x|z + f
2E
(2)
x|z + . . . , (3.26)
P˜x|z(x, z, t) = f
2 · (f 0P (0)x|z + f 1P (1)x|z + f 2P (2)x|z + . . . ). (3.27)
This results in an infinite hierarchy of coupled wave equations for the transverse and
longitudinal field components. The paraxial approximation, based on the separation
of time and length scales together with the consistent expansion ansatz, represents a
systematic method to truncate this hierarchy and to decouple the transverse from the
longitudinal problem. In lowest order, a closed equation for the transverse field dynamics
3This is achieved in DFB or DBR laser diodes by an additional periodic grating structure.
3.2 Paraxial Approximation—Transverse Wave Equation 61
will be obtained. The usual paraxial approach [114] considers the propagation of a light
beam with a Gaussian profile of width w, and introduces a characteristic longitudinal
length of diffraction l = k0neffw
2 and a characteristic propagation time t = lneff/c. The
expansion parameter is the ratio w/l = 1/(k0neffw) ≪ 1, and we associate the various
terms in the wave equation with different powers of f : The electric field amplitude as
envelope of a wave packet starts with power f 0. The derivation ∂x of field amplitudes are
considered to be order f 1, and the derivations ∂z, ∂t of field amplitudes are associated
with f 2. The amplitudes are assumed to vary little on the time scale of the period of
one optical cycle and on the length scale of the wavelength of the longitudinal mode.
Variations in transverse direction are one order higher, but still smaller than the fast-
oscillating phase factors. This type of slowly varying amplitude approximation (SVAA)
capitalises on the separation of the time and length scales of the slow amplitude and fast
phase dynamics
∣∣∂tE˜x|z∣∣≪ ∣∣ωE˜x|z∣∣, ∣∣∂zE˜x|z∣∣≪ ∣∣k0neffE˜x|z∣∣. (3.28)
In the limit f → 0 or in the order O(f 0) travelling plane waves have to fulfil the system
of wave equation (3.15)+(3.16) and the auxiliary divergence-free condition (3.17). This
justifies to start the expansion of the polarisation (3.27) with f 2. In addition, we impose
the following ansatz for the dielectric constant, which is a sum of contributions only
dependent on a single coordinate
ǫ(x, y, z) ≈ ǫ(y) + f 2ǫ(x) ≈ (n(y) + δnpas(x))2 ≈ ǫ(y) + f 22n(y)δnpas(x). (3.29)
We apply the paraxial approximation and insert the above expansions into the equations
(3.15)–(3.17), and obtain for the transverse part up to the order O(f 3)
O(f 0) : 0 = −E(0)x ∂2yφφ + k20n2effE(0)x − ǫ(y)c2 ω2E(0)x
O(f 1) : + ik0neff∂xE(0)z − E(1)x ∂2yφφ + k20n2effE(1)x − ǫ(y)c2 ω2E(1)x
O(f 2) : + ik0neff∂xE(1)z − E(2)x ∂2yφφ + k20n2effE(2)x − i2k0neff∂zE(0)x − ǫ(y)c2 ω2E(2)x
O(f 2) : − ǫ(y)
c2
i2ω∂tE
(0)
x −
ǫ(x)
c2
ω2E(0)x −
1
ǫ0c2
ω2P (0)x Θ(y)
O(f 3) : + ik0neff∂xE(2)z + ∂x∂zE(0)z − E(3)x ∂2yφφ + k20n2effE(3)x − i2k0neff∂zE(1)x
O(f 3) : − ǫ(y)
c2
ω2E(3)x −
ǫ(y)
c2
i2ω∂tE
(1)
x −
ǫ(x)
c2
ω2E(1)x −
1
ǫ0c2
ω2P (1)x Θ(y), (3.30)
62 LIGHT FIELD DYNAMICS IN LASER CAVITIES
and for the longitudinal part of the wave equation
O(f 0) : 0 = −E(0)z ∂2yφφ − ǫ(y)c2 ω2E(0)z
O(f 1) : + ik0neff∂xE(0)x − E(1)z ∂2yφφ − ǫ(y)c2 ω2E(1)z
O(f 2) : + ik0neff∂xE(1)x − E(2)z ∂2yφφ − ∂2xE(0)z − ǫ(y)c2 ω2E(2)z − ǫ(y)c2 i2ω∂tE(0)z
O(f 2) : − ǫ(x)
c2
ω2E(0)z −
1
ǫ0c2
ω2P (0)z Θ(y)
O(f 3) : + ik0neff∂xE(2)x + ∂z∂xE(0)x − E(3)z ∂2yφφ − ∂2xE(1)z − ǫ(y)c2 ω2E(3)z
O(f 3) : − ǫ(y)
c2
i2ω∂tE
(1)
z −
ǫ(x)
c2
ω2E(1)z −
1
ǫ0c2
ω2P (1)z Θ(y). (3.31)
The auxiliary transversality condition divD = 0 (3.17) gives
O(f 0) : 0 = ǫ(y)ik0neffE(0)z
O(f 1) : + ǫ(y)∂xE(0)x + ǫ(y)ik0neffE(1)z
O(f 2) : + ǫ(y)∂xE(1)x + ǫ(y)ik0neffE(2)z + ǫ(y)∂zE(0)z + ǫ(x)ik0neffE(0)z
O(f 2) : + (∂xǫ(x))E(0)x + 1ǫ0 ik0neffP (0)z Θ(y)
O(f 3) : + ǫ(y)∂xE(2)x + ǫ(x)∂xE(0)x + ǫ(y)ik0neffE(3)z + ǫ(y)∂zE(1)z + ǫ(x)ik0neffE(1)z
O(f 3) : + (∂xǫ(x))E(1)x + 1ǫ0∂xP (0)x Θ(y) +
1
ǫ0
ik0neffP
(1)
z Θ(y). (3.32)
In order O(f 0) we extract a one-dimensional eigenvalue problem with the eigenvalues
neff and eigensolutions φ(y)[
∂2yφ(y)
φ(y)
+ k20
(
ǫ(y)− n2eff
)]
= 0. (3.33)
Assuming that the refractive index of the active core zone is higher than the one of
the surrounding cladding layers, nact > neff > ncl, the solution of the above eigenvalue
problem describes a concentration of the electric field amplitude onto the active semi-
conductor layers due to optical guiding by total internal reflection. A simple double
heterostructure with a thickness dact of the active zone confines both the optical mode
and the charge carriers to the same layer. The smallest eigenvalue is determined by the
characteristic equation
√
n2act − n2eff tan
(
k0dact/2
√
n2act − n2eff
)−√n2eff − n2cl = 0. The re-
lated eigenfunction φ(y) within the active layer is given by cos
(
k0
√
n2act − n2effy
)
, and the
exponential decaying fields in the cladding layers by exp
(− k0√n2eff − n2cl(|y| − dact/2)).
Considering the carrier confinement, the reduction of the thickness of the active layer
3.2 Paraxial Approximation—Transverse Wave Equation 63
and of the width of the quantum wells appears advantageous in order to improve de-
vice performance parameters, e.g. lower the lasing threshold. This, however, reduces
the coupling of the gain medium to the optical mode as quantified by the confinement
factor Γy of the optical waveguide, which will be defined later on. Due the fact that
the optical confinement is associated with a length scale of the magnitude of the opti-
cal wavelength, and the carrier quantum confinement with the nanometer scale, more
complicated heterostructures, e.g. the separate confinement heterostructure and large
optical cavity configuration, have been proposed. This allows the optical confinement to
be separated from the quantum confinement of the active laser centres (see Figure 2.2).
It is possible to generalise the eigenvalue problem to more spatial dimensions. As long
as an ansatz ǫ(x, y, z) ≈ ǫ(x) + ǫ(y) + ǫ(z) is reasonable, a product of one-dimensional
confinement eigenfunctions will be the solution. With the split-off of the stationary lat-
eral mode profile, the dynamical problem is reduced to an effective two-dimensional one
in (x, z). The lateral layer structure is accounted for by an effective refractive index neff
and a lateral confinement factor Γy. The amplitude E
(0)
x is still undetermined, and we
set E
(0)
z = 0. This exactly corresponds to the situation of a TE linear polarised plane
wave propagating in z-direction as required in the f → 0 limit.
An evaluation of the series expansion in order O(f 1) couples transverse with longitu-
dinal field components
∂xE
(0)
x + ik0neffE
(1)
z = 0. (3.34)
E
(1)
x remains undetermined. In the next order we are able to derive a closed equation
for the transverse field dynamics. The transverse part of the wave equation in O(f 2) is
then given by
ik0neff∂xE
(1)
z φ(y)−
(
i2k0neff∂z + i
2ǫ(y)k0
c
∂t + ǫ(x)k
2
0
)
E(0)x φ(y) =
k20
ǫ0
P (0)x Θ(y)φ(y).
(3.35)
By multiplying this equation with
∫∞
−∞
dyφ(y) and integrating over the lateral direction,
the lateral confinement factor Γy appears
Γy =
∫∞
−∞
dyΘ(y) |φ(y)|2∫∞
−∞
dy |φ(y)|2 . (3.36)
Γy is a measure for the overlap of the optical mode or confinement function φ(y) with the
active zone, which is the region where the electromagnetic field is amplified or absorbed.
We employ the coupling of the longitudinal with the transverse field components (3.34)
and can truncate the infinite hierarchy of the coupled wave equations. We introduce the
notations E˜ = E
(0)
x and P˜ = P
(0)
x . The transverse wave equation for the forward and
backward propagating fields, in paraxial approximation, for a longitudinal single-mode
edge-emitting laser structure may now be expressed by(
n2
cneff
∂t ± ∂z − i 1
2k0neff
∂2x − ik0
n
neff
δnpas(x)
)
E˜±(x, z, t) = i
k0Γy
ǫ02neff
NQW
Lref
P˜±(x, z, t).
(3.37)
64 LIGHT FIELD DYNAMICS IN LASER CAVITIES
The coupling to the active medium has been redefined ΓyP˜
± → ΓyNQW/LrefP˜± (see
Figure 2.6). The field amplitudes are defined as complex-valued quantities (R2×R→
C). To sum up, the above partial differential equation (3.37) describes the temporal
evolution, propagation and diffraction of a transverse electric field amplitude E˜, with a
possible ridge waveguide structure in transverse direction, and driven by the nonlinear
dynamical induced electric polarisation P˜ . A typical value for the refractive index-
guiding profile is δnpas(x) ≈ 0.003. Although small, this quantity has a strong influence
on the stability and transverse mode dynamics. The transverse mode profile is not only
determined by the transverse waveguide structure but also by the carrier pump profile
(gain-guiding) [2], see equation (5.3). The carried out approximations to derive the above
paraxial wave equation (3.37) impose some limits on this model: The assumption of plane
wave expansion functions as solution of the passive optical problem and the application
of the slowly varying amplitude and rotating wave approximations are contradictory to
the longitudinal multi-mode operation, micrometer-sized active devices, or ultrashort
(sub-picosecond) pulse propagation.
A solution of (3.37) requires the specification of initial conditions and of boundary
conditions, for example the reflection of the optical fields at the facets of the resonator
E˜+(0) =
√
RlE˜
−(0), (3.38)
E˜−(L) =
√
RrE˜
+(L). (3.39)
L denotes the length of the cavity, Rl and Rr are the intensity reflectivities of the res-
onator structure. One may apply a further approximation called the mean-field approxi-
mation [115] by introducing averaged field amplitudes within the cavity. The longitudinal
averaged field amplitudes are then defined by
E˜(x, t) =
〈
E˜±(x, z, t)
〉
z
=
1
L
∫ L
0
dz |ζ(z)|2 E˜±(x, z, t)→ 1
L
∫ L
0
dzE˜±(x, z, t).
(3.40)
The longitudinal resonator structure has been taken into account by introducing a spa-
tially averaged decay term
± ∂zE˜±(x, z, t)→ γresonatorE˜(x, t), γresonator = − ln(
√
RlRr)
2L
. (3.41)
A longitudinal confinement factor Γz = ΓrVz as well as a filling factor Vz =
∫ L
0
dzΘ(z)/L
(Vz = Lact/L) express the longitudinal overlap between field and active gain sections.
3.3 Full Time-Domain Maxwell Equations
In the last section we have derived a model that is suitable to describe the transverse
multi-mode dynamics in a longitudinal single-mode high-power broad area laser or am-
plifier structure. In this section we aim to derive an optical model which allows us to
3.3 Full Time-Domain Maxwell Equations 65
Figure 3.2: Simplified schematic (from [12]) of a vertical cavity surface-emitting laser
(VCSEL): VCSEL are engineered by sandwiching active optical gain elements, such as
semiconductor multi quantum wells, between highly reflecting dielectric multi-layered
structures, so-called distributed Bragg reflectors (DBR). This laser structure may also
be characterised as a one-dimensional photonic crystal [20] with a n · λ/2, n ∈ N, de-
fect cavity. Advantages of this design are the circular output beams (easy in-coupling
of optical signals into fibres), low divergence, longitudinal single-mode operation (one
strong localised laser or defect mode), and the relatively inexpensive manufacturing
costs. The compact size of these devices and the small lengths of the gain or amplifica-
tion regions cause the rather low-power output of some mW and explain the importance
of spontaneous emission as additive process [11, 108]. From a modelling point of view,
the sub-wavelength refractive index structures and the discrepancy of the defect (cold-
cavity) or laser mode from the delocalised plane wave assumption of the previous section
demand for a modification of the paraxial wave model [12] and the development of novel
approaches, such as models based on the full time-domain Maxwell equations. VCSEL
devices with periodically structured defects are investigated in Section 6.1.
66 LIGHT FIELD DYNAMICS IN LASER CAVITIES
Figure 3.3: Vertical external cavity surface-emitting laser structure (VECSEL): One
Bragg mirror stack of a VCSEL is replaced by an external resonator [16]. The structure
is optically pumped (no doping required). The aim is to combine the near-diffraction-
limited beam quality of a VCSEL with the scalability to a high-power output laser
device in continuous wave operation [17]. Moreover, the possibility to place additional
nonlinear elements into the external cavity configuration, e.g. semiconductor saturable
absorber mirrors (SESAM) for passive mode-locking [32,116], or nonlinear crystals for
frequency conversion, is attractive. Starting from the three-dimensional full-vectorial
Maxwell equations, we focus on the longitudinal propagation direction and assume
stationary profiles in the perpendicular directions. The curved mirror results in a
highly preferred transverse TEM00 beam profile. The spatio-temporal dynamics of
the electromagnetic fields can thus be qualified by the Maxwell curl equations in one
dimension. The other directions are effectively included via confinement factors Γx,y(z),
which are the overlap integrals of the transverse mode profiles with the active media,
and by loss terms. Optically pumped VECSEL are explored in Section 6.2.
consider problems or devices which are characterised by a very broad range of relevant
frequencies, longitudinal multi-mode behaviour, and spatially strong localised cold-cavity
modes as typical for nanometer- or micrometer-sized cavities [13,14] or refractive index
structures [77, 117]. Examples of such active structures are vertical (external) cavity
surface-emitting lasers (Figure 3.2 and Figure 3.3). It is obvious that approximations
like the rotating wave or the slowly varying amplitude approximation can no longer
be applied as these assumptions are only correct for a small spectral range around a
central frequency. A model based on the two Maxwell curl equations and the mate-
rial response (3.5) allows us to explore characteristics of novel laser structures such as
vertical extended cavity surface emitting lasers (VECSEL), or processes of femtosecond
pulse propagation, reshaping and generation in semiconductor structures. The layer
or refractive index structure ǫ(z) is an important factor in such problems and will be
3.4 Conclusion 67
taken into consideration without any approximations. For devices with simple geometry,
the influence of the two perpendicular directions can be disregarded in the dynamical
interplay of the various longitudinal frequency components. Thus, we can reduce our
dynamical problem to sets of coupled differential equations with one spatial degree of
freedom, the longitudinal propagation z-direction. For the optical fields (R× R → R2)
we obtain from (3.1)–(3.4)
∂zEx + ∂tBy = 0, (3.42)
1
µ0
∂zBy + ∂tDx = 0; (3.43)
∂zEy − ∂tBx = 0, (3.44)
1
µ0
∂zBx − ∂tDy = 0. (3.45)
The material response is given by
D(z, t) = ǫ0ǫ(z)E(z, t) + Γx,y(z)
NQW(z)
Lref
P(z, t). (3.46)
The two Maxwell div equations are satisfied with the assumption of transverse polar-
isation. In the case of linear polarised light we can identify two decoupled subsets
(Ex|Dx|Px, By) and (Ey|Dy|Py, Bx). Indeed, in configurations with a pronounced prop-
agation direction and with a rotational symmetry in the perpendicular plane, a formu-
lation in left and right circular polarised optical fields4 is much more adapted [118].
Typically absorbing boundary conditions are used. This means, that at the boundary
of the spatial region of interest there should be no reflection of an out-going wave. Mur
suggested a simple scheme for such boundary conditions [111,119]. For more details see
the Appendix A.
In summary, the one-dimensional equations (3.42)–(3.46) are, in contrast to the parax-
ial transverse wave equation, formulated using real-valued variables. Thus, the electric
polarisation, which describes the dynamical response of the gain material, has to be
transformed to a formulation in R (see Section 2.5). The presented model links a
very comprehensive mapping of the optoelectronic properties of the semiconductor gain
medium, e.g. the asymmetric gain spectra and a diversity of optical nonlinearities, with
a description of the sub-wavelength passive structure and the optical fields including the
fast-oscillating carrier wave. The approach is valid for a huge spectral range, however,
due to computational effort, the model is limited to micrometer-sized active devices.
3.4 Conclusion
The propagation, diffraction and reflection of electromagnetic fields inside laser devices
can be either modelled by the use of the macroscopic Maxwell curl equations or by some
4We can generate these fields by superposing the two linear polarised field components of equal am-
plitude with a constant relative phase shift of ±π/2. This is only possible for a single frequency.
68 LIGHT FIELD DYNAMICS IN LASER CAVITIES
derived approximated wave equations. The induced macroscopic polarisation acts as
source term of the laser fields (see Chapter 2). In this chapter we have developed two
different models to account for the optical problem: Applying the slowly varying ampli-
tude, rotating wave and effective index approximations, and utilising the separation of
time and length scales together with a consistent expansion ansatz (paraxial approxima-
tion), a transverse scalar wave equation model has been obtained. This model is suitable
to the numerical simulation of high-power and large area edge-emitting semiconductor
laser and amplifier structures. A second model based on the full time-domain Maxwell
curl equations is a full-vectorial wave equation model. It allows us to study problems
which imply a very broad range of relevant frequencies or spatially strong localised laser
modes. However, this approach is limited to micrometer-sized active devices. To our
knowledge, a model combining the Maxwell curl equations with a band-resolved de-
scription of the semiconductor gain dynamics within the finite-difference time-domain
framework has not been reported and implemented before.
4 Semiconductor Laser Fundamentals
4.1 Calculation of Scattering Rates
4.1.1 Microscopic Scattering Rates in Semiconductor Quantum
Wells
The screened Coulomb interaction between the charge carriers and the coupling of the
carriers to the various quantised modes or excitations of the lattice (phonons) are two
of the relevant scattering processes for semiconductor two-component plasmas. A mi-
croscopic approach is based on the generalised quantum Boltzmann collision integrals
(2.83)+(2.86) in second Born approximation, which takes into account out-scattering as
well as in-scattering contributions [4]
∂t n
e
ii,k
∣∣
relax
= −2Γee,outii,k {n◦k}neii,k + 2Γee,inii,k {n◦k}
(
1− neii,k
)
= −2
(
Γee,outii,k {n◦k}+ Γee,inii,k {n◦k}
)
neii,k + 2Γ
ee,in
ii,k {n◦k} (4.1)
≈ −γeii,k{f ◦k}
(
neii,k − f eii,k
) ≈ −γe(ph)ii,k {f ek, nph} (neii,k − f eii,k(Tlat))
− γee(cc)ii,k {f ek}
(
neii,k − f eii,k(T epl)
)− γeh(cc)ii,k {f ◦k} (neii,k − f eii,k(T hpl)) ,
(4.2)
∂t pji,k|relax ≈ −
(
Γee,outii,k {f ◦k}+ Γee,inii,k {f ◦k}+ Γhh,outjj,k {f ◦k}+ Γhh,injj,k {f ◦k}
)
pji,k
= −γpij,k{f ◦k}pji,k = −
γeii,k{f ◦k}+ γhjj,k{f ◦k}
2
pji,k. (4.3)
The above equations and the definitions of the scattering matrices (2.84)+(2.85) and
(2.87)+(2.88) [51, 64] reflect the highly complex and nonlinear dependence of the mi-
croscopically calculated relaxation rates on the carrier distribution functions. For more
details see Section 2.8. Here, individual scattering processes are analysed and in the
summation the single interaction process is weighted with the appropriate measure, e.g.
scattering with small momentum transfer is favoured. This is in contrast to the simple
approach of phenomenologically defined scattering times, which is associated with the
concepts of baths and reservoirs to describe the exponential relaxation of a system [4].
Semiconductor-based gain materials incorporate a hierarchy of carrier relaxation times,
which may cause a loss of coherence. Slower relaxation times are connected with re-
combination and loss channels of the macroscopic carrier density, but the carrier-carrier
scattering is the most dominant mechanism [120] responsible for the dephasing of the
interband polarisation (4.3).
70 SEMICONDUCTOR LASER FUNDAMENTALS
Figure 4.1: Calculated momentum relaxation rates of the electron (top) and hole (mid-
dle) distribution functions and dephasing rates of the interband polarisation (bottom)
versus the carrier sheet density N ∈ [5 ·1010 cm−2, 5 ·1013 cm−2] and the carrier momen-
tum vector k ·aBohr ∈ [0, 30]. Depicted are the scattering rates due to the carrier-carrier,
screened Coulomb interaction (left column) and for the Fro¨hlich interaction with lon-
gitudinal optical phonons (right column). The dielectric function is specified by the
Lindhard formula (2.94) in the effective plasmon pole approximation (2.97)+(2.98).
4.1 Calculation of Scattering Rates 71
Figure 4.2: Same as Figure 4.1, but applying the simple, free-carrier dispersion model
(2.96) to consider plasma screening.
72 SEMICONDUCTOR LASER FUNDAMENTALS
Figure 4.3: Same as Figure 4.1, but the temperature T = Tpl = Tl-o = Tlat has been
increased from 300K to 400K. Microscopic scattering rates reveal complex functional
dependencies on (k,N, T ).
4.1 Calculation of Scattering Rates 73
0 5 10 15 20 25 30
momentum vector [1/aBohr]
0
5
10
15
20
25
30
35
ca
rr
ie
r-c
ar
rie
r s
ca
tte
rin
g 
ra
te
s [
ps
-
1 ]
0 5 10 15 20 25 30
momentum vector [1/aBohr]
0
5
10
15
ca
rr
ie
r-p
ho
no
n 
sc
at
te
rin
g 
ra
te
s [
ps
-
1 ]
0 5 10 15 20 25 30
momentum vector [1/aBohr]
0
5
10
15
20
25
30
35
ca
rr
ie
r-c
ar
rie
r s
ca
tte
rin
g 
ra
te
s [
ps
-
1 ]
0 5 10 15 20 25 30
momentum vector [1/aBohr]
0
5
10
15
ca
rr
ie
r-p
ho
no
n 
sc
at
te
rin
g 
ra
te
s [
ps
-
1 ]
Figure 4.4: The graphs give a more detailed picture of the various contributions to
the carrier-carrier (left) and carrier-phonon (right) scattering rates at T = 300K and a
density of N = 2 · 1012 cm−2 (screening model as in Figure 4.1) of the electron (circle)
and hole (square) distribution functions and the interband polarisation (top graphs,
no symbol). (Top) For higher q the out-scattering processes (dashed lines) are more
significant than the in-scattering contributions (dot-dashed). (Bottom left) We compare
electron-electron and hole-hole scattering (dashed) with electron-hole interactions (dot-
dashed lines). (Bottom right) Carrier-phonon scattering under absorption (dashed) and
emission (dot-dashed) of a longitudinal optical phonon with ~ωl-o = 36meV.
74 SEMICONDUCTOR LASER FUNDAMENTALS
Due to the computational effort involved in the calculation of correlation contribu-
tions, the evaluation of scattering integrals together with a spatially resolved modelling
of laser structures on a time window of several nanoseconds is not feasible. Instead, we
evaluate the scattering matrices with quasi-equilibrium distribution functions {f ◦k, nph},
compute the characterising relaxation rates (see (4.2)+(4.3)) for various parameter sets
γk(N, Tpl, Tlat), and use these precalculated values in the time-domain simulations. This
approach of lookup tables [36, 53] is extended and modified to describe multi-subband
quantum well structures [107]. In quantum confined structures the effective rates also
depend on structural parameters. The relaxation rate approximation neglects scattering
mediated by polarisation terms and applies a linearisation (see (4.2)+(4.3)) around the
quasi-equilibrium state using the principle of detailed balance. This approach is suffi-
ciently accurate, as long as the electromagnetic fields are not too strong, and scattering
processes are studied on time scales longer than a characteristic relaxation time.
Figures 4.1–4.4 show results of the calculation of the scattering rates for a GaAs
quantum well of 5 nm width (single-subband case). For every parameter tuple (k,N),
the scattering matrices (2.84)+(2.85) and (2.87)+(2.88), respectively, are calculated for
each individual scattering process, and subsequently the summation over all possible
processes is evaluated. The sharp structures in the graphs of the scattering rates due
to the interaction of carriers with longitudinal optical phonons are induced by the onset
of the possibility of out-scattering processes associated with the emission of a phonon
(or in-scattering processes under absorption of a phonon). The threshold conditions are
given by k ≥√2m/~2 · ~ωl-o and thus dependent on the mass of the scattered particles.
As can be clearly seen, the dephasing rates of the interband polarisation correspond to
the average values of the relaxation rates of the electron and hole distribution functions.
The diverse constituent parts in the calculations of the scattering matrices exhibit very
different, partly contrariwise shifts with changing (k,N, T ): The static inverse screening
length κ (2.95) increases and hence the strength of the interaction, the screened po-
tential (2.93) decreases with increasing density and decreasing temperature. Scattering
processes with small momentum transfers are favoured. Other important factors are the
number of scattering partners (initial states), unoccupied final states have to be available,
and band filling effects. In the case of the Fro¨hlich interaction the phonon occupation
number rises with increasing temperature. Figures 4.4 compares the different scatter-
ing contributions: We see that γ
hh(cc)
k > γ
he(cc)
k , γ
eh(cc)
k > γ
ee(cc)
k . Momentum relaxation
processes which cause a redistribution of energy and establish quasi-equilibria charac-
terised by N, T epl for the electrons and by N, T
h
pl for the holes and scattering processes
which couple the two subsystems occur on time scales of the same order of magnitude.
This relation suggests a common plasma temperature Tpl, see Section 2.11.
4.1.2 Extension to the Multi-Subband Case
Intrasubband and intersubband relaxation rates play an important role in the optoelec-
tronic properties of semiconductor laser devices. The electrical pumping of quantum
confined active structures by carrier capture [63, 104] or intersubband transitions me-
diated by many-body interactions can be modelled by the multi-subband Boltzmann
4.1 Calculation of Scattering Rates 75
collision integrals [121,122]. These processes are essential for the operation of novel de-
vices such as quantum cascade lasers [34] or infrared detectors. The scattering matrices
for the screened Coulomb interactions (in the electron-electron case) are defined by
Γee,outi1i1,k =
π
~
∑
k′,q
∑
i2,i3,i4
δ
(−Eei1,k − Eei2,k′ + Eei3,k′+q + Eei4,k−q) ∣∣V sci1i2i3i4(q)∣∣2
× nei2i2,k′
(
1− nei3i3,k′+q
) (
1− nei4i4,k−q
)
, (4.4)
Γee,ini1i1,k =
π
~
∑
k′,q
∑
i2,i3,i4
δ
(−Eei1,k − Eei2,k′ + Eei3,k′+q + Eei4,k−q) ∣∣V sci1i2i3i4(q)∣∣2
× (1− nei2i2,k′)nei3i3,k′+qnei4i4,k−q. (4.5)
The out-scattering and in-scattering processes (for electrons) due to the Fro¨hlich inter-
action can be calculated by
Γee,outi1i1,k =
π
~
∑
q,qz
∑
i2
∑
±
δ
(−Eei1,k + Eei2,k−q ± ~ωl-o) |γi1i2(q, qz)|2
× (1− nei2i2,k−q)
(
nph,q,qz +
1
2
± 1
2
)
, (4.6)
Γee,ini1i1,k =
π
~
∑
q,qz
∑
i2
∑
±
δ
(−Eei1,k + Eei2,k−q ∓ ~ωl-o) |γi1i2(q, qz)|2
× nei2i2,k−q
(
nph,q,qz +
1
2
± 1
2
)
. (4.7)
To understand the carrier and energy relaxation from non-equilibrium distribution func-
tions, it is beneficial to investigate the strength of the interactions, which are mainly de-
termined by the form factors (see Figure 4.5), and to group the scattering processes into
different categories. Figure 4.6 shows the different types of scattering processes. The
symmetry of the electronic envelope functions in a symmetric confinement potential re-
sults in a reduction of the number of nonzero form factors (2.59). The relevant processes
are those weighted with
∣∣V sci1i2i3i4(q)∣∣2 in which the combinations of involved subband
numbers satisfy the restrictive condition mod(i1 + i2 + i3 + i4, 2) = 0. Thereby, q is the
transfer momentum vector between two interacting particles i1, i2 with i1 → i4, i2 → i3,
Ei1,k − Ei4,k−q the transferred energy. A total of 8 categories of electron-electron inter-
actions (in two subbands) and 16 different scattering contributions in equations (4.4)
and (4.5) have to be considered [124]: (A) V sc1111, V
sc
2222 describe purely intrasubband pro-
cesses with an occupation and energy redistribution within a subband. (B) In V sc1221, V
sc
2112
electrons from different subbands interact, which reflects intrasubband transitions. This
allows for an energy density equalisation. (C) In V sc1212, V
sc
2121 the particles are initially
in different subbands (intersubband interactions) and undertake intersubband, exchange
transitions. (D) Here, the intrasubband interactions V sc1122, V
sc
2211 result in intersubband
transitions. There is both, an energy and a net occupation transfer between the sub-
bands. Electron-hole and hole-hole scattering processes can be classified in an analogous
way, and the consideration of intersubband and intrasubband scattering rates mediated
76 SEMICONDUCTOR LASER FUNDAMENTALS
0 5 10 15 20 25 30
momentum transfer vector [1/aBohr]
0
0.2
0.4
0.6
0.8
1
fo
rm
 fa
ct
or
s
Figure 4.5: The form factors as defined in (2.59) determine the relative strength of the
different classes of scattering processes in semiconductor quantum wells, primarily there
is a distinction (for small q) between intrasubband transitions with F (q → 0) = 1, such
as F1111 (dashed), F1221 (dot-dashed) or F2222 (dotted), and intersubband transitions
with F (q → 0) = 0, weighted with the factor F1122. At this i1 → i4 in Fi1i2i3i4(q) labels
the transition we are considering.
Figure 4.6: Classification of allowed out-scattering processes mediated by the screened
Coulomb interaction [123, 124] involving the lowest two electron subbands. The inves-
tigated structure is a single, symmetric GaAs quantum well of 20 nm width.
4.1 Calculation of Scattering Rates 77
0 5 10 15 20 25 30
momentum vector [1/aBohr]
0
2
4
6
8
sc
at
te
rin
g 
ra
te
s [
ps
-
1 ]
0 5 10 15 20 25 30
momentum vector [1/aBohr]
0
1
2
3
4
5
6
sc
at
te
rin
g 
ra
te
s [
ps
-
1 ]
0 5 10 15 20 25 30
momentum vector [1/aBohr]
0
2
4
6
8
10
12
sc
at
te
rin
g 
ra
te
s [
ps
-
1 ]
0 5 10 15 20 25 30
momentum vector [1/aBohr]
0
2
4
6
8
10
sc
at
te
rin
g 
ra
te
s [
ps
-
1 ]
Figure 4.7: Momentum scattering rates of the electron (top) and hole (bottom) distri-
bution functions at T = 300K and at a carrier density of N = 4 · 1012 cm−2 in a single
quantum well of 20 nm width. Plotted are the microscopically calculated rates for the
lower (left column) and upper (right column) subbands. In the graphs we analyse
the various contributions, to specify we distinguish between carrier-carrier interactions
involving intrasubband (solid lines) and intersubband (dotted) transitions, as well as in-
trasubband (dashed) and intersubband (dot-dashed) transitions caused by the Fro¨hlich
interaction.
by the interaction with phonons is straightforward. The rates of the different categories
of scattering processes characterise and give a measure of the time scales of the redistri-
bution of carriers and energy in a multi-subband system starting from non-equilibrium
distributions. The processes (A) lead to an equalisation separately for each subband,
(B) and (C) couple the different subbands and involve an energy transfer. The scattering
processes described by (D) are the only ones which make possible a relaxation to a single
overall Fermi-Dirac distribution.
The scattering times of the different categories of scattering processes are calculated
in Figure 4.7. Compared to the scattering rates calculated for the single-subband case
in Figure 4.4 (with a quantum well width of 5 nm) the relaxation times in the multi-
subband case are increased by a factor of approximately 3. This can be explained by
78 SEMICONDUCTOR LASER FUNDAMENTALS
the reduced interaction strengths mainly given by the form factors and by the increased
effect of screening for higher densities; in a broader quantum well the electronic wave
functions are less localised. A comparison of the influence of different screening models in
the calculations of the scattering rates, from the dynamical dielectric function in random
phase approximation to the static, long-wavelength approach, can be found in [124].
The threshold conditions for the sharp structures in the carrier-phonon relaxation rates
are modified by the offsets in intersubband transitions. The graphs clearly show that
intraband scatterings due to the screened Coulomb interactions represent the dominant
contributions. Typical rates are 200 fs (intrasubband), 1 ps (interaction with longitudinal
phonons) and 2 ps (intersubband) for the electrons and 100 fs, 500 fs, 1 ps for the holes,
respectively. These results conflict with the concept of separate temperatures for the
carriers and the lattice in multi-subband or multi-level systems. We estimate the typical
carrier capture time from the electrically or optically pumped wetting structures into a
quantum confined active structure, for instance a quantum well or a quantum dot, to be
a few picoseconds. The interaction strengths are weakened due to the reduced overlap
of wetting layer states and the final state in the quantum confined structure.
4.2 Calculations of Quantum Well Laser Gain
The optoelectronic properties of semiconductor quantum well structures are modelled
in this work within the semiclassical framework [10]. Quantum mechanically derived
expressions of the semiconductor gain response are coupled to (and are the sources
of) the optical waves and propagating laser fields by adding a term for the induced
polarisation or complex susceptibility. The investigation of these local gain properties
of active media and the calculation of important laser parameters allow quantitative
predictions of the performance characteristics of diode laser structures [3, 4, 25].
Gain properties and typical laser parameters, as computed from a realistic description
of the quantum confined semiconductor amplifier medium including energy and field
renormalisations as well as correlation contributions [23, 24, 49–51, 125], are shown and
discussed in Figures 4.15–4.20. An approximation to these spectra and a mapping
of typical behaviours to a n-oscillator model with few fitting parameters [12, 41, 54, 55]
in combination with a spatially resolved model for the electromagnetic fields can be
a numerically efficient tool to investigate the dynamics of lasers [96, 126] and ampli-
fiers [55]. Other approaches are more suitable to simulate large area active laser devices,
such as broad area lasers [37, 127] or tapered amplifiers [106], along with the effects of
optical nonlinearities, spectral hole burning or nonlinear gain saturation, but require
some approximations of the dynamical and band-resolved description of the gain ma-
terial. Figures 4.8–4.14 show optical gain spectra and important laser parameters as
implemented in the latter approach.
In order to calculate the quantum well laser gain spectra, we start with a mode expan-
sion of the electric field and the induced macroscopic polarisation in forward travelling
4.2 Calculations of Quantum Well Laser Gain 79
plane waves
E(z, t) =
1
2
Eenv(z;ω)e
iΦ(z;ω)e−iωteik0neffz + conjugate complex, Eenv,Φ ∈ R, (4.8)
Penv(z;ω) =
∑
i,j
2
A
∑
k
M ehij p˜ji,k, Eenv ←
ΓNQWdQW
Lref
1
dQW
Penv
ǫ0
. (4.9)
Applying the slowly varying amplitude approximation and the rotating wave approxi-
mation, the amplitude and phase dynamics is given by (3.37) for the complex-valued
electric field amplitude, or equivalently by
∂zEenv(z) = − k0
ǫ02neff
ΓNQWdQW
Lref
1
dQW
Im
(
Penv(z)
)
= gmod(ω)Eenv(z), (4.10)
Eenv(z)∂zΦ(z) =
k0
ǫ02neff
Γ˜
1
dQW
Re
(
Penv(z)
)
, Γ˜ =
ΓNQWdQW
Lref
(4.11)
for the real-valued electric field envelope Eenv and phase shift ∂zΦ. Note that by the
expansion ansatz (4.8) a partial transformation to Fourier space has been performed. The
optical field and semiconductor gain medium are coupled by the modal gain gmod(ω) and
carrier-induced refractive index change δn(ω) as properties of the dielectric susceptibility
response. The refractive index change on the other hand is related to the phase shift by
eiΦ(z;ω)eik0neffz = eik0(neff+δn(ω))z → ∂zΦ(ω) = k0δn(ω). (4.12)
We note that not the phase shift itself but phase shift changes influence the light field
dynamics of broader laser structures. This is quantified by the linewidth enhancement
or anti-guiding factor1 α
α = −d (∂zΦ) /dN
dgmod/dN
. (4.13)
The modal amplitude gain, which in the simplest case is the amplification of a monochro-
matic travelling plane wave with frequency ω and k0n, is connected to the semiconductor
material gain by gmod(ω) ∝ ΓrVzΓ˜g(ω)|n→neff. The coupling term VzΓ˜ filters out active
regions2. The gain characteristics are now computed in the quasi-equilibrium approxima-
tion: The band-resolved, k-dependent carrier populations neii,k, n
h
jj,−k and the interband
1A dominant feature of semiconductor-based active materials is the strong coupling between amplitude
and phase dynamics [1, 2, 4, 48], quantified by the α factor. Consequences are a rather large laser
linewidth (a typical value is given by 100MHz), poor beam quality [3, 8, 36], and the occurrence
of filamentation effects in semiconductor lasers [128–132]. In contrast to (asymmetric) multi-level
or band-resolved descriptions, for a homogeneously broadened two-level active gain material we
obtain a density independent value of α = (Ω0 − 2πc/λ)/γP (or in our full time-domain approach
α =
(
Ω20+(γ
P )2−ω2)/(2γPω), respectively). Thus, for commonly applied parameters index-guiding
is suppressed in phenomenological models as α ≈ 0.
