Development of a Passively Q-switched Microchip Laser Operating at 914 nm for Automotive Lidar Applications A thesis accepted by the Faculty of Aerospace Engineering and Geodesy of the University of Stuttgart in partial fulfilment of the requirements for the degree of Doctor rerum naturalium (Dr. rer. nat.) by Marco Nägele born in Künzelsau Main referee: Prof. Dr. rer. nat. Thomas Dekorsy Co-referee: PD Dr. habil. Christian Kränkel Date of defense: 11. July 2022 Institute of Aerospace Thermodynamics University of Stuttgart Institute of Technical Physics DLR, Stuttgart 2022 C O N T E N T S Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Laser crystals and energy structure . . . . . . . . . . . . . 11 2.1.1 Rare-earth ions . . . . . . . . . . . . . . . . . . . . . 12 2.1.2 Transition metal ions . . . . . . . . . . . . . . . . . 13 2.1.3 Host materials and importance of phonons . . . . 14 2.1.4 Transitions and selection rules . . . . . . . . . . . . 16 2.2 Lasers and optical resonators . . . . . . . . . . . . . . . . . 18 2.2.1 Three-level lasers . . . . . . . . . . . . . . . . . . . . 18 2.2.2 Optical resonators and modes . . . . . . . . . . . . 22 2.2.3 Thermal effects . . . . . . . . . . . . . . . . . . . . . 26 2.3 Passive Q-switching . . . . . . . . . . . . . . . . . . . . . . 30 2.3.1 General operation principle . . . . . . . . . . . . . 30 2.3.2 Features and properties of passively Q-switched laser . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 Saturable absorbers for passive Q-switching . . . . . . . . 38 3. Simulation and system design . . . . . . . . . . . . . . . . . . 45 3.1 Rate-equation model . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Second threshold condition . . . . . . . . . . . . . . . . . . 50 3.3 Matrix formalism - cavity stability . . . . . . . . . . . . . . 55 3.4 Pump-induced heat and temperature investigations . . . . 63 4. Q-switched laser experiments . . . . . . . . . . . . . . . . . . . 73 4.1 State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 Experimental setup I - Insertable saturable absorber . . . . 81 iii 4.2.1 Conversion and slope efficiency . . . . . . . . . . . 84 4.2.2 Mode selection in passively Q-switched lasers . . . 87 4.3 Experimental setup II - Bonded crystals . . . . . . . . . . . 93 4.3.1 Pump power investigations and instabilities . . . . 94 4.3.2 Polarization of laser emission . . . . . . . . . . . . 114 4.3.3 Cavity length dependencies . . . . . . . . . . . . . 117 4.3.4 Output coupler reflectivity . . . . . . . . . . . . . . 126 4.3.5 Long-term stability . . . . . . . . . . . . . . . . . . 132 4.4 Experimental setup III - Discrete crystals . . . . . . . . . . 137 4.4.1 Temperature-dependency of output parameters . . 138 4.4.2 Temperature-dependent wavelength shift . . . . . 146 4.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . 150 5. Passively Q-switched microchip laser for lidar systems . . . 155 5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 155 5.2 Experimental investigations . . . . . . . . . . . . . . . . . . 158 5.2.1 Output characteristics . . . . . . . . . . . . . . . . . 162 5.3 Future lidar system design . . . . . . . . . . . . . . . . . . 169 5.4 Summary and outlook . . . . . . . . . . . . . . . . . . . . . 172 6. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . 175 7. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.1 Setup and devices for characterization . . . . . . . . . . . . 181 7.2 Quasi-continuous pulse train analysis . . . . . . . . . . . . 182 7.3 Fiber-coupled pump laser characterization . . . . . . . . . 185 7.4 Free-space pump laser characterization . . . . . . . . . . . 188 Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 iv A B S T R A C T Since most solid-state lasers have emission wavelengths above one mi- crometer, they cannot be used for contemporary lidar systems in combi- nation with inexpensive and widely established silicon-based detector technology. Therefore, the aim of this work is to investigate and realize a passively Q-switched Nd3+:YVO4 laser at a wavelength of 914 nm for the application in an automotive lidar sensor. To investigate the laser parameters relevant for the lidar application, a total of three experimental resonator configurations is used. Thereby, the configurations are chosen in such a way that the laser parame- ters can be analyzed as decoupled as possible from the overall system. Experimental investigations demonstrate that the quasi-continuously pumped Nd3+:YVO4 laser can achieve pulse durations in the single- digit nanosecond range and pulse energies of almost 40 µJ. Moreover, for a possible lidar application, the repetition rate can be scaled up to about 60 kHz via the applied pump power without significantly affect- ing the other parameters. Theoretical comparison via a rate equation model provides close agreement with the experiments and enables an assessment of the future potential for possible applications of the laser. Across all studies, a very good beam quality was observed, which sug- gests an excellent resolution capability in lidar applications. In addition to the consideration of various system relationships using the experimental configurations, a passively Q-switched monolithic res- onator approach is presented operating in a single-pulse regime at a repetition rate of 200 Hz. The system is pumped by a 808 nm single broad area laser diode, which offers a more compact overall system and a narrower