2A typical number for an edge-emitting, quantum well-based lasing structure is 0.05. For quantum dot
or other nanometer-sized active structures the coupling and consequently amplification are reduced
due to much smaller filling factors Vz in propagation direction.
80 SEMICONDUCTOR LASER FUNDAMENTALS
polarisation amplitudes p˜ji,k are assumed to adiabatically follow temporal variations
of the slower macroscopic variables N, T,Eenv according to the slaving principle [10].
The driving physical processes are the ultrafast scattering and dephasing with typical
relaxation times of about 50 fs. In the small signal or linear regime, where nonlinear
saturation and spectral hole burning can be neglected, these relaxation processes result
in electron and hole quasi-equilibrium Fermi-Dirac distributions3 f eii,k, f
h
jj,−k. From the
free-carrier susceptibility [4] we can derive
gmod(λ) = ΓzΓ˜
π
ǫ0neffλ
1
dQWA
∑
k
γpk
(Ωk − 2πc/λ)2 + (γpk)2
1
~
∣∣M eh∣∣2 (f ek + fh−k − 1) ,
(4.14)
∂zΦ(λ) = −ΓzΓ˜ π
ǫ0neffλ
1
dQWA
∑
k
Ωk − 2πc/λ
(Ωk − 2πc/λ)2 + (γpk)2
1
~
∣∣M eh∣∣2 (f ek + fh−k − 1) ,
(4.15)
where we only consider the transitions p˜11,k. Gain spectra strongly depend on macro-
scopic parameters like the carrier density or temperature and on the optical frequency
~Ωk(N, Tpl, Tlat) =
(
Eek + E
h
−k + Egap
)
(Tlat) + δE
SX
k (N, Tpl) + δE
CH(N, Tpl),
f ek(N, Tpl), f
h
−k(N, Tpl), γ
p
k(N, T ). (4.16)
Some results of calculations of quantum well laser gain, applying the free-carrier the-
ory, are presented in the following Figures 4.8–4.14. The investigated structure is
a single GaAs/AlGaAs quantum well of 5 nm width with only one relevant interband
polarisation term p˜11,k. However, a generalisation, as required for broader quantum con-
fined structures with several subbands or bound states, is straightforward by adding a
summation over all oscillators with non-vanishing dipole matrix elements or oscillator
strengths.
From the simulations in Figure 4.8 we can extract a transparency carrier density, that
is gmod(N > Ntransparency) > 0, of approximately 1.54 · 1012 cm−2. The slowly decaying
tails in the gain spectra, and the fact that the material is not transparent below the
band gap is related to the treatment of the dephasing. The correct microscopic inclusion
of nondiagonal collision contributions would remove this problem [4] as the dielectric
susceptibility response is not just the summation over independent, driven harmonic
oscillators, but the various terms will cancel each other out. In our simulations of spatio-
temporal dynamics of active semiconductor structures a term
∑
k
∑
k′ is numerically too
expensive, and we have also disregarded these contributions in the computation of local
properties of the gain material. We proceed as follows: For each tuple (N, T ) we solve
the electronic confinement problem, which determines the eigensolutions of the quantum
3The spectrally local depletion of the band-resolved inversion through spectral hole burning [37,40,106]
or the spectrally selective carrier generation through optical pumping [15–17, 133, 134] play impor-
tant roles in the propagation of ultrashort pulses in semiconductor optical amplifiers (see Sections
6.4+6.5), in high-speed modulation of lasers, or in multi-mode operation. In these situations non-
linear saturation and non-equilibrium gain cannot be disregarded [73].
4.2 Calculations of Quantum Well Laser Gain 81
740 760 780 800 820 840 860 880
wavelength [nm]
-3000
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.451.51.551.61.65
energy [eV]
Figure 4.8: Small signal amplitude gain g(λ,E;N, T ) versus optical wavelength or
energy for various carrier sheet densities N ∈ {0.5 · 1012 cm−2 . . . 4 · 1012 cm−2} at room
temperature T = 300K and Γ˜ = 1, Γz = 1, neff = nbackground.
740 760 780 800 820 840 860 880
wavelength [nm]
-1500
-1000
-500
0
500
1000
1500
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.451.51.551.61.65
energy [eV]
740 760 780 800 820 840 860 880
wavelength [nm]
-1500
-1000
-500
0
500
1000
1500
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.451.51.551.61.65
energy [eV]
Figure 4.9: Comparison of gain spectra calculated from a model comprising micro-
scopically determined scattering and dephasing rates γpk(N, T ) as input parameters with
the more simple free-carrier approach with a constant phenomenological homogeneous
broadening (thin line) of the individual transitions, that means of the Lorentzian oscil-
lators.
82 SEMICONDUCTOR LASER FUNDAMENTALS
740 760 780 800 820 840 860 880
wavelength [nm]
1000
1500
2000
2500
3000
3500
4000
4500
ph
as
e 
sh
ift
 d
Φ
/d
z 
[cm
-
1 ]
1.451.51.551.61.65
energy [eV]
Figure 4.10: Induced phase shift or change of the refractive index. For energies near
the band gap and λ = 820 nm, we obtain from the calculation of ∂zΦ applying the
relation (4.12) as typical value of the refractive index change ∆(δn)(1.5 · 1012 cm−2 →
3 · 1012 cm−2) = −0.0136 (with a coupling factor of ΓrVzΓ˜ = 1).
740 760 780 800 820 840 860 880
wavelength [nm]
-2
0
2
4
6
8
10
12
14
16
18
lin
ew
id
th
 e
nh
an
ce
m
en
t f
ac
to
r
1.451.51.551.61.65
energy [eV]
Figure 4.11: Linewidth enhancement or anti-guiding factor. At λ = 820 nm, we can
extract α(1.5 · 1012 cm−2) = 1.38 and α(3 · 1012 cm−2) = 2.36.
4.2 Calculations of Quantum Well Laser Gain 83
750 770 790 810 830 850 870 890
wavelength [nm]
-3000
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.41.451.51.551.61.65
energy [eV]
750 770 790 810 830 850 870 890
wavelength [nm]
-3000
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.41.451.51.551.61.65
energy [eV]
760 780 800 820 840 860 880 900
wavelength [nm]
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.41.451.51.551.6  1.65
energy [eV]
760 780 800 820 840 860 880 900
wavelength [nm]
-2500
-2000
-1500
-1000
-500
0
500
1000
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.41.451.51.551.6  1.65
energy [eV]
770 790 810 830 850 870 890 910
wavelength [nm]
-2500
-2000
-1500
-1000
-500
0
500
1000
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.41.451.51.551.6
energy [eV]
300 320 340 360 380 400
temperature [K]
1.5
1.6
1.7
1.8
1.9
2
tr
an
sp
ar
en
cy
 c
ar
rie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
Figure 4.12: Modification of gain spectra with increasing plasma and lattice tem-
perature T ∈ {320K . . . 400K}: The transparency carrier density increases linearly
with temperature. Other properties of these small signal gain curves are depicted and
discussed in Figures 4.13+4.14.
84 SEMICONDUCTOR LASER FUNDAMENTALS
2 3 4 5 6 7 8 91.5
carrier density [1012cm-2]
0
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
pe
ak
 g
ai
n 
[cm
-
1 ]
300K
320K
340K
360K
380K
400K
Figure 4.13: Peak gain versus carrier sheet density (of the quantum well) for various
temperatures T = Tpl = Tlat ∈ {300K . . . 400K}. Depicted are the microscopically cal-
culated gain measures (symbols) and the logarithmic curve fits, which take into account
the sublinear dependence of the small signal gain on the inversion N − Ntransparency at
higher densities, according to gpeak(N, T ) = g0(T ) · ln (N/Ntransparency(T )) with the fit-
ting parameters transparency density Ntransparency(T ) and the slope g0(T ) (which only
slightly varies with temperature). We can extract g0(300K) = 1210 cm
−1.
2 3 4 5 6 7 8 9
carrier density [1012cm-2]
1.45
1.5
1.55
1.6
en
er
gy
 [e
V]
300K
320K
340K
360K
380K
400K
760
780
800
820
840
w
av
el
en
gt
h 
[n
m]
  1.5 2 2.5 3 3.5 4
carrier density [1012cm-2]
0
50
100
150
200
250
300
350
ga
in
 b
an
dw
id
th
 [m
eV
]
300K
320K
340K
360K
380K
400K
Figure 4.14: Frequency at the gain peak versus carrier density for various temperatures
(left); the shift to higher frequencies with increasing density is caused by the effect of
band filling. The available positive gain bandwidth is depicted on the right and we can
estimate [4] a dependence of the spectral width of positive gain on ∝ µ = µe + µh.
4.2 Calculations of Quantum Well Laser Gain 85
well structure, calculate microscopic scattering rates γpk(N, T ) (see Section 4.1), and
then weight the various contributions or oscillators in (4.14)+(4.15) with the electronic
density of states ρ2Dk of the quasi two-dimensional system.
Figure 4.9 illustrates the importance of relaxation rates and homogeneous broad-
enings in the calculations of the laser gain. These critical input parameters strongly
determine the shape and magnitude of the calculated gain spectra [23]. The peak gain
decreases with increasing rates γp. The results obtained from calculations with constant
rates of 10 ps−1 (left) and 20 ps−1 (right) considerably differ from the calculations (4.14).
The inclusion of realistic microscopic dephasing times and broadenings hence represents
a significant improvement for the prediction of gain and carrier-induced refractive index
change spectra.
In evaluating equation (4.15) there is a problem with the convergence of the integral.
Absolute phase shifts are not of importance, and we have the freedom to choose a
reference value. Thus, we could reinterpret the formula and include the shifts from
the term −1 into the background refractive index of the host material [36]. Another
possibility is to relate the changes of the refractive index ∆(δn) at two different carrier
densities to the variations ∆gmod by the Kramers-Kronig transformation [5, 69,70].
The graphs in Figure 4.11 reveal the strong dependence of α on the frequency and
the carrier density. The implementation of this coupling between amplitude and phase
dynamics by the use of a constant factor, like it is common in phenomenological semi-
conductor laser models [52,73,113,135,136], is therefore a problematic approach.
By incorporating renormalisation effects such as the variation of the transition energies
and the field renormalisation (shift of the optical Rabi frequencies) in the screened
Hartree-Fock approximation we obtain
gmod(λ) + i∂zΦ(λ) = i
k0
ǫ02neff
ΓrVzΓ˜ 1
dQW
Penv
Eenv
=
1
A
∑
k
(
gmod,k + i (∂zΦ)k
)
(4.17)
= ΓrVzΓ˜ π
ǫ0neffλ
1
dQWA
∑
k
γpk − i (Ωk − 2πc/λ)
(Ωk − 2πc/λ)2 + (γpk)2
1
~
∣∣M eh∣∣2 (f ek + fh−k − 1) · 11− qk .
Many-body Hamiltonians, more precisely the screened Coulomb interaction, lead to
a change of transition energies δESXk = δE
e,SX
k + δE
h,SX
−k , δE
CH, ~Ωk and a modified
material susceptibility function. The induced polarisation can be expanded in orders
of V sc. The Pade´ approximation [48] reinterprets this as a geometrical series followed
by a resummation. This high density approximation introduces the complex-valued
dimensionless factors qk(λ), Qk(λ) = 1/(1− qk(λ)) which are given by
qk(λ) = − i
~
∑
k′
V sck,k′
γpk′ − i (Ωk′ − 2πc/λ)
(Ωk′ − 2πc/λ)2 + (γpk′)2
(
f ek′ + f
h
k′ − 1
)
. (4.18)
The angular integrations of the screened Coulomb potential to V sck,k′(N, Tpl) have to
be evaluated numerically. The small signal gain spectra in the screened Hartree-Fock
approximation can be related to the results computed by the free-carrier theory, see
86 SEMICONDUCTOR LASER FUNDAMENTALS
800 820 840 860 880 900 920 940 960
wavelength [nm]
-3500
-3000
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
2000
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.31.351.41.451.51.55
energy [eV]
Figure 4.15: Same as Figure 4.8, but with renormalisation effects from many-body
interactions as per (4.17). The signatures of the Coulomb effects are the existence
of resonances below the band gap (excitons), plus an enhancement of the continuum
optical spectrum [4].
equations (4.14)+(4.15), to obtain
gmod(λ) + i∂zΦ(λ) =
1
A
∑
k
(
gFCTmod,k · Re(Qk)− (∂zΦ)FCTk · Im(Qk)
)
+ i
1
A
∑
k
(
(∂zΦ)
FCT
k · Re(Qk) + gFCTmod,k · Im(Qk)
)
. (4.19)
Some results of the calculation of quantum well laser gain in the screened Hartree-
Fock approximation including energy and field renormalisations are shown in Figures
4.15–4.20. In addition to the steps described above for a non-interacting electron-hole
plasma, we do have to evaluate the plasma screening, calculate the Coulomb potentials
V sck,k′(N, Tpl), and compute the integrals of the renormalisation terms δE
SX
k , δE
CH, qk, Qk.
4.2 Calculations of Quantum Well Laser Gain 87
0 5 10 15 20 25 30
momentum vector [1/aBohr]
-60
-50
-40
-30
-20
-10
0
ex
ch
an
ge
 sh
ift
 e
le
ct
ro
ns
 [m
eV
]
0 5 10 15 20 25 30
momentum vector [1/aBohr]
-60
-50
-40
-30
-20
-10
0
ex
ch
an
ge
 sh
ift
 h
ol
es
 [m
eV
]
Figure 4.16: Exchange shifts (renormalisation of particle energies) δEe,SXk (left graph)
and δEh,SX−k (right) versus the carrier momentum vector for various carrier sheet densities
N ∈ {0.5 · 1012 cm−2 . . . 4 · 1012 cm−2}. Moreover, spectral positions are modified by the
Debye shift δECH(1.5 · 1012 cm−2) = −22.3meV, δECH(3 · 1012 cm−2) = −27meV.
740 760 780 800 820 840 860 880 900 920 940 960
wavelength [nm]
-2000
-1500
-1000
-500
0
500
1000
1500
2000
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.31.351.41.451.51.551.61.65
energy [eV]
800 820 840 860 880 900 920 940 960
wavelength [nm]
-2000
-1500
-1000
-500
0
500
1000
1500
2000
sm
al
l s
ig
na
l (
am
pli
tud
e) 
ga
in 
[cm
-
1 ]
1.31.351.41.451.51.55
energy [eV]
Figure 4.17: The interband Coulomb enhancement results in a reshaping and an
increase of the magnitude of amplification or absorption, most noticeable at low carrier
densities. Energy renormalisation causes a spectral shift. This can be seen in the left
picture in comparison to results neglecting many-body effects (thin line). (Right) Gain
calculations assuming a constant broadening of the Lorentzian line shapes of 20 ps−1
clearly differ from our numerical results which incorporate microscopically determined
dephasing times.
88 SEMICONDUCTOR LASER FUNDAMENTALS
800 820 840 860 880 900 920 940 960
wavelength [nm]
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
ph
as
e 
sh
ift
 d
Φ
/d
z 
[cm
-
1 ]
1.31.351.41.451.51.55
energy [eV]
800 820 840 860 880 900 920 940 960
wavelength [nm]
-4
-2
0
2
4
6
8
10
lin
ew
id
th
 e
nh
an
ce
m
en
t f
ac
to
r
1.31.351.41.451.51.55
energy [eV]
Figure 4.18: Same as Figure 4.10 (left diagram) and Figure 4.11 (right graph),
but including renormalisation effects. The spectra are more structured. At an optical
wavelength of 870 nm we can extract a refractive index change of ∆(δn)(1.5·1012 cm−2 →
3 · 1012 cm−2) = −0.0128 (ΓrVzΓ˜ = 1), and anti-guiding factors of α(1.5 · 1012 cm−2) =
0.62, α(3 · 1012 cm−2) = 0.81.
300 320 340 360 380 400
temperature [K]
1.5
1.6
1.7
1.8
1.9
2
tr
an
sp
ar
en
cy
 c
ar
rie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
2 3 41.5
carrier density [1012cm-2]
0
200
400
600
800
1000
1200
1400
1600
pe
ak
 g
ai
n 
[cm
-
1 ]
300K
320K
340K
360K
380K
400K
Figure 4.19: (Left) Transparency carrier density versus temperature. The thin line
gives a comparison to results obtained from free-carrier theory. (Right graph) As Fig-
ure 4.13: The effect of Coulomb enhancement or interband attraction leads to an
increased peak gain, the slope measures are given by g0(300K) = 1464 cm
−1 (an in-
crease of approximately 20%) and g0(400K) = 1321 cm
−1. Band filling effects cause
the gain rollover and reduction of the differential gain as the carrier density increases.
4.3 Relaxation Oscillations 89
1.5 2 2.5 3 3.5 4
carrier density [1012cm-2]
1.35
1.4
1.45
1.5
en
er
gy
 [e
V]
300K
320K
340K
360K
380K
400K
820
840
860
880
900
920
w
av
el
en
gt
h 
[n
m]
  1.5 2 2.5 3 3.5 4
carrier density [1012cm-2]
0
50
100
150
200
250
300
350
400
ga
in
 b
an
dw
id
th
 [m
eV
]
300K
320K
340K
360K
380K
400K
Figure 4.20: As Figure 4.14: Both, band filling and Coulomb enhancement cause a
blue shift of the frequencies at peak gain which exceeds the opposing red shift due to
energy renormalisations (left). The available gain bandwidth is not affected significantly
when when many-body interactions are considered (right).
4.3 Relaxation Oscillations
In this section high frequency relaxation oscillations, which are typical in semiconduc-
tor laser systems, are investigated, and important laser parameters and terminology
are introduced [1–3]. The damped relaxation oscillations are a transient dynamical be-
haviour [9] which dominate when a semiconductor laser diode is switched on or off, or
perturbed during operation by the modulation or fluctuations of parameters, e.g. the
pump current, by optical injection [137] or by optical feedback [52]. The starting point
of our considerations are the semiconductor laser rate equations4, a set of equations for
the complex-valued electric field amplitude in slowly varying amplitude approximation
(3.37) and for the carrier density (2.122)
n2
cneff
∂tE˜ = −γresonatorE˜ + iκP˜ + ik0 n
neff
δnpasE˜, κ =
k0
ǫ02neff
ΓNQW
Lref
ΓrVz, (4.20)
∂tN = − 1
2~
Im(P˜ ∗ · E˜)− ΓnrN − ΓspN − ΓAugerN + Λ. (4.21)
Spatial degrees of freedom and physical effects like diffraction and transverse index-
and gain-guiding, spatial hole burning (self-focussing) or carrier diffusion will not be
explicitly considered. Instead, they are treated by introducing effective parameters,
such as a phenomenological optical loss rate γresonator (or a resonator Q-factor), and the
confinement and filling factors as measures of the spatial overlap between field mode and
4A derivation starting from Maxwell or the laser cavity equations [14] shows the basic assumptions
and thus limitations of this approach: The cold-cavity modes, which are the eigensolutions of the
associated Helmholtz eigenvalue problem, have to form an orthonormal, complete basis system. This
is essential for superposition and projection operations and implies metallic or periodic boundary
conditions. The field profiles for the active structure are assumed to be identical with the cold-cavity
modes, and only longitudinal single-mode operation is considered.
90 SEMICONDUCTOR LASER FUNDAMENTALS
quantum confined active structures. The semiconductor susceptibility response can be
obtained by adiabatically eliminating the induced polarisations, which leads to
P˜ = ǫ0ǫχ(ω,N)E˜, G(ω,N) = −ǫ0ǫIm
(
χ(ω,N)
)
, (4.22)
ǫ0ǫχ = −
∑
i,j
1
A
∑
k
1
i (Ωij,k − ω) + γpij,k
i
~
∣∣M ehij ∣∣2 (f eii,k + fhjj,−k − 1) ·Qij,k. (4.23)
We can separate the amplitude and phase dynamics5 by the ansatz E˜ = Eenv exp(iΦ)→
∂tΦ = cneff/n
2κǫ0ǫRe (χ(ω,N))+ c/nk0δnpas, and end up with a reduced set of variables
|E˜(t)|2 and N(t) and the rate equations of a class-B laser system
∂t|E˜|2 = −2cneff
n2
(γresonator − κG) |E˜|2, (4.24)
∂tN = Λ− ΓnrN − ΓspN − ΓAugerN −
1
2~
G|E˜|2. (4.25)
The condition ’gain equals optical losses’ determines the threshold carrier density and to-
gether with the rates of the various carrier loss channels the for lasing required threshold
current. Above laser threshold, the steady state solution of the inversion is independent
of the pump power. This phenomenon is called density pinning or gain clamping and
prevents multi-mode lasing operation (see Chapter 5)
G(ω,Nthreshold) =
γresonator
κ
→ Nst = Nthreshold ≥ Ntransparency, (4.26)
Λthreshold = Γ
nrNst + Γ
sp
Nst
+ ΓAugerNst , Λ˜ = Λ− Λthreshold. (4.27)
This means that above threshold all additional pumping is converted into light emission.
The slope efficiency of the light-current curve, which is the conversion factor injected
carriers to emitted photons, is thus constant
|E˜|2st =
2~Λ˜
G(Nthreshold)
=
2~Λ˜κ
γresonator
, ∂Λ˜|E˜|2st =
2~κ
γresonator
. (4.28)
A linear stability analysis of the fixed point attractor (Nthreshold, |E˜|2st) in reduced phase
space quantifies how the laser system reacts to perturbations and gives the oscillation
period Tro = 2π/
√
Ω2ro − Γ2ro and damping Γro of the relaxation oscillations
λ± = −Γro ± i
√
Ω2ro − Γ2ro, (4.29)
Γro =
1
2
(
Γnr + ∂NΓ
sp
N + ∂NΓ
Auger
N +
Λ˜κ
γresonator
∂NG
)∣∣∣∣∣
Nst
, (4.30)
Ωro =
√
2
cneff
n2
Λ˜κ ∂NG|Nst . (4.31)
5In broad area lasers the coupling between amplitude and phase dynamics, measured by the α factor,
leads to filamentation and instabilities as the result of competing gain- and induced index-guiding.
For more details see Chapter 5. These and other phase sensitive effects cannot be neglected in
spatially extended systems.
4.3 Relaxation Oscillations 91
In the following we will compare this theoretical results with simulations of the spatio-
temporal dynamics in edge-emitting lasers applying the paraxial model (see Figures
4.21+4.22), and in vertical cavity surface-emitters modelled within the full time-domain
approach (see Figures 4.23+4.24). In both cases, the gain medium is formed of several
active layers of GaAs (quantum wells) and will be modelled by the microscopic, band-
resolved semiconductor Bloch equations. We see that starting from noise or spontaneous
emission the electric field amplitude is rising exponentially as energy stored in the carrier
subsystem is transformed into the optical laser field. Then the carrier density is eaten up
by the light field and eventually falls below threshold (δN = N −Nst < 0). The lasing
breaks down. The electrical pumping pushes the inversion back above the threshold
value, and the behaviour repeats: Intensity and the carrier density oscillate around their
steady state solution with a constant phase shift of π/2. In the main, our simulations
and the theoretical calculations show a very close agreement apart from small deviations
due to the spatial degree of freedom and diffraction and diffusion processes (in edge-
emitting structures, see Figures 4.21+4.22). Also there are difficulties in determining
the effective parameters neff, γresonator, Γz = ΓrVz in surface-emitters or microcavity
lasers due to the problem to define an effective length of the structure. In Figure
4.23 the switch-on dynamics of a VECSEL is numerically studied. In order to fit the
transient dynamical behaviour, which is shown in our full time-domain simulations with
values Γro = 2.54 · 109 s−1 and Tro = 260 ps, we would set κ = 6.9 · 1023VC−1m−1
(ΓNQW/LrefΓz = 5.4 · 106m−1 and neff = 3.41) and γresonatorc/neff = 8.3 · 1010 s−1.
Relaxation oscillations are a transient dynamical feature before steady state lasing
operation sets in within a cavity. In an amplifier structure we treat the field intensity
(or envelope amplitude) not as a dynamical variable but as given parameter. Here, we
introduce the saturation intensity
|E˜|2sat =
2~
(
Γnr + ∂NΓ
sp
N + ∂NΓ
Auger
N
)
∂NG
∣∣∣∣∣∣
Ntransparency
(4.32)
and solve for the stationary steady state value of the carrier density
N −Ntransparency =
Λ− ΓnrNtransparency − ΓspNtransparency − ΓAugerNtransparency(
Γnr + ∂NΓ
sp
N + ∂NΓ
Auger
N
)∣∣∣
Ntransparency
1
1 + |E˜|2/|E˜|2sat
.
(4.33)
This large signal effect is visible both in absorption and amplification. The gain com-
pression or saturation [4] increases with increasing field intensity as the available carriers
and gain are reduced.
For a more simple characterisation of the gain material we consider a homogeneously
broadened two-level system (or macroscopic semiconductor gain model) with the sus-
ceptibility response ǫ0ǫχ = −1/
(
i(Ω− ω) + γP ) i/~ ∣∣M eh∣∣2 2 (N −Ntransparency), which
assumes a linear dependence of the gain on the carrier density ∂NG(ω,N) = G0(ω),
92 SEMICONDUCTOR LASER FUNDAMENTALS
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
time [ps]
1.6
1.7
1.8
1.9
2
2.1
2.2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 0
2
4
6
8
10
12
14
16
18
(el
ec
tri
c f
iel
d)2
 
[1
01
2 V
2 m
-
2 ] 
Figure 4.21: Damped relaxation oscillations in an edge-emitting, single-mode semi-
conductor laser structure; depicted are the electric field amplitude squared |E˜|2 (at
the top) and the carrier density (at the bottom) recorded at the centre of the de-
vice versus time. The lasing structure is characterised by the waveguide properties
neff = 3.51 (with nact = 3.59, dact = 250 nm, ncl = 3.45), ΓNQW/Lref = 5.2 ·106m−1 and
γresonatorc/neff = 2.68 · 1010 s−1, the coupling factor κc/neff = 5.5 · 1031VC−1s−1, Γz = 1,
Γnr = 2·108 s−1 and Ntransparency(815 nm) = 1.54·1012 cm−2. With the above derived the-
ory (4.20)–(4.31) we can estimate for the stationary solution Nthreshold = 1.7 ·1012 cm−2,
Λthreshold = 7.935 · 1020 cm−2s−1, |E˜|st = 1.85 · 106Vm−1, and for the damped relaxation
oscillations at a pumping of two times threshold Γro = 2.64 · 109 s−1 and Tro = 406 ps.
G(ω,N) = G0(ω) (N −Ntransparency). Furthermore, we sum up all carrier loss channels
to an effective loss rate Γeff. The stationary solution above lasing threshold is given by
Nthreshold = Ntransparency +
γresonator
κG0
, Λthreshold = Γ
effNthreshold, |E˜|2st =
2~Λ˜κ
γresonator
.
(4.34)
The decay rate and frequency of the damped relaxation oscillations are quantified by
Γro =
1
2
(
Γeff +
Λ˜κG0
γresonator
)
, Ωro =
√
2
cneff
n2
Λ˜κG0, (4.35)
and the gain saturation behaviour is described by
|E˜|2sat =
2~Γeff
G0
, G(ω, |E˜|2) = G0(ω)(Λ− Γ
effNtransparency)
Γeff
1
1 + |E˜|2/|E˜|2sat
. (4.36)
The physical cause of the gain compression is the depletion or the decrease of the macro-
scopic carrier density due to hole burning in an amplifier structure or the increase of
N in an absorber, respectively (see Sections 6.4+6.5). We note, that an applied light
intensity (optical pumping) cannot induce a transition from absorption to amplification
as |E˜|2 →∞ : N → Ntransparency, G→ 0.
4.3 Relaxation Oscillations 93
1.7 1.8 1.9 2 2.1 2.2
carrier density [1012cm-2]
0
2
4
6
8
10
12
14
16
18
(el
ec
tri
c f
iel
d a
mp
litu
de
)2  
[1
01
2 V
2 m
-
2 ]
1.78 1.79 1.8
2
2.2
2.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
-10 -8 -6 -4 -2 0 2 4 6 8 10
transverse position [µm]
 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
(el
ec
tri
c f
iel
d a
mp
litu
de
)2  
[1
01
2 V
2 m
-
2 ] 
Figure 4.22: Reduced phase space diagram [14] (left graph), and transverse electric
field amplitude and carrier density profiles at time 10 ns (right). The variations of the
simulation results (based on a set of partial differential equations) from the theoretical
calculations are caused by the processes of diffraction of the optical fields, a transverse
gain- and index-guiding (by a dielectric structure), for more details see equation (3.37),
and the diffusion of carriers in the laser with a 5µm broad current contact. Therefore,
crossings of the trajectory in the reduced phase space occur, the stationary transverse
profiles are not been reached even after 10 ns, and the threshold carrier density and
necessary current for lasing are higher (by reason that diffraction can be reinterpreted
as additional optical loss channel). However, applying Nthreshold and γresonator as fitting
parameters we do get a nearly perfect match of theory and simulation.
0 500 1000 1500 2000 2500 3000
time [ps]
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 0
2
4
6
8
10
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ] 
Figure 4.23: Damped relaxation oscillations in a vertical cavity surface-emitting semi-
conductor laser structure (VCSEL); depicted are the absolute value of the electric field
including the carrier wave as well as the electric field envelope Eenv = |E˜| (at the top)
and the carrier sheet density (at the bottom) versus time. The fixed point attrac-
tor of the nonlinear (class-B laser) system is specified by Nst = 2.167 · 1012 cm−2 and
|E˜|st = 2.10 · 106Vm−1. Other relevant parameters include Λ = 2 · Λthreshold and the
carrier loss channels Γnr = 1 · 109 s−1 and ΓspN according to (2.48).
94 SEMICONDUCTOR LASER FUNDAMENTALS
1
1.5
2
2.5
3
3.5
4
re
fra
ct
iv
e 
in
de
x
0 1 2 3 4 5 6 7 8 9
position [µm]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
3.1
3.2
3.3
3.4
3.5
3.6
3.7
re
fra
ct
iv
e 
in
de
x
4 4.2 4.4 4.6 4.8 5 5.2 5.4
position [µm]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
Figure 4.24: Refractive index structure and longitudinal electric field profile (snapshot
at time point 3 ns). Between a top and a bottom highly reflective Bragg mirror a defect
of length λ is enclosed, at the maxima of the modes active quantum well layers are
positioned. The electromagnetic field is enhanced over the active quantum confined
structures, for this reason the laser mode strongly differs from a delocalised plane wave.
The calculations of the overlap or filling factor, average refractive index and the optical
losses are critical because thereby it is necessary to identify an effective length of the
lasing structure (in particularly of the DBR mirror).
4.4 Quantisation of Light
Together with the semiclassical approach, presented in Chapter 2, a phenomenological
loss term accounting for spontaneous emission has been introduced into the semicon-
ductor carrier dynamics. A full quantum theoretical treatment of the electromagnetic
fields will motivate this loss factor and analyse the impact of spontaneous emission on
the dynamics of the optical fields itself. In analogy to the evaluations in Chapter 2,
we work in lowest order of the Taylor expansion of the Wigner distribution functions.
By applying a plane wave expansion one can specify the Hamilton operators in second
quantisation by
Hlight =
∑
p,q
Ebp,qb
†
p,qbp,q, E
b
p,q = ~ωp,q, ωp,q = ωq =
c
n
q, (4.37)
E =
∑
p,q
i
√
~ωp,q
2ǫ0ǫV
(
bp,qe
iq·r − b†p,qe−iq·r
)
ep,q, gp,q = i
√
~ωp,q
2ǫ0ǫV
Mehij · ep,q, (4.38)
Hcarrier-light = −
∑
i,j,k
∑
p,q
[
gp,qbp,qc
†
i,kd
†
j,−k + g
∗
p,qb
†
p,qdj,−kci,k
]
, (4.39)
where only the resonant terms in the interaction Hamiltonian have been considered.
Then, the following dynamical equations can be deduced for the carrier correlations
4.4 Quantisation of Light 95
neii,k, n
h
jj,−k, pji,k and for 〈bp,q〉 and the photon distribution nbp,q
∂tn
e
ii,k =
i
~
∑
j
∑
p,q
(
gp,q
〈
bp,qc
†
i,kd
†
j,−k
〉
− g∗p,q
〈
b†p,qdj,−kci,k
〉)
, (4.40)
∂tpji,k = −iΩij,kpji,k − i
~
∑
p,q
gp,q
〈
bp,q
(
c†i,kci,k + d
†
j,−kdj,−k − 1
)〉
, (4.41)
∂t 〈bp,q〉 = −iωp,q 〈bp,q〉+ i
~
∑
i,j
g∗p,q
∑
k
pji,k, (4.42)
∂tn
b
p,q = ∂t
〈
b†p,qbp,q
〉
=
i
~
∑
i,j,k
(
g∗p,q
〈
b†p,qdj,−kci,k
〉
− gp,q
〈
bp,qc
†
i,kd
†
j,−k
〉)
. (4.43)
The semiclassical description would be the result of a factorisation of the above dy-
namical equations (4.40)–(4.43). Instead, we solve the field-dipole correlations dynamics
∂t
〈
bp,qc
†
i,kd
†
j,−k
〉
by a formal integration similar to the methods applied in Section 2.86.
An equivalent formulation of the quantum noise dynamics and spontaneous emission
based on a space-space representation of these field-dipole correlations is given in [138].
In order to truncate the hierarchy and to close the system of equations, the expecta-
tion values must be factorised. This leads to the following additional terms entering
the dynamical equations for the carrier and photon distributions and accounting for the
influence of spontaneous emission
∂tn
e
ii,k : −
∑
j
γspij,kn
e
ii,kn
h
jj,−k, γ
sp
ij,k =
2π
~
∑
p,q
1
~
δ (∆Ωij,k) gp,qg
∗
p,q =
2n (Ωij,k)
3 |M ehij |2
3~ǫ0πc3
,
(4.44)
∂tn
b
p,q : +γ
sp
p,q, γ
sp
p,q =
2π
~
∑
i,j
∑
k
1
~
δ (∆Ωij,k) gp,qg
∗
p,qn
e
ii,kn
h
jj,−k, (4.45)
∑
i,j
1
A
∑
k
γspij,kn
e
ii,kn
h
jj,−k =
1
A
∑
p,q
γspp,q =
∑
p
∫
dωSp(ω) =
∑
p
∫
dλS˜p(λ), (4.46)
Sp(ω) =
∑
i,j
n (ωp,q)
3
3~ǫ0π2c3
1
A
∑
k
|M ehij |2
γpij,k
(Ωij,k − ωp,q)2 +
(
γpij,k
)2neii,knhjj,−k. (4.47)
A steady state solution for the photon distribution is given by the amplified sponta-
neous emission spectra (4.48). Contrary to the semiclassical approach the gain gmodp,q is
smaller than the optical losses of the resonator γ˜resonator. Indeed, spontaneous emission
occurs with a rate γspp,q into the laser mode p,q which we assumed to be a plane wave
with wavenumber q and polarisation p. Microcavity structures show a large fraction of
spontaneous emission emitted into the laser mode, qualified by a high β coupling factor.
6∂t
〈
bp,qc
†
i,kd
†
j,−k
〉
= −i/~ (Ebp,q − ~Ωij,k) 〈bp,qc†i,kd†j,−k〉 + i/~∑i′,j′,k′ g∗p,q〈c†i,kd†j,−kdj′,−k′ci′,k′〉+
i/~
∑
p′,q′ g
∗
p′,q′
〈
b†p′,q′bp,q
(
c†i,kci,k + d
†
j,−kdj,−k − 1
) 〉
specifies the time evolution of this correlation
as the sum of expectation values of products of four operators.
96 SEMICONDUCTOR LASER FUNDAMENTALS
0.1 1 10
carrier density [1012cm-2]
1018
1019
1020
1021
1022
sp
on
ta
ne
ou
s e
m
iss
io
n 
ra
te
 [c
m-
2 s
-
1 ]
Figure 4.25: Total spontaneous emission or radiative recombination rate ΓspN versus
carrier density. At low carrier densities the usually assumed ansatz ∂tN = −BN2 [88]
represents a relatively good fit (dashed line), and we can extract B = 1.95 ·10−4 cm2s−1.
For higher densities there are considerable deviations as a result of the Pauli exclusion
principle for fermionic particles (as described by the Fermi-Dirac distribution functions).
The radiative recombination rate ΓspN is obtained from the spontaneous emission spectra,
see Figure 4.26, by ΓspN =
∑
p=TE,TM
∫
dλS˜p(λ).
550 600 650 700 750 800 850 900 950 1000 1050
wavelength [nm]
1
2
3
4
5
6
7
8
9
sp
on
ta
ne
ou
s e
m
iss
io
n 
sp
ec
tra
 [1
01
8 c
m
-
2 s
-
1 n
m
-
1 ]
Figure 4.26: Spontaneous emission or luminescence spectra S˜TE(λ)dλ for TE-
polarisation for various carrier sheet densities N ∈ {0.5 · 1012 cm−2 . . . 4 · 1012 cm−2}
at room temperature T = 300K. Many-body effects are included via microscopically
calculated scattering or dephasing rates γpk(N, T ), Coulomb renormalisation effects are
incorporated by renormalised transition frequencies Ωij,k and Coulomb enhancement,
within the framework of the Pade´ approximation, by Qij,k. Only bound quantum well
states are considered (system is a single GaAs/AlGaAs quantum well of 5 nm width).