linewidth compared to a fiber-coupled diode laser module. v Consequently, the entire laser system can be operated without active temperature control in a temperature range of 20-50 °C merely by ad- justing the pump laser current. Besides short pulse durations, the short resonator of the monolithic laser crystal allows operation on a single longitudinal mode and consequently a spectral emission bandwidth of a few picometers only. This results in excellent stability of the spectral properties, the pulse energy, and the pulse duration for long-term mea- surements over 60 minutes. vi Z U S A M M E N FA S S U N G Die meisten Festkörperlaser besitzen Emissionswellenlängen oberhalb eines Mikrometers und können deshalb nicht für moderne Lidarsyste- me in Kombination mit günstiger und weit etablierter siliziumbasier- ten Detektortechnologie verwendet werden. Ziel dieser Arbeit ist da- her die Untersuchung und Realisierung eines passiv gütegeschalteten Nd3+:YVO4 Lasers bei einer Wellenlänge von 914 nm für die Anwen- dung in einem automobilen Lidar Sensor. Zur Untersuchung der für die Lidaranwendung relevanten Laserpara- meter werden insgesamt drei experimentelle Resonatorkonfigurationen verwendet. Die Konfigurationen sind dabei so gewählt, dass die La- serparameter möglichst entkoppelt vom Gesamtsystem analysiert wer- den können. Experimentelle Untersuchungen zeigen, dass der quasi- kontinuierlich gepumpte Nd3+:YVO4 Laser Pulsdauern im einstelligen Nanosekundenbereich und Pulsenergien von knapp 40 µJ erreichen kann. Zudem lässt sich für eine mögliche Lidaranwendung die Repeti- tionsrate bis ungefähr 60 kHz über die verwendete Pumpleistung ska- lieren. Der Vergleich mit der Theorie basierend auf Ratengleichungen zeigt eine gute Übereinstimmung zum Experiment, woraus sich das zu- künftige Potential des Lasers für mögliche Anwendungen abschätzen lässt. Über alle Untersuchungen hinweg konnte eine sehr gute Strahl- qualität beobachtet werden, was in der Lidaranwendung ein hervorra- gendes Auflösevermögen verspricht. Neben der Betrachtung verschiedener Systemzusammenhänge mittels experimenteller Konfigurationen wird ein kompakter, monolithischer, passiv gütegeschalteter Demonstratoraufbau im Einzelpulsbetrieb bei einer Wiederholrate von 200 Hz präsentiert. Hierbei kommt als Pum- vii plaser ein 808 nm Breitstreifen-Diodenlaser zum Einsatz, welcher ver- glichen mit einem fasergekoppelten Laserdiodenmodul nicht nur ein deutlich kompakteres Gesamtsystem verspricht, sondern ebenfalls eine schmalere Linienbreite besitzt. Folglich kann das Gesamtsystem allein durch Anpassung des Pumplaserstroms und ohne aktive Temperatur- stabilisierung in einem Temperaturbereich von 20-50 °C stabil betrieben werden. Zudem liefert der kurze Resonator des monolithischen Laser- kristalls nicht nur kurze Pulsdauern, sondern ermöglicht ebenfalls den Betrieb auf einer einzelnen longitudinalen Mode und folglich spektrale Emissionsbandbreiten von wenigen Pikometern. Hierdurch ergibt sich für Langzeitmessungen über 60 Minuten eine hervorragende Stabilität der spektralen Eigenschaften, der Pulsenergie und der Pulsdauer. viii 1I N T R O D U C T I O N Nowadays, light detection and ranging (lidar) systems are indispens- able. They play an important role in metrology, where weather and cli- mate data are collected [1–3], in agriculture, where deforestation [4], ter- rain profiles [5], and the targeted distribution of fertilizers are mapped, but also find application in the measurement of air pollution [6, 7]. Like- wise, lidar sensors can be used in the automotive industry [8] and take an important role in the mobility of the future where these systems are considered as one of the key technologies for automated driving [9]. While the applications of lidar systems are extremely diverse, the un- derlying principle of all sensors is the same. Light is emitted from the sensor via a laser, reflected at objects in the field, and then imaged via a receive path onto the detector to sense the environment. Thereby, the distance determination to the object can be conducted either using an indirect coherent method or a direct time-of-flight (TOF) measurement. With the coherent detection method, the emitted signal is compared with a local oscillator which is tunable in time. Using this technique, both the phase and the intensity of the received signal can be measured. However, the major challenge with this method is that an optically het- erodyne system is required. For this reason, the continuously operated laser must have a coherence length as large as possible, at which the amplitude or frequency can be tuned quickly and, especially, indepen- dent of environmental influences. Regardless of which of the two detection methods is used, the laser, along with the detector and electronics, plays an important role related to the overall performance of the lidar sensor. 