4.5 Thermodynamics of Semiconductor Lasers 97
In theory this could offer the possibility of thresholdless lasers. Generally, there is a
modification of spontaneous emission by the dielectric structure or cavity [11,42,43,139]
∂tn
b
p,q = 2
(
gmodp,q − γ˜resonator
)
nbp,q + γ
sp
p,q, γ˜resonator =
cneff
n2
γresonator,
nbp,q =
γspp,q
2
(
γ˜resonator − gmodp,q
) , (4.48)
gmodp,q = ΓzΓ˜
1
~2
∑
i,j,k
gp,qg
∗
p,q
γpij,k
(Ωij,k − ωp,q)2 +
(
γpij,k
)2 (neii,k + nhjj,−k − 1) . (4.49)
An in-depth consideration of resonator effects requires the incorporation of phase sensi-
tive quantities, such as
〈
bp,qbp,q
〉
and
〈
b†p,qb
†
p,q
〉
. In the set of equations of motion, this
phase sensitivity is mediated by the coupling to the interband polarisation terms pji,k. In
the end, the resonator structure spectrally redistributes the electromagnetic fields orig-
inating from spontaneous emission γspp,q itself [108]. In a one-dimensional Fabry-Pe´rot
cavity the steady state solution for the photon distribution nbp,q can be calculated from
the condition [36]
Ep,q− = fp,qEp,q+ cos (ϕp,q) + E
sp
p,q → nbp,q = γspp,q · F
[
gmodp,q , ϕp,q, γresonator, Tresonator
]
.
(4.50)
The feedback coefficient fp,q and the phase shift ϕp,q are computed for one round trip with
period Tresonator. Furthermore, the quantum mechanical fluctuation-dissipation theorem
allows for the description of spontaneous emission by the addition of Langevin noise
operators or forces (and connected Gaussian-Markovian stochastic processes) in spatially
homogeneous systems7.
4.5 Thermodynamics of Semiconductor Lasers
In microcavity lasers such as V(E)CSEL structures the heat dissipation from the active
nanostructures to the ambient is seriously hindered. Heating and thermal effects become
more pronounced and problematic in microlasers and ever more complicated structured
active devices as the thermal rollover limits the maximum output power [12, 91, 106].
Because of the time scales of the macroscopic balancing processes, we do not work within
a time-domain approach but rather use iterative solvers to obtain the thermal steady
state solution: On basis of a thermodynamic model we balance the carrier density and
energy density flows according to the various sources and sinks. Applied approximations
comprise the assumption of Fermi-Dirac distributions (no spectral hole burning), an
7A generalisation of the quantum theory of light for open or spatially inhomogeneous systems is rather
critical [44,47]. One approach is to make an expansion in the exact eigenmodes of the total system,
including resonator and external electromagnetic fields, that is to say, system and bath [45,46]. The
required mode projection of the electromagnetic fields however conflicts with the application of the
Langevin approach [8] in a full time-domain scheme or in a mathematical description build on partial
differential equations.
98 SEMICONDUCTOR LASER FUNDAMENTALS
adiabatic elimination of the interband polarisations, single optical polarisation, and no
explicit treatment of the spatial degrees of freedom. In dependence on the macroscopic
variables carrier sheet density N , common plasma temperature Tpl for the electrons and
the holes, and lattice temperature Tlat, the electronic band structure is calculated by
Egap(Tlat, N, Tpl) (Debye shift included), E
e,SX
ii,k (N, Tpl), E
h,SX
jj,−k(N, Tpl), E
SX
ij,k(N, Tpl). The
diverse carrier loss channels and the coupling of the carrier subsystem to the laser fields
are balanced by a pumping term ΛN (for more details see Section 2.11)
ΛN = Γ
nrN +
∑
i,j
1
A
∑
k
γspij,k(N, T )f
e
ii,k(N, Tpl)f
h
jj,−k(N, Tpl) + Γ
Auger
N (T )
+
∑
i,j
1
A
∑
k
∑
q
g˜qij,kn
b
q. (4.51)
Here, γspij,k is calculated using the Weißkopf-Wigner theory (2.48). A crucial point is not
to treat Auger recombination [2] by phenomenological parameters, but rather address
these Coulomb mediated scattering processes on a microscopic level (similar to the con-
siderations (2.83)–(2.85)) by ΓAugerN (T ), Γ
Auger
U (T ) [87,108]. The coupling of the inversion
to the photon distribution is quantified by
g˜qij,k =
ωq
ǫ0ǫV
|Mehij |2
~
γpij,k
(Ωij,k − ωq)2 +
(
γpij,k
)2 (f eii,k + fhjj,−k − 1) . (4.52)
A similar equation can be obtained for the plasma energy density U(N, Tpl) = U
e + Uh
ΛU = Γ
nrU +
∑
i,j
1
A
∑
k
γspij,k(N, T )f
e
ii,k(N, Tpl)f
h
jj,−k(N, Tpl)E
SX
ij,k(N, Tpl)
+ Γe,AugerU (T ) + Γ
h,Auger
U (T ) +
∑
i,j
1
A
∑
k
∑
q
g˜qij,kE
SX
ij,k(N, Tpl)n
b
q
+
∑
i
1
A
∑
k
γ
e(ph)
ii,k (N, T )
(
f eii,k(N, Tpl)− f eii,k(N, Tlat)
)
Ee,SXii,k (N, Tpl)
+
∑
j
1
A
∑
k
γ
h(ph)
jj,−k(N, T )
(
fhjj,−k(N, Tpl)− fhjj,−k(N, Tlat)
)
Eh,SXjj,−k(N, Tpl). (4.53)
Finally, we have to balance the energy density flows into and out of the lattice subsys-
tem. The energy density transfer (thermalisation) plasma to the lattice is mediated by
phonons
σth
Tlat − Tamb
dambdact
= Γnr
(
U(N, Tpl) +NEgap(Tlat, N, Tpl)
)NQW
Lref
+
∑
i
1
A
∑
k
γ
e(ph)
ii,k (N, T )
(
f eii,k(N, Tpl)− f eii,k(N, Tlat)
)
Ee,SXii,k (N, Tpl)
NQW
Lref
(4.54)
+
∑
j
1
A
∑
k
γ
h(ph)
jj,−k(N, T )
(
fhjj,−k(N, Tpl)− fhjj,−k(N, Tlat)
)
Eh,SXjj,−k(N, Tpl)
NQW
Lref
.
4.6 Conclusion 99
The behaviour of the solution can now be studied by variation of the independent pa-
rameter N . An evaluation of the balance equations for the density flows (4.51), and
the energy density flows (4.53)+(4.54) determines ΛN → Ipump, Tpl and Tlat. As final
result we obtain the photon distribution in dependence on the state of the active semi-
conductor structure nbq(N, Tpl, Tlat)
8. In our approach, the pumping fields (an optical
pumping scheme is implemented [16, 33]) are coupled to the active gain structures in a
semiclassical way
ΛN = −
∑
i,j
1
A
∑
k
|Mehij · E˜pump|2
2~2
γpij,k
(Ωij,k − ωpump)2 +
(
γpij,k
)2 (f eii,k + fhjj,−k − 1) ,
(4.55)
and similar for the energy density pump ΛU . The pump intensities are given by Ipump =
ǫ0
√
ǫc|E˜pump|2/2. This explains the thermal rollover in V(E)CSEL or general microcavity
laser structures. The previously described operating principles of gain clamping and
density pinning are not justified. With a rise in temperature the available gain is reduced,
which has to be compensated by a higher carrier density. However, with increasing
temperature and carrier density (or pumping) the carrier loss processes of spontaneous
emission and particularly Auger recombination are of more and more relevance, and the
thermal load per absorbed photon rises. The active nanostructures in microlasers are
comparatively small-sized compared with millimeter-sized edge-emitting structures. A
higher carrier density and pumping is necessary to compensate the reduced gain as the
amplification ∝ exp(gmodLact) is determined by the length of the active structures. With
that, losses and temperature effects are strongly enhanced.
In addition to the intrinsic temperature dependence of the gain, thermally induced
refractive index changes will contribute to a further reduction of the effective or modal
gain and eventually cause thermal rollover [15,33]. The shift of the optical thickness and
consequently of the microcavity resonance with a typical value of +0.1 nmK−1 results in a
mismatch between the antinodes of the mode and the active gain elements (in a resonant
periodic gain structure), and in an increasing offset between resonance frequency and
gain peak. A typical value of the band gap and gain shift is +0.3 nmK−1. A detuning at
room temperature of gain peak and microcavity resonance can improve the high-power,
high temperature operation of V(E)CSEL [105].
4.6 Conclusion
In this chapter we have considered the local properties of the active laser gain medium.
The resonator structure and spatial effects are accounted for by the introduction of
8In contrast to the quantum mechanical approach presented above, there is no contribution from spon-
taneous emission to the laser intensity in the semiclassical limit. The lasing wavelength is determined
by the spectral position for which first the constraint gmodql (N,Tpl, Tlat) = γ˜
ql
resonator is fulfilled. The
coupling between the classical electric field amplitude and a semiconductor gain material is charac-
terised by the replacement
∑
q g˜
q
ij,kn
b
q → g˜ij,k (ωql) (with an expression for g˜ij,k (ωql) ∝ |E˜|2 similar
to (4.55)).
100 SEMICONDUCTOR LASER FUNDAMENTALS
effective parameters. We calculate realistic input parameters for the analysis of spatio-
temporal laser dynamics, which improves our models quantitatively. The following in-
sights have been gained applying this approach:
• Microscopic scattering and dephasing rates for the screened Coulomb and the Fro¨hlich
interactions in GaAs quantum well systems have been calculated for various carrier
sheet densities and temperatures. These effective measures enter our simulations of
spatio-temporal laser dynamics in form of lookup tables. The different contributions
have been analysed, and we have specified typical values for each process. We have
justified the concept of a common plasma temperature and identified carrier-carrier
scattering as dominant contribution to the ultrafast optical dephasing.
• We have computed the optical gain properties and important laser parameters of
semiconductor amplifier media. We have performed the analyses for a realistic de-
scription of the quantum confined semiconductor gain medium including many-body
interactions and correlation contributions (as starting point for our fitting ansatz of
the susceptibility), as well as for a more simple approach in which renormalisation
effects have been neglected (as implemented in the dynamical band-resolved models).
The numerical results are in good agreement with experimental findings. The inclu-
sion of realistic microscopic dephasing times and broadenings provide a significant
improvement for the prediction of gain and carrier-induced refractive index change
spectra.
• We have analysed the strongly damped high frequency relaxation oscillations in semi-
conductor laser systems. This transient dynamical behaviour is dominant when a laser
is switched-on or perturbed during operation by modulation or fluctuations of param-
eters or by optical injection. We have compared simulations of the spatio-temporal
dynamics in edge-emitters and in surface-emitting structures with theoretical calcu-
lations. We have found the concept of density pinning and a classification as class-B
laser system to be well justified.
• We have touched the problem of a quantum theoretical treatment of the electromag-
netic field dynamics in semiconductor laser systems and discussed the microscopic
evaluation of the processes of spontaneous emission.
• We have developed a thermodynamic model, which balances the flows (the various
sinks and sources) of the macroscopic carrier density, the common plasma tempera-
ture and for the lattice temperature. This allows us to investigate the performance
characteristics and to analyse thermal rollover of diode lasers. We have identified
the invalidity of the concept of the pinning of the density (above lasing threshold) in
microcavity lasers, and subsequent the rapid increase of Auger recombination with
increasing density and temperature as the main contributions to thermal rollover. We
also show that the process of spontaneous emission is an important factor in these
laser structures.
5 Transverse Multi-Mode Laser
Dynamics
5.1 Introduction—Effects of Spatial Degrees of Freedom
The in Chapter 4 discussed semiconductor laser principles of density pinning and gain
clamping (at their threshold values) and the selection mechanism due the spectral depen-
dence of the material gain prevent the coexistence of laser modes within the semiclassical
framework. Single-mode operation and a stable, steady state output signal are expected.
The competition between the various modes for the available carrier inversion and gain
is won, according to the principle of survival of the fittest as derived from nonlinear
dynamics [9] and synergetics [10], by the fundamental mode. This mode is the first to
fulfil the condition ’modal gain equal optical losses’ on increasing the pump and carrier
density. In contrast to this analysis, experimental and numerical studies [112,132,140] of
wide-aperture edge-emitting diode lasers (broad area lasers1) show several mode families
from the interplay between longitudinal and higher order transverse modes2. This coexis-
tence of multiple modes in semiconductor lasers is due to the (in the above considerations
neglected) spatial degrees of freedom and inhomogeneous broadenings. Multi-mode be-
haviour originates from the mechanisms of spatial and spectral hole burning, as well as
due to spontaneous emission [3, 112]. The interaction of the field with different spatial
regions of the inversion distribution (as the various modes are driven by separated and
independent carrier baths) allows multi-mode behaviour. In analogy, for a sufficiently
large frequency spacing between modes, the different laser modes can couple to individual
1The scaling of the output power of edge-emitting lasers (up to around 10 Watt in continuous wave
operation) is achievable by the broadening of the current contact and of the aperture of the waveguide
(typical stripe widths are 50–500µm). But, this broad area lasers suffer from the deterioration
of coherence properties, beam quality and profile, broad multi-lobed far-field patterns, and from
dynamical optical filamentation and spatio-temporal instabilities in the near-field [36, 37, 106, 130,
132]. That is caused by the excitation of higher order transverse modes and self-focussing effects
in large-aperture structures. Concepts to overcome this (geometrical) instability are proposed by
multi-stripe laser arrays [37, 112], by current profiling (inhomogeneous pumping) [36, 131], or the
control and stabilisation of spatio-temporal emission by delayed optical feedback [132].
2In a waveguide or cavity structure a laser mode is defined as the spatial pattern of the electromagnetic
field or the intensity distribution, characterised by its eigenfrequency and a decay rate, from the
coupling to the surrounding free space modes, and driven by a source term (the induced electric
polarisation or material gain). Demanding stationarity, due to the fact that (integrated over the
spatial coordinates) the optical losses are balanced by the gain, active or laser modes can be computed
from the Maxwell equations specifying applicable, absorbing boundary conditions. These modes do
not necessarily form a complete set of orthonormal functions. A less strict definition understands a
laser mode as the spatial field profile at a specified frequency.
102 TRANSVERSE MULTI-MODE LASER DYNAMICS
polarisations and electron-hole pairs of the inhomogeneously broadened gain material.
The spontaneous formation of spatio-temporal patterns (in the quasi-periodic or regu-
lar regime) develops as the result of the superposition or interference of all excited laser
modes. Nonlinear mode coupling, optical nonlinearities (self-focussing) and time-varying
carrier and refractive index profiles lead to higher spatio-temporal complexity.
The mode structure of a typical edge-emitting diode laser device can be estimated as
follows. The frequency spacing of standing waves in a Fabry-Pe´rot laser cavity with the
application of metallic (or periodic) boundary conditions is specified by
∆klo =
π
L
→ ∆ωlo = c
neff
∆klo =
πc
neffL
→ ∆λlo = λ
2
2πc
∆ωlo =
λ2
2neffL
. (5.1)
Typical values (using the parameters L = 1mm, neff = 3.5122, λ = 815 nm) for the
longitudinal mode spacing are ∆ωlo = 268 · 109 s−1, ∆λlo = 0.1 nm. A similar condition
for the higher order transverse modes, corresponding to the same longitudinal mode
resonance, gives the estimate
k =
√
k2lo + k
2
tr ≈ klo +
k2tr
2klo
, ∆ktr =
π
W
→ ∆ωtr = cλ
4πn2eff
∆
(
k2tr
)→ ∆λtr. (5.2)
Typical values (with W = 50µm) for the mode separation of the first four transverse
modes are ∆ωtr = 18.7 · 109 s−1, 31.1 · 109 s−1, 43.6 · 109 s−1. Higher order transverse
modes possess higher frequencies than the fundamental mode. Each mode creates a
carrier (and intensity) grating with a lattice constant of 2π/(2klo) = λlo/(2neff) and
λtr/(2neff), respectively.
We have seen that multi-mode behaviour originates from hole burning and the separa-
tion of carrier baths. Consequently, processes which are coupling the different spatial (or
spectral) regions counteract the mechanisms behind multi-mode operation. For spectral
hole burning such processes include scattering and homogeneous broadening. Spatial
hole burning, carrier depletion and inversion inhomogeneities are balanced by carrier
diffusion3.
The electric field (envelope) distributions and lasing frequencies are determined by
gain-guiding as well as by the effective refractive index profile neff+ δnpas(x) + δn[N(x)]
3For an edge-emitting diode laser structure, applying the one-dimensional Maxwell semiconductor
Bloch model in full time-domain, we have computed the output spectra with a realistic value of the
ambipolar carrier diffusion constant of DN = 10 cm
2s−1 and with artificially switched-off diffusion,
respectively: The laser spectrum of the device with switched-off diffusion (this means that the
inversion baths of the different modes are decoupled) shows a multitude of peaks, connected with
the different longitudinal modes, with constant ∆ωlo. Because of the high spatial frequency of the
intensity and carrier grating associated with longitudinal modes, carrier inhomogeneities created
by longitudinal spatial hole burning are washed out on a picosecond time scale (∝ DN (2klo)2).
Consequently, our simulations reveal that carrier diffusion drastically reduces the number of excited
longitudinal modes, for shorter structures even to single-mode operation.
5.2 Transverse Instabilities in Broad Area Lasers 103
(both quantified by the specified inversion distribution N(x)) [2, 141]
∂2xE˜(x;ωmode) = 2k0neff
(
i (−κǫ0ǫIm(χ(ωmode))[N(x)]− γresonator)
− (κǫ0ǫRe(χ(ωmode))[N(x)] + n
neff
k0δnpas(x))
)
E˜(x;ωmode). (5.3)
In a self-consistent approach, the transverse carrier profile is the result of diffusion,
electrical pumping, carrier loss channels, and stimulated recombination, depletion by
spatial hole burning and gain saturation. At regions where the optical intensity is high,
the gain and carrier-induced refractive index change are saturated as a result of spatial
hole burning. With decreasing carrier density the effective index increases, governed
by the anti-guiding factor α. Due to these optical nonlinearities, a mode creates its
own confining waveguide. The importance of the local dynamics (in contrast to the
mode picture) is rising for increasing stripe width and/or pumping. The process of self-
focussing creates tightly focussed spots in the intensity, i.e. so-called optical filaments.
5.2 Transverse Instabilities in Broad Area Lasers
This section investigates the transverse instability, multi-mode behaviour and the for-
mation of optical filamentation in semiconductor lasers. In particular, we analyse above
which stripe width the regime of single-mode, continuous wave operation in the funda-
mental mode becomes instable. We also identify the main control parameters and the
physical mechanisms that lead to these instabilities. We calculate the growth rates of
the self-modulated solutions (with transverse wavenumber ktr = k). Optical nonlineari-
ties due to carrier depletion by hole burning, and the nonlinear interaction between the
electromagnetic field and the gain material (qualified by g(x) and δn(x)), are included
in our calculations. The starting point for our analysis are the transversally resolved
Maxwell semiconductor Bloch equations in paraxial and mean-field approximation. We
apply an adiabatic elimination of the induced polarisation variables (see (4.20)+(4.21))
and obtain
n2
cneff
∂tE˜(x, t) =
(− γresonator − κǫ0ǫIm(χ(N)))E˜ + i 1
2k0neff
∂2xE˜
+ i
(
κǫ0ǫRe(χ(N)) +
n
neff
k0δnpas(x)
)
E˜, (5.4)
σ =
cneff
n2
1
2k0neff
, E˜ = Eenve
iΦ,
∂tN(x, t) =
ǫ0ǫ
2~
Im(χ(N))
∣∣E˜∣∣2 − ΓeffN + Λ(x) +DN∂2xN, (5.5)
Γeff = ΓeffNst = Γ
nr + ∂NΓ
sp
N + ∂NΓ
Auger
N
∣∣∣
Nst
.
A linear stability analysis of the stationary and spatially homogeneous (translation in-
variance assumed) solution (Est,Φ = −ωΦt+Φ0, Nst) is performed. The exponential de-
cay or growth rates in time of oscillatory perturbations (δE, δE∗, δN), characterised by
104 TRANSVERSE MULTI-MODE LASER DYNAMICS
a wavevector k (and with an intensity and carrier density grating of 2k, and D = 4DN),
are calculated from
∂tδE = −iσk2δE + cneff
n2
κǫ0ǫ
(− ∂N Im(χ) + i∂NRe(χ))∣∣Nst EstδN, (5.6)
∂tδN =
ǫ0ǫ
2~
Im(χ(Nst))Est (δE + δE
∗) +
(ǫ0ǫ
2~
∂N Im(χ)|Nst E2st − Γeff −Dk2
)
δN.
(5.7)
Using the ansatz δE, δE∗, δN ∝ exp(λt) an eigenvalue problem is derived, which leads
to the following characteristic cubic equation
(
σ2k4 + λ2
) (ǫ0ǫ
2~
∂N Im(χ)|Nst E2st − Γeff −Dk2 − λ
)
+ 2
ǫ0ǫ
2~
Im(χ(Nst))E
2
st
cneff
n2
κǫ0ǫ
(−λ∂N Im(χ) + σk2∂NRe(χ))∣∣Nst = 0, (5.8)
with the eigenvalues λ1 ∈ R and λ2,3 ∈ C (conjugate complex). Analysis of equation (5.8)
[36, 37] reveals that for realistic semiconductor diode laser parameters Re(λ2,3) < 0,∀k
and limk→∞Re(λ2,3) = 0. Thus, in order to ensure the stability of transverse modes
with high wavenumbers, a numerical diffusion term Dx∂
2
xE˜ [36] of an elliptic character
is added to (5.4), as we discuss in Appendix A. Instabilities occur for Re(λ) > 0; a
conditional equation for kcrit follows from Re(λ1) = 0
σ2k4
(ǫ0ǫ
2~
∂N Im(χ)|Nst E2st − Γeff −Dk2
)
+ 2σk2
ǫ0ǫ
2~
Im(χ(Nst))E
2
st
cneff
n2
κǫ0ǫ ∂NRe(χ)|Nst = 0, (5.9)
k2crit =
ǫ0ǫ/(2~) ∂N Im(χ)|Nst E2st − Γeff
2D
(5.10)
+
√√√√(Γeff − ǫ0ǫ/(2~) ∂N Im(χ)|Nst E2st)2
(2D)2
+
(ǫ0ǫ)
2 Im(χ)E2stcneff/n
2κ∂NRe(χ)|Nst
~Dσ
.
Instable eigenvalues are found for a band of wavenumbers with 0 < k < kcrit. The
interpretation of this is that there is no stable stationary and spatially homogeneous
solution and that, independent of the electrical pumping and the width of the lasing
structure, spatio-temporal instabilities arise. However, transverse boundary conditions
and the unpumped margin stripes stabilise the system, and the oscillatory perturbations
can only develop in structures withW > Wcrit. For typical semiconductor gain materials
and laser parameters we may approximate (5.10) by
kcrit ≈
√
2Λ˜cneff/n2κα
(−ǫ0ǫ ∂N Im(χ)|Nst)
σ
(
Γeff − ǫ0ǫ/(2~) ∂N Im(χ)|Nst E2st
) . (5.11)
For an edge-emitting diode laser operating at Λ = 2Λthreshold, Λ˜ = Λ − Λthreshold, and
assuming the parameters specified in Figure 4.21, the critical wavevector and critical
5.3 Simulations—Different Dynamic Emission Regimes 105
width (of the transverse spatially extended structure) are given by
kcrit = 2 · 10−4 nm−1 → Wcrit ≈ π
kcrit
= 15.7µm.
The main control parameters are given by the stripe width, the electrical pumping Λ, the
anti-guiding factor α, the differential gain and the various carrier loss channels. Wcrit is
decreasing for increasing electrical pumping but remains nearly constant for higher values
of the pump [36]. The transverse instability is promoted by increasing the width of the
current contact or likewise by increasing the pumping. Another important dependency
is represented by the linewidth enhancement factor α; carrier-induced refractive index
changes are destabilising and breaking up the transverse modes and producing filaments.
Quantum dot gain systems with reduced amplitude-phase coupling are known to improve
laser beam quality as optical filamentation is suppressed in these structures [29,93,95].
5.3 Simulations—Different Dynamic Emission Regimes
In this section we numerically verify the analytical predictions (5.2) and (5.10) by apply-
ing a microscopic (transverse) spatially resolved model. The model incorporates a density
matrix approach to the optoelectronic properties of the quantum well gain medium, and
incloses a diversity of time scales and optical nonlinearities. The electromagnetic field
dynamics is calculated by integrating the Maxwell wave equation in paraxial approxi-
mation. Different degrees of spatio-temporal complexity and several dynamic emission
regimes can be distinguished depending on the stripe width as the main control pa-
rameter. In particular, we identify continuous wave operation, regular or quasi-periodic
operation and irregular behaviour.
We note that the analysis of spatio-temporal emission dynamics is carried out in anal-
ogy to the experimental streak camera technique, which permits the measurement of
integrated field intensity traces with a maximum resolution of 10 ps [132,140]. Numeri-
cally, we calculate the instantaneous optical intensity as well as on-the-fly computations
of (over few ten picoseconds) time-averaged transverse profiles
〈|E˜(x, t)|2〉
T
=
1
N + 1
t0+N∆t∑
t=t0
|E˜(x, t)|2, 〈N(x, t)〉
T
=
1
N + 1
t0+N∆t∑
t=t0
N(x, t). (5.12)
This averaging allows to eliminate the mode beating dynamics with characteristic fre-
quencies of ∆ωmode − ∆ωmode’. Furthermore, the placement of a spectrometer into the
experimental setup for near-field measurements allows a spectral decomposition of the
spatio-temporal laser emission dynamics. Our theoretical approach in the same way
assumes a linear expansion in transverse modes E˜mode(x;ωmode) with the coefficients
cmode ∈ C. We interpret transverse modes as spectrally filtered spatial near-field pat-
terns or field distributions, which can be extracted from the electric field amplitude time
106 TRANSVERSE MULTI-MODE LASER DYNAMICS
series by projection or spectral filtering (with ωmode = ω −∆ωmode)
e−iωt · E˜(x, t) = e−iωt ·
∑
mode
e+i∆ωmodetcmodeE˜mode(x;ωmode), (5.13)
cmodeE˜mode(x;ωmode) =
1
N + 1
t0+N∆t∑
t=t0
e−i∆ωmodetE˜(x, t). (5.14)
In this context, the Karhunen-Loe`ve decomposition should be mentioned, which extracts
a complete set of orthonormal laser modes from the eigenvalue problem of the Hermi-
tian covariance matrix (space-space correlations) Cij = 1/T
∫ t0+T
t=t0
dtδE˜∗(xi, t)δE˜(xj, t),
δE˜(x, t) = E˜(x, t)− 〈E˜(x, t)〉
T
[37, 112].
The most important simulation parameters are summarised in the following table:
important simulation parameters (GaAs/AlGaAs quantum well lasers)
electronic properties of the semiconductor gain material [60]:
band gap energy Egap(T = 300K) 1.424 eV
Varshni band gap parameters:
Egap(T = 0K), α, β 1.517 eV, 5.5 · 10−4 eVK−1, 225K
effective electron mass me 0.067m0
effective hole mass mh 0.330m0
width of quantum wells dQW 5 nm
Bohr radius aBohr 13.777 nm
dipole matrix element M eh 0.3 enm
effective polarisation dephasing constant γp 20 ps−1
nonradiative recombination time 1/Γnr 5 ns
Auger recombination coefficient ΓAuger 2.5 · 10−18 cm4s−1
energy of longitudinal optical phonons ~ωl-o 36meV
(electrical) pump efficiency η 0.5
pumping Λ 2Λthreshold
ambipolar carrier diffusion constant DN 10 cm
2s−1
optical and geometrical properties of the dielectric waveguide:
laser wavelength λ 815 nm
resonator/laser cavity length L 1mm
width of current contact, stripe width W 5 . . . 100µm
width of unpumped margin stripes Wmargin 7.5µm
thickness of active layer dact 250 nm
transverse index-guiding δnpas(x) −3 · 10−3
intensity reflectivities Rl, Rr 0.95, 0.3
resonator loss γresonatorc/neff 0.0268 ps
−1
dielectric constant ǫ 12.89
effective refractive index neff (nact, ncl) 3.5122 (3.590, 3.452)
confinement factor ΓyNQW/Lref 5.2µm
−1
5.3 Simulations—Different Dynamic Emission Regimes 107
-2.5
0
2.5
x
 [µ
m
]
  600 1200 1800 2400 3000 3600 4200
-7.5
-5
-2.5
0
2.5
5
7.5
x
 [µ
m
]
0 1000 2000 3000 4000 5000
time [ps]
1.6
1.7
1.8
1.9
N
 [1
01
2 c
m
-
2 ]
0
3
6
9
12
E2
 
[1
01
2 V
2 m
-
2 ]
N
I
Figure 5.1: Transverse single-mode switch-on dynamics of a 5µm broad edge-emitting
semiconductor diode laser structure: Depicted are the optical near-field intensity
(top), the carrier sheet density (middle), and the (over the stripe width) spatially
averaged electric field amplitude squared (solid line; bottom) and carrier inversion
(dashed; bottom). The transverse field profiles (x, z = L, t) are linearly coded in
grey scales with dark/black corresponding to high values and white to low inten-
sities or carrier densities, respectively. Device parameters are specified in Figure
4.21; other important characteristics quantify the index-guiding by a ridge waveguide
structure δnpas(stripe) = 0, δnpas(margin) = −3 · 10−3, the ambipolar carrier diffusion
DN = 10 cm
2s−1, and the width of the unpumped margin stripes Wmargin = 7.5µm.
Our simulations show stable transverse single-mode operation after a few nanoseconds.
Figures 5.1+5.2 give an insight into the transient dynamic regime of damped high
frequency relaxation oscillations and transverse single-mode operation (characterised by
the Gaussian fundamental mode) for a broad area laser structure with W = 5µm. We
find a very close agreement between these numerical results, applying the paraxial trans-
verse wave equation model with a microscopic approach to the dynamical polarisation
response, and the theoretical description of the transient dynamic regime of damped
relaxation oscillations, for more details see Section 4.3.
Increasing the stripe width to 16µm (see Figures 5.3+5.4), and to 50µm (Figures
108 TRANSVERSE MULTI-MODE LASER DYNAMICS
-10 -5 0 5 10
transverse position [µm]
0
0.5
1
1.5
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
 0
0.5
1
1.5
2
2.5
(el
ec
tri
c f
iel
d a
mp
litu
de
)2  
[1
01
2 V
2 m
-
2 ]
Figure 5.2: (Transverse) Electric field amplitude and carrier density profiles at 5 ns.
An electric field amplitude squared of |E˜|2 = 2·1012V2m−2 translates in the investigated
laser structure with GaAs/AlGaAs quantum wells to an intensity of ≈ 1MWcm−2.
4800 4900 5000 5100 5200 5300 5400
-8
-4
0
4
8
x
 [µ
m
]
0 1000 2000 3000 4000 5000
time [ps]
1.6
1.7
1.8
1.9
N
 [1
01
2 c
m
-
2 ]
0
6
12
18
E2
 
[1
01
2 V
2 m
-
2 ]
I
Figure 5.3: Spatio-temporal emission dynamics of a 16µm wide diode laser structure
for moderate pumping Λ = 2Λthreshold.
5.3 Simulations—Different Dynamic Emission Regimes 109
-15 -10 -5 0 5 10 15
transverse position [µm]
0
0.5
1
1.5
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
-2
-1
0
1
2
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
-15 -10 -5 0 5 10 15
transverse position [µm]
0
1
2
3
-15 -10 -5 0 5 10 15
transverse position [µm]
(el
ec
tri
c f
iel
d)2
 
[1
01
2 V
2 m
-
2 ]
0
1
2
3
4
(el
ec
tri
c f
iel
d)2
 
[1
01
2 V
2 m
-
2 ]
Figure 5.4: (Left graph) Transverse profiles of the carrier density
〈
N(x, t)
〉
T
(dashed)
and of the electric field amplitude squared
〈|E˜(x, t)|2〉
T
(solid line) averaged over the
time interval t ∈ [3.6 ns,5.4 ns] (T = 1.8 ns). The real and imaginary components
(top; solid and dashed lines, respectively) and the absolute values squared (bottom)
of the transverse active laser modes cmodeE˜mode(x;ωmode), as numerically determined
from spectral filtering (projection) operations according to (5.14), are depicted: funda-
mental, symmetric mode with the oscillation frequency ω0 = 2.31066 · 1015 s−1, λ0 =
815.2005 nm (right column), and first order transverse eigenmode (antisymmetric)
ω1 = 2.31081 · 1015 s−1, λ1 = 815.1469 nm (middle).
5.5+5.6), higher order modes show up supported by the carrier, gain and refractive
index profiles. The spatio-temporal emission can be characterised in the mode picture
as superposition and beating of several transverse modes (as spectrally filtered spatial
near-field distributions). Spatio-temporal patterns evolve on picosecond time scales.
The optical field dynamics can be projected, using (5.14), onto laser eigenmodes. The
simulations shown in Figure 5.3 reveal an alternating right-left pulsing of the optical
intensity with a time period of 41.3 ps. We also note that this quasi-periodic or regular
spatio-temporal dynamics is a non-transient behaviour; a constant electrical pumping
level is applied, and we see from the graph at the bottom that after ≈ 3 ns the integrated
intensity as well as density are constant. The spectral decomposition of the near-field
pattern (cf. Figure 5.4) shows that in addition to the (Gaussian) fundamental mode a
higher order transverse mode (with a higher eigenfrequency) is excited. The beating and
superposition of these two modes with different spatial distributions (and hence carrier
baths), symmetries and oscillation frequencies (because of the additional component ktr)
110 TRANSVERSE MULTI-MODE LASER DYNAMICS
result in the high frequency switching of the output intensity. Figure 5.4 shows the
carrier and spectrally filtered field profiles. The coexistence and interaction of two modes
with ∆λ = 0.0536 nm, ∆ν = 24.19GHz and |c0|/|c1| = 1.537 induce a high frequency
oscillation of the optical intensity between the left-hand and right-hand edges of the
pumped region (stripe widthW = 16µm), for more details see Figure 5.3. We see that
the two modes interact with different spatial regions of the inversion distribution. Our
simulations demonstrate the dependence of spatio-temporal emission dynamics on the
stripe width of a semiconductor broad area laser. The dynamic regime of single-mode
operation and damped relaxation oscillations (class-B laser system) becomes instable
at a critical parameter of Wcrit = 16µm, which is in close agreement with theoretical
predictions from (5.11). The mode analysis allows for the characterisation of spatio-
temporal complexity and an estimate of the number of dynamical degrees of freedom.
We note that in our simulations of transverse multi-mode laser dynamics, the induced
polarisation variables are not adiabatically eliminated, unlike in [12, 36, 37, 140], as a
consequence the spectral variation of the gain (and the dispersion) acts as natural mode
selection mechanism.
The results shown in Figure 5.6 further support the validity of the mode picture. We
find (for W = 50µm) seven dominant modes (and associated peaks in the laser output
spectrum
∑
x |E˜(x,∆ω)|2). The numerically computed properties of these higher or-
der transverse modes—field distributions, laser frequencies ∆ωmode, and symmetries (in
respect of the transverse coordinate x)—confirm in the main the simple analytical esti-
mates derived in (5.2). We note small asymmetries and deviations of the extracted modes
from perfect distributions (see e.g. Re(c1E˜1(x;ω1))), and that
〈|E˜(x, t)|2〉
T
slightly dif-
fers from the summed modal intensities
∑
mode |cmode|2|E˜mode(x;ωmode)|2. We note that
the mode analysis assumes the linear superposition of the various spectral contributions,
and it is based on spectral filtering using Fourier transformation. The transverse den-
sity, gain and refractive index profiles, supporting the various laser modes, can be well
approximated as flat (constant) over the pumped, active area [141]. Our time-domain
approach enables the investigation of temporal changes in the mode structure as the
effect of the slow carrier and/or temperature dynamics (e.g. spatial transport processes)
and therefore changing gain g[N(x, t), T (x, t)] and index profiles δn[N(x, t), T (x, t)] [12].
Figure 5.11 gives an insight into the microscopic carrier and gain dynamics of a 50µm
wide broad area laser with two optically active electron and two hole subbands. As the
imprints of the optical field dynamics we see spectral hole burning in the carrier distribu-
tion functions δn◦k(t) = n
◦
k(t)− f ◦k [N(t)], and interband polarisations |p11,k(t)|, |p22,k(t)|.
In Figures 5.7+5.8—important laser parameters includeW = 16µm, Λ = 4Λthreshold,
and δnpas = 0—a spontaneous symmetry breaking occurs and the laser intensity gradu-
ally migrates to one edge of the stripe. We interpret this irregular dynamic behaviour as
a consequence of the high linewidth enhancement factor and the strong carrier-induced
changes of the refractive index δn[N(x)] in semiconductor-based active structures. We
find a spatio-temporal instability resulting in the spontaneous breaking of the symmetry
and a drift of the intensity profile to one edge of the stripe in combination with con-
5.3 Simulations—Different Dynamic Emission Regimes 111
-20
-10
0
10
20
x
 [µ
m
]
3600 3900 4200 4500 4800 5100 5400
-20
-10
0
10
20
x
 [µ
m
]
0 1000 2000 3000 4000 5000
time [ps]
1.7
1.8
1.9
N
 [1
01
2 c
m
-
2 ]
0
6
12
18
E2
 
[1
01
2 V
2 m
-
2 ]
I
N
Figure 5.5: Same as Figure 5.1, but for a 50µm wide edge-emitting semiconductor
laser. The calculation of the spatially resolved output spectrum |E˜(x,∆ω)|2 and a
spectral decomposition of the electric field amplitude time dynamics, based on equation
(5.14) (see Figure 5.6), shows that the observed spatio-temporal behaviour may be
explained in the mode picture by the coexistence of several transverse field patterns,
i.e. active laser modes (quasi-periodic regime).
tinuously decreasing integrated output power [112,140]. The laser emission jumps back
to the centre of the stripe, and a large pulse can be seen simultaneously in the laser
output. The gradual migration of the intensity distribution and the asymmetric emis-
sion demonstrate the impact of the strong amplitude-phase coupling and high linewidth
enhancement factor in active semiconductor devices. An accumulation of carriers not
only increases the local optical gain but also leads to a repulsion of the optical field
because of counteracting anti-guiding by the carrier-induced refractive index changes of
the waveguide properties.
As another result of the strong amplitude-phase coupling, mediated by the α factor,
a spot of the intensity creates its own confining waveguide (self-focussing) induced by
optical nonlinearities. With increasing stripe width (W = 100µm) the influence of local
dynamics is rising, and spatio-temporal instabilities and dynamical optical filamentation
deteriorate the laser emission characteristics (see Figures 5.9+5.10). With increasing
stripe width we note an increase in the spatio-temporal complexity and a gained im-
112 TRANSVERSE MULTI-MODE LASER DYNAMICS
-30 -20 -10 0 10 20 30
transverse position [µm]
0
0.5
1
1.5
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
0
1
2
3
4
5
(el
ec
tri
c f
iel
d)2
 
[1
01
2 V
2 m
-
2 ]
-30 -20 -10 0 10 20 30
transverse position [µm]
0
0.5
1
-1
-0.5
0
0.5
1
-30 -20 -10 0 10 20 30
transverse position [µm]
-30 -20 -10 0 10 20 30
transverse position [µm]
(el
ec
tri
c f
iel
d)2
 
[1
01
2 V
2 m
-
2 ]
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
-30 -20 -10 0 10 20 30
transverse position [µm]
0
0.5
1
-1
-0.5
0
0.5
1
-30 -20 -10 0 10 20 30
transverse position [µm]
-30 -20 -10 0 10 20 30
transverse position [µm]
-30 -20 -10 0 10 20 30
transverse position [µm]
0
0.5
1
1.5
2
2.5
(el
ec
tri
c f
iel
d)2
 
[1
01
2 V
2 m
-
2 ]
-1
-0.5
0
0.5
1
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
Figure 5.6: As Figure 5.4, but for a broad area laser structure with a stripe width of
50µm. Spatio-temporal emission is characterised by time-averaged transverse profiles
and by the projection of the (complex) electric field amplitude dynamics onto the laser
eigenmodes (for more details see (5.13)+(5.14)).
5.3 Simulations—Different Dynamic Emission Regimes 113
10800 11400 12000 12600 13200 13800 14400
-12
-8
-4
0
4
8
12
x
 [µ
m
]
10000 12000 14000 16000 18000
time [ps]
1.8
2
2.2
N
 [1
01
2 c
m
-
2 ]
0
10
20
30
E2
 
[1
01
2 V
2 m
-
2 ]
I
Figure 5.7: As in Figure 5.3, but at higher pumping levels Λ = 4Λthreshold and without
a built-in rectangular index-guiding (by a ridge waveguide structure).
-15 -10 -5 0 5 10 15
transverse position [µm]
0
0.5
1
1.5
2
2.5
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
0
2
4
6
(el
ec
tri
c f
iel
d a
mp
litu
de
)2  
[1
01
2 V
2 m
-
2 ]
Figure 5.8: Transverse electric field amplitude and inversion profiles at t = 13.2 ns.
114 TRANSVERSE MULTI-MODE LASER DYNAMICS
-50 -40 -30 -20 -10 0 10 20 30 40 50
transverse position [µm]
30
40
50
60
70
80
90
100
110
120
130
∆ν
 
[G
Hz
]
815.1
815.15
815.2
815.25
w
av
el
en
gt
h 
[n
m]
Figure 5.9: Spatially resolved output spectrum of a 100µm wide diode laser computed
by Fourier transforms of the electric field amplitude time series (applying a Hann win-
dow filter [142]); depicted is the spectral electric field |E˜(x, λ)| (ν = c/815 nm−∆ν).
-50 -40 -30 -20 -10 0 10 20 30 40 50
transverse position [µm]
0
0.5
1
1.5
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
0
1
2
3
4
5
6
(el
ec
tri
c f
iel
d a
mp
litu
de
)2  
[1
01
2 V
2 m
-
2 ]
-50 -40 -30 -20 -10 0 10 20 30 40 50
transverse position [µm]
0
0.5
1
1.5
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
0
3
6
9
12
15
(el
ec
tri
c f
iel
d a
mp
litu
de
)2  
[1
01
2 V
2 m
-
2 ]
Figure 5.10: Depicted are the transverse profiles of the carrier sheet density (dashed)
and of the electric field amplitude squared (solid line) averaged over the time interval
t ∈ [3.6 ns,5.4 ns] (left graph), and at the time point t = 5ns (right).
portance of local dynamics. The spatio-temporal emission dynamics becomes irregular
and the mode resonances spectrally strongly broadened (Figure 5.9). The noticeable
asymmetries in the left graph of Figure 5.10 implies that for a 100µm broad area
laser, optical filamentation and local dynamics dominate global correlations or modes.
A two-dimensional visualisation of the spatial distribution of the intracavity intensity
I(x, z, t) ∝ |E˜+(x, z, t)|2 + |E˜−(x, z, t)|2 shows the dynamics of these filamentary struc-
tures. We see the formation, interaction, breakup and longitudinal propagation (diagonal
and mainly parallel to the optical axis) of optical filaments [37, 112]. Our simulations
feature the intrinsic time scale of the cavity round trip time τ = 2Lneff/c = 23.43 ps.
5.3 Simulations—Different Dynamic Emission Regimes 115
    4800 4950 5100 5250 5400
time [ps]
0
2.5
5
7.5
10
m
o
m
en
tu
m
 v
ec
to
r [
1/a
B
oh
r]
    4800 4950 5100 5250 5400
time [ps]
0
2.5
5
7.5
10
m
o
m
en
tu
m
 v
ec
to
r [
1/a
B
oh
r]
Figure 5.11: Microscopic, band-resolved carrier dynamics in a semiconductor broad
area laser. Depicted are the non-equilibrium Wigner distributions for the electron (top)
as well as hole subbands (middle), and of the induced interband polarisations (at the
bottom); recorded at the position x = 0, z = L. The properties of the lower electron
and hole subbands are plotted on the left, for the upper, optically active subbands
at the right. The investigated structure contains 20 nm wide GaAs/AlGaAs quantum
wells (in all other graphs in this section dQW = 5nm). Other simulation parameters
include kcut-off = 30/aBohr and a resolution of 201 momentum grid points.
116 TRANSVERSE MULTI-MODE LASER DYNAMICS
5.4 Conclusion
In this chapter we have applied the paraxial wave equation model in order to theo-
retically and numerically investigate the transverse multi-mode dynamics and optical
filamentation in broad area edge-emitting laser structures. We demonstrate that with
increasing stripe width and/or higher pumping powers, the gain and effective refractive
index profiles become large enough to support multiple transverse laser modes. Lo-
cal structures, so-called optical filaments, are focussed spots in the intensity and arise
from the nonlinear optical effect of self-focussing. This multi-mode and filamentation
behaviour together with the self-organised, spontaneous formation of spatio-temporal
patterns, a known phenomenon in nonlinear dynamics, originate from optical nonlin-
earities due to hole burning, the carrier and gain dynamics, and the selective depletion
of the density by the various transverse modes. The interactions of the optical field
with different domains of the gain material, with different spatial (transverse degree of
freedom) and spectral (inhomogeneous broadening) regions of the population inversion,
result in the coexistence of multiple modes. A quantitative understanding of the im-
pacts of carrier diffusion, gain- and index-guiding, gain saturation and self-focussing is
achieved. We have presented a linear stability analysis of the transverse instability to
identify the main control parameters as the stripe width, the pumping and the linewidth
enhancement factor. Numerical simulations of spatio-temporal dynamics support this
interpretation. Depending on the stripe width, different degrees of spatio-temporal com-
plexity and several dynamic emission regimes have been distinguished. An eigenmode
analysis of the numerically calculated complex optical field envelopes enables us to ex-
tract the lasing frequencies and the active modes, by means of discrete, spatially resolved
Fourier transforms of the near-field.
6 Longitudinal Multi-Mode Laser
Dynamics
6.1 VCSEL with Resonant Periodic Gain and Refractive
Index Structures
Microcavity [13] or more generally micrometer-sized active laser and amplifier structures
offer a couple of advantages compared to commonly used (sub-) millimeter-sized active
optical elements. Examples are the strong optical confinement, large quality factors
and long photon lifetimes, the low lasing thresholds, and the strong coupling of active
materials with photons. These features make them attractive as compact coherent laser
sources and efficient small amplifiers in photonic integrated circuits. However, most of
these structures have very limited output powers and very low single-pass amplification
due to the relative small size of the active gain sections. For example, a typical VCSEL
structure only contains few quantum wells. In order to improve the performance of
VCSEL, the overlap between the quantum-confined gain elements and the quasi-standing
wave pattern can be optimised by placing the quantum wells at the antinodes of the
cavity mode. The number and position of the antinodes is thereby controlled by the
optical thickness of the microcavity [11, 15]. Using a resonant periodic gain structure a
maximum relative longitudinal confinement factor of Γr = 2 can be achieved [11]. The
longitudinal confinement factor Γz is defined in (6.3). Emod(z) describes the electric field
profile as the stationary solution of the wave equation or of the Helmholtz equation
D(z, t) = ǫ0ǫ(z)E(z, t) +
(
Γx,y
NQW
∆z
)
(z)P (z, t), (6.1)(
∂2z −
ǫ(z)
c2
∂2t
)
E(z, t) =
1
ǫ0c2
(
Γx,y
NQW
∆z
)
(z)∂2t P (z, t), (6.2)
Γz =
∫
L
dzE2mod(z)Θ(z)∫
L
dzE2mod(z)
= VzΓr = Lact
L
Γr. (6.3)
A further improvement of active micrometer-sized laser and optical amplifier elements
can be achieved by incorporating photonic band gap materials or photonic crystal struc-
tures with high refractive index contrasts [20]. This allows to control modes and photons
at the length scale of the optical wavelength [18,19].
In the following we study the photonic band edge band gap laser, which employs
gain enhancement using photonic crystal structures: We analyse asymmetric (one-
118 LONGITUDINAL MULTI-MODE LASER DYNAMICS
   850 900 950 1000 1050 1100 1150
wavelength [nm]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
re
fle
ct
iv
ity
Figure 6.1: Reflectivity spectra of an asymmetric VCSEL structure (thick line) com-
posed of a 13·λ/2 point defect (microcavity) embedded in dielectric Bragg mirror stacks
of λ/4 layers (the top DBR mirror consists of 15 pairs and the bottom DBR, grown on
the substrate, of 25 pairs), and spectra of two lasing structures with an additional pe-
riodic refractive index and gain structure configuration (dashed and dot-dashed). The
edges of the photonic stop band of the Bragg mirror with periodicity aBragg are for com-
parison plotted as thin lines (for more details see Figure 6.2). The active laser device
operates at frequencies of resonant cavity or defect modes (within the cavity dips).
dimensional) VCSEL structures which comprise dielectric Bragg mirror stacks of λ/4
layers, an embedded resonant periodic gain structure with quantum wells placed at the
antinodes of the defect modes, and an additional periodic refractive index structure
n(z) = n(z + Z). The device structure is depicted in Figure 6.7. Graded index layers
(GRIN) are used to improve the carrier capture from barrier layers into the quantum
wells [63]. The passive refractive index structures are characterised by the reflectivity
spectra, see Figure 6.1. The first stop band generated by distributed Bragg reflectors
and dips at wavelengths of resonant cavity modes are clearly visible.
To gain a deeper insight into the gain enhancement mechanism, we analyse the indi-
vidual elements of the laser device and identify the functions of the various components.
The infinite long Bragg mirror (see Figure 6.2) as well as periodic gain and refractive
index structures (in Figures 6.3+6.4) with periodicity aBragg and apgs, respectively, are
reinterpreted as one-dimensional photonic crystals. A common approach in discussing
the properties of periodic photonic systems is to normalise the eigenfrequencies and
wavevectors by the fundamental length of the crystal (because of the linearity of the ba-
sic equations, the results are scalable with the geometry). The frequencies are depicted
in units of 2πc/a [20]. In a homogeneous material of uniform refractive index naverage
the light cone is specified (for the dimensionless quantities wave number and frequency)
by the relation ω = k/(2naverage). At the edges of the Brillouin zone the modes in a ho-
mogeneous refractive index configuration show a twofold degeneracy, the perturbation
through a periodic refractive index structure causes a splitting and the occurrence of a
6.1 VCSEL with Resonant Periodic Gain and Refractive Index Structures 119
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
wavevector [pi/aBragg]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
fre
qu
en
cy
 [2
pi
c/
a B
ra
gg
] =
 a B
ra
gg
λ-
1
Figure 6.2: Band structure diagram calculated for an infinite long AlGaAs/GaAs
Bragg mirror structure [143]: The unit cell of the photonic crystal consists of two layers
(with refractive indices of n = 2.96, 3.55) and is characterised by a lattice constant of
aBragg = 152 nm. Considering the symmetry of the layer system and the fact that for
k > π/aBragg modes may be folded back by a reciprocal lattice vector into the first
Brillouin zone, it is sufficient to consider k ∈ [0, π/aBragg] [20]. Similar to the properties
of electrons in a solid described by electronic band structure diagrams [5] band gaps
occur at π/aBragg (i.e. there are no propagating solutions for frequencies within the band
gaps). The periodic index structure acts as a frequency-selective filter, i.e. it functions
as mirror for wavelengths within the band gaps.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
wavevector [pi/apgs]
0
0.05
0.1
0.15
0.2
0.25
0.3
fre
qu
en
cy
 [2
pi
c/
a p
gs
] =
 a p
gs
λ-
1
Figure 6.3: Band structure diagram of infinite long periodic (graded) index structures
with apgs = 141 nm and different refractive index contrasts ∆n [143]. In an actual
device these structures are enclosed within an asymmetric Bragg mirror configuration.
A more detailed graph of the band structure near π/apgs (showing flat dispersion curves
near the photonic band edges) is given in Figure 6.4.
120 LONGITUDINAL MULTI-MODE LASER DYNAMICS
0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
wavevector [pi/apgs]
0.13
0.14
0.15
0.16
fre
qu
en
cy
 [2
pi
c/
a p
gs
] =
 a p
gs
λ−
1
0.06
0.12
0.18
0.24
Figure 6.4: Resonant periodic gain and GRIN structures, and Bragg mirror stop band
(plotted as thin lines).
frequency interval where the optical density of states is zero. In the lower branch the
maxima of the standing waves are in the higher index material, whereas in the upper
branch the modes are mainly localised in the lower dielectric material [20]. Figure
6.4 shows an increasing photonic band gap and a shift of the eigenfrequencies to higher
values on decreasing the average dielectric constant naverage and increasing ∆n. We con-
sider structures with different index contrasts (as the depths of the GRIN is altered).
At the band edges the dispersion is flat associated with a zero group velocity. Of special
interest is the region near the long-wavelength side of the band gap, where the maxima
of the standing wave patterns are localised in the high index material at active gain
elements. Thus, this photonic crystal provides the effective gain enhancement for the
photonic band edge modes. The slow down of propagating waves is associated with more
gain per unit length. Figure 6.2 shows the band structure of an infinite long dielectric
Bragg mirror. For a range of frequencies the eigenvalues and the group velocities are
purely imaginary, which corresponds to evanescent waves. For normalised frequencies of
the photonic band edge modes equal to apgs/λ ≈ 0.142, this photonic crystal acts as a
mirror. A photonic band edge band gap laser is composed of two photonic crystal struc-
tures: In the finite-sized VCSEL the flat dispersion curves near the photonic band edges
in the band structure diagrams of the inner GRIN are used for lasing operation. The
photonic band gap provided by the surrounding photonic crystal minimises the losses of
the laser mode.
In summary, operating the laser at frequencies of apgs/λ ≈ 0.142 and controlling the
optical modes by exploiting the special properties of photonic band gap materials im-
proves the performance of the laser device. The periodic GRIN pattern stimulates the
band edge enhancement of the effective gain and an optimum overlap between the laser
mode and the resonant periodic gain structure. The DBR suppresses the optical losses
of photonic band edge modes [18,19,144]. We note that the refractive index contrasts of
the analysed periodic structures are relatively low. Thus, to achieve the special proper-
6.1 VCSEL with Resonant Periodic Gain and Refractive Index Structures 121
8 13 18 23
number of quantum wells, of periods with apgs
945
950
955
960
965
970
975
980
985
990
995
1000
w
av
el
en
gt
h 
[n
m]
0.06
0.12
0.18
0.24
Figure 6.5: Operating wavelength of the laser device: Plotted are the edges of the
band structure diagram Figure 6.3 (with the lower branches as thick lines) and the
active laser frequencies as computed using the FDTD method (symbols). With an
increasing number of quantum wells and periods of the index structure with a lattice
constant of apgs the frequencies of the lower band edges as the limits are approached.
The concept of the photonic band edge laser [18, 144] is to capitalise on the special
properties of the singularities in the band structure diagrams, and it is based upon gain
enhancement by an increased localisation of the modes over the active quantum confined
structures [11, 15] and a more efficient interaction of photons with an active material
in a photonic crystal structure. The two deviating wavelengths around 950–955 nm are
linked to a different cavity dip.
ties offered by photonic band gap materials and to reach the limits of the infinitely long
structures a certain number of periods of the crystals are required.
We next test the above discussed control of photons by periodic refractive index
structures and the mode and gain enhancement in these novel VCSEL structures. The
numerical simulation of real, finite-sized samples and the calculations of the operating
wavelengths (see Figure 6.5) and threshold carrier densities (see Figure 6.6) demon-
strate the significant improvements. Figure 6.7 shows that the specified concepts of
effective gain enhancement become more distinctive with increasing number of periods
of the finite structure and increasing refractive index contrasts.
Recently, a two-dimensional photonic crystal slab laser structure based on the concept
of photonic band edge lasing [144] assisted by a photonic band gap has been demon-
strated [18]. Its design is based on the enhancement of the effective gain in periodic
systems ǫ(r) = ǫ(r+R) (R describes a lattice vector) by the singularities of the optical
density of states at points of zero group velocity at the photonic band edges [19], ei-
ther from confinement or periodicity. As a consequence, the photon dispersion relations
ω(k) close to the band gaps become more flat, and the group velocity is reduced. The
122 LONGITUDINAL MULTI-MODE LASER DYNAMICS
0 500 1000 1500 2000 2500 3000
time [ps]
0
2
4
6
8
10
el
ec
tri
c 
fie
ld
 e
nv
el
op
e 
[1
06
V
m
-
1 ]
2
2.5
3
3.5
4
4.5
5
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
∆n of GRIN structure2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
av
er
ag
e 
ca
rri
er
 sh
ee
t d
en
sit
y 
[1
01
2 c
m
-
2 ]
Figure 6.6: Carrier sheet density (dashed) and electric field envelope (solid line) at
the position of the gain elements versus time (in picoseconds). The simulations show
typical relaxation oscillations of the semiconductor-based class-B laser system (for more
details see Section 4.3) towards the steady state characterised by Nst and (Eenv)st. In
the right diagram we show the averaged stationary (threshold) carrier density in the
quantum wells for a periodic gain and refractive index structure with 13 (circles) and
18 (squares) periods. The photonic band edge laser design optimises the mode and gain
enhancement [11,18,144] and consequently reduces the threshold for lasing.
thresholds for lasing decrease as the photons in the photonic crystal can interact with
the active quantum well material more efficiently (i.e. for a lengthy period) and the
coupling between cavity mode and gain material is enhanced (see Figures 6.6+6.7).
To recapitulate, the wave is slowed down and experiences an increased gain per unit
length. In a real sample, the periodic photonic crystal structures are finite-sized, and
the group velocities at the band edges are small but nonzero. Consequently, the laser
modes represent quasi-standing waves and optical losses occur. An outer photonic crys-
tal region maximises the quality factor of the photonic band edge mode. Both types of
laser, utilising 1D (our VCSEL device) and 2D photonic crystal structures, are examples
of photonic band edge band gap lasers [18]. Finite-difference full time-domain simula-
tions of the lasing frequencies and thresholds and of the field profiles fully support the
theory of gain enhancement. We note that some of the functional ingredients are also
implemented in conventional distributed feedback (DFB) lasers, which are structures
with very low index contrasts.
6.1 VCSEL with Resonant Periodic Gain and Refractive Index Structures 123
el
ec
tri
c 
fie
ld
 [a
. u
.]
0 1 2 3 4 5 6 7 8
position [µm]
1
1.5
2
2.5
3
3.5
4
re
fra
ct
iv
e 
in
de
x
el
ec
tri
c 
fie
ld
 [a
. u
.]
2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
position [µm]
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
re
fra
ct
iv
e 
in
de
x
el
ec
tri
c 
fie
ld
 [a
. u
.]
0 1 2 3 4 5 6 7 8
position [µm]
1
1.5
2
2.5
3
3.5
4
re
fra
ct
iv
e 
in
de
x
(el
ec
tri
c f
iel
d e
nv
elo
pe
)2  
[a.
 u.
]
1 2 3 4 5 6 7
position [µm]
Figure 6.7: Electric field profile and refractive index structure for a configuration with
∆n = 0.12 (at the top) and for ∆n = 0.24 (bottom). The laser mode is becoming
more strongly confined over the active quantum wells with increasing contrast of the
GRIN structure (see bottom right graph). In the left column we compare the electric
field profiles of the laser system simulated using the finite-difference full time-domain
model (dashed lines) with transfer matrix calculations [11,139,145] (dotted lines) of the
passive structure. We note the very close agreement between laser mode and cold-cavity
mode. The discrepancies of the profiles in the left Bragg mirror are due to different
boundary conditions and excitations. In the full time-domain solver sources of the
electromagnetic field are placed in the defect cavity and absorbing boundary conditions
are applied. The transfer matrix method starts with the constraint of an out-going
wave at the right boundary, a to the right travelling plane wave, and calculates the
reflected and transmitted components at every layer surface. Consequently, for passive
structures and real-valued dielectric constants there are at the left boundary to the left
and to the right travelling wave components. We note that in finite length structures
the mode enhancement is not uniform over the pattern, but highest near the centre of
the periodic index structure [144].
124 LONGITUDINAL MULTI-MODE LASER DYNAMICS
6.2 Optically Pumped VECSEL
Amodified version of the VCSEL structure is the vertical external cavity surface-emitting
laser (VECSEL). A simplified schematic of this novel type of laser structure is shown
in Figure 6.8. VECSEL are bridging the gap between semiconductor diode lasers and
optically pumped solid-state lasers [33]: An external laser cavities enforces the optical
output in low divergence, circular TEM00 beams. Optical pumping techniques offer
power scalability by the increase of the active, pumped area. With this, a combination
of continuous wave high-output power and near-diffraction-limited beam quality can be
achieved [16, 17]. The use of semiconductor gain materials renders the possibilities of
band gap engineering and semiconductor processing technologies. Also, complex optical
layer structures can be grown. The optical pumping scheme and external mirror config-
uration permit high-power operation by an increase in the size of the active region (a
uniform carrier distribution over a large aperture is achieved) without the occurrence
of transverse multi-mode dynamics. Mode beating effects, instabilities and filamen-
tation and large beam divergence are known to limit the performance of broad area
lasers [36,37,106,130], and wide-aperture surface-emitters [12,131]. The open structure
of the VECSEL allows to introduce nonlinear optical elements into the cavity. The ready
access to the lasing mode is exploited for applications such as passive mode-locking using
semiconductor saturable absorber mirrors (SESAM) or frequency doubling by nonlinear
crystals [16,17,33]. An inherent drawback of the external mirror design is that the optical
setup is less compact and requires the precise alignment of multiple discrete components.
A monolithically integrated device structure, a so-called microchip VECSEL, may offer a
Figure 6.8: Simplified schematic of an optically pumped vertical external cavity
surface-emitting semiconductor laser structure [15–17, 33, 86, 133, 134, 146, 147]. The
optoelectronic device (coherent light source) comprises a resonator, defined by a dielec-
tric Bragg reflector and an external cavity with a curved mirror, a resonant periodic gain
structure, an optical pumping mechanism, and intracavity nonlinear optical elements.
6.2 Optically Pumped VECSEL 125
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
position [µm]
1
1.5
2
2.5
3
3.5
4
re
fra
ct
iv
e 
in
de
x
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
Figure 6.9: Refractive index structure of a realistic vertical extended cavity surface-
emitting laser device, and snapshot of the electric field profile including the carrier
wave (lasing has not yet set in). The points of discontinuity in the electric field profile
are connected to the positions where the external optical fields (that are to the right
travelling plane wave components) are injected into the active structure: The contin-
uous wave optical pump is realised at −2.5µm (with parameter Eenv = 2 · 106Vm−1,
which translates to an intensity of approximately 2MWcm−2 and, assuming a Gaussian
transverse intensity profile with a FWHM of 50µm, to a pump power of few 10W).
A delayed optical feedback boundary condition is implemented at ≈ −5µm to mimic
external cavities of some cm length. We note the exponential decay of the electric
field (the pump) over the periodic gain and the graded index layer structure (with
∆n = 0.12, for more details see Figure 6.7).
solution to this problem [134]. A more detailed diagram of a realistic VECSEL structure
plus the numerical implementation of optical pumping and of the external cavity (via the
longitudinal delayed optical feedback boundary conditions, see Appendix A) is depicted
in Figure 6.9. We note the resonant periodic gain and graded refractive index struc-
tures as analysed in Section 6.1. The actual laser sub-cavity is coupled to the external
resonator by means of an anti-reflection coating (for the pump). The Bragg mirror is not
highly reflective for the pump wavelength of 808 nm. In transverse or radial direction
static field profiles are assumed. In our effective longitudinal, one-dimensional model,
the structures and physical processes in radial direction are accounted for by radial aver-
aging and the introduction of effective parameters, e.g. the transverse confinement factor
Γx,y(z). In conclusion, the increased power performance (scalable to continuous wave
output powers of some 10W) of this novel type of surface-emitting semiconductor laser
structure is accomplished by the adaptation of optical pumping technologies originally
developed for solid-state laser systems. Optically pumped VECSEL convert (with rea-
sonable efficiency) low beam quality optical pump power from multi-mode high-power
diode laser bars into near-diffraction-limited, circularly symmetric output beams of the
fundamental transverse mode [33]. This combination of high-output power and good
beam quality (in continuous wave operation or mode-locked) together with the compact
126 LONGITUDINAL MULTI-MODE LASER DYNAMICS
semiconductor-based design is attractive for a wide range of industrial applications such
as materials processing, leading edge projection and display technologies (laser TV),
spectroscopy or high-bandwidth telecommunication.
In the following, the various functional constituent parts of this novel laser type will
be specified [16,17,33,133,134]. Capabilities, key design aspects and operating principles
will be analysed.
Semiconductor wafer design and external laser cavity: A vertical extended cavity
surface-emitting laser basically resembles a VCSEL structure (see Figure 6.7), how-
ever, one of the DBR mirrors is replaced by a transparent (wide band gap) window
layer and an external spherical mirror. Typical values for the radius of curvature are
50mm, a cavity length of few cm, and an optical mode diameter on the chip of some
100µm. A VECSEL works like a diode-pumped thin disk solid-state laser, but replaces
the disk with a surface-emitting semiconductor active mirror structure, which comprises
of a Bragg mirror and an active region consisting of an array of quantum wells. The
utilisation of semiconductor gain materials offers a flexible choice of laser emission wave-
length. Epitaxy and processing technologies allow for the realisation of sub-wavelength
index and gain structures. The quantum wells can be positioned periodically (at half-
wavelengths intervals) in the antinodes of the laser standing wave. A resonant periodic
gain structure is formed with Γr ≈ 2. A one-dimensional photonic crystal involves the
further enhancement of the effective gain for photonic band edge modes, for more details
see Section 6.1. However, the enhancement of the effective modal gain by the longi-
tudinal confinement factor proceeds at the expense of narrow spectral filter windows, as
Γr(λ) is peaked at the etalon resonances of the sub-cavity between Bragg mirror and
the air surface. To be more tolerant to variations of the device structure and less tem-
perature sensitive, and for applications that require a very broad gain bandwidth (e.g.
mode-locking), the spectrum of gmod(λ) can be flattened as the etalon resonances are
broadened and weakened by anti-reflection coatings1, by double quantum well resonant
periodic gain structures (wells are distributed in pairs) [15], or by using a relatively short
active structure.
Optical pumping scheme: A main advantage of optical pumping is the possibility to
define a uniform distribution of pump power over a broad active region (e.g. 200µm
diameter). This creates a controlled, homogeneous carrier profile over a large aperture.
Also, undoped semiconductor materials can be used which reduces optical losses due to
free carrier absorption and heat deposition. The optical pumping (pump optics is AR
coated) occurs via interband transitions and the generation of electrons and holes in
the pump-absorbing barrier material. Because of the broad pump bandwidth and the
high densities, the pump light is converted into carriers within the barriers in a single
pass. The carriers are diffusing and captured into the lasing states of the quantum wells
and participate in stimulated emission [16]. An alternative approach is in-well pumping,
1AR coatings are composed of transparent thin films, with multi-layers of contrasting refractive index
and different thickness. The functionality is based on destructive interference. For a single-layer AR
coating of thickness λ/(4nAR) the optimum value is given by the geometric mean of the surrounding
indices nAR =
√
nairnsub. AR coating is also important for optical pumping.
6.2 Optically Pumped VECSEL 127
800 820 840 860 880 900 920 940 960 980 1000
wavelength [nm]
sp
ec
tra
l e
le
ct
ric
 fi
el
d 
[a.
 u.
]
965 970 975 980 985 990
Figure 6.10: Spectral distribution of the electric field: The peaks in the spectrum are
generated by optical pumping at λpump = 808 nm (from reflections) and by several lon-
gitudinal, external cavity laser modes at ≈ 975 nm. The simulated external resonator
structure has a length of Lcav = 0.5mm and 1mm (in inset), respectively, which trans-
lates to a longitudinal mode spacing of ∆k = π/Lcav → ∆λ ≈ λ2/(2Lcav). For the 1mm
long external cavity configuration the multi-mode, continuous wave solution consists of
approximately 20 longitudinal modes with mode spacing of ≈ 0.5 nm.
which reduces the heat generated within the chip as the quantum defect is minimised
[146]. Due to low absorption multi-passing and a rather complicated optical setup with a
resonant cavity for the pump light is required. An example of the optical intracavity field
dynamics is given in Figure 6.10: The laser operates in a group of longitudinal, external
cavity modes that fit in an etalon resonance within the mirror bandwidth. The optical
pumping, carrier relaxation and lasing due to stimulated emission are characterised in
Figure 6.11.
Thermal Design: High-power continuous wave operation of VECSEL requires the
efficient heat removal from the resonant periodic gain structure or pumped region on a
large area. As long as one can assume one-dimensional heat flow into the heatsink, the
laser structure is power scalable. The heat can be either extracted through the DBR
mirror, or a heat spreading plate technique can be applied [17]. Indeed, real optically
pumped VECSEL devices suffer from the strong intrinsic temperature dependence of
the semiconductor gain material2, and the shift of the relative longitudinal confinement
factor with temperature further reduces the effective gain. The maximum output power
is ultimately limited by thermal rollover.
Intracavity nonlinear optical elements: Intracavity circulating powers in VECSEL de-
vices frequently exceed the output power by a factor of approximately 50. For this reason,
2With higher pumping of the active region and rising temperature, the available material gain dimin-
ishes. To compensate for that the inversion has to increase (no density pinning from gain clamping).
With higher carrier densities and temperatures the loss channels, mainly Auger recombination, be-
come more important, and the thermal load per absorbed pump photon increases.
128 LONGITUDINAL MULTI-MODE LASER DYNAMICS
0 2 4 6 8 10 12 14 16
momentum vector [1/aBohr]
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
de
vi
at
io
n 
fro
m
 e
qu
ili
br
iu
m
 fu
nc
tio
n 
[1
0-4
]
Figure 6.11: Spectral hole burning and spectral selective carrier generation originat-
ing from lasing and optical pumping, respectively; depicted are deviations of the actual
distribution functions for the electrons (dashed) and holes (solid line) from the quasi-
equilibrium Fermi-Dirac distributions. After a few nanoseconds the transient relaxation
oscillations are damped out, and a quasi-stationary state is achieved. Carrier genera-
tion due to pumping into high energy states of the band structure (in-well pumping),
momentum relaxation of carriers towards the Fermi-Dirac distribution and stimulated
recombination from states near the band edge are in a dynamical balance. The active
mirror contains a periodic structure of 8 nm wide (high gain, compressively strained)
InGaAs/GaAs quantum wells (plus spacer, GRIN and stress compensation layers) and
a high contrast GaAs/AlAs DBR. In this section, we assume γek = γ
h
k . The pumping
wavelength is 808 nm.
physical processes based on the placement of nonlinear optical elements into the extended
cavity are comparatively efficient, examples are nonlinear frequency conversion and the
saturation of absorber elements. Applications include frequency doubling of infrared
surface-emitters using nonlinear optical crystals to attain powerful and compact sources
of coherent, single-mode light in the green and blue wavelength regions (which can be
used for projection and display technologies). VECSEL structures are also attractive for
ultrafast optical pulse generation using the the concept of passive mode-locking. This
can be achieved by integrating semiconductor saturable absorber mirrors3 as developed
for pulse generation in solid-state lasers. The broad gain bandwidth (the active laser sub-
cavity has to be AR coated at the operating wavelength) and the high intracavity powers
support the mode-locking process. Pulses generated by surface-emitting structures are
typically less chirped than pulses from edge-emitters due to the comparatively short
3As the absorber section has to saturate at lower pulse intensities than the gain section, the effect
of any resonant enhancement for the SESAM may also be taken into account, see Section 6.4. A
cavity mode designed to be more tightly focussed on the SESAM than on the gain structure (in our
one-dimensional model the mode area determines the coupling or overlap factor Γx,y(z)) helps to
achieve the conditions for stable, self-starting mode-locking.
6.3 Small Signal Gain Calculations 129
length of the highly nonlinear waveguide [17]. The self-starting pulse generation and
pulse shaping mechanisms in passive mode-locked lasers take advantage of the various
temporal regimes that can be distinguished in the evolution of non-equilibrium distribu-
tions generated by optical excitations (pulses) in semiconductor absorber materials (see
Sections 6.4+6.5).
6.3 Small Signal Gain Calculations
In this section we analyse the passing of an optical pulse through an active medium
[14,148], in order to verify the accuracy of the presented finite-difference full time-domain
model including the Maxwell curl equations and the band-resolved semiconductor Bloch
equations. Firstly, we probe a semiconductor optical amplifier structure (GaAs quantum
wells) with a weak femtosecond pulse that is spectrally broad enough to scan the optical
polarisation response of the gain medium. In the linear pulse propagation regime the
probe pulses are sampling the optoelectronic properties of the semiconductor gain mate-
rial without changing the state of the active gain medium. Hole burning effects, carrier
depletion and saturation can therefore be neglected. Using a travelling wave approach
the propagation of a pulse can be described by the following set of equations
E(z, t) =
1
2
Eenv(z, t)e
iΦ(z,t)e−iωteik0neffz + conjugate complex, Eenv,Φ ∈ R, (6.4)(
n2
cneff
∂t + ∂z
)
E˜(z, t) = i
k0Γ
ǫ02neff
NQW
Lref
P˜ (z, t), E˜ = Eenve
iΦ, P˜ ∈ C, (6.5)
E˜out(z;ω) = E˜in(ω)e
gmod(ω)zei(∂zΦ)(ω)z = E˜in(ω)e
gmod(ω)zeik0δn(ω)z. (6.6)
The different pulse Fourier components are amplified or attenuated by a factor of
exp(gmod(ω)z) and experience a phase shift quantified by exp(i(∂zΦ)(ω)z). The physical
effects that result in changes of the pulse spectrum and shape include the linear gain, the
gain dispersion, and the dispersion of the waveguide structure neff(ω) and the chromatic
dispersion of the host or background material nbackground(ω) [41]. The latter arises from
high energy excitations and is typically qualified by empirical formula, e.g. the Sellmeier
equation. Dispersion results in different spectral components of the optical pulse travel-
ling at different velocities and manifests itself as temporal effect causing an increase of
the length of an initially unchirped ultrashort pulse in time-domain. A measure for these
phenomena is given by the group velocity dispersion vpulse = ∂kω = vphase/(1−λ/n∂λn).
In the following simulations, although, we will solely consider the carrier-induced refrac-
tive index change dispersion. To calculate spectra within a model in frequency-/time-
domain we would have to run a series of simulations with different central wavelengths
(as the ansatz only gives correct results for a rather small bandwidth around the split-
off frequency). Within our full time-domain approach we inject a probe pulse, record
a time series of the electric field at a position zL and subsequently perform a Fourier
transformation. We note that, based on the fact that we have not split-off a reference
phase, the electromagnetic field at a single time point does not reveal any information.