1 2 introduction The key requirements of an automotive lidar sensor include a suffi- ciently long detection range, an accurate depth resolution, a large field of view (FOV), a precise angular resolution, and a high measurement speed. In addition, the sensor should be as compact as possible, inex- pensive, and insensitive to external environmental influences and back- ground light. Since in many cases the different requirements influence each other, a certain trade-off has to be taken into account, and conse- quently requirements must be prioritized according to the application. Even with comparatively low pulse energies and poorly reflecting tar- gets, high sensor ranges can be achieved using single-photon detec- tors and evaluation electronics based on photon-timing (time-to-digital converter, TDC) and photon-counting (digital gated counter) [10]. In addition to photomultiplier tubes, modern silicon-based single-photon avalanche diode (SPAD) arrays are suitable as single-photon detectors [11–13]. These detectors are available in large quantities, have a high pixel count up to the megapixel range [14, 15], and can reliably detect even single reflected photons. Furthermore, with a typical dead time of a few nanoseconds, they can be used at repetition rates up to 100 MHz [16]. To suppress background light and maximize the sensing range, a his- togram can be computed from incoming photons. Further, spectrally narrowband laser light sources can be combined with narrow optical bandpass filters. However, the challenge here is to ensure that the laser linewidth is in the sub-nanometer range and, in addition, the central wavelength should shift only slightly with an alternating temperature of the gain medium. Relative to the temperature requirement range of the lidar sensor, this typically results in bandpass filters with a spectral transmission window between 5 nm to 40 nm width [10]. Besides the choice of detector and suppression of background light, the straightforward route to extend the sensor range is to increase the emit- ted pulse energy. On one hand, this requires that the laser is capable introduction 3 of increasing the pulse energy at the required repetition rate and pulse duration. On the other hand, eye safety considerations also have to be ensured for lidar sensors in the automotive sector. According to the international standard, the maximum permissible ex- posure (MPE) is the maximum permissible energy density and depends strongly on the laser wavelength used. Although significantly higher energy densities are possible at wavelengths above 1400 nm compared with 900 nm as defined by the MPE, this is by no means the only crite- rion for the choice of laser [17]. Other important factors are the avail- ability, overall performance, as well as the cost of detectors and lasers. Moreover, external environmental conditions play a non-negligible role in the choice of laser wavelength, as absorption bands of air molecules, fog, rain, and snow can have a significant influence on the range of the lidar sensor [18]. Alternatively, the energy density can be reduced by widening the beam diameter or the utilization of a micropulse lidar approach. Using a mi- cropulse lidar approach, low energy pulses in the microjoule regime are averaged at repetition rates of several kilohertz to improve the signal- to-noise ratio. Due to the low individual pulse energies and sufficient beam expansion, the lidar system remains eye-safe. When this approach was invented in 1993, it was still limited by comparatively insensitive detector technology and the maximum repetition rate of the lasers [19]. However, the aforementioned single-photon detectors and in combina- tion with passively Q-switched lasers with repetition rates in the sub- 100 kHz range and pulse energies in the mid-double-digit µJ range, open up completely new possibilities. With short pulse durations of a few nanoseconds, good longitudinal resolution can be achieved, while high repetition rates provide sufficient signal at the detector and high acquisition rates. For lidar sensors, different possibilities for the illumination of the scene can be considered to detect objects in the field [20]. Besides point scan- 4 introduction ning of the field using micro or galvo mirror systems and the flash approach, where the whole scene is illuminated at once, illuminating the scene with a laser line offers a good compromise. Using a laser line it is possible to achieve both a relatively large FoV while maintain- ing a high frame rate, as well as acquisition speed and comparatively low pulse energy [10]. At the same time, when illuminating the scene using a scanning or rotating laser line, it is possible to use the laser in a quasi-continuous wave (QCW) operation mode. Thereby the laser is modulated periodically and the illumination of the field is synchro- nized with the laser to achieve the desired FoV. For QCW operation, the duty cycle indicates the ratio of the lasing duration to the total periodic duration. Thus a low duty cycle can reduce the thermal load in the laser itself, but also in the entire sensor. The laser types most frequently used in lidar systems are based on semiconductor technology. Reasons for this are the wide market avail- ability, the emission wavelength that can be mostly selected by design, and an attractive price at least for low pulse energies and high volumes. In this context, semiconductor edge and surface emitters are used in equal measure. But both of these lasers have their own characteristics and limitations. For edge emitters, the output power scales with the width of the ac- tive zone. In addition, depending on the output power, the ratio of the exit aperture varies greatly, resulting in a very asymmetrical beam pro- file with a slow and fast optical axis. In contrast, surface emitters have a comparatively low output power but good beam quality. To achieve sufficient pulse energies, they can be arranged as an array, but this, in turn, requires the use of elaborate lens assemblies to