130 LONGITUDINAL MULTI-MODE LASER DYNAMICS
760 780 800 820 840 860 880 900 920 940
wavelength [nm]
-2000
-1500
-1000
-500
0
500
1000
sm
al
l s
ig
na
l a
m
pl
itu
de
 g
ai
n 
[cm
-
1 ]
1.351.41.451.51.551.61.65
energy [eV]
Figure 6.12: Small signal (electric field amplitude) gain spectra gmod for various car-
rier sheet densities N ∈ {0.5 · 1012 cm−2 . . . 4 · 1012 cm−2} calculated from the full time-
domain approach (lines) compared with results obtained from standard free-carrier
theory (circles) according to equation (4.14) [149]. We inject a probe pulse (with pa-
rameters E0 = 1 · 10−3Vm−1, TFWHMpulse (Eenv(t)) = 10 fs) into the semiconductor optical
amplifier structure, and record the electric fields at the position zL. Important simu-
lation parameters comprise the dephasing constant γp = 20 ps−1, the electronic band
gap Egap(GaAs) = 1.424 eV and the waveguide properties neff = nbackground = 3.6,
ΓNQW/Lref = 1/∆z = 1/6.556 nm. We assume a in propagation direction unstructured
device, that is ΓrVz = 1.
It is necessary to calculate a time series. For a semiconductor optical amplifier of length
Lact we obtain the small signal amplitude gain from the relation
gmod =
1
Lact
ln
(
Eact(zL, λ,N)
Epas(zL, λ)
)
, (6.7)
where Eact is the spectral electric field of the amplified pulse, and Epas of a reference op-
tical pulse (probing the passive structure). The induced refractive index change spectra
can be extracted by
∂zΦ =
1
Lact
mod
(
Φact(zL, λ,N)− Φpas(zL, λ), 2π
)
. (6.8)
We note that the electrical pumping term and the carrier loss processes are switched-off.
The relevant spectral information is extracted from the simulation data (that are the
time series of the amplified and of a reference pulse) by discrete Fourier transformations.
In summary, we can show that the full time-domain simulations (see Figures 6.12–
6.14) agree well with analytical calculations. Furthermore, we extend our investigations
to the propagation of sub-picosecond pulses in the nonlinear or saturation regime where
theoretical approaches applying the slowly varying amplitude and the rotating wave
approximations are known to be invalid. This will be discussed in the next sections.
6.3 Small Signal Gain Calculations 131
760 780 800 820 840 860 880 900 920 940
wavelength [nm]
1000
1500
2000
2500
3000
3500
ph
as
e 
sh
ift
 d
Φ
/d
z 
[cm
-
1 ]
1.351.41.451.51.551.61.65
energy [eV]
Figure 6.13: Induced phase shift ∂zΦ or change of the refractive index k0δn = ∂zΦ
versus optical wavelength or energy calculated from the (finite-difference) full time-
domain model according to (6.8).
760 780 800 820 840 860 880 900 920 940
wavelength [nm]
-2
0
2
4
6
8
10
12
14
16
18
20
lin
ew
id
th
 e
nh
an
ce
m
en
t f
ac
to
r
1.351.41.451.51.551.61.65
energy [eV]
780 800 820 840 860 880
0
2
4
6
1.451.51.55
Figure 6.14: Anti-guiding or linewidth enhancement factor α. The very close agree-
ment between the spectra calculated with the Maxwell semiconductor Bloch equations
in full time-domain and the calculations depicted in Figures 4.9–4.11 is a verification
for our novel approach.
132 LONGITUDINAL MULTI-MODE LASER DYNAMICS
6.4 Nonlinear Pulse Propagation
Semiconductor optical amplifiers (SOA) are essentially pumped, active structures with-
out end mirrors but with fibres4 attached to both facets for the in- and out-coupling of
electromagnetic fields. SOA are key components of all-optical networks, e.g. attractive
as optical switches and regenerators. The amplification of optical signals is achieved
through stimulated emission which causes transfer of the energy stored in the carrier
subsystem of a semiconductor to the optical fields. However, the mode of operation
is limited by pulse-induced changes of the gain medium and the carrier dynamics, and
the large amplitude and phase nonlinearities as changes of the semiconductor gain feed
back to the trailing part of the optical pulse. In this nonlinear or saturation regime, the
amplitude gain or induced refractive index change becomes dependent on characteristics
of the pulse, such the pulse shape, pulse width TFWHMpulse (Eenv(t)) or spectral bandwidth
∆νFWHMpulse (E(t)) and the peak amplitude E0. In addition, parameters of the amplifier
structure, e.g. the length Lact or coupling strength to the active waveguide ΓNQW/Lref,
have a strong influence. In the following, we investigate the (sub-) picosecond pulse
interactions in SOA structures, namely the propagation and amplification of ultrashort
optical pulses with hyperbolic secant pulse envelopes5. We model the injected pulses by
E(t) = Eenv(t) · cos
(
ω(t− t0) + Φ0
)
, Eenv(t) = E0 · sech
(
γpulse(t− t0)
)
, (6.9)
TFWHMpulse (Eenv(t)) =
2 ln
(
2 +
√
3
)
γpulse
, ∆νFWHMpulse (E(t)) =
2 ln
(
2 +
√
3
)
γpulse
π2
, (6.10)
TFWHMpulse
(
E2env(t)
)
=
ln
(
3 +
√
8
)
γpulse
, νFWHMpulse
(
E2(t)
)
=
ln
(
3 +
√
8
)
γpulse
π2
, (6.11)∫ ∞
−∞
dtEenv(t) = E0 · π
γpulse
,
∫ ∞
−∞
dtE2env(t) = E
2
0 ·
2
γpulse
. (6.12)
In addition to the above discussed linear mechanisms of small signal gain and the various
dispersions, and amplified spontaneous emission as broad bandwidth noise background,
there are further important physical processes which cause changes of the pulse shape
and the pulse spectrum during propagation in semiconductor amplifier structures for
high electric fields and optical intensities. These nonlinear optical effects comprise the
inelastic light scattering with different phonons (Brillouin, Raman) and the optical Kerr
effect n(ω, I) = n(ω) +
∫
dω′n2(ω, ω
′)I(r, t;ω′), which states that the index change of
the host or background material is directly proportional to the electric field amplitude
squared. The second order nonlinear refractive index n2 is attributed to an instanta-
neously occurring nonlinear response χ(3). A nonlinear polarisation is generated, which
4Laser sources and amplifier materials operating in the near infrared at wavelengths near to the (data)
transmission windows of the silica fibres are preferable, more precisely that is at 1310 nm (minimal
dispersion), 1550 nm (absorption minimal) and at around 850 nm (local minimum).
5Pulses with a Gaussian pulse shape, i.e. Eenv(t) = E0 · exp(−γ2pulse(t − t0)2), are characterised in
time- and frequency-domain by TFWHMpulse (Eenv(t)) = 2
√
ln 2/γpulse, ∆ω
FWHM
pulse (E(t)) = 4
√
ln 2γpulse
and
∫∞
−∞
dtEenv(t) = E0 ·
√
π/γpulse.
6.4 Nonlinear Pulse Propagation 133
in turn modifies a propagating pulse. It manifests itself temporally as self-phase modu-
lation (dynamical frequency shift), and spatially as self-focussing counteracting optical
diffraction. In the subsequent simulations we concentrate on the resonant optical tran-
sitions. The consideration of the distribution functions and of the macroscopic carrier
density as dynamical variables in the band-resolved microscopic and macroscopic Bloch
equations, respectively, and the physical effects of spectral and spatial hole burning rep-
resent an optical Kerr-type nonlinearity6. Our goal is to correlate the changes of the
pulse shape and spectrum to the modifications of the microscopic and macroscopic state
of the semiconductor gain medium. The pulse-induced changes of the gain material
by induced emission or absorption generate non-equilibrium distributions. These and
other relevant interactions and recovery mechanisms lead to a dynamical gain gmod(t)
and phase shift ∂zΦ→ δn(t), which in turn modify the temporal and spectral properties
of the propagating optical pulse. Characteristic time scales of the various interaction
processes in momentum and real space have to be compared to the pulse duration. This
helps to determine which terms have a strong influence. For (sub-) picosecond optical
pulses the carrier loss channels nonradiative recombination, spontaneous emission and
Auger recombination, an electrical carrier pumping term Λ, and the transverse spatial
degree of freedom7 and processes like diffraction and carrier diffusion may be neglected
as these terms have only a marginal influence on the state of the active gain material
on the sub-picosecond time scale. The mechanisms which essentially govern the (sub-)
picosecond pulse interactions in quantum well-based SOA structures include hole burn-
ing or more general, the generation of non-equilibrium states of the gain medium by
stimulated transitions, the propagation of the optical fields, and the partial recovery
caused by Coulomb and phonon interactions.
In Figures 6.15–6.18 we analyse the interaction of 10 fs hyperbolic secant optical
pulses with semiconductor amplifier and absorber elements in the saturation regime.
The results are obtained by calculating averaged gain and carrier-induced refractive
index change spectra according to (6.7) and (6.8) (Figures 6.15+6.17), and by char-
acterising the input and output pulses in time- and frequency-domain (see Figures
6.16+6.18). An increasing electric field amplitude E0 reads to higher instantaneous op-
tical intensities and a larger time-integrated pulse energy of the excitation pulse. The
spectra are flattening and the absolute value of the amplitude gain is reduced due to
6By applying an adiabatic elimination of the nonlinear polarisation we can specify the pulse-induced
change of the inversion by ∂tN |carrier-light = −1/(2~)G|E˜|2. Assuming a linear gain model G =
G0(N −Ntransparency), this depletion of the carrier density results in a change of the refractive index
of ∆N → ∆(δn) ∝ G0αI(r, t). For most frequencies the refractive index increases with decreasing
inversion.
7Taking into consideration the transverse spatial degree of freedom and transverse pulse profiles would
have obvious implications: For high-intensity pulses the centre area of the pulse operates in the
nonlinear regime connected with gain saturation and hole burning effects. At the (transverse) edges
the pulse sees the linear gain. Consequently, an optical pulse is higher amplified at the edges and
the spatial width of the transverse pulse profile is increasing. The refractive index depends on the
inversion and thus on the optical intensity. The field components at the pulse edges will be, as a
result of individual optical path lengths and group indices and self-focussing, time-shifted compared
to the pulse centre [106].
134 LONGITUDINAL MULTI-MODE LASER DYNAMICS
760 780 800 820 840 860 880 900 920 940
wavelength [nm]
-200
-150
-100
-50
0
50
100
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
1.351.41.451.51.551.6   1.65
energy [eV]
760 780 800 820 840 860 880 900 920 940
wavelength [nm]
-500
-450
-400
-350
-300
-250
-200
-150
ph
as
e 
sh
ift
 d
Φ
/d
z 
[cm
-
1 ]
1.351.41.451.51.551.6   1.65
energy [eV]
Figure 6.15: Propagation of 10 fs hyperbolic secant pulses (see (6.9)) in an active
semiconductor optical amplifier (SOA) structure. Pulse and device parameters are:
central wavelength λ = 850 nm, N = 3 · 1012 cm−2, ΓNQW/Lref = 1/50 nm, neff = 3.6
and Lact = 0.1mm. The amplitude gain (left) and induced refractive index change
spectra (right) are calculated for various electric field amplitudes and pulse energies of
the input pulse: E0 ≤ 1 ·107Vm−1 (small signal gain or linear regime; displayed as thick
line), 5 · 107, 1 · 108, 2 · 108, 3 · 108, 5 · 108Vm−1 (nonlinear saturation regime with carrier
depletion due to hole burning effects; thin lines) and E0 = 1 · 109, 2 · 109Vm−1 (pulse
areas M eh/~E0 · π/γpulse exceed π; dashed).
hole burning and carrier depletion by induced emission in an amplifier element or as
the result of an increased inversion by stimulated absorption within the saturable ab-
sorber. This effect is referred to as pulse-induced gain saturation. The generation of
non-equilibrium distribution functions, the partial saturation recovery and the resulting
ultrafast gain dynamics of gmod(t) and δn(t) can be experimentally measured by pump-
probe techniques. In the next section we implement these measurements numerically.
To summarise, gain saturation is caused by the coupling of photons with the carrier
system and the pulse-induced changes of n◦k in momentum space and N in real space.
In an amplifier the instantaneous available resonant carrier inversion is reduced. In an
absorber (and in the electronic multi-level picture) the possible initial states of the pump
transitions are depleted, whilst the final states are occupied due to stimulated absorp-
tion. A realistic modelling of few-cycle femtosecond pulse interactions in semiconductor
materials gives a quantitative description of gain saturation and recovery. An in-depth
treatment of the intraband relaxation and redistribution of carriers, more precisely of
the scattering processes mediated by the Coulomb interaction must be included. Evi-
dently, a description of carrier relaxation by the generalised quantum kinetic Boltzmann
equations in the Markovian approximation (no memory effects) which also neglects non-
linear terms in the interband polarisations is deficient. A full treatment of many-body
interactions and of the correlation contributions in the semiconductor Bloch equations
without approximations is beyond an approach with microscopically precalculated re-
laxation rates. Such a treatment in combination with a spatially extended time-domain
6.4 Nonlinear Pulse Propagation 135
1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44
time [ps]
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
    700 750 800 850 900 950 1000 1050
wavelength [nm]
0
1
2
3
4
5
6
sp
ec
tra
l i
nt
en
sit
y 
[a.
 u.
]
1.21.31.41.51.61.7
energy [eV]
1.28 1.3 1.32 1.34 1.36 1.38 1.4e 1.42 1.44
time [ps]
0
0.5
1
1.5
2
2.5
3
el
ec
tri
c 
fie
ld
 [1
08
V
m
-
1 ]
    700 750 800 850 900 950 1000 1050
wavelength [nm]
0
1
2
3
sp
ec
tra
l i
nt
en
sit
y 
[a.
 u.
]
1.21.31.41.51.61.7
energy [eV]
Figure 6.16: Depicted are the input or more precisely a reference pulse (thin line),
which is probing the passive (M eh = 0) waveguide structure, and the amplified pulse
(as thick line) for pulse propagation experiments in SOA in the linear (top) and in
the saturation regime (bottom). Within the time-domain picture the amplification of
few femtosecond pulses is realised by a pronounced elongation of the trailing part of
the optical pulse (caused by the fact that TFWHMpulse < 1/γ
p) and by an increase of the
optical pulse energy ∝ ∫ dt|Eenv(z, t)|2. The maximum value of the pulse envelope is
not increased. We can also notice a small time delay of the amplified pulse by approx-
imately 5 fs as a result of the noninstantaneous gain response to the pulse excitation.
In frequency-domain the different spectral components of the optical pulse experience
absorption (for high frequencies), amplification or transparency (for energies below the
band edge), compare to Figure 6.15.
136 LONGITUDINAL MULTI-MODE LASER DYNAMICS
760 780 800 820 840 860 880 900 920 940
wavelength [nm]
-300
-250
-200
-150
-100
-50
0
50
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
1.351.41.451.51.551.6   1.65
energy [eV]
760 780 800 820 840 860 880 900 920 940
wavelength [nm]
-500
-450
-400
-350
-300
-250
-200
-150
-100
ph
as
e 
sh
ift
 d
Φ
/d
z 
[cm
-
1 ]
1.351.41.451.51.551.6   1.65
energy [eV]
Figure 6.17: Same as Figure 6.15, but for a semiconductor absorber structure with
N = 0.5 · 1012 cm−2 (and we replace E0 = 5 · 107Vm−1 → 4 · 108Vm−1).
1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44
time [ps]
0
0.2
0.4
0.6
0.8
1
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
    700 750 800 850 900 950 1000 1050
wavelength [nm]
0
0.2
0.4
0.6
0.8
1
sp
ec
tra
l i
nt
en
sit
y 
[a.
 u.
]
1.21.31.41.51.61.7
energy [eV]
1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44
time [ps]
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
el
ec
tri
c 
fie
ld
 [1
08
V
m
-
1 ]
    700 750 800 850 900 950 1000 1050
wavelength [nm]
0
0.2
0.4
0.6
0.8
1
sp
ec
tra
l i
nt
en
sit
y 
[a.
 u.
]
1.21.31.41.51.61.7
energy [eV]
Figure 6.18: Same as Figure 6.16, but for the optical absorber structure (quantum
wells). Pulse propagation in the linear and saturation regime is characterised in time-
domain (left column) and in a frequency-domain picture (right). Here, spectral windows
showing absorption and transparency can be identified.
6.4 Nonlinear Pulse Propagation 137
approach involves a huge computational effort and is difficult to realise.
Figures 6.19–6.22 show the propagation and interaction of 1 ps electromagnetic
pulses in semiconductor gain structures. When we cut an optical pulse into tempo-
ral segments we see that earlier parts of the pulse induce changes to the active gain
material, such as modifications of the carrier distributions, the macroscopic carrier den-
sity and the available gain gmod(t) (dynamical saturation). Later segments of the pulse
experience these modifications of the gain medium. Consequently, the amplification or
absorption is larger at the leading edge of the pulse than for the trailing edge. This
results in asymmetry of the amplified pulse envelope. Because of the large α factor in
active semiconductor materials and the dependence of the refractive index on the in-
tensity (optical Kerr-type nonlinearity) spectral changes occur as a result of self-phase
modulation, and temporal changes are due to changes of the group index n + ω∂ωn.
The temporal variation of the nonlinear phase leads to dynamical modulations of the
instantaneous frequency ωinst(t) = ∂tΦ. In other words, due to nonlinearities and disper-
sion during propagation a chirp of the optical pulse arises. New frequency components
are generated, and the spectrum becomes asymmetric and broadened. The cause of
self-phase modulation is the density dependence of the refractive index change δn, the
physical effects behind it are hole burning by stimulated emission and relaxation dy-
namics. Self-phase modulation (SPM) is classified according to the recovery times: Slow
SPM depends on the time-integrated power of the input pulse, whereas fast SPM de-
pends on the pulse and amplifier characteristics TFWHMpulse (Eenv(t)), ∆ν
FWHM
pulse (E(t)), E0
and Lact. This fast nonlinearity causes an inhomogeneous saturation of the modal gain
due to the effect of spectral hole burning, but is only significant for very high field am-
plitudes or intensities. For moderate field amplitudes, there is no clear hole due to the
large, dominant homogeneous broadening. In two-level or macroscopic gain descriptions,
this nonlinear gain saturation is typically specified by the following phenomenological
ansatz
gmod =
gmod, small signal (λ,N)
1 + ǫ˜satE20
→ g(z, t) = gmod, small signal (λ,N)
1 + ǫsatS(z, t)
, (6.13)
S(z, t) =
1
2
ǫ0ǫ
|E˜(z, t)|2
~ω
, ǫsat =
2~ω
ǫ0ǫ
ǫ˜sat, (6.14)
with the nonlinear gain coefficient ǫsat [73,113] and the saturated gain being function of
the instantaneous optical intensity. In Figures 6.19+6.21 we investigate nonlinear gain
suppression and determine from our simulation data effective nonlinear gain coefficients
ǫsat (by curve fitting). Results for optical amplifier structures, in which spectral deple-
tion of resonant carriers saturates the gain, are shown in Figure 6.19: The picosecond
pulses test the spectrally local (at around 850 nm) available gain. The saturation be-
haviour is hence determined by the intraband carrier dynamics (kinetics), mainly how
fast a spectral hole in the distribution functions is refilled. Consequently, increasing
the time constants of these intraband carrier relaxation processes (e.g. Coulomb scat-
tering) by a factor of two will change the nonlinear gain coefficient. We find (dashed
line) ǫ˜sat = 1.4 · 10−15m2V−2 → ǫsat = 5.8 · 10−24m3. Other in literature discussed
138 LONGITUDINAL MULTI-MODE LASER DYNAMICS
0.01 0.1 1 10 100
electric field [106Vm-1]
0
10
20
30
40
50
60
70
80
90
100
110
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
Figure 6.19: Nonlinear gain suppression in semiconductor optical amplifiers due to
spectral (and spatial) hole burning, as result of the depletion of available resonant
carriers for stimulated recombination processes. The propagation of 1 ps pulses in a
structure as described in Figure 6.15 is simulated with the full time-domain model.
Depicted are the saturated amplitude gain at 850 nm versus the electric field amplitude
E0. The phenomenological ansatz according to (6.13) (thick line) is compared with
the simulation data (circles), we can extract from nonlinear curve fitting ǫ˜sat = 1.0 ·
10−15m2V−2 → ǫsat = 4.2 · 10−24m3.
mechanisms behind the gain power saturation [73] comprise the reduced instantaneous
gain (as the consequence of finite effective recovery or scattering times of the phonon
scattering processes) associated with carrier heating and heat-induced changes of the
amplifier medium, and due to the depletion of the macroscopic carrier density through
the coupling of carriers with photons. The associated recovery time scales of the pro-
cesses balancing the various depletion mechanisms determine the thresholds from which
saturation behaviour sets in. Typical values of these physical parameters (effective life-
times) are given by τshb = 100 fs, τch = 1ps and τmacr = 500 ps. For saturable absorber
elements, in which the saturation mechanism is the spectral selective carrier generation,
simulation results of gain saturation are displayed in Figure 6.21.
In Figure 6.20 the simulated results of the propagation and amplification of 1 ps
pulses in a SOA structure are shown: In the nonlinear regime asymmetries in the pulse
shape and spectrum are observed. The dynamical saturation leads to a small shift of the
pulse spectrum to longer wavelengths, and we see a pulse narrowing in frequency-domain.
In an amplifier structure the leading edge of the optical pulse will experience more am-
plification than the trailing segments. The connected reshaping of the pulse shape (or
envelope) results effectively in a temporal shift of the pulse peak (here by ≈ 50 fs) which
can be reinterpreted as pulse advancement. In an analogous manner, we find a pulse
slowdown for saturable semiconductor absorber elements (of ≈ 100 fs), see Figure 6.22.
As this time advancement or delay is dependent on the carrier density and on the optical
pulse properties, the propagation time is controllable and tunable [152]. The propagation
6.4 Nonlinear Pulse Propagation 139
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
time [ps]
0
0.5
1
1.5
2
2.5
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
846 847 848 849 850 851 852 853 854
wavelength [nm]
0.01
0.1
1
sp
ec
tra
l e
le
ct
ric
 fi
el
d 
[a.
 u.
]
1.4551.461.465
energy [eV]
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
time [ps]
0
1
2
3
4
5
6
7
el
ec
tri
c 
fie
ld
 [1
07
V
m
-
1 ]
846 847 848 849 850 851 852 853 854
wavelength [nm]
0.01
0.1
1
sp
ec
tra
l e
le
ct
ric
 fi
el
d 
[a.
 u.
]
1.4551.461.465
energy [eV]
Figure 6.20: Amplification of picosecond pulses in an active semiconductor struc-
ture. In the linear regime (top) the output pulse is connected with the input pulse
by the simple multiplication with a factor exp(gmod(850 nm, N)Lact) (or in other words
ln(Eout(ν)) = ln(Ein(ν)) + gmod(850 nm, N)Lact), pulse shape and spectrum remain un-
changed. Possible processes which may change the pulse shape and spectrum in the
linear regime include gain dispersion plus the dispersion linked with the background (or
host) material or caused by the waveguide, refractive index structure. For high-intensity
pulses (bottom) the effects of dynamical saturation behaviour and fast self-phase mod-
ulation (the modification of the refractive index by the optical pulse itself) [106], and
consequently a time-dependent phase shift and temporally varying instantaneous fre-
quency lead to a strong reshaping of pulses in time- and frequency-domain.
140 LONGITUDINAL MULTI-MODE LASER DYNAMICS
0.01 0.1 1 10 100
electric field [106Vm-1]
-160
-140
-120
-100
-80
-60
-40
-20
0
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
Figure 6.21: Same as Figure 6.19, but for a semiconductor optical absorber structure
(N = 0.5 ·1012 cm−2). From nonlinear curve fitting to the phenomenological description
(6.13) we can extract ǫ˜sat = 6.7 · 10−16m2V−2 → ǫsat = 2.7 · 10−24m3. Semiconductor
saturable absorber elements, embedded into a Bragg mirror structure, so-called semi-
conductor saturable absorber mirrors (SESAM) [32, 150], or as part of a multi-section
laser device [113, 136], are the key elements of ultrashort laser pulse generators based
on the concept of passive mode-locking [116,151].
in an active semiconductor can also induce a strong chirp of the pulse [153]. The various
segments of the pulse see different gain gmod(t) and individual refractive index changes
δn(t) due to spectral carrier depletion and subsequent intraband scattering processes,
and because of hole burning and recovery by an effective macroscopic carrier pump term.
Thus, the microscopic and macroscopic carrier state is varying over time, and likewise
gain and refractive index are time-dependent. As consequence of this dynamical satu-
ration behaviour we notice the following changes to the pulse shape and spectrum (the
effects are more pronounced with increasing power): In time-domain the reshaping of
the pulse envelope results in the sharpening of the trailing edge of the pulse, the pulse
broadening and a shift towards negative times (pulse advancement). The pulse spec-
trum exhibits an asymmetric shift towards longer wavelengths, the propagation in the
saturation regime imposes a chirp with the leading edge redshifted and the trailing edge
of the pulse blueshifted. In a similar way, the nonlinear pulse interaction in a saturable
absorber (Figure 6.22) shows a sharpening of the leading edge of the pulse, a pulse
shortening in time-domain and a time delay (or pulse slowdown). In frequency-domain
we find an asymmetric broadening to higher frequencies, the dynamical saturation im-
poses a positive (followed by a negative) chirp.
We note that the quantitative understanding of nonlinear pulse shaping in semiconduc-
tors is critical to applications such as passive mode-locking. The generation of ultrashort
optical pulses by mode-locking of many longitudinal laser modes can be realised by active
(loss modulation), passive (self-amplitude modulation) and hybrid techniques [1,73,151].
6.4 Nonlinear Pulse Propagation 141
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
time [ps]
0
0.2
0.4
0.6
0.8
1
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
846 847 848 849 850 851 852 853 854
wavelength [nm]
0.01
0.1
1
sp
ec
tra
l e
le
ct
ric
 fi
el
d 
[a.
 u.
]
1.4551.461.465
energy [eV]
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
time [ps]
0
1
2
3
4
5
6
el
ec
tri
c 
fie
ld
 [1
07
V
m
-
1 ]
846 847 848 849 850 851 852 853 854
wavelength [nm]
0.01
0.1
1
sp
ec
tra
l e
le
ct
ric
 fi
el
d 
[a.
 u.
]
1.4551.461.465
energy [eV]
Figure 6.22: Propagation of picosecond pulses in a semiconductor optical absorber,
analogue to Figure 6.20. The nonlinear dynamical saturation process in the absorber
structure effectively sharpens the leading edge of the electromagnetic pulse and leads
to a pulse shortening of the propagated pulse (thick line) compared to the input pulse.
Contrariwise, in a semiconductor optical amplifier the leading edge of the pulse is
stronger amplified and thus effectively the trailing edge sharpened. The combination
of saturable absorber elements and gain sections offers the possibility of ultrafast pulse
generation by passive mode-locking [33]. Furthermore, we notice as a result of pulse-
induced changes of the gain medium a pulse broadening in frequency-domain and a
shift of the pulse spectrum to shorter wavelengths.
142 LONGITUDINAL MULTI-MODE LASER DYNAMICS
The realisation of the concept of passive mode-locking based upon semiconductor sat-
urable absorber elements capitalises on the existence of a variety of saturation energies
and of very different time scales of saturation recovery in semiconductor-based absorber
materials. For more details on the gain dynamics and recovery see Section 6.5. The
physical processes with a lower saturation intensity (for a part of the absorption) and
characterised by longer recovery times (the carriers are removed by interband recom-
bination or macroscopic carrier loss processes) enable the self-starting of mode-locking
from spontaneous emission or small fluctuations. The mechanism with the faster time
scale, due to a partial recovery of the absorption when carriers thermalise by intraband
relaxation or scattering, and higher saturation energy provide a reshaping mechanism of
sub-picosecond pulses [32, 33]. To summarise, conditions for stable pulse generation by
mode-locking in (monolithic) multi-section diode lasers can be specified [32,33,73,151]:
1) The small signal gain of the amplifier section has to be bigger than the unsaturated
losses of the absorber elements (self-starting). 2) The absorber elements must saturate
faster, at lower pulse intensities than the gain section. This may be implemented by
an adequate choice of cavity configurations and the dielectric multi-layer design. To
enhance the optical fields over the quantum well-based (or more general quantum con-
fined) absorber structures the saturable absorbers are often embedded into a Bragg
mirror structure (SESAM) and thus the light focussed by the absorber cavity. 3) The
loss recovery must take place more quickly than that of the gain section. With defect
engineering, which involves the generation of defect states in the band gap which give
rise to ultrafast carrier trapping into deep levels to deplete the bands [33], this goal can
be achieved. In all, the slow absorber in conjunction with the dynamical gain saturation,
the absorption exceeds the pulse amplification everywhere except near the peak of the
optical pulse. As a consequence, the pulse will shorten due to amplification of the peak
and attenuation of the edges as there is just an ultrashort net-gain window from the
combined saturation of absorber and gain [33,73]. Our approach to combine a material
model which implies a variety of different time regimes and saturation mechanisms with
the direct solution of Maxwell equations represents an improvement to known theoretical
descriptions of passive mode-locking [96,113,116,126,136] and permits the optimisation
of pulsed monolithic semiconductor lasers.
6.5 Ultrafast Gain Dynamics
In this section we analyse the femtosecond and picosecond dynamics of the gain gmod(t)
and of the refractive index change δn(t) by simulating and numerically rebuilding pump-
probe experiments in which (high-intensity) pump and (weak) probe pulses with a vari-
able, tunable time delay copropagate in the semiconductor optical amplifier structure. In
our simulations we calculate the pulse interactions of 200 fs optical pulses with the SOA
structure in the nonlinear or saturation regime. At the points in time of interest the whole
microscopic and macroscopic state of the gain structure is recorded: n◦k(z, t), N(z, t),
pk(z, t), ∂tpk(z, t), P (z, t). In an additional step, we impose a time delay, prepare the
system into the appropriate recorded state and sample with a weak femtosecond probe
6.5 Ultrafast Gain Dynamics 143
-1 -0.5 0 0.5 1 1.5 2
time delay [ps]
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
n
o
rm
al
ise
d 
ga
in
 [d
B]
0 50 100 150 200
time delay [ps]
40
50
60
70
80
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
Figure 6.23: Pump-probe experiment of a semiconductor gain structure with car-
rier density of N = 3 · 1012 cm−2 showing the ultrafast dynamics (recovery) of the
gain. The pump pulse (central wavelength λ = 850 nm, TFWHMpulse = 200 fs, E0 =
5 · 107Vm−1, hyperbolic secant pulse shape) generates an excitation, i.e. a modifi-
cation of the state of the semiconductor amplifier medium (non-equilibrium distribu-
tions). Probe pulses sample the gain dynamics. Depicted are the normalised am-
plitude gain 10 · log(gmod(t)/gmod, small signal) [dB] (left picture) or the gain (right) at
850 nm versus the delay time between pump and probe pulse. The power satura-
tion of the gain originates from several depletion mechanisms of the available reso-
nant carriers. Assuming an exponential decay of the various carrier excitations and
exponential recovery of the gain, and the mechanisms to be independent, that is
gmod, small signal−gmod(t) = ∆gshb exp(−t/τshb)+∆gch exp(−t/τch)+∆gmacr exp(−t/τmacr),
we can identify the different time scales on which the gain recovers. The balancing of
spectral depletion of carriers and spectral hole burning by Coulomb scattering occurs
with a rate of approximately 1/τshb = 11 ps
−1 (see fit, dashed line in left graph). The
macroscopic carrier density recovers on the nanosecond scale. The rate is given by the
effective pumping, the electrical pump minus the carrier loss channels (right picture).
In Figure 6.24 we analyse the instantaneous gain at time points marked with circles
(left) and with squares (right).
pulse the non-equilibrium carrier state of the amplifier medium to obtain gmod(t) and
δn(t). We calculate the time series of the probe pulse after passing through the gain
material and perform a Fourier transformation to gain the instantaneous spectra.
The microscopic origin of the fast nonlinearities are spectral hole burning or the spec-
tral selective generation of carriers, respectively, and the many-body Coulomb dynamics
by intraband carrier relaxation and scattering (Figures 6.24+6.27). The recovery of
the gain saturation is achieved by various mechanisms and physical effects, the different
recovery rates are calculated for semiconductor optical amplifier structures in Figure
6.23 and for saturable absorbers in Figure 6.26. In Figure 6.25 the dependence of
the gain dynamics and recovery on the central pulse frequency and peak amplitude,
the pumping or macroscopic carrier density, and on the microscopic scattering rates is
analysed in detail.
144 LONGITUDINAL MULTI-MODE LASER DYNAMICS
0 5 10
-0.1
-0.05
0
0.05
δn
0 5 10 5 10 0 5 10
800 850 900
-150
-100
-50
0
50
100
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
800 850 900 800 850 900 800 850 900
0
momentum vector [1/aBohr]
0
wavelength [nm]
0 5 10 15
0
0.2
0.4
0.6
0.8
1
di
str
ib
ut
io
n 
fu
nc
tio
n
5 10 15
800 850 900
-150
-100
-50
0
50
100
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
800 850 900
0
momentum vector [1/aBohr]
0
wavelength [nm]
Figure 6.24: Gain saturation by spectral hole burning (leads to a reduced spectral
inversion) and recovery due to carrier redistribution by Coulomb scattering processes
(left, at the time points marked with circles in Figure 6.23). Slower gain dynamics by
the depletion of the macroscopic inversion (N = 3 · 1012 cm−2 → 2.6 · 1012 cm−2) and
the recovery due to an increase of the macroscopic density by carrier injection (right,
squares). We plot the electron (dashed) and hole (solid line) distribution functions
or the spectral deviations δn◦k = n
◦
k − f ◦k (at top) and the instantaneous gain spectra
(bottom) for the six time points marked in Figure 6.23. For sub-picosecond pulse
interactions the gain dynamics and the induced microscopic and macroscopic changes
of the active material are determined by spectral carrier generation gk or spectral hole
burning, and by intraband carrier relaxation.
6.5 Ultrafast Gain Dynamics 145
-1 -0.5 0 0.5 1 1.5 2
time delay [ps]
-6
-5
-4
-3
-2
-1
0
n
o
rm
al
ise
d 
ga
in
 [d
B]
-1 -0.5 0 0.5 1 1.5 2
time delay [ps]
-6
-5
-4
-3
-2
-1
0
n
o
rm
al
ise
d 
ga
in
 [d
B]
-1 -0.5 0 0.5 1 1.5 2
time delay [ps]
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
n
o
rm
al
ise
d 
ga
in
 [d
B]
-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
time delay [ps]
-5
-4
-3
-2
-1
0
n
o
rm
al
ise
d 
ga
in
 [d
B]
Figure 6.25: The excitation of the SOA medium as the response to the pump pulse,
characterised in Figure 6.23, and the subsequent gain recovery is depicted as wide
solid lines. The influence of parameters of the pulse and of the SOA structure on the
gain recovery dynamics is analysed. (Top left) The gain saturation is more distinct with
increasing electric field amplitude and energy of the pump pulse: E0 = 3 · 107Vm−1
(dashed) and 7·107Vm−1 (dotted). (Top right) We change the electrical pump or carrier
density of the SOA,N = 2·1012 cm−2 (dashed) and 4·1012 cm−2 (dotted), and the central
wavelength of the pump pulse (bottom left) to λ = 870 nm (near the band edge, dashed
line) and 810 nm (near transparency, dotted). (Bottom right) Shows the influence of
the time scales of intraband relaxation processes. The Coulomb scattering rates are
reduced by a factor of two (dotted). An additional relaxation process with a time
constant of τch = 2ps, associated with the interaction of the electron-hole subsystem
with phonons (with the lattice) and with carrier temperature relaxation, is taken into
account.
146 LONGITUDINAL MULTI-MODE LASER DYNAMICS
-1 -0.5 0 0.5 1 1.5 2
time delay [ps]
-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
n
o
rm
al
ise
d 
ga
in
 [d
B]
0 50 100 150 200
time delay [ps]
-140
-120
-100
-80
-60
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
Figure 6.26: Same as Figure 6.23, but for a semiconductor absorber element (initial
carrier density N = 0.5 · 1012 cm−2 < Ntransparency). Curve fitting allows to extract
effective rates of the gain (or better absorption) recovery 1/τshb = 9.5 ps
−1 (left, dashed
line) and 1/τmacr = 3.5 ns
−1 (right).
0 5 10
-0.05
0
0.05
0.1
δn
0 5 10 5 10 0 5 10
800 850 900
-250
-200
-150
-100
-50
0
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
800 850 900 800 850 900 800 850 900
0
momentum vector [1/aBohr]
0
wavelength [nm]
0 5 10 15
0
0.2
0.4
0.6
0.8
1
di
str
ib
ut
io
n 
fu
nc
tio
n
5 10 15
800 850 900
-250
-200
-150
-100
-50
0
am
pl
itu
de
 g
ai
n 
[cm
-
1 ]
800 850 900
0
momentum vector [1/aBohr]
0
wavelength [nm]
Figure 6.27: As Figure 6.24 the formation and the relaxation of optically excited,
non-equilibrium distributions (left graph) by spectrally selective absorption (and carrier
generation) and by the Coulomb interaction (which does not change the macroscopic
carrier density), respectively, is shown. (Right) The macroscopic carrier generation
and the absorber recovery due to various macroscopic carrier loss channels such as
nonradiative and Auger recombination, spontaneous emission plus diffusion processes
are analysed.
6.6 Chirped Pulse Amplification 147
6.6 Chirped Pulse Amplification
In this section we extend the investigation of (sub-) picosecond pulse interactions with
semiconductor gain materials to the propagation and amplification of pulses with a
temporally varying nonlinear phase and time-dependent instantaneous frequency. More
specifically, we simulate optical input pulses with a positive or negative linear chirp8
E(t) = Eenv(t) · cos
(
(ω +∆ω(t− t0))(t− t0) + Φ0
)
, (6.15)
ωinst(t) = ∂tΦ(t) = ω + 2∆ω(t− t0). (6.16)
So far we have characterised an optical pulse both in frequency-domain by calculating the
spectrum and in time-domain by simulating the time series of the electric field envelope.
This corresponds to time-integrated and spectrally integrated measurements, respec-
tively. To gain a better understanding of the interaction of pulses with gain materials
it is preferable to use an hybrid time-/frequency-domain approach. This involves both
resolutions simultaneously, that is the spectrum or the instantaneous frequency versus
time delay. The experimental technique to implement such measurements, the so-called
frequency-resolved optical gating (FROG) method [153], measures the spectrogram
|Eg(ω, τ)|2 ∝
∣∣∣∣
∫ ∞
−∞
dtEsignal(t, τ)e
−iωt
∣∣∣∣
2
, Esignal(t, τ) = E(t) · g(t− τ), (6.17)
by applying a gate function g(t − τ) with variable time delay τ . For ultrashort optical
pulses the only available, however unknown gate function is the pulse itself. From our full
time-domain simulations, in which the evolution of the electric field including the carrier
wave is calculated, we can reproduce the various FROG measurements with different
gate functions, for example the PG-FROG by using
Esignal(t, τ) = E(t) |Eenv(t− τ)|2 . (6.18)
We have seen that dynamical gain saturation, the intensity dependence of the refrac-
tive index, fast SPM and nonlinearities are of particular importance with decreasing
pulse width and increasing peak power and instantaneous intensities. The amplifica-
tion of pulses with femtosecond durations to extremely high energies and peak powers
is crucial for a lot of applications, such as frequency conversion by means of nonlinear
optical crystals. However, nonlinearities and the excess of the threshold of catastrophic
optical damage, where the semiconductor gain material and the facets of the device take
damage, limit the mode of operation. One proposal to overcome this problem is given
by the chirped pulse amplification technique: The femtosecond pulses are chirped and
consequently temporally stretched to pulse durations of several ten or more picosec-
onds by dispersive elements. This reduces the peak electric field amplitude E0 and the
8A frequency chirp always increases the spectral width (for a given pulse length), or in other words,
involves the increment of the time-frequency bandwidth product beyond the Fourier transform limit.
148 LONGITUDINAL MULTI-MODE LASER DYNAMICS
-80 -60 -40 -20 0 20 40 60 80
delay [fs]
830
840
850
860
870
w
a
v
e
le
ng
th
 [n
m
]
2.16
2.18
2.2
2.22
2.24
2.26
2.28
fr
eq
ue
nc
y 
[1
01
5 s
-
1 ]
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
delay [ps]
840
845
850
855
860
865
w
a
v
e
le
ng
th
 [n
m
]
2.18
2.19
2.2
2.21
2.22
2.23
2.24
2.25
fr
eq
ue
nc
y 
[1
01
5 s
-
1 ]
Figure 6.28: Calculated PG-FROG spectrograms |Eg(ω, τ)|2, applying a gate function
as specified in (6.18), of the amplified optical pulses for the 100 fs, unchirped input pulse
(left) with E0 = 1·108Vm−1, and for a 10 ps, positive linearly chirped pulse (right) with
an identical pulse energy and spectrum. The relative spectrograms are linearly colour-
coded in grey scales, dark/black corresponds to high values and white to low spectral
intensities. The propagation and amplification of pulses in an active semiconductor
material has induced asymmetries and small chirps. We note that the gradient of
the PG-FROG spectrograms with time delay does not necessarily correspond to the
instantaneous frequency ωinst(τ), but is dependent on the pulse shape and the applied
gate function g(t− τ). (Right diagram) In the saturation regime there is a shift of the
spectra to longer wavelengths (by approximately 2 nm) compared to the spectrum of
the optical input pulse. For more details see Figure 6.30.
intensities. After the pulses have passed through the active media a compressor ele-
ment (with opposite dispersion) temporally compresses the pulse by removing the chirp.
Figure 6.29 illustrates that the fast Kerr-type nonlinearity is partially avoided and
the electrical to optical energy conversion, the extraction of the in the carrier system
stored energy, becomes more efficient: We have numerically analysed the amplification
of the pulse energy, which is proportional to
∫∞
−∞
dtE2env(z, t) =
∫∞
−∞
dν|Eν(z, ν)|2, and
the gain saturation in semiconductor-based amplifiers. A 10 ps pulse E0 = 1 · 107Vm−1
is characterised by a maximal intensity of I0 = 0.5ǫ0ǫc/n · E20 ≈ 50MWcm−2, a power
of P0 =
∫
dAI0(x, y) ≈ 50W (FWHM 10µm), pulse energy of
∫
dtP (t) ≈ 0.4 nJ and∫
dtE2env(t) ≈ 750V2m−2s. Plotted is the amplification of the pulse energy which corre-
sponds to twice the amplitude gain gmod. From nonlinear curve fitting to (6.13) we can
extract the nonlinear saturation coefficient in the case of the 100 fs pulse (circles, dashed)
as 2.31 · 10−3m2V−2s−1. The application of the concept of chirped pulse amplification
utilises the in the amplifier structure stored energy more effectively, suppresses nonlinear
saturation effects, and prevents optical damage. An improvement by a factor of 2 to
3 is realised, and we find a saturation coefficient (solid line) of 9.85 · 10−4m2V−2s−1.
The main reasons for the improved performances are given by the less distinctive hole
burning and gain saturation for smaller optical peak powers and by a partial recovery
of the spectral inversion and of the gain for pulse lengths of the stretched pulses greater
6.6 Chirped Pulse Amplification 149
0.1 1 10 100 1000
pulse ’energy’ [V2m-2s]
0
25
50
75
100
125
150
175
pu
lse
 e
ne
rg
y 
am
pl
ifi
ca
tio
n 
[cm
-
1 ]
Figure 6.29: Amplification of optical pulses with Gaussian pulse shapes and a pulse
duration of TFWHMpulse (Eenv(t)) = 100 fs (unchirped) or 10 ps and showing a positive linear
chirp (see (6.15)) of ∆ω = 2.78 · 1024 s−2, respectively, in a SOA. Important parameters
are λ = 850 nm, N = 3 · 1012 cm−2 ≈ 2Ntransparency, ΓNQW/Lref = 1/50 nm, neff = 3.6
and Lact = 0.05mm. Using dispersive optical elements, such as diffraction grating-pairs
for pulse stretching and compression, we can convert the two pulses into one another
as both pulses feature the same pulse energy and spectrum.
than the time constants of recovery. Obviously, in a realisation of the chirped pulse am-
plification method the recompression step demands some considerations as not only the
chirp from pulse stretching but also the extra dispersions arising from the interactions
in the SOA structure have to be compensated.
As the intensities of the stretched and chirped pulses and the gain approach saturation,
the amplified pulse spectrum will be frequency-shifted. For a positive chirped pulse, with
the leading edge extracting energy preferentially and the trailing edge experiencing less
amplification, the central wavelength is redshifted (Figure 6.30).
150 LONGITUDINAL MULTI-MODE LASER DYNAMICS
830 840 850 860 870
wavelength [nm]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
re
la
tiv
e 
sp
ec
tra
l i
nt
en
sit
y
1.4251.451.4751.5
energy [eV]
Figure 6.30: Relative spectral intensity |Eν |2 versus wavelength (or energy): Depicted
are the spectrum of the linearly chirped input pulse (as solid line) and the charac-
terisations of various amplified (chirped) pulses in frequency-domain. With increasing
pulse energy or electric field amplitude of the input pulse, E0 = 5 · 106Vm−1 (dashed),
1 · 107Vm−1 (dot-dashed) and 2 · 107Vm−1 (dotted), we recognise an increasing shift
of the spectra to longer wavelengths. This occurs because in the nonlinear or gain
saturation regime the leading red edge of a pulse with positive linear chirp extracts en-
ergy from the semiconductor amplifier structure preferentially. The trailing blue edge
is less amplified and consequently the spectrum is effectively redshifted. For a negative
linearly chirped input pulse we observe a blueshift of the spectrum (long-dashed line).
6.7 Ultrashort Coherent Optical Pulse Interactions
The availability of high-intensity ultrashort optical pulses allows for the study of transient
nonlinear coherent pulse propagation phenomena in active media [82, 83], such as self-
induced transparency [154], pulse area theorem [81] or soliton solutions [155]. Common
assumptions comprise the neglect of pumping terms and of decay or dephasing processes.
Lifetimes greatly exceed the pulse duration, and the induced polarisation has a more
or less infinite memory. Consequently, a coherent medium is considered with the sole
relevant interaction being the electric dipole transitions. The interaction of an optical
pulse with a resonant absorber (or amplifier) medium within the semiclassical framework
can be analysed by solving the Maxwell and the exactly resonant two-level (or equivalent
macroscopic) Bloch equations. However, an extension to inhomogeneous broadening of
the two-level atomic system (with ω0 ≫ ∆ωinh ≫ ∆ωpulse) is possible [155]. The full
time-domain model consists of the one-dimensional Maxwell curl equations (3.42)–(3.46)
6.7 Ultrashort Coherent Optical Pulse Interactions 151
and the macroscopic Bloch equations [14,76]
∂zEx + ∂tBy = 0,
1
µ0
∂zBy + ∂tDx = 0, Dx = ǫ0ǫEx + Γx,y
NQW
Lref
Px, (6.19)
∂2t P + 2γ
P∂tP + ω
2P = −4Ω(M
eh)2
~
E (N −Ntransparency) , E ← ΓNQW
Lref
P
ǫ0
, (6.20)
∂tN =
1
~Ω
E
(
∂tP + γ
PP
)− ΓnrN. (6.21)
The results can then be compared to analytical descriptions of ultrashort coherent optical
pulse propagation in a resonant medium [36,154,155]. A summary of common notations
and known analytical results is presented below. We introduce the real-valued phase
Φ(z, t) and envelope variables Eenv(z, t), which may also become negative, and rewrite
the electric field as
E(z, t) =
1
2
Eenv(z, t)e
iΦ(z,t)e−iω0teik0neffz + conjugate complex, Eenv,Φ ∈ R. (6.22)
We imply a similar expression for the induced polarisation. ω0 is the carrier frequency
and we assume Ω − ω0 = 0 and ω0Tpulse ≫ 1. By applying the slowly varying envelope
approximation (SVEA), the dynamics is given by the reduced Maxwell equations for a
forward travelling wave, by the following partial differential equations [36](
n2
cneff
∂t + ∂z
)
Eenv(z, t) = − k0Γ
ǫ02neff
NQW
Lref
Im (Penv(z, t)) , (6.23)
Eenv(z, t)
(
n2
cneff
∂t + ∂z
)
Φ(z, t) =
k0Γ
ǫ02neff
NQW
Lref
Re (Penv(z, t)) , (6.24)
which we consider in combination with the macroscopic Bloch equations9 in rotating
wave approximation (RWA) (see Sections 2.4+2.5).
The state of the two-level system is characterised by the Bloch pseudo-vector. We
introduce a new variable θ(z, t) which is the net angle that this pseudo-vector has been
turned through at a time t by an applied electric field envelope Eenv(z, t). It is also an
expression for the area of the pulse developed up to time t (no explicit knowledge of
Eenv(z, t) is required)
θ(z, t) =
∫ t
−∞
dt′ΩRabi(z, t
′) =
M eh
~
∫ t
−∞
dt′Eenv(z, t
′)→ ∂tθ(z, t) = M
eh
~
Eenv(z, t).
(6.25)
9The above described macroscopic or two-level Bloch equations model may also be applied as efficient
simulation tool [55, 126] to analyse millimeter-sized active semiconductor-based devices. To cover
important properties of semiconductor gain materials we perform some changes to the atomic-
like two-level approach: In (6.20) a nonlinear gain model (as for quantum wells) is introduced by
N − Ntransparency = Ntransparency(N/Ntransparency − 1) → Ntransparency ln(N/Ntransparency) and gain
or absorption suppression, nonlinear saturation by E → E/(1 + ǫsatS), with the photon density
S ∝ E2env [73]. In (6.21) additional loss channels and carrier diffusion may be implemented.
152 LONGITUDINAL MULTI-MODE LASER DYNAMICS
The idea is to search for unique pulse shapes, for analytical solutions of the nonlinear
problem (6.23)+(6.24) with dependency (z, t) = t− z/vpulse. This condition reduces the
problem to an ordinary differential equation, the so-called Mathieu equation ∂2t θ(t) =
γ2pulse sin(θ(t)), which also governs the motion of a pendulum initially angled in the non-
equilibrium upward position (the solution describes exactly one rotation). The solution
of this equation specifies a temporal soliton
Eenv(z, t) = Eenv
(
t− z
vpulse
)
, ∂tΦ(z, t) = constant, (6.26)
Eenv(z, t) = E0 · sech
(
γpulse
(
t− z
vpulse
))
, E0 = 2γpulse
~
M eh
, (6.27)
vpulse =
(
n2
cneff
+
k0Γ
ǫ02neff
NQW
Lref
4~
E20
2 (Ntransparency −N0)
)−1
< vphase. (6.28)
The hyperbolic secant solution (6.27) is unattenuated and retains its shape. The pulse
area theorem describes general properties of pulse propagation [81]. In the nonlinear
regime the pulse area Θ(z) develops according to (6.31)
Θ(z) = lim
t→∞
θ(z, t), (6.29)
α = πδ(Ω− ω0) k0Γ
ǫ02neff
NQW
Lref
(M eh)2
~
2 (Ntransparency −N0) , (6.30)
∂zΘ(z) = −α sin
(
Θ(z)
)
, (6.31)
lim
z→∞
Θ(z) =
{
2nπ, n ∈ N0 α > 0
(2n+ 1)π, n ∈ N0 α < 0 . (6.32)
Consequently, in optical pulse configurations with Θ0 = mπ,m ∈ N0 the pulse area is
conserved in the coherent propagation problem, which means ∂zΘ(z) = 0. In an absorber
medium with α > 0 the asymptotic solutions (attractors) are given as even multiples of
π where the branch number n is determined by the initial pulse area. Pulses with larger
areas divide into n separate 2π pulses, and n solitons given by (6.27) are formed. In an
amplifier the stable solutions are specified by odd multiples of π.
Figure 6.31 shows the propagation of a 2π pulse with hyperbolic secant pulse en-
velope shape (6.27) in an absorber: This optical pulse gives a complete transition of
the carrier system from the ground state (in the terminology of a two-level system), i.e.
N0 = Ntransparency − δN in our approach, to the excited state Ntransparency + δN . The
properties of the optical excitation pulse are maintained. These transitions occur within
one Rabi period, the optical pulse solution propagates distortionless as a solitary wave as
though it were unaffected by the presence of the active nonlinear medium, as though the
medium were transparent [76,154,155]. In the full time-domain simulations, a propagat-
ing sech-pulse enters a 360µm long active, resonantly absorbing medium. The temporal
evolutions of the electric field including the carrier wave (top) and of the macroscopic
carrier density (bottom) are recorded at positions 0µm, 90µm, 180µm, 270µm and
6.7 Ultrashort Coherent Optical Pulse Interactions 153
-1
0
1
el
ec
tri
c 
fie
ld
 [1
09
V
m
-
1 ]
0.12 0.14
1
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
0.42 0.44 0.72 0.74
time [ps]
1.03 1.05 1.33 1.35
Figure 6.31: Self-induced transparency or propagation of a temporal soliton. Simula-
tion parameters are set to γP = 1 ·109 s−1, ΓNQW/Lref = 1 ·107m−1, Ntransparency = 1.5 ·
1012 cm−2, N0 = 1 ·1012 cm−2, n = neff = 1, and γpulse = 2.2 ·1014 s−1 → TFWHMpulse = 12 fs.
360µm (from the left). The full time-domain Maxwell Bloch simulations [76, 80] essen-
tially reproduce the known analytical results of self-induced transparency and pulse area
theorem [81,154,155].
Figure 6.32 shows the resonant coherent interaction of pulses with different pulse
areas Θ0 with a two-level atomic system: A 2π pulse generates a complete excitation and
subsequent deexcitation of the medium (top left) [14]. The coherent induced absorption
of pulse energy during the first half of the pulse, the leading edge of the pulse putting
energy into the medium by flipping the Bloch pseudo-vector, ∝ ∫ z0/vpulse
−∞
dtE2env(z0, t) is
followed by the coherent induced emission of exactly the same amount of energy back
into the light field during the second half [154]. The trailing edge of the optical pulse is
flipping the Bloch vector back to its original position (Rabi flopping). The carrier density
is displayed as wide solid line, the electric field with carrier wave (thin solid line) is phase
shifted by π/2 versus the polarisation (dashed), as expected for a harmonic oscillator
driven by a resonant excitation. The time-derivatives of the electric field play an essential
role in the nonlinear evolution of the system, and the flattenings in the density profile
appear where |∂tE| takes its maximum value [76]. Apart from this additional cubic,
polynomial-like features the numerical full time-domain simulations agree well with the
154 LONGITUDINAL MULTI-MODE LASER DYNAMICS
0.11 0.12 0.13 0.14 0.15 0.16
time [ps]
-1
-0.5
0
0.5
1
el
ec
tri
c 
fie
ld
 [1
09
V
m
-
1 ]
1
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 
0.11 0.12 0.13 0.14 0.15 0.16
time [ps]
-2
-1
0
1
2
el
ec
tri
c 
fie
ld
 [1
09
V
m
-
1 ]
1
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 
0.11 0.12 0.13 0.14 0.15 0.16
time [ps]
-6
-4
-2
0
2
4
6
el
ec
tri
c 
fie
ld
 [1
09
V
m
-
1 ]
1
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 
0.11 0.12 0.13 0.14 0.15 0.16
time [ps]
-4
-2
0
2
4
el
ec
tri
c 
fie
ld
 [1
08
V
m
-
1 ]
1
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 
Figure 6.32: Resonant coherent interaction of pulses with different pulse areas Θ0 (E0
in (6.27) is varied) with a semiconductor structure described by the macroscopic gain
model (6.20)+(6.21), which is equivalent to a two-level atomic system. Depicted are
details of the temporal profiles at position 0µm of pulse and active material (simulation
parameters same as in Figure 6.31).
analytical results [14, 76, 80]. The same is true in the case of 4π pulses in an absorber
medium with two symmetric transitions (top right), and for π pulse propagation in an
amplifier medium with N0 = 2 · 1012 cm−2, displayed at bottom right. For larger pulse
areas, we have numerically calculated a 12π pulse (bottom left), incomplete Rabi flops
occur instead of an integer number. In [80] this is called the breakdown of the pulse
area theorem due to carrier wave Rabi flopping, which manifests itself in strong carrier
reshaping, electric field time-derivative effects and a subsequent production of higher
spectral components in the travelling pulse. Figures 6.33+6.34 show further tests of
the pulse area theorem.
An extension to an inhomogeneously broadened two-level atomic system is given in
Figure 6.35, where we consider a band-resolved semiconductor absorber medium: The
coupling between the different energy states of the band structure has been artificially
switched-off (top). This means, we treat the absorber material as a purely inhomo-
geneously broadened ensemble of two-level systems. The temporal profiles are given
at 0µm (left) and at the end of the active region at 360µm (right). The simulations
show strong signatures of the above discussed nonlinear coherent resonant effects [155].
The impacts of the time-derivatives of the electric field seem to have changed, and the
value of the carrier density at the upper reversal point of the Rabi flop is not equal
6.7 Ultrashort Coherent Optical Pulse Interactions 155
-3
-2
-1
0
1
2
3
el
ec
tri
c 
fie
ld
 [1
09
V
m
-
1 ]
0.12 0.14
1
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
0.42 0.44 0.46 0.73 0.75 0.77
time [ps]
1.03 1.05 1.07 1.34 1.36 1.38
Figure 6.33: Test of the pulse area theorem: We analyse the propagation of a 4π
pulse in a resonantly absorbing medium. The pulse divides into two separate 2π soliton
pulses of the specific envelope shape given by (6.27) (having different amplitudes and
durations Tpulse) which propagate at different group velocities (6.28). The velocity may
be considerably less than the phase velocity of light in the medium cneff/n
2. With the
objective to shorten the required simulation time, the coupling parameter ΓNQW/Lref
has been increased by a factor of 5.
-1
0
1
el
ec
tri
c 
fie
ld
 [1
09
V
m
-
1 ]
0.1 0.15
1
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
0.45 0.5 0.8 0.9
time [ps]
1.1 1.2 1.3 1.4 1.5
Figure 6.34: Propagation of a π pulse in a semiconductor amplifier: According to the
area theorem both an increase of the pulse energy ∝ ∫ dtE2env(z, t) and maintaining the
pulse area is possible if with increasing envelope the pulse duration is shortened (pulse
compression). For a few-cycle pulse coherent stimulated emission from the medium
back into the pulse may happen behind the main pulse (longer tail effects) and the
pulse would split [156]. This pulse splitting, where sections with negative electric field
envelopes and areas occur, is a another possible mechanism of conserving the pulse
area.
156 LONGITUDINAL MULTI-MODE LASER DYNAMICS
0.11 0.12 0.13 0.14 0.15 0.16
time [ps]
-1
-0.5
0
0.5
1
el
ec
tri
c 
fie
ld
 [1
09
V
m
-
1 ]
1
2
3
4
5
6
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 
1.34 1.35 1.36 1.37 1.38 1.39
time [ps]
-1
-0.5
0
0.5
1
el
ec
tri
c 
fie
ld
 [1
09
V
m
-
1 ]
1
2
3
4
5
6
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 
0.11 0.12 0.13 0.14 0.15 0.16
time [ps]
-1
-0.5
0
0.5
1
el
ec
tri
c 
fie
ld
 [1
09
V
m
-
1 ]
1
2
3
4
5
6
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 
Figure 6.35: Interaction of 2π self-induced transparency pulses with a band-resolved
semiconductor absorber medium.
Ntransparency+ δN . These deviations from analytical predictions can be explained by the
huge inhomogeneous broadening associated with the band-resolved material model (the
SVEA and RWA approximations are not applicable any more) and by the thereby im-
plemented nonlinear gain characteristics 6∝ (N −Ntransparency). We also study coherent
pulse interactions in the case of dephasing processes, other relaxation and loss channels,
and consequently homogeneous broadening represented by realistic parameters (bottom).
Incomplete Rabi flops occur, ultrafast nonlinear propagation effects become less distinc-
tive and disappear entirely for longer pulses. Depicted are the temporal evolutions at
0µm, contrary to all other simulations we have chosen n = neff = 3.6 and the length of
the absorber region as 100µm.
6.8 Conclusion
In this chapter we have employed the full time-domain model, which combines the
Maxwell curl equations and the semiconductor Bloch equations, in order to investigate
the longitudinal multi-mode dynamics in novel surface-emitting laser structures and the
nonlinear interaction of femtosecond and picosecond pulses with an optical gain medium.
6.8 Conclusion 157
In more detail:
• We have analysed VCSEL with embedded periodic gain and refractive index struc-
tures. The concept of the photonic band edge laser is to exploit the special properties
of the singularities in the optical band structure diagrams, i.e. the band edges with
flat dispersion. This laser is based on gain enhancement by an increased localisation
of the modes over the active quantum confined structures and by the more efficient
interactions of photons with the gain medium. The surrounding photonic band gap
region suppresses optical losses. Our simulations of lasing frequencies, thresholds and
field profiles quantitatively analyse photonic band edge band gap diode lasers and
numerically validate the gain enhancement for band edge modes.
• Realistic optically pumped external cavity surface-emitting laser structures have been
modelled. Our microscopic approach reveals the dynamical balance between carrier
generation or pumping at high energy states in the band structure diagram, mo-
mentum relaxation of carriers towards the Fermi-Dirac distribution and stimulated
recombination or lasing from states near the band edge. We show that the longitu-
dinal multi-mode behaviour is composed of several external cavity modes.
• By probing the semiconductor gain medium with weak electromagnetic test pulses,
we have found the analogy between these propagation experiments and small signal
gain calculations assuming quasi-equilibrium.
• For high-intensity pulses, that is in the nonlinear or saturation regime, the pump
pulses are modifying the gain material. We numerically investigate (sub-) picosec-
ond pulse interactions with semiconductor structures and identify the microscopic
origin of the fast nonlinearities, and specify the physical effects behind various sat-
uration mechanisms. We also compute nonlinear gain coefficients and the different
recovery rates. Group velocity dispersion, and the fast optical nonlinearities and self-
phase modulation are the main causes of asymmetries in pulse temporal and spectral
shape. The most critical parameters of gain recovery are the time constants of intra-
band scattering and carrier redistribution. The reshaping of the pulse envelope (e.g.
pulse slowdown or advancement, sharping of edges, and change of pulse duration)
and asymmetries in the pulse spectrum are discussed. To investigate the dynamics
of the gain and the refractive index change pump-probe experiments are numeri-
cally reproduced. One technique for amplifying high-intensity pulses, which avoids
these nonlinearities and increases the extractable energy from the active medium, is
proposed by the chirped pulse amplification method.
• We have investigated nonlinear coherent pulse propagation phenomena in active gain
media, such as the pulse area theorem or self-induced transparency. Our numerical
full time-domain simulations with adjusted constants of decay and in particular of the
dephasing processes agree well with analytical predictions, apart from the propagation
of optical pulses with large pulse areas, or few-cycle pulses in amplifying materials.
Coherent ultrafast nonlinear propagation effects become less distinctive if we apply
a more realistic model of the quantum well gain material, loss channels, dephasing
processes and homogeneous broadening.
A Discretisation Schemes and
Numerical Implementations
The numerical treatment of partial differential equations implies a finite-difference dis-
cretisation scheme of the incorporated field amplitudes and differential operators in time
and space. We are solely working on uniformly spaced (and rectangular) grids with
the spatial and temporal steppings ∆x,∆z,∆t and introduce the following abbreviatory
notation for a macroscopic field F
F (x, z, t) = F (x0 + i∆x, z0 + j∆z, t0 + n∆t) = F
n
i,j. (A.1)
The various differential operators (up to second order in the investigated models) have
to be transformed to finite-difference operators, for instance for the derivations with
respect to the transverse coordinate by
∂xF (x, z, t) = ∂xF
n
i,j →
1
∆x
(
F ni+1/2,j − F ni−1/2,j
)→ 1
2∆x
(
F ni+1,j − F ni−1,j
)
, (A.2)
∂2xF (x, z, t) = ∂
2
xF
n
i,j →
1
(∆x)2
(
F ni+1,j − 2F ni,j + F ni−1,j
)
, (A.3)
and for the temporal differential operator by
∂tF (x, z, t+∆t/2) = ∂tF
n+1/2
i,j →
1
∆t
(
F n+1i,j − F ni,j
)
. (A.4)
The correct centring of the time points and of the positions is absolute essential, which
may demand for an additional averaging process.
For the active points on the spatial grid, the dynamical material response is represented
on basis of the band-resolved semiconductor Bloch equations. Thereby, microscopic
quantities f(k, t) have to be discretised on an extra, equally spaced momentum grid
f(k, t) = f(k0 + k∆k, t0 + n∆t) = f
n
k . (A.5)
To evaluate integrals over the momentum grid (this can be found in the connection of
macroscopic with the respective microscopic variables as a spectral summation, or in the
scattering integrals) we do approximate the integrals as a sum of trapezoidal areas. In
integrations of functions with singularities an adaptive scheme based on Gauss-Kronrod
quadrature rules is applied [142].
160 DISCRETISATION SCHEMES AND NUMERICAL IMPLEMENTATIONS
A.1 Discretisation Scheme of the Transverse Wave
Equation Model
In numerically solving a multi-dimensional partial differential equation the stability and
accuracy, plus the complexity of the algorithm, that is to say, the number of iteration
steps required to calculate the new field values, together with the spatial grid (this quan-
tity chiefly determines the memory requirements) and temporal stepping are the most
crucial parameters as one is still at the very limit of calculating capacities even on su-
percomputers for these extremely time-critical (and multi-physics, multi-scale) problems.
In the transverse scalar paraxial wave equation model, associated with an ambipolar
diffusion equation for the carrier sheet density and a dynamical equation for the in-
duced electric polarisation, the equation (3.37) for the complex optical field amplitude
E˜+(x, z, t),R2 × R→ C, is the most critical part concerning numerical stability
n2
cneff
∂tE˜
+(x, z, t) =
(
−∂z + i 1
2k0neff
∂2x + ik0
n
neff
δnpas(x)
)
E˜+ + i
k0Γy
ǫ02neff
NQW
Lref
P˜+
→ a∂tE =
(−∂z + ib∂2x + ic(x))E + idP = L[E] +N [E]. (A.6)
On the right-hand side of the differential operator equation we have separated the linear
terms in the electric field amplitude, denoted by L[E], from the nonlinear functional
N [E] = ik0/(ǫ02neff)ΓyNQW/LrefP [E] which turns out to be not too critical for the
stability of the numerical method. The different numerical schemes can now be classified
by the way the linear part of the right-hand side of the equation is treated [142]. A simple
explicit scheme in which one uses the field values at the old time n is straightforward
a
En+1i,j − Eni,j
∆t
= L[En] +N [Eni,j], (A.7)
but the algorithm is not stable for too large time steps ∆t. On the other hand, one
could evaluate this linear term fully implicit at the new time point n+ 1, consequently
one calculates L[En+1] (En+1 represents a vector with Ncells = Nx · Nz elements). This
numerical scheme is unconditionally stable but it would require in the case of a two-
dimensional grid the inversion of a huge matrix (Ncells × Ncells) at every time step.
Evidently one has to find a compromise between stability and accuracy and the simplicity
of the algorithm. The proposed Hopscotch method [157] is introducing some implicit
character into a predominantly explicit scheme. For a two-dimensional spatial grid,
which can be visualised by a chessboard with white and black squares, all white fields
are solved explicitly. The remaining (in-between) black fields are then solved implicitly
a
En+1i,j − Eni,j
∆t
= −E
n+1
i,j+1 − En+1i,j−1
2∆z
+ ib
En+1i+1,j − 2En+1i,j + En+1i−1,j
(∆x)2
+ iciE
n+1
i,j +N [Eni,j].
(A.8)
As the electric fields on four of the five integration stencil points have already been
calculated in the explicit half step (as all neighbours of a black square are white squares)
A.1 Discretisation Scheme of the Transverse Wave Equation Model 161
Figure A.1: Hopscotch method: The diagram (from [12]) shows the two-dimensional
spatial grid with white and black squares (the colour is linked to the explicit and implicit
scheme, respectively) and the discretisation of the various differential operators of the
scalar transverse wave equation. The ambipolar diffusion equation of the macroscopic
charge carrier density is numerically solved in synchronism with the wave equation. An
additional momentum grid is attached to each active field to compute the band-resolved
semiconductor gain and carrier dynamics.
the nonlocal elements in the above expression are in the end evaluated explicitly (see
Figure A.1). In the following integration step the colours are inverted. With the
Hopscotch method one does not have to solve a matrix-valued problem, the implicit
character (and for this reason necessary inversion) arises only locally. The method is
very efficient and stable, reasonable accurate, and memory requirements are low due
to the fact that there is no need for introducing helping fields in the algorithm. One
remaining problem occurs because of the hyperbolic character of the propagation term
−∂zE which can lead to numerical instabilities [36]. This problem can be solved be
adding a very small numerical diffusion, terms with elliptic character which are stabilising
the equation1
+Dx∂
2
xEi,j +
n2
cneff
Dz
∆t
(Ei,j+1 − 2Ei,j + Ei,j−1) . (A.9)
Before numerically solving the partial differential equations for the macroscopic fields
1Another ansatz called the Lax-Wendroff discretisation scheme suggests a spatial averaging of the
field amplitudes Eni,j = 1/2(E
n
i,j+1 + E
n
i,j−1). This induces an additional (dissipative) term of
elliptic character on the right-hand side which stabilises the algorithm but also strongly changes and
thus corrupts the solution of the partial differential equation, for example a propagating pulse will
numerically elapse.
162 DISCRETISATION SCHEMES AND NUMERICAL IMPLEMENTATIONS
Ei,j, Ni,j we do consider the stiff system of ordinary differential equations which models
the dynamical material response. For the white squares we choose an explicit method,
for the black fields an implicit discretisation is opted for, and it turns out to be sufficient
to treat all nonlinear terms (in the dynamical variables) in a fully explicit way without
losing accuracy. Most of the computing time is spent in the time integrations of the
semiconductor material part, but the algorithm for the system of ordinary differential
equations is straightforward and numerically very stable.
At the transverse boundary of the spatial region of interest boundary conditions of
Dirichlet or von Neumann type are specified for the macroscopic fields electric field and
carrier density. In longitudinal direction the boundary condition is given by the reflec-
tions of the optical fields at the facets of the cavity.
As summarisation of the discretisation scheme and the numerical algorithm typical
values and the sequence of iteration steps are indicated in the following table:
Hopscotch method, paraxial transverse wave equation model
typical values of numerical parameters:
spatial stepping ∆x 250 nm
spatial stepping2∆z 2.5µm
time step ∆t 0.5 fs
numerical diffusion Dx 0.01 · 1/(2k0neff)
numerical diffusion term3Dz 0.005
transverse index guiding4δnpas(x) 0.003
sequence of time integration steps:
1. explicit half step (white) pk, n
◦
k → spectral summation → Ei,j → Ni,j, f ◦k
2. implicit half step (black) SBE → (2.114)+(2.117) → (A.8) → diffusion eq.
3. exchange of colours t+∆t → back to 1.
A.2 Discretisation Scheme of the Longitudinal Full
Time-Domain Model
The numerical implementation of the one-dimensional Maxwell curl equations model
(3.42)–(3.46) together with the dynamical nonlinear material response (realised by ordi-
nary differential equations for the band-resolved polarisation and the distribution func-
tions) within the finite-difference time-domain (FDTD) framework builds on an adequate
choice of discretisation schemes and the correct centring of fields in time and space [58].
2This is a typical value in the case of a continuous wave operating laser, the numerical treatment of
pulse propagation in semiconductor optical amplifiers requires a higher (by a factor of approximately
20) resolution.
3The chosen value is 100 times smaller than the elliptic (and strongly dissipative) stabilisation term
suggested by Lax-Wendroff.
4This very small ridge waveguiding term is needful to avoid unphysical numerical solutions.
A.2 Discretisation Scheme of the Longitudinal Full Time-Domain Model 163
Figure A.2: Yee discretisation scheme (from [14]): Yee suggested a by half steps
(in time and space) displaced arrangement of the various vector components of the
electric and magnetic field quantities, what in a perfect way meets the natural rotational
structure of the first order Maxwell curl equations.
A central finite-difference approximation of derivatives of the fields is applied
∂tF (z, t) = ∂tF
n
i →
F
n+1/2
i − F n−1/2i
∆t
, ∂zF
n
i →
F ni+1/2 − F ni−1/2
∆z
, (A.10)
and the fundamental idea of the Yee discretisation scheme to arrange the electric and
magnetic field quantities on staggered grids, on temporal and spatial grids with relative
offsets of half steps [158] (see Figure A.2), is taken on. This matches perfectly with the
special type of first order Maxwell curl equations. This fully explicit leap-frog algorithm
is non-dissipative and of second order accuracy in time and space, but only of around
first order in discontinuous media. The scheme is adopted to discretise the two Maxwell
curl equations (E|D|P,H = Ey|Dy|Py, Hx,R× R→ R)
Hni+1/2 = H
n−1
i+1/2 +
c∆t
∆z
(
E
n−1/2
i+1 − En−1/2i
)
, (A.11)
D
n+1/2
i = D
n−1/2
i +
c∆t
∆z
(
Hni+1/2 −Hni−1/2
)
(A.12)
→ En+1/2i = En−1/2i +
1
ǫi
c∆t
∆z
(
Hni+1/2 −Hni−1/2
)− 1
ǫi
∆t · (Γx,y)i
(NQW)i
∆z
(∂tP
n
i ) .
The above field equations have been transformed to the Heaviside-Lorentz unit system to
avoid the imbalance in the electric and magnetic field amplitudes (ǫ0 has a very different
164 DISCRETISATION SCHEMES AND NUMERICAL IMPLEMENTATIONS
magnitude from µ0). The material parameter ǫ(z) generally features step discontinu-
ities (in refractive index structures like DBR mirrors or photonic crystals), thus, when
mapping ǫ(z)→ ǫi, we do use weighted average values in some cells.
At the left and right boundary of our finite analysis window (for the core region, the
spatial region of interest, we solve the above discretised curl equations) boundary con-
ditions have to be specified. Metallic (or perfectly reflecting) boundary conditions are
very simple to implement by setting the tangential electric field component Eboundary = 0
at every time step (Dirichlet type), but in most cases a more realistic model would be
to simulate a system with the structure of interest embedded into infinite expanded
space. Absorbing or open boundary conditions have to be defined in a way that an out-
going wave (across the interface) is completely absorbed without any reflection back.
According to Mur [119] the simulation region can be closed (artificially) by demand-
ing exclusively for the electric field the constraint, compatible to the wave operator
equation with plane wave solutions travelling to the left or to the right respectively,(
∂z ∓√ǫboundary/c∂t
)
Eboundary = 0, written in a discretised formulation by (with the
abbreviation C = c∆t/(√ǫboundary∆z))
E
n+1/2
0 = E
n−1/2
1 +
C − 1
C + 1
(
E
n+1/2
1 − En−1/20
)
, (A.13)
E
n+1/2
imax+1
= E
n−1/2
imax
+
C − 1
C + 1
(
E
n+1/2
imax
− En−1/2imax+1
)
. (A.14)
In order to optically pump, excite or probe the active laser or amplifier structure well
defined external optical fields (plane waves) are injected into the system at the grid
point iext (and travel e.g. to the right). With the total-field/scattered-field (TFSF)
technique [58] we can insert external plane wave components. The name arises from the
fact that the total electric field Etotal (and the equivalent for the magnetic field) may be
superposed by an incident part Eext and the response, the scattered field Escat. Having
solved (A.11)+(A.12) for all grid points in the core region with the generic Yee scheme
just at two positions the field values have to be corrected (or updated)
Hniext−1/2 = {Hniext−1/2} −
c∆t
∆z
E
n−1/2
ext,iext
, (A.15)
D
n+1/2
iext
= {Dn+1/2iext } −
c∆t
∆z
Hnext,iext−1/2. (A.16)
Targeting to simulate novel semiconductor laser structures with enclosed external mirror
configurations (like VECSEL) an efficient method to model these external cavities, which
is fast and has low memory requirements, is proposed by the longitudinal delayed optical
feedback boundary conditions [12]. The concept behind it is outlined in the following:
Light leaves the actual active laser structure, propagates to the mirrors some distance
Ll, Lr away, is reflected there (with the intensity reflectivities Rl, Rr and with a phase
shift of around π) and is finally feed backed (with a delay of the round trip times
τl = 2Llnext,l/c, τr = 2Lrnext,r/c) from the passive external cavity into the active laser
or amplifier structure. We pick up the out-going part of the light fields (this can be done
A.2 Discretisation Scheme of the Longitudinal Full Time-Domain Model 165
by the correct choice of grid points) and put it into delay line buffers with τl/∆t + 1,
τr/∆t+ 1 elements
Hnext,l =
√
Rl
2
(
Hn1+1/2 +H
n
2+1/2
)
, E
n+1/2
ext,l = −
√
Rl
2
(
E
n+1/2
1 + E
n+1/2
2
)
, (A.17)
Hnext,r =
√
Rr
2
(
Hnimax−1+1/2 +H
n
imax−2+1/2
)
, E
n+1/2
ext,r = −
√
Rr
2
(
E
n+1/2
imax
+ E
n+1/2
imax−1
)
.
(A.18)
The fields stored in the buffers are inserted back into the active device with delay
times of Nl = τl/∆t, Nr = τr/∆t time steps taking use of the above explained total-
field/scattered-field method
Hn3+1/2 = {Hn3+1/2} −
c∆t
∆z
E
n−1/2−Nl
ext,l , D
n+1/2
4 = {Dn+1/24 } −
c∆t
∆z
Hn−Nlext,l , (A.19)
Hnimax−3+1/2 = {Hnimax−3+1/2}+
c∆t
∆z
E
n−1/2−Nr
ext,r , D
n+1/2
imax−3
= {Dn+1/2imax−3}+
c∆t
∆z
Hn−Nrext,r .
(A.20)
As next step we have to incorporate the dynamical equations for the active material
part, which are formulated in real space, into the stable finite-difference time-domain
solver in a self-consistent way. The induced macroscopic polarisation P
n+1/2
i needs to be
available to close the set of equations (A.11)+(A.12), this can be done by centring the
(second order differential) equations of the set of damped harmonic oscillators around
n− 1/2
∂2t pk + 2γ
p
k∂tpk + ω
2
kpk
∣∣n−1/2
i
= −ΩkM
ehE
~
(
nek + n
h
−k − 1
)∣∣∣∣
n−1/2
i
→ pn+1/2i
(
1
(∆t)2
+
γp
∆t
)
=
(
2
(∆t)2
− ω2
)
p
n−1/2
i +
(
− 1
(∆t)2
+
γp
∆t
)
p
n−3/2
i
− ΩM
eh
~
E
n−1/2
i
(
n
e,n−1/2
i + n
h,n−1/2
i − 1
)
. (A.21)
After calculating the spectral summation of the microscopic quantities pk → P , and
then the electric field for the new time step n+ 1/2 by
E
n+1/2
i =
1
ǫi
(
D
n+1/2
i − (Γx,y)i
(NQW)i
∆z
P
n+1/2
i
)
, (A.22)
we can evaluate the dynamical distribution functions at the new time step by centring the
differential equations around n. At this, nek, n
h
−k, N are fixed to the same grid positions
with the electric field quantities as suggested in [148]
n
n+1/2
i
(
1
∆t
+
γnr + γ + Λ
n−1/2
i
2
)
=
(
1
∆t
− γ
nr + γ + Λ
n−1/2
i
2
)
n
n−1/2
i + γf
n−1/2
i
+ . . .
∣∣∣n−1/2 + M eh
~Ω
(
E
n+1/2
i + E
n−1/2
i
)(pn+1/2i − pn−1/2i
∆t
+
γp
2
(
p
n+1/2
i + p
n−1/2
i
))
.
(A.23)
166 DISCRETISATION SCHEMES AND NUMERICAL IMPLEMENTATIONS
As some terms in the above equation (for instance Λi or the quasi-equilibrium distribu-
tion functions fi) cannot be evaluated at the new time step we use the values at n− 1/2
without losing stability. The corresponding equation for the macroscopic carrier density
N
n+1/2
i is solved afterwards, since some physical effects (e.g. carrier diffusion) may only
be qualified on the macroscopic level. The crucial property of our discretisation and
integration scheme is the fact that the combined active material and passive refractive
index structure model in full time-domain can be implemented in a fully explicit manner.
The computational realisation turns out to be stable and pretty efficient [74] (because no
iterative predictor-corrector step as required in another proposed numerical scheme [76]
or costly Runge-Kutta integration scheme [77] has to be performed). Connecting the
Maxwell curl equations written in the Heaviside-Lorentz unit system with the semicon-
ductor gain dynamics formulated in the SI unit system some conversions are necessary.
As summarisation of the discretisation scheme and implementation typical numerical
values and the sequence of integration steps (this sequence is determined by the chosen
time centrings) are indicated in the following table:
Maxwell semiconductor Bloch equations in full time-domain
typical values of numerical parameters:
spatial stepping5∆z λ/(max{n} · 24)
time step6∆t c∆t/∆z = 0.999
aliasing of material parameters linear weighted average values
sequence of iteration steps:
1. Hni+1/2 Maxwell curl equation (A.11) →
fields injected by (A.15) + delayed feedback
2. D
n+1/2
i Maxwell curl equation (A.12) →
fields corrected by (A.16) + delayed feedback
3. end of passive part buffers updated → at boundaries (A.13)+(A.14)
4. active gain material → En+1/2i pk (A.21), P n+1/2i → En+1/2i (A.22) →
n◦k (A.23), N
n+1/2
i , (f
◦
k )
n+1/2
i
5In the medium with the highest refractive index in the investigated system the (shortest) optical
wavelength is resolved with 24 grid points (Nyquist criterion), a typical value for a InGaAlAs-based
structure operating at around 850 nm would be 10 nm. A much higher resolution (as sometimes
discussed in literature) does not make sense: Smaller structures (than the above given ∆z) are
optically not active (are not seen by the electromagnetic fields) and, even more important, the
macroscopic Maxwell equations with the material properties ǫ(z) are not the adapted concept for
(sub-) nanometer structures, the vacuum Maxwell equations with all the bound charges and sources
would be the correct choice.
6The algorithm is stable according to the Courant condition [58] for time steps ∆t ≤ ∆s/(c√Dim)
(∆s/(c
√
Dim) is called the magic time step), a typical value for the needed temporal resolution
would be 0.03 fs.
A.3 Numerical Analysis of the Scattering Integrals 167
A.3 Numerical Analysis of the Scattering Integrals
The numerical treatment of the carrier-phonon and in particular of the carrier-carrier
scattering is exceedingly time-critical based on the fact that we have to sum or integrate
over all possible interaction events. This implies in the case of Coulomb scattering to
integrate over the momentum transfer vector q and the momentum vector k′ of the
scattering partner, plus a summation over all potential subband indices combinations.
In the case of the Fro¨hlich interaction we integrate over the carrier momentum transfer,
and add up the possible scattering incidents (that is to say, scattering under absorp-
tion and under emission of a longitudinal optical phonon, and various subband indices
combinations). Consequently, the aim has to be to carry out analytically as many of
the integrations as possible, in doing so we follow the numerical implementation scheme
suggested in [64]7.
We start with the carrier-phonon scattering and indicate the integration algorithm
considering as example the scattering matrix Γee,outi2i5,k (which occurs for instance in the de-
scription of the relaxation dynamics ∂tn
e
i1i2,k
∣∣
relax
=
∑
i5
[−Γee,outi2i5,knei1i5,k − Γee,out∗i1i5,k ne∗i2i5,k]
of the electron expectation values)
∑
q,qz
∑
i4,i6
∑
±
π
~
δ
(−Eei5,k + Eei6,k−q ± ~ωl-o)
× γ∗i2i4(q, qz)γi5i6(q, qz)
(
nph +
1
2
± 1
2
)(
δi4,i6 − f e∗i4i6,k−q
)
. (A.24)
We replace the summation by a continuous integral formulation and make the coordinate
transformation q→ k′ = k− q. Assuming isotropy in momentum space we obtain (for
one subband indices combination)
∫
d2k′
A
(2π)2
π
~
δ
(
Eei5,k=0 − Eei6,k′=0 +
~
2
2me
(
k2 − k′2)∓ ~ωl-o
)
×
(
nph +
1
2
± 1
2
)(
δi4,i6 − f e∗i4i6,k′
)(∫
dqz
L
2π
γ∗i2i4(q, qz)γi5i6(q, qz)
)
. (A.25)
The integration over qz can be carried out analytically. After introducing polar coordi-
nates for the momentum vector by
∫
d2k′ → ∫ dk′k′ ∫ dϑ, ϑ = ∠(k,k′), the delta function
(guaranteeing the energy conservation of each single scattering process) is used in the
integration over the radius k′
∫
dϑk′
A
(2π)2
π
~
me
~2k′
(
nph +
1
2
± 1
2
)(
δi4,i6 − f e∗i4i6,k′
) (2π)2
A
e2~
ǫ20(4π)
2ωl-oγ
1
q
Fi4i5i6i2(q),
(A.26)
7Another approach to compute the manifold integrals in the scattering matrices is to try to decouple
the integrations with respect of the amplitude and the angle (they occur in the arguments of the
distribution functions such as k− q) [36,64].
168 DISCRETISATION SCHEMES AND NUMERICAL IMPLEMENTATIONS
thereby the solutions are given by
k′ = k′(i5, i6,∓) =
√
k2 ∓ 2me
~2
(
~ωl-o ∓ (Ei5,0 − Ei6,0)
)
, (A.27)
q = |k− k′| =
√
k2 + k′2 − 2kk′ cosϑ. (A.28)
The threefold integral we started with is reduced to an integration over one coordinate8,
but that remaining integration over the angle ϑ has to be evaluated numerically
e2meωl-o
16π~2ǫ0
(
1
ǫ∞
− 1
ǫs
)∫ 2pi
0
dϑ
1
q
Fi4i5i6i2(q)
(
nph +
1
2
± 1
2
)(
δi4,i6 − f e∗i4i6,k′
)
. (A.29)
The upper sign is associated with the emission of a phonon, the lower sign is linked to an
absorption event. A screening model may be added in the evaluation of the scattering
contributions by the replacement 1/qF (q)→ 1/ǫ(q, ω) · 1/qF (q).
For the carrier-carrier scattering we again look as example at the matrix Γee,outi2i5,k∑
k′,q
∑
i4,j7,j6,i8,j3,j2
π
~
δ
(−Eei5,k − Ehj7,−k′ + Ehj6,−k′−q + Eei8,k−q)Vi5j7j6i8(q)Vi1j3j2i4(q)
× fhj3j7,−k′
(
δj2,j6 − fhj6j2,−k′−q
) (
δi8,i4 − f ei8i4,k−q
)
. (A.30)
In the following abbreviations for an energy expression Es = E
e
i5,0
+Ehj7,0 −Ehj6,0 −Eei8,0
and for the reduced mass of the two-body system µ = memh/(me+mh) are introduced.
Undertaking a coordinate transformation for the momentum coordinates
g =
2µ
me
k− 2µ
mh
k′ → k′(k,g) (A.31)
and a transition from sums to integrals over momentum vectors we obtain (a fourfold
integral)
∫
d2g
A
(2π)2
(
mh
2µ
)2
· 2
∫
d2q
A
(2π)2
π
~
δ
(
Es +
~
2
2µ
(g · q− q2)
)
Vi5j7j6i8(q)Vi1j3j2i4(q)
× fhj3j7,k′
(
δj2,j6 − fhj6j2,|k′+q|
) (
δi8,i4 − f ei8i4,|k−q|
)
. (A.32)
A factor of two as the number of scattering partners because of the degeneracy of elec-
tron and hole functions with regard to the spin degree of freedom has been added in the
above equation (the Coulomb interaction of two charged particles is independent of the
spin degree of freedom, and does not represent a spin-flip process). As next step in eval-
uating the integrals over momentum vectors polar coordinates
∫
d2g → ∫ dgg ∫ dϕ, ϕ =
∠(k,g) = ∠(x − axis,g) and ∫ d2q → ∫ dqq ∫ dϑ, ϑ = ∠(g,q) are introduced. In the
8For bulk semiconductor structures (with the usual phenomenological screening models) the threefold
integrals can be fully evaluated analytically.
A.3 Numerical Analysis of the Scattering Integrals 169
integration over the angle ϑ the delta function, the total energy conservation of the
many-body interaction, is used to filter out the two possible solutions
ϑ1 = arccos
(
q2 − 2µ/~2Es
gq
)
∈ [0, π], ϑ2 = 2π − ϑ1 (A.33)
(and with it the momentum transfer q is determined). The constraints for the existence
of real-valued solutions (plus the fact that radii have to be positive numbers) give the
limits of the integration areas, namely
qmin = max
[
g −√g2 + 8µ/~2Es
2
,
−g +√g2 + 8µ/~2Es
2
]
, (A.34)
qmax =
g +
√
g2 + 8µ/~2Es
2
, (A.35)
gmin =
√
max
[
0,−8µ
~2
Es
]
. (A.36)
All these coordinate transformations and analyses result in the following expression of
nested integrals∫ ∞
gmin
dgg
∫ 2pi
0
dϕ
A
(2π)2
(
mh
2µ
)2
· 2
∫ qmax
qmin
dqq
A
(2π)2
π
~
2µ
~2
√
g2q2 − (q2 − 2µ/~2Es)2
×
(
e2
2ǫ0ǫA
)2
1
q2
Fi5j7j6i8(q)Fi1j3j2i4(q)f
h
j3j7,k′
(
δj2,j6 − fhj6j2,|k′+q|
) (
δi8,i4 − f ei8i4,|k−q|
)
.
(A.37)
The fourfold integrals we started with are reduced to a threefold integrals formulation9.
The sequence of the integrations is crucial (g → q → ϕ) by reason that the integral
boundaries or the arguments of the integrands of the inner loops are dependent on the
values of the variables defined in outer loops10. In the end we have derived the following
term
m2he
4
64π3~3µǫ20ǫ
2
∫ ∞
gmin
dgg
∫ qmax
qmin
dqq
1√
g2q2 − (q2 − 2µ/~2Es)2
1
q2
Fi5j7j6i8(q)Fi1j3j2i4(q)
×
∫ 2pi
0
dϕfhj3j7,k′
(
δj2,j6 − fhj6j2,|k′+q|
) (
δi8,i4 − f ei8i4,|k−q|
)
. (A.38)
9For bulk semiconductor structures the sixfold integrals can be simplified to twofold integrals which
have to be treated numerically.
10Because of the complicated structure of the nested loops and the fact that still manifold integrals
have to be carried out numerically, it is just impossible to perform dynamical simulations of the
carrier-carrier scattering combined with a spatially resolved modelling of the laser fields. Therefore
we replace in the scattering matrices the distribution functions by quasi-equilibrium (Fermi-Dirac)
distribution functions. Scattering rates depending on the macroscopic active material state (that
is to say, dependent on macroscopic field variables like carrier density or temperatures) as effective
measures (of the importance of the interaction) enter our spatially resolved semiconductor laser
descriptions.
170 DISCRETISATION SCHEMES AND NUMERICAL IMPLEMENTATIONS
The effects of screening of the interaction may be accounted for in the scattering inte-
grals by V (q)→ V sc(q) = 1/ǫ(q, ω) · V (q).
As summary of the algorithms implemented to numerically evaluate the (many-body)
scattering events necessary analyses and the sequence of integrations are specified in the
following table:
numerical analysis of scattering integrals
carrier-phonon scattering:
1. (k, subband indices, ±) → calculate k′ (A.27)
2. integrate over angle ϑ → determine q(k, k′, ϑ) (A.28)
3. evaluate (A.29) → back to 2.
carrier-carrier scattering:
1. (k, subband indices) → compute gmin (A.36)
2. integrate over g (outer loop) → calculate qmin(g) (A.34), qmax(g) (A.35)
3. integrate over q → find solutions ϑ(g, q) (A.33)
4. integrate over angle ϕ → k′(k, g, ϕ) = k′(k,g),q(g, ϑ) → k′, |k′ + q| , |k− q|
5. evaluate (A.38) → back to 4. → back to 3. → back to 2.
A.4 Conclusion
In this chapter we have discussed fundamental properties of the numerical implementa-
tions of our theoretical time-domain models, that are the applied discretisations of the
fields and differential operators on regular grids and the integration schemes. As we aim
to numerically solve multi-dimensional partial differential equations in combination with
a band-resolved description of the semiconductor material (a multi-physics and multi-
scale problem) the numerical complexity of the algorithm, characterised by the number
of iteration steps, the grid resolutions and memory requirements, represents besides the
accuracy and the stability the very critical parameter. In more detail:
• For the paraxial transverse wave equation model, a frequency-/time-domain approach,
we have proposed the Hopscotch method, which is very efficient and stable and reason-
able accurate. This numerical scheme involves the partitioning of the two-dimensional
grid into two groups of grid points (with stride two) plus an integration scheme with
alternating explicit and implicit discretisations. That results in not having to solve a
matrix-valued problem, the implicit character and for this reason necessary inversion
arises only (spatially) locally. Adequate boundary conditions are introduced, and we
have discussed the addition of terms of elliptic character to stabilise the equations.
• The finite-difference time-domain (FDTD) method solves the first order Maxwell curl
equations by arranging the electric and magnetic field quantities on staggered grids
in time and space according to the Yee scheme. This allows for the correct cen-
tring of time points and positions of the fields and differential operators in central
finite-difference approximation. At the boundaries of our finite analysis window ab-
sorbing or open boundary conditions (Mur) are defined. External optical fields (plane
A.4 Conclusion 171
wave components) are injected using the total-field/scattered-field (TFSF) technique,
and external homogeneous cavities are considered by the longitudinal delayed optical
feedback boundary conditions. The additional field variables carrier density, induced
polarisation and the microscopic distribution functions and interband polarisations
share the same grid with the electric field quantities. The combined dynamical model
in full time-domain of the (active) nonlinear material response and the passive refrac-
tive index structure can be numerically implemented in a fully explicit manner. Due
to the time-domain character of the FDTD method a quite extensive effort of data
extraction, basically Fourier transforms, is required to gain physical results.
• The computational realisations of our theoretical descriptions of carrier-carrier and
carrier-phonon scattering processes and other loss mechanisms like Auger recombi-
nation involve the evaluation of nested manifold integrals (as the summation over all
possible interaction events). The aim has to be to carry out analytically as many of
the integrations as possible (and to utilise the conservation laws of the single interac-
tion process for some physical quantities such as the energy or the total momentum).
In the case of the Coulomb interaction we are left with the numerical calculation of
threefold integrals, in the Fro¨hlich scattering with an integral over one coordinate.
B Dynamical Treatment of the
Scattering Contributions
In Section 4.1 we have investigated the Boltzmann scattering integrals (4.2) assum-
ing quasi-equilibrium Fermi-Dirac distributions. For these functions the collision terms
become identically zero (the principle of detailed balance), but we can extract non-
vanishing relaxation rates for the out-scattering and in-scattering processes [4]. In this
chapter the relaxation of initially prepared non-equilibrium distributions and the way
thermal equilibrium is approached are modelled (on sub-picosecond time scales). In the
hydrodynamic approach [103], the non-equilibrium carrier state is characterised by the
first moments of the distributions, a number of these physical quantities are conserved
in the scattering processes1 ∂t|relax
[
1/V
∑
kW
(n)(k)n◦k
]
= 0. The carrier-carrier scat-
tering interactions involve the conservation of the carrier density (W (0) = 1), of the
total momentum density (corresponding toW (1) = k), and of the kinetic energy density,
that is to say the plasma temperature (W (2) ∝ k2). Whereas the collision with other
quasi-particles, namely with phonons (the most efficient process is given by the Fro¨hlich
interaction with longitudinal optical phonons), leads to a relaxation of the carrier kinetic
energy density, to plasma cooling [91,106]. Because of the facts that the numerical anal-
ysis of the scattering integrals for bulk semiconductor structures can be executed much
more efficient than for quantum well structures [64] (and also cf. Appendix B) and also
the carrier density conservation is numerically better fulfilled [108], in order to under-
stand in principle the characteristics of the relaxation of non-equilibrium distributions
we do consider in this chapter bulk semiconductor structures. In doing so, the strengths
of the scattering processes are specified by the interaction potentials
V (q) =
e2
ǫ0ǫ
1
V
1
q2
, γe(ph)(q) = − i
ǫ0
1
q
√
e2~
2γωl-oV
, (B.1)
and the dielectric function and screening constants
κ2 =
e2
ǫ0ǫπ2~2
[
me
∫ ∞
0
dkf ek +mh
∫ ∞
0
dkfhk
]
, ǫ(q→ 0, ω → 0) = 1 + κ
2
q2
, (B.2)
ω2pl =
e2N
ǫ0ǫmr
, ω2q = ω
2
pl
(
1 +
q2
κ2
)
+
C
4
(
~q2
2mr
)2
. (B.3)
1These conservation rules can be derived from the quantum Boltzmann collision terms. However, the
representation of scattering processes in the relaxation rate approximation (2.92) with k-dependent
scattering rates fails to preserve the moments of the distribution, such as the carrier density.
174 DYNAMICAL TREATMENT OF THE SCATTERING CONTRIBUTIONS
0 5 10 15
momentum vector [1/aBohr]
0
0.2
0.4
0.6
0.8
1
el
ec
tro
n 
di
str
ib
ut
io
n 
fu
nc
tio
n
0 50 100 150 200 250
time [fs]
0.05
0.1
-
δn
e
0 5 10 15 20
momentum vector [1/aBohr]
0
0.1
0.2
0.3
0.4
0.5
ho
le
 d
ist
rib
ut
io
n 
fu
nc
tio
n
0 50 100 150 200 250
time [fs]
0.01
0.1
-
δn
h
Figure B.1: Relaxation of a kinetic hole, due to screened carrier-carrier scattering and
the interaction with longitudinal optical phonons, in the electron (top) and hole (bot-
tom) distributions: The initially prepared non-equilibrium carrier distribution functions
show strong spectral hole burning effects which may arise as the result of the interaction
of the inhomogeneously broadened carrier system with the lasing modes in a resonator
configuration (that is to say, from spectrally selective stimulated electron-hole recom-
bination processes). Plotted are the distribution functions obtained from the direct
numerical integration [36, 64] of the generalised Boltzmann equations [25, 159] at 0 fs
(solid line), 50 fs, 100 fs, 200 fs and at 500 fs (dotted). The insets in the graphs give
the decay of the (maximum) deviations from quasi-equilibrium (as symbols) over time
plus an exponential fit of these data (we assume the relaxation of the system to be
of a reservoir coupling type [4]). We find an electron relaxation time of 181 fs and an
effective hole scattering time of 95 fs (in a GaAs bulk laser structure and a density of
N = N e(t = 0) = Nh(t = 0) = 3 · 1018 cm−3).
175
0 5 10 15
momentum vector [1/aBohr]
0
0.2
0.4
0.6
0.8
1
el
ec
tro
n 
di
str
ib
ut
io
n 
fu
nc
tio
n
0 50 100 150 200 250
time [fs]
0.05
0.1
δn
e
0 5 10 15 20
momentum vector [1/aBohr]
0
0.1
0.2
0.3
0.4
0.5
ho
le
 d
ist
rib
ut
io
n 
fu
nc
tio
n
0 50 100 150 200 250
time [fs]
0.01
0.05
0.1
δn
h
Figure B.2: Equivalent to Figure B.1, but this time the carrier and energy relax-
ation of optically excited distributions (as the result of the spectrally selective carrier
generation by an optical pumping scheme [16,17,133,134] as applied in novel laser struc-
tures, e.g. VECSEL) is investigated. An exponential fit of the scattering enables the
identification of effective relaxation rates [92]. The comparison of dynamical scattering
calculations with simulations utilising the effective scattering times demonstrates a good
qualitative agreement. In general, hole relaxation times are faster than electron scatter-
ing times 1/γh ≈ 100 fs ≈ 0.5 · 1/γe. Lookup tables of scattering times and dephasing
rates extracted from calculations for different carrier densities and temperatures, ex-
citation parameters (and structural parameters in quantum confined structures) make
it possible to microscopically model correlation contributions within spatially resolved
time-domain simulations of semiconductor laser structures. These effective scattering
times can be extracted from an exponential fit to the temporal decay of an excitation,
at first suggested in [92], or with a quasi-equilibrium approach, for more details see
Section 4.1.
176 DYNAMICAL TREATMENT OF THE SCATTERING CONTRIBUTIONS
0 20 40 60 80 100 120 140 160 180 200
energy [meV]
0
0.1
0.2
0.3
0.4
0.5
ho
le
 d
ist
rib
ut
io
n 
fu
nc
tio
n
Figure B.3: Details of Figure B.2: The model of the correlation contributions of
the many-body interactions separates the carrier momentum and energy relaxation into
two processes. The carrier-carrier interaction by the screened Coulomb potential relaxes
the system towards thermal quasi-equilibrium characterised by the plasma temperature
Tpl. The carrier-phonon scattering, mediated by the Fro¨hlich interaction, establishes
Fermi-Dirac distributions at the lattice (phonon) temperature Tlat. The carrier-phonon
scattering proceeds (in a microscopic model) via the successive emission and absorption
of longitudinal optical phonons, which leads to the occurrence of sidebands (of the
excitations) in the carrier distribution at energy values Eexc ± n · ~ωl-o, n ∈ N. We
have plotted the hole distribution function versus energy at time points 0 fs, 25 fs, 50 fs,
100 fs and 200 fs. As can be clearly seen, two sidebands with an energy distance (to the
excitation energy) of ~ωl-o =36meV occur, higher order sidebands are hindered by the
strong screening effects and the dominant carrier-carrier scattering contributions.
Conclusion
In this part of the work we have analysed, by direct numerical integration of the (Marko-
vian) quantum kinetic Boltzmann collision integrals, the relaxation of non-equilibrium
distribution functions due to the screened carrier-carrier scattering and the interaction
with longitudinal optical phonons in a bulk semiconductor structure. We have consid-
ered as the initially prepared non-equilibrium distribution functions two in laser systems
relevant situations, namely the relaxation of a kinetic hole (spectral hole burning) and
an optically excited distribution (spectrally selective carrier generation), and extracted
effective scattering rates by exponential fits of the decay of these excitations. The results
from dynamical scattering calculations agree well with microscopically calculated relax-
ation times assuming quasi-equilibrium. Consequently, an approach with lookup tables
of scattering and dephasing rates allows for a microscopic treatment of correlation con-
tributions along with spatially resolved time-domain simulations of semiconductor laser
structures.
C Zusammenfassung
C.1 Kurzzusammenfassung
Diese Arbeit behandelt die Wechselwirkung von Licht und Materie und optische Nicht-
linearita¨ten in Halbleiter-Nanostrukturen und pra¨sentiert eine detaillierte numerische
Analyse der raumzeitlichen Dynamik von neuartigen Hochleistungsdiodenlasern. Wir
leiten ein mikroskopisches, ra¨umlich aufgelo¨stes Modell ab, welches eine Dichtematrix-
Beschreibung der optoelektronischen Eigenschaften von Quantenfilm-Gewinnmedien mit
den makroskopischen Maxwell Gleichungen fu¨r die elektromagnetische Felddynamik
kombiniert: die Maxwell Halbleiter-Bloch Gleichungen in Full Time-Domain. Diese bein-
halten Vielteilchen-Wechselwirkungen, eine Vielzahl an Zeitskalen und an Gewinnsa¨tti-
gungsmechanismen und schließen die schnell oszillierende Tra¨gerwelle und eine Subwellen-
la¨ngen-Auflo¨sung ein. Mikroskopisch berechnete Streuraten sind in den ra¨umlich aufge-
lo¨sten Simulationen eingebunden.
Unsere Arbeit konzentriert sich auf ultraschnelle Ladungstra¨gereffekte, das quantita-
tive Versta¨ndnis von optischen Nichtlinearita¨ten, die gezielte Manipulation der Moden-
struktur in Mikrokavita¨ten, und deren Einfluß auf die Kenngro¨ßen der Laserstrahlung.
Optische Dephasierung und die Ladungstra¨ger- und Energieumverteilung aufgrund der
abgeschirmten Coulomb-Wechselwirkung und der Streuung mit Phononen werden im
Detail behandelt.
Wir untersuchen die technologisch wichtige Struktur eines kantenemittierenden Breit-
streifenlasers im Rahmen der paraxialen Wellenna¨herung. Die Anregung von ho¨heren
transversalen Moden und das Auftreten von instabilen optischen Filamenten werden
quantitativ analysiert. Wir zeigen, wie transversale Instabilita¨ten von ra¨umlichem Loch-
brennen, Gewinn- und Index-Fu¨hrung und von Selbstfokussierung herru¨hren. Wir un-
tersuchen die Abha¨ngigkeit der Emissionsdynamik von Kenndaten des Gewinnmaterials
(z. B. dem Amplituden-Phasen-Kopplungsfaktor), der Streifenbreite, Pumpe und La-
dungstra¨gerdiffusion. Abha¨ngig von der Breite der Laserstruktur ko¨nnen unterschiedliche
dynamische Emissionsregimes festgestellt werden. Ferner projizieren wir die raumzeitli-
che Dynamik auf die Lasermoden.
Wir untersuchen VCSEL mit periodisch strukturiertem Defekt als Beispiel fu¨r einen
photonischen Bandkanten-Laser. Insbesondere erforschen wir die Ausnutzung von photo-
nischen Kristallstrukturen: die Gewinnu¨berho¨hung fu¨r Bandkanten-Moden aufgrund der
wirksameren Wechselwirkung von Photonen mit dem Gewinnmedium und der sta¨rkeren
Lokalisierung u¨ber den aktiven Schichten, und die Reduzierung von optischen Verlusten.
Wir besta¨tigen numerisch, daß photonische Kristalleffekte fu¨r endliche Kristallstruktu-
ren erzielt werden ko¨nnen, und zeigen, daß diese zu einer wesentlichen Optimierung der
178 ZUSAMMENFASSUNG
Laserleistung, beispielsweise reduzierten Laserschwellen, fu¨hren.
Optisch gepumpte VECSEL sind ein Konzept entwickelt zur Steigerung der Aus-
gangsleistung von oberfla¨chenemittierenden Lasern in Verbindung mit nahezu beugungs-
begrenzter Strahlqualita¨t. Wir untersuchen das komplexe Wechselspiel in VECSEL-
Strukturen zwischen resonatorinternen optischen Feldern und dem Quantenfilm-Gewinn-
material. Unsere Simulationen machen das dynamische Gleichgewicht zwischen Ladungs-
tra¨gererzeugung in Zusta¨nden hoch im Band (durch Pumpen), Relaxation von Ladungs-
tra¨gern und stimulierter Rekombination an Zusta¨nden nahe der Bandkante deutlich. Wir
zeigen, daß das longitudinale Multimode-Verhalten zusammengesetzt ist aus mehreren
externen Resonatormoden.
Außerdem betrachten wir die Wechselwirkung von Femtosekunden- und Pikosekunden-
Pulsen hoher Intensita¨t mit Halbleiterstrukturen. Wir bestimmen den mikroskopischen
Ursprung der schnellen Nichtlinearita¨ten und untersuchen die physikalischen Effekte
hinter den verschiedenen Sa¨ttigungsmechanismen. Außerdem gewinnen wir die nichtli-
nearen Gewinnkoeffizienten und Erholungsraten. Es wird gezeigt, daß die Dispersion der
Gruppengeschwindigkeit, dynamische Gewinnsa¨ttigung und schnelle Selbstphasenmodu-
lation die Hauptursachen fu¨r A¨nderungen und Asymmetrien in Form und Spektrum der
versta¨rkten Pulse sind. Wir zeigen, daß die Zeitkonstanten der Intraband-Streuprozesse
kritisch fu¨r die Gewinnerholung sind. Unsere Ergebnisse sind grundlegend fu¨r die Inter-
pretation und das quantitative Versta¨ndnis von nichtlinearer Pulsformung in optischen
Halbleiterversta¨rkern und -absorbern.
Die exakte und spektral breite Modellierung von Halbleiter-Gewinn und komplex
strukturierten Laserkavita¨ten, die in dieser Arbeit dargestellt wird, erweitert die wis-
senschaftliche Diskussion von Halbleiterlasersystemen. Unser mikroskopischer Ansatz in
der Zeitdoma¨ne ist, aufbauend auf effizienten numerischen Algorithmen und der zuneh-
menden Verfu¨gbarkeit kostengu¨nstiger High Performance Computing-Ressourcen, gut
geeignet fu¨r die Entwicklung und Designoptimierung von modernen nano-strukturierten
Hochleistungsdiodenlasern.
C.2 Einleitung und Motivation
Lichtversta¨rkung durch stimulierte Emission von Strahlung (d. h. LASER-Betrieb [1])
wurde in Halbleiter-Gewinnmaterialien zum ersten Mal im Jahr 1962 demonstriert. Ei-
nem theoretischen Vorschlag von Basov folgend haben drei Labors unabha¨ngig voneinan-
der Laserta¨tigkeit in Verbindungshalbleitern mit direkter Bandlu¨cke und in gepulstem
Betrieb bei tiefen Temperaturen berichtet [2–4]. Die stimulierte Emission von Strah-
lung wurde in diesen Halbleiterdiodenlasern durch die strahlende Rekombination von
Elektron-Loch-Paaren, injiziert in einen pn-U¨bergang, umgesetzt. Die Besetzungsinver-
sion, die fu¨r den Laserbetrieb no¨tig ist, wurde in der Verarmungszone einer GaAs(P)-
Homojunction erzeugt. Die Einfu¨hrung von Heterostrukturen in 1970 war ein bahnbre-
chender Beitrag zur Entwicklung von effizienten optoelektronischen Bauteilen und ein
Durchbruch in Richtung auf industrielle Anwendungen. Heterostrukturen sind aufgebaut
aus mehreren Schichten von unterschiedlichen Verbindungshalbleitern. Insbesondere ha-
C.2 Einleitung und Motivation 179
ben Doppelheterostrukturen bestehend aus einer intrinsischen, aktiven GaAs-Schicht,
einem du¨nnen Film fu¨r die Lichterzeugung und -versta¨rkung, umgeben von dotierten
AlGaAs-Mantelschichten einen Dauerstrichbetrieb bei Raumtemperatur ermo¨glicht [3].
Die verbesserten Leistungskennzahlen (z. B. hohe Effizienz, niedrige Umwandlung von
Energie in Wa¨rme) und die reduzierte Laserschwelle in solchen Strukturen wurde durch
erho¨hten Einschluß der Ladungstra¨ger und optischen Feldmode erreicht [2–4].
Hochentwickelte Kristallwachstumsverfahren (z. B. Molekularstrahlepitaxie, metall-
organisch chemische Gasphasenabscheidung, Flu¨ssigphasenepitaxie, selbstorganisierte
Wachstumsmethoden) [5] und Prozess- und A¨tztechnologien ermo¨glichen die Herstel-
lung von komplexen Halbleiter-Nanostrukturen (wie zum Beispiel GRIN-Strukturen,
U¨bergitter, Quantenfilme und Quantenpunkte oder photonische Kristalle). Diese Tech-
nologien erlauben eine bessere Kontrolle u¨ber die elektronischen Eigenschaften des Ge-
winnmaterials durch die Anwendung der Konzepte der Size-Quantisierung und reduzier-
ten Dimensionalita¨t, und die gezielte Manipulation der optischen Modenstruktur und
Zustandsdichte in funktionalen photonischen Materialien. Gleichzeitig haben sich die
Theorie und Simulation der Laserdynamik zu einem Thema von mehr als nur akademi-
schem Interesse entwickelt. Wegen der Vorhersagekraft stellt die Computermodellierung
ein unscha¨tzbares Werkzeug zur Entwicklung neuartiger Laserstrukturen und zur De-
signoptimierung dar. Durch die Forderung nach ho¨heren Ausgangsleistungen, anderen
Betriebswellenla¨ngen und Hochfrequenzmodulation wurde die Entwicklung neuartiger
Laser- und Versta¨rkerstrukturen und die Einfu¨hrung von neuen Gewinnmaterialien und
Laserkonzepten vorangetrieben. Dies ist begleitet durch die fortlaufende Miniaturisie-
rung von koha¨renten Lichtquellen und Versta¨rkern (mit dem Ziel einer On-Chip Reali-
sierung). Intensive Forschungsaktivita¨ten auf dem Gebiet der Halbleiterlaser haben den
U¨bergang von einem Gegenstand im Labor zu einem Massenprodukt mit Anwendun-
gen in unserem ta¨glichen Leben angeregt. Halbleiterdiodenlaser weisen einige Vorteile
gegenu¨ber anderen koha¨renten Lichtquellen auf: kompakte Gro¨ße, hohe Umwandlungs-
effizienz von Pumpenergie in koha¨rentes Licht, Abstimmbarkeit und Manipulation der
optoelektronischen Eigenschaften, direkte Signalmodulation bis zu einigen zehn GHz
und kostengu¨nstige Produktion. Dies hat einen riesigen Markt fu¨r Diodenlaser generiert,
mit einem gescha¨tzten weltweiten Marktvolumen fu¨r Halbleiterlaser von 3.2 Milliarden
US-$ in 2005. Die wichtigsten industriellen Anwendungen von Diodenlasern umfassen
optische Datenspeicherung, Glasfaserkommunikation, Hochleistungs- (so wie zum Bei-
spiel das Pumpen von Festko¨rperlasern) und medizinische Anwendungen. Neben der
technologischen Bedeutung von Diodenlasern, machen die Komplexita¨t der physikali-
schen Wechselwirkungen, die nichtlineare Kopplung der verschiedenen Teilsysteme und
eine Vielzahl an Zeit- und La¨ngenskalen Halbleiterlaser zu idealen Laboren zur Untersu-
chung von optisch nichtlinearen [6, 7] und quantenoptischen Pha¨nomenen [8], sowie von
ultraschnellen Prozessen in Halbleitern, und von Konzepten der nichtlinearen Dynamik
und Synergetik [9, 10].
Neue und hochinteressante Entwicklungen zielen auf die Kontrolle und Manipulation
der Wechselwirkung von Licht und Materie, Lichtemission und -propagation und auf die
gezielte Beeinflussung der optoelektronischen Eigenschaften von Halbleiter-Gewinnme-
180 ZUSAMMENFASSUNG
dien. Im Folgenden geben wir einen U¨berblick u¨ber neueste Forschungsbemu¨hungen, die
im Zusammenhang mit dieser Arbeit stehen.
Laser- und Versta¨rkerstrukturen: Vertikal emittierende Laserdioden (VCSEL) [11,12]
sind aufgebaut aus hochreflektiven Bragg-Spiegeln mit eingeschlossener Defektkavita¨t
und einem zusa¨tzlichen transversalen Einschluß (durch Oxid-Apertur). Insbesondere zei-
gen VCSEL eine starke Kopplung zwischen Gewinnmaterial und dem Lichtfeld. Mikro-
kavita¨ten [13,14] versprechen hohe Qualita¨tsfaktoren, kleine Modenvolumen und niedri-
ge Laserschwellen. Sie sind Schlu¨sselwerkzeuge zur Untersuchung und Maßschneiderung
von Lichtquellen und erlauben die Modifizierung der Wechselwirkung von Licht und Ma-
terie, z. B. der spontanen Emission, da die lokale optische Zustandsdichte durch die di-
elektrische Struktur bestimmt ist. Optische Halbleiterversta¨rker (SOA) sind attraktiv als
Zwischenversta¨rker und funktionelle Einheiten in optischen Netzwerken. Resonante peri-
odische Gewinnstrukturen (PGS) [11,15] bieten eine U¨berho¨hung des effektiven modalen
Gewinns. Oberfla¨chenemitter mit vertikaler externer Kavita¨t (VECSEL) [16,17] sind ein
Designschema zur Erho¨hung der Laserleistung in Strahlen von hoher Qualita¨t und geben
leichten Zugang zu nichtlinear-optischen Effekten (aufgrund der hohen Innenintensita¨-
ten), wie zum Beispiel dynamische Gewinnsa¨ttigung und Pulsformung oder Frequenz-
umwandlung. Photonische Kristalle (PC) und Materialien mit photonischer Bandlu¨cke
ermo¨glichen die Lichtausbreitung mit sehr kleinen Gruppengeschwindigkeiten [18, 19].
Sie ko¨nnen als frequenzselektive Spiegel oder optische Mikrokavita¨ten fungieren und bie-
ten einen Fu¨hrungsmechanismus basierend auf Bragg-Vielfachinterferenz und nicht auf
Totalreflexion [20]. Diese Fu¨hrung im Niederindex-Kern (Luft) wird in photonischen
Kristallfasern angewandt.
Halbleiter-Gewinnmaterialien: Neue bina¨re, terna¨re und quaterna¨re Verbindungshalb-
leitermaterialien ko¨nnen entworfen werden fu¨r eine Vielfalt von optischen Emissionswel-
lenla¨ngen von Infrarot bis Ultraviolett. Neueste Forschung konzentriert sich auf II-VI
Verbindungen, Gruppe III-Nitride mit großer Bandlu¨cke [21–23] und verdu¨nnte Nitride,
z. B. GaInNAs [24], fu¨r hochentwickelte Anwendungen in der Telekommunikation. Akti-
ve Nanostrukturen mit Quanten-Confinement [25] und reduzierter Dimensionalita¨t, wie
zum Beispiel Quantenfilme [26], Nanodra¨hte und Quanten-Dashes [27, 28], und Quan-
tenpunkte [29], vera¨ndern grundlegend die elektronische Bandstruktur und optoelek-
tronischen Eigenschaften durch Anwendung des Konzepts der Size-Quantisierung, was
zu effizienterer Ladungstra¨gerinversion fu¨hrt. Die Anwendung von Zug-/Druckspannung
ermo¨glicht eine weitere A¨nderung von Bandstruktur und Gewinn. Außerdem stellen or-
ganische Halbleiter [30] mo¨gliche neue Gewinnmaterialien fu¨r Festko¨rperlaser im gru¨nen
Spektralbereich fu¨r Bildschirm- und Beleuchtungstechnologien dar. Die einfache Herstel-
lung und Formbarkeit dieser Filme aus konjugierten Polymeren sind attraktiv [31]. Die
meisten Laser dieser Art verwenden Strukturen mit Korrugation, welche als Resonator
basierend auf dem Prinzip der verteilten Ru¨ckkopplung wirken.
Laser- und Gewinn-Konzepte: Optische Pumpschemata stellen eine Alternative zu
elektrischem Pumpen dar. Sie bieten die Mo¨glichkeit, die ra¨umliche Verteilung der Be-
setzungsinversion u¨ber große Bereiche festzulegen. Die Erzeugung ultrakurzer optischer
Pulse durch passive Modenkopplung mit Hilfe von sa¨ttigbaren Absorbern [32,33] baut auf
eine gezielte Ausnutzung nichtlinearer Effekte und das komplexe dynamische Wechsel-
C.2 Einleitung und Motivation 181
spiel zwischen Gewinn und Lichtfeld. Quantenkaskaden-Laser beruhen auf Intersubband-
U¨berga¨ngen (d. h. Intraband-Polarisationen). Diese Intersubband-Natur (nur eine einzel-
ne Ladungstra¨gerart beteiligt) fu¨hrt zu zahlreichen Vorteilen verglichen mit Diodenlasern
basierend auf stimulierter Elektron-Loch-Rekombination [34]. Zum Beispiel Abstimm-
barkeit aufgrund der Size-Quantisierung in diesen Vielfach-Quantenfilm Heterostruktu-
ren (d. h. keine Beschra¨nkung durch die Energiebandlu¨cke gegeben) und ho¨here Effizienz.
Das Erzielen von optischem Gewinn fu¨r Rekombinationsprozesse in Halbleitermaterialien
mit indirekter Bandlu¨cke ist schwierig, was eine Untersuchung nach anderen physikali-
schen Wechselwirkungen und Gewinnmechanismen motiviert. Ku¨rzlich wurde Lichtver-
sta¨rkung und Laserta¨tigkeit in Silizium demonstriert [35], wo die Versta¨rkung durch
stimulierte Raman-Streuung erzielt wurde.
Die numerische Simulation von Halbleiterlasern spielt eine wichtige Rolle in der Ent-
wicklung von neuartigen Strukturen und zur Designoptimierung. Des Weiteren sind sie
wertvoll in der Untersuchung zugrunde liegender physikalischer Grenzen und der opti-
schen und elektronischen Eigenschaften der verschiedenen Teilsysteme. Aufgrund der
Komplexita¨t des Problems und der nichtlinearen Kopplung der Teilsysteme (Optik,
Ladungstra¨ger, Phononen) ist eine analytische Behandlung schwierig. Mit nur weni-
gen Beschra¨nkungen bezu¨glich Geometrie und physikalischen Wechselwirkungen bieten
Methoden im Zeitbereich ein flexibles und ausbaufa¨higes Werkzeug, um die Lo¨sung
der gekoppelten Dynamik von optischem Feld innerhalb der Kavita¨t und dem akti-
ven Gewinnmaterial anzugehen. Die einzige Einschra¨nkung in der Umsetzung solcher
Methoden ist durch die beno¨tigten Computerressourcen gegeben. Der Ausgangspunkt
fu¨r Halbleiterlaser-Modelle sind die Lasergleichungen, abgeleitet von Haken und Lamb,
Jr. [36]: Der Basissatz dynamischer Variablen ist gegeben durch das optische Feld, die
induzierte Polarisation im aktiven Material und die Ladungstra¨gerinversion. Einen U¨ber-
blick ha¨ufig verwendeter Modelle in der Zeitdoma¨ne zur Simulation von Halbleiterlasern
im Rahmen der semiklassischen Na¨herung (d. h. eine Verknu¨pfung klassischer elektroma-
gnetischer Felder mit den quantenelektronischen Eigenschaften des aktiven Materials)
wird in [3, 37] pra¨sentiert. Wir fassen die verschiedenen Zuga¨nge zur Modellierung der
Halbleiter-Gewinndynamik und der Dynamik der optischen Felder in Abbildung 1.1
bzw. Abbildung 1.2 zusammen. Dabei definieren wir auch die Anwendungsmo¨glichkei-
ten und Vorteile unserer neuentwickelten Modelle gegenu¨ber anderen Ansa¨tzen.
Im Unterschied zu anderen Laserstrukturen sind Halbleiterdiodenlaser durch einige
besondere Eigenschaften gekennzeichnet, welche in realistischen Computermodellen be-
ru¨cksichtigt werden mu¨ssen [2, 4, 37]: Aufgrund der hohen Dichten sind Vielteilchen-
Wechselwirkungen von großer Bedeutung, vor allem in Gewinnstrukturen mit Quanten-
Confinement, und es tritt eine schnelle Dephasierung der induzierten Polarisation ein.
Halbleiter-Gewinnmaterialien sind gekennzeichnet durch ein breites Gewinnspektrum
und eine starke Kopplung von Amplituden- und Phasendynamik (quantitativ beschrie-
ben durch den α Faktor). Diodenlaser schließen eine Vielzahl relevanter Zeitskalen, von
wenigen Femtosekunden (fu¨r die Coulomb Intraband-Streuung) bis einigen Nanosekun-
den (fu¨r makroskopische Transportprozesse wie Ladungstra¨gerdiffusion), ein. Außerdem
sind starkes ra¨umliches und spektrales Lochbrennen, Nichtlinearita¨ten und Sa¨ttigungs-
182 ZUSAMMENFASSUNG
effekte von Bedeutung. Moderne Diodenlaser sind aus komplex strukturierten Kavita¨ten
aufgebaut. Die Lasermode kann stark von den Moden der kalten Kavita¨t abweichen,
da Laserbetrieb und Ladungstra¨gerdynamik (z. B. Lochbrennen und thermische Effek-
te) die Brechungsindexstruktur vera¨ndern. Die Ausgangsleistung und Kenndaten von
Diodenlasern sind durch die nichtlinearen Eigenschaften des Gewinnmaterials limitiert.
Wichtig dabei sind die Ladungstra¨gerdynamik in den Quantenfilmen (z. B. spektrales
Lochbrennen und Streuprozesse), ra¨umliche Effekte (z. B. ra¨umliches Lochbrennen und
Selbstfokussierung) und thermisch-induzierte A¨nderungen des Gewinnmediums. Zusam-
menfassend gesagt, die Theorie von Halbleiterlasern muß kontinuierlich u¨berarbeitet und
angepaßt werden an die in den vorherigen Abschnitten beschriebenen neuartigen Struk-
turen, Konzepte und Gewinnmaterialien.
Die Hauptzielsetzung dieser Arbeit ist die Entwicklung eines theoretischen Modells
der raumzeitlichen Dynamik von neuartigen Hochleistungshalbleiterlasern. Wir stre-
ben außerdem ein quantitatives Versta¨ndnis des nichtlinearen Wechselspiels zwischen
Halbleiter-Gewinndynamik und der resonatorinternen Lichtfelddynamik an. Hierzu ver-
bessern und erweitern wir vorhandene theoretische Beschreibungsansa¨tze der Halbleiter-
laserdynamik, um den Anforderungen auferlegt durch die obig diskutierten Forschungs-
aktivita¨ten zu genu¨gen. Besondere Beru¨cksichtigung finden Gewinnstrukturen mit Quan-
ten-Confinement, komplexe dielektrische Mikrokavita¨ten und photonische Strukturen
(strukturiert auf einer Subwellenla¨ngen-Skala). Zur realistischen Computermodellierung
von Diodenlasern kombiniert unser Ansatz im Rahmen der semiklassischen Na¨herung die
ra¨umlich aufgelo¨sten Maxwell Gleichungen, oder die paraxiale Wellengleichung, mit ei-
ner energieaufgelo¨sten Beschreibung der Materialantwort, gegeben durch die Halbleiter-
Bloch Gleichungen. Zwei Modelle wurden erarbeitet: 1) ein transversales Modell fu¨r
die Untersuchung der transversalen Multimode-Dynamik in Breitstreifenlasern, und 2)
ein longitudinales Modell zur Beschreibung der Propagation ultrakurzer Pulse in opti-
schen Halbleiterversta¨rkern. Wir verwenden das letztere Modell auch zur Analyse kom-
plex strukturierter aktiver Laser, z. B. Oberfla¨chenemitter mit vertikaler Kavita¨t und
periodisch moduliertem Defekt, und optisch gepumpte oberfla¨chenemittierende Laser
mit einem externen Resonator. Zur gleichzeitigen Lo¨sung der Modellgleichungen fu¨r die
Lichtfelddynamik (partielle Differentialgleichungen) und der Polarisations- und Ladungs-
tra¨gerdynamik in aktiven Halbleiterstrukturen wandeln wir die Gleichungen in finite
Differenzengleichungen um, welche auf homogenen Gittern integriert werden.
C.3 U¨berblick
Diese Arbeit zielt auf ein verbessertes quantitatives Versta¨ndnis von Lasersystemen mit
technologischer oder grundlegender Bedeutung. Wir streben einen Vergleich mit Experi-
menten an, d. h. wir mu¨ssen Material-, Struktur- und Kontrollparameter realistisch ab-
bilden. Auf Basis eines mikroskopischen Gewinnmodells einschließlich von Vielteilchen-
Wechselwirkungen konzentrieren wir uns auf die Untersuchung von Pha¨nomenen auf der
Femtosekunden- und Pikosekunden-Skala und analysieren die ra¨umliche Musterbildung
C.3 U¨berblick 183
im optischen Nahfeld. Wir untersuchen den mikroskopischen Ursprung von optischen
Nichtlinearita¨ten und den Einfluß der Ladungstra¨gerdynamik auf die Ausgangskennzah-
len von Diodenlasern. Wir konzentrieren uns auf Hochleistungsstrukturen wie Breitstrei-
fenlaser und optische Halbleiterversta¨rker. Wir besta¨tigen numerisch neuartige Konzep-
te zur Erho¨hung der Ausgangsleistung von oberfla¨chenemittierenden Lasern, z. B. durch
die gezielte Manipulation der optischen Modenstruktur. Genauer gesagt, wir untersu-
chen photonische Bandkanten-Bandlu¨cken-Laser und optisch gepumpte VECSEL. Diese
Arbeit ist folgendermaßen aufgebaut:
In Kapitel 2 leiten wir die Multisubband-Halbleiter-Bloch Gleichungen ab, welche
eine quantenmechanische Beschreibung der ultraschnellen Gewinndynamik in Quanten-
filmen basierend auf dem Dichtematrix-Formalismus darstellen. Unser Ansatz, formuliert
im Impulsraum, umfaßt die wichtigsten Wechselwirkungen in optisch aktiven Halbleiter-
Gewinnmaterialien und eine Vielzahl an Zeitskalen. Die Wechselwirkung von Licht mit
Halbleiter-Gewinnmedien ist im Rahmen der semiklassischen Na¨herung unter Anwen-
dung der elektrischen Dipolna¨herung modelliert. Vielteilchen-Wechselwirkungen, na¨m-
lich die abgeschirmte Coulomb-Wechselwirkung und die Streuung mit Phononen, ha-
ben die Renormierung von U¨bergangsenergien, das Coulomb-Enhancement der Rabi-
Frequenzen und Relaxations- und Dephasierungsprozesse zur Folge. Wir geben voll mi-
kroskopische Vielteilchen-Berechnungen an. Wir betrachten Gleichungen fu¨r die Dyna-
mik der Wigner-Funktionen von diagonalen und nichtdiagonalen Elementen der Einteil-
chen-Dichtematrix:
∂tpji,k = −
(
iΩij,k + γ
p
ij,k
)
pji,k − i
~
Uij,k
(
neii,k + n
h
jj,−k − 1
)
, (C.1)
∂tn
e
ii,k =
i
~
∑
j
(
Uij,kp
∗
ji,k − U∗ij,kpji,k
)− γnrneii,k −∑
j
γspij,kn
e
ii,kn
h
jj,−k − γe,Augerii,k
+ Λ
f eii,k(Tlat)
(
1− neii,k(T epl)
)
∑
i 1/A
∑
k f
e
ii,k
(
1− neii,k
) − γe(ph)ii,k (neii,k(T epl)− f eii,k(Tlat))
− γee(cc)ii,k
(
neii,k(T
e
pl)− f eii,k(T epl)
)− γeh(cc)ii,k (neii,k(T epl)− f eii,k(T hpl)) . (C.2)
Eine Formulierung vollsta¨ndig in der Zeitdoma¨ne wird ebenfalls abgeleitet (Halbleiter-
Bloch Gleichungen in Full Time-Domain). Unser Ansatz setzt die u¨bliche Rotating Wave
Na¨herung (RWA) nicht voraus, und stellt somit eine genaue und spektral breite Model-
lierung des Gewinnmediums dar. Das Modell ermo¨glicht die Simulation ultraschneller,
nichtlinearer Pulswechselwirkungen in Halbleiterlasern und -versta¨rkern. Die Antwort
des Versta¨rkermediums auf optische Anregungen wird quantitativ beschrieben und be-
stimmt durch die induzierte makroskopische Polarisation P(r, t) (ein nichtlineares Funk-
tional), die Ladungstra¨gerdichte und die Materialdispersion nbackground(r;ω). Die relative
Permittivita¨t kennzeichnet den dielektrischen Resonator. Außerdem schlagen wir das Fit-
ting und die Parametrisierung der komplexen dielektrischen Suszeptibilita¨t durch wenige
Oszillatoren als effektiven Weg einer Nachbildung der optoelektronischen Eigenschaften
eines komplexen Halbleiter-Gewinnmediums vor.
Kapitel 3 betrachtet die Beschreibung der Dynamik von Lichtfeldern in Laserka-
vita¨ten. Die Propagation, Beugung, Reflexion und Fu¨hrung von elektromagnetischen
184 ZUSAMMENFASSUNG
3.1
3.2
3.3
3.4
3.5
3.6
3.7
re
fra
ct
iv
e 
in
de
x
4 4.2 4.4 4.6 4.8 5 5.2 5.4
position [µm]
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
el
ec
tri
c 
fie
ld
 [1
06
V
m
-
1 ]
0.11 0.12 0.13 0.14 0.15 0.16
time [ps]
-4
-2
0
2
4
el
ec
tri
c 
fie
ld
 [1
08
V
m
-
1 ]
1
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ] 
Abbildung C.1: Beispielsysteme zu Modell (C.3)–(C.4): Oberfla¨chenemittierende La-
serstruktur; Lasermode in einem VCSEL; Propagation von Femtosekunden-Pulsen.
Feldern in Brechungsindex-Strukturen und optischen Kavita¨ten von Laserresonatoren
wird modelliert u¨ber die makroskopischen Maxwell (Rotations-) Gleichungen oder eine
davon abgeleitete (gena¨herte) Wellengleichung. Die spektral aufgelo¨ste induzierte ma-
kroskopische Polarisation, welche wir in Kapitel 2 berechnen, wirkt als Quellterm des
optischen Laserfelds. Wir entwickeln zwei unterschiedliche Modelle zur Beschreibung des
passiven Problems: 1) Die vollen Maxwell Gleichungen zielen auf Problemstellungen ab,
welche ein breites Spektrum an relevanten Frequenzen oder ra¨umlich stark lokalisier-
te Moden mit sich bringen (siehe Abbildung C.1). Der Ansatz ist jedoch beschra¨nkt
auf wenige Mikrometer große aktive Strukturen. Wir untersuchen Mikrokavita¨ten-Laser
und die nichtlineare Pulswechselwirkung in Halbleiterversta¨rkern. Wir verknu¨pfen die
Maxwell Rotationsgleichungen mit einer band-aufgelo¨sten Beschreibung der Dynamik
des Halbleiter-Gewinnmaterials im Rahmen der Methode der Finiten Differenzen im
Zeitbereich (FDTD):
curlE+ ∂tB = 0, curlH− ∂tD = 0, (C.3)
∂zEx + ∂tBy = 0,
1
µ0
∂zBy + ∂tDx = 0, ∂zEy − ∂tBx = 0, 1
µ0
∂zBx − ∂tDy = 0,
D(z, t) = ǫ0ǫ(z)E(z, t) + Γx,y(z)
NQW(z)
Lref
P(z, t). (C.4)
Dieser numerisch herausfordernde, generische Zugang wurde in dieser Arbeit zum ersten
Mal umgesetzt. 2) Abha¨ngig von den Eigenschaften des Resonators werden zur Ver-
einfachung der Modellgleichungen die paraxiale, Slowly Varying Amplitude und RWA
Na¨herungen angewandt:(
n2
cneff
∂t ± ∂z − i 1
2k0neff
∂2x − ik0
n
neff
δnpas(x)
)
E˜±(x, z, t) = i
k0Γy
ǫ02neff
NQW
Lref
P˜±(x, z, t).
(C.5)
Das skalare, transversale Wellengleichungsmodell wurde entwickelt zur Untersuchung
von großfla¨chigen kantenemittierenden Halbleiterlasern und Versta¨rkerstrukturen hoher
Leistung und zur Analyse transversaler Multimode-Dynamik (siehe Abbildung C.2).
In Kapitel 4 berechnen wir mit Hilfe ra¨umlich nicht-aufgelo¨ster Modelle charak-
teristische Gro¨ßen von Laser-Gewinnmedien. Der Resonator wird dabei u¨ber effektive
C.3 U¨berblick 185
4800 4900 5000 5100 5200 5300 5400
-8
-4
0
4
8
x
 [µ
m
]
0 1000 2000 3000 4000 5000
time [ps]
1.6
1.7
1.8
1.9
N
 [1
01
2 c
m
-
2 ]
0
6
12
18
E2
 
[1
01
2 V
2 m
-
2 ]
I
-50 -40 -30 -20 -10 0 10 20 30 40 50
transverse position [µm]
0
0.5
1
1.5
2
ca
rr
ie
r d
en
sit
y 
[1
01
2 c
m
-
2 ]
0
3
6
9
12
15
(el
ec
tri
c f
iel
d a
mp
litu
de
)2  
[1
01
2 V
2 m
-
2 ]
Abbildung C.2: Beispielsysteme zu (C.5): Breitstreifenlaser; transversale Multimode-
Dynamik in Kantenemittern.
Parameter beru¨cksichtigt. Mikroskopische Streu- und Dephasierungsraten aufgrund der
Coulomb- und Fro¨hlich-Wechselwirkung werden in GaAs-Quantenfilmen fu¨r verschiedene
Ladungstra¨gerdichten und Temperaturen berechnet. Die Berechnungen erfolgen u¨ber die
Quantum-Boltzmann Gleichung. Wichtig ist, daß individuelle Streuprozesse analysiert
werden. Wir verwenden die mikroskopischen Relaxationsraten in unseren Simulationen
der raumzeitlichen Laserdynamik (als Lookup Tabellen) und zeigen, daß dies das Mo-
dell quantitativ verbessert. Wir zeigen, daß die Relaxationsraten entscheidend sind fu¨r
die Modellierung von Gewinn-Parametern, nichtlinearer Pulspropagation und das dyna-
mische Sa¨ttigungsverhalten von Versta¨rker- und Absorberelementen. Eine dynamische
Behandlung der Streuprozesse ist in Anhang B dargestellt. Außerdem bestimmen wir
Gro¨ßen, die die Wechselwirkung von Licht und Materie charakterisieren. Die wichti-
gen optischen Gewinngro¨ßen von Halbleiter-Versta¨rkermedien (Kleinsignalversta¨rkung,
differentieller Gewinn), die Spektren der spontanen Emission und strahlende Rekombi-
nationsraten werden mit einem mikroskopische Ansatz, in Screened Hartree-Fock Na¨he-
rung, berechnet. Grundkonzepte von Halbleiterlasern, wie zum Beispiel Density Pinning
und Gain Clamping, Gewinnsa¨ttigung und thermisches U¨berrollen werden im Rahmen
unserer Modelle diskutiert. Die transiente dynamische Antwort, wenn ein Laser einge-
schaltet oder wa¨hrend des Betriebs gesto¨rt wird, d. h. Relaxationsoszillationen, werden
untersucht. Wir gewinnen realistische Eingangsparameter fu¨r die Analyse raumzeitlicher
Laserdynamik, was den Ansatz quantitativ verbessert. Die gute U¨bereinstimmung der
numerischen Ergebnisse mit experimentellen Resultaten, wie auch mit den Vorhersagen
einfacherer theoretischer Modelle, besta¨tigt unseren Ansatz.
In Kapitel 5 untersuchen wir Breitstreifenlaser mit Hilfe des paraxialen optischen
Modells, wie in Kapitel 3 vorgestellt. Dieser Kantenemitter ist charakterisiert durch
einen einfachen Fabry-Pe´rot Resonator und einen dielektrischen Wellenleiter. Wir zeigen,
daß mit zunehmender Breite der aktiven Zone oder Pumpleistung mehr transversale Mo-
186 ZUSAMMENFASSUNG
den angeregt werden, ra¨umliche wie zeitliche Fluktuationen im Nahfeld auftreten und
instabile optische Filamente geformt werden. Wir weisen die Abha¨ngigkeit des trans-
versalen Multimode- und Fluktuationsverhalten von der Ladungstra¨ger- und Gewinn-
Dynamik und von der selektiven Verarmung der Ladungstra¨ger aufgrund verschiedener
transversaler Lasermoden nach. Wir zeigen, daß die Koexistenz mehrerer Moden un-
terstu¨tzt wird durch die Wechselwirkung der verschiedenen Moden mit getrennten Be-
reichen des Gewinnmaterials, ra¨umlich aufgrund des transversalen Freiheitsgrades und
spektral als Resultat der inhomogenen Verbreiterung. Fu¨r breitere Strukturen unter-
stu¨tzen Gewinn-, Dichte- und Indexprofile neben der Grundmode ho¨here transversa-
le Moden. Der Einfluß von optischen Nichtlinearita¨ten hervorgerufen durch ra¨umliches
Lochbrennen und von den physikalischen Effekten Ladungstra¨gerdiffusion, Gewinn- und
Index-Fu¨hrung, Selbstfokussierung und Beugung werden quantitativ analysiert. Wir ge-
ben eine lineare Stabilita¨tsanalyse der transversalen Instabilita¨t an. Dies ermo¨glicht es,
die Hauptkontrollparameter zu identifizieren: die Streifenbreite, Pumpe und der Linien-
verbreiterungsfaktor. Wir zeigen numerisch, daß abha¨ngig von der Breite der Struktur
unterschiedliche Maße raumzeitlicher Komplexita¨t und dynamische Emissionsregimes
festgestellt werden ko¨nnen. Wir bestimmen zeitgemittelte Profile, verwenden eine Mo-
denanalyse der komplexen Felder zur numerischen Berechnung optischer Feldamplitu-
den und Laserfrequenzen, und projizieren das ra¨umliche Nahfeld auf die Lasermoden.
Im quasi-periodischen Regime la¨ßt sich das nicht-transiente Verhalten als U¨berlagerung
verschiedener Moden mit unterschiedlichem Profil, Symmetrie und Frequenz deuten.
In Kapitel 6 wenden wir das Full Time-Domain Modell, d. h. die Maxwell Gleichun-
gen gekoppelt mit den Halbleiter-Bloch Gleichungen, zur Untersuchung von longitudi-
naler Multimode-Dynamik in neuartigen Laserstrukturen und der Wechselwirkung von
Femtosekunden- und Pikosekunden-Pulsen mit optisch aktiven Medien an. VCSEL mit
eingeschlossener periodischer Gewinn- und Brechungsindex-Struktur werden betrachtet.
Zur Erho¨hung der Ausgangsleistung von Oberfla¨chenemittern werden die Quantenfilme
periodisch in Resonanz mit der Defektmode angeordnet (optimaler U¨berlapp). Die Be-
stimmung der Lasermoden, Laserfrequenzen und -schwellen beweist numerisch, daß PC-
Effekte fu¨r endliche Kristalle erzielt werden ko¨nnen und zu einer merklichen Verbesserung
der Laserleistung fu¨hren. Wir zeigen, wie in photonischen Bandkanten-Lasern speziel-
le Eigenschaften von Singularita¨ten im optischen Bandstruktur-Diagramm ausgenu¨tzt
werden, z. B. die flache Dispersion an Bandkanten. Unsere Ergebnisse zeigen die Ge-
winnerho¨hung u¨ber eine versta¨rkte Lokalisierung der Moden u¨ber den aktiven Quanten-
strukturen und die effizientere Wechselwirkung von Photonen mit dem Gewinnmaterial
(erho¨hter Gewinn per Unit Length). Das letztere tritt aufgrund der reduzierten Gruppen-
geschwindigkeit auf. Wir untersuchen die Unterdru¨ckung optischer Verluste der inneren
endlichen photonischen Struktur durch die Verwendung einer umgebenden photonischen
Struktur mit Bandlu¨cke. Realistische optisch gepumpte oberfla¨chenemittierende Laser
mit externer Kavita¨t werden untersucht. Das optische Pumpschema und die externe Ka-
vita¨t ermo¨glichen eine Skalierung der Leistung bei guter Strahlqualita¨t in der TEM00
Grundmode. Wir zeigen, daß das longitudinale Multimode-Verhalten in VECSELn aus
mehreren Moden der externen Kavita¨t aufgebaut ist. Eine mikroskopische Analyse deckt
das dynamische Gleichgewicht zwischen Ladungstra¨gererzeugung durch Pumpen ho-
C.3 U¨berblick 187
her Energiezusta¨nde, Relaxation gegen die Fermi-Dirac Verteilung und Laserta¨tigkeit
an Zusta¨nden nahe der Bandkante auf. Außerdem betrachten wir die Propagation von
Femtosekunden- und Pikosekunden-Pulsen in aktiven Halbleiterstrukturen und analysie-
ren die puls-induzierten A¨nderungen des Halbleitermaterials. Wir fu¨hren umfangreiche
numerische Pulspropagationsexperimente durch, in denen Probe-Pulse die Polarisations-
antwort und den Nichtgleichgewichtszustand des Gewinnmaterials abtasten. Wir iden-
tifizieren den mikroskopischen Ursprung der schnellen Nichtlinearita¨ten und diskutieren
die physikalischen Effekte hinter den verschiedenen Sa¨ttigungsmechanismen, z. B. die
Verarmung an vorhandenen resonanten Ladungstra¨gern fu¨r stimulierte Emission. Wir
berechnen auch die nichtlinearen Gewinnkoeffizienten und die verschiedenen Erholungs-
raten (u¨ber Pump-Probe-Experimente). Die Zeitkonstanten der Intraband-Streuprozesse
sind kritische Parameter der dynamischen Gewinnsa¨ttigung und -erholung. Die Disper-
sion der Gruppengeschwindigkeit, dynamische Sa¨ttigung und schnelle Selbstphasenmo-
dulation sind die Hauptursachen fu¨r die beobachteten Asymmetrien in Pulsform und
-spektrum. Diese nichtlineare Pulsformung, z. B. Pulse Slowdown oder Advancement,
A¨nderung von Pulsbreite und Pulsenvelope, Chirp, wird im Detail analysiert. Außerdem
untersuchen wir nichtlineare koha¨rente Pulspropagationspha¨nomene in aktiven Gewinn-
medien, insbesondere das Pulsfla¨chentheorem (die Pulsfla¨che, definiert als das zeitliche
Integral u¨ber die Rabi-Frequenz, strebt gegen bestimmte Attraktoren, Vielfache von π),
Rabi-Oszillationen und selbstinduzierte Transparenz (zeitliches Soliton fu¨r eine Pulsfla¨-
che von 2π). Unsere Simulationen zeigen, daß bis auf kleine Abweichungen durch die
Beru¨cksichtigung der schnell oszillierenden Tra¨gerwelle die numerischen Ergebnisse mit
der Theorie u¨bereinstimmen. In einem mikroskopischen Halbleitermodell mit realisti-
schen Parametern fu¨r Dephasierungsprozesse und die homogene Verbreiterung dagegen
treten unvollsta¨ndige Rabi-Flops auf, der Puls wird geda¨mpft. Diese Ergebnisse zeigen,
daß ultraschnelle koha¨rente nichtlineare Propagationseffekte in Halbleiter-Quantenfilm
Gewinnmaterialien und bei Raumtemperatur wenig stark ausgepra¨gt sind.
Anhang A gibt Einzelheiten zu den numerischen Umsetzungen der theoretischen
Modelle in der Zeitdoma¨ne, wie zum Beispiel die Diskretisierung der Felder und Dif-
ferentialoperatoren auf regula¨ren Gittern und die verwendeten Integrationsschemata.
Außerdem diskutieren wir die numerische Komplexita¨t, Genauigkeit und Stabilita¨t un-
serer Algorithmen. Fu¨r das paraxiale transversale Wellengleichungsmodell schlagen wir
die Hopscotch Methode als effizienten, stabilen und hinreichend genauen Ansatz vor.
Wir wenden einen numerischen Algorithmus an, der die Partitionierung des Gitters in
zwei Gruppen von Gitterpunkten und ein Integrationsschema mit abwechselnder ex-
pliziter und impliziter Diskretisierung einschließt. Der Vorzug dieser Methode ist, daß
kein matrixwertiges Problem gelo¨st werden muß. Wir diskutieren die Finite Differen-
zen im Zeitbereich (FDTD) Methode, welche die Maxwell Rotationsgleichungen erster
Ordnung durch Anordnen der elektrischen und magnetischen Feldkomponenten auf ver-
setzten Gittern in Zeit und Raum gema¨ß dem Yee-Schema lo¨st. Eine voll explizite nu-
merische Implementierung unserer Full Time-Domain Modelle, von aktiver nichtlinearer
Materialantwort und der passiven Brechungsindex-Struktur, wird entwickelt.
188 ZUSAMMENFASSUNG
C.4 Ausblick
Zusammenfassend gesagt, in dieser Arbeit entwickeln und pra¨sentieren wir Modelle in der
Zeitdoma¨ne, welche sich als wertvolle Werkzeuge fu¨r die Untersuchung der gekoppelten
Dynamik von resonatorinternen optischen Feldern und der band-aufgelo¨sten Antwort des
Halbleitermaterials erweisen. Unsere neuartigen Ansa¨tze werden erfolgreich angewandt
zur Untersuchung von eindimensionalen photonischen Bandkanten-Bandlu¨cken-Lasern
und optisch gepumpten VECSELn. Außerdem betrachten wir Hochleistungsstrukturen,
wie zum Beispiel Breitstreifenlaser und optische Halbleiterversta¨rker, in welchen optische
Nichtlinearita¨ten und ultraschnelle Prozesse von großer Bedeutung sind. Das vorgestellte
theoretische Geru¨st kann auf komplexere Gewinnsysteme und Laserstrukturen erweitert
werden. Ihre Umsetzung allerdings ist begrenzt durch die verfu¨gbaren Kapazita¨ten auf
Ho¨chstleistungsrechnern.
Eine mo¨gliche Erweiterung des Modells und Thema fu¨r zuku¨nftige Forschung sind
Mikrokavita¨ten- und PC-Laser, was die Betrachtung der dreidimensionalen vektoriellen
Maxwell Gleichungen erfordern wu¨rde [14,38]. Die Untersuchung von elektrischen Dipol-
u¨berga¨ngen σ-Typs, d. h. zirkulare Polarisation, kann mit Hilfe komplexwertiger elektro-
magnetischer und Polarisationsfelder umgesetzt werden [39]. Die nichtlineare Wechsel-
wirkung von Femtosekunden-Pulsen mit Halbleitermaterialien ko¨nnte analysiert werden.
Hierzu sind eine mikroskopische Behandlung der Ladungstra¨gerstreuung und Energieum-
verteilung [40] und die Beru¨cksichtigung von Gewinn- und Materialdispersion [41] von
großer Bedeutung. Mit der Miniaturisierung von Laserstrukturen (z. B. Mikrokavita¨-
ten) wird die Untersuchung der spontanen Emission ein wichtiges Thema, da strahlende
Lebensdauern u¨ber die dielektrische Struktur und das Design der Kavita¨t kontrolliert
werden ko¨nnen und Schwellenlose Laser mo¨glich sind [20,42–44]. Wir schlagen somit vor,
unser Full Time-Domain Modell zu erweitern durch die Beru¨cksichtigung von Quanten-
rauschen, spontaner Emission und Cavity-Quantenelektrodynamik [45–47].
BIBLIOGRAPHY 189
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Acknowledgements
A number of people have contributed to this work in one way or another. I gratefully
acknowledge the help and advice of all of them.
First of all, I would like to thank Prof. Dr. O. Hess for the opportunity to carry out
this research project in his group, for the support and constant interest in the progress
of the work, and for providing access to an excellent computing infrastructure.
I thank Prof. Dr. G. Mahler for writing the second report on my dissertation.
Many thanks go to Joachim Hamm and Eldad Yahel for proofreading the manuscript
and correcting my English, and to Andreas Klaedtke and Dietmar Preißer for the assis-
tance in the numerical implementation of the various semiconductor laser models.
I also want to thank all my friends and family, and my colleagues in the Theoretical
Quantum Electronics group at the Institute of Technical Physics (German Aerospace
Center, DLR Stuttgart) and in the Theory and Advanced Computation group at the
Advanced Technology Institute (University of Surrey, Guildford, United Kingdom) for
helping and supporting me over the past years.
Guildford (United Kingdom), May 2007 Klaus Bo¨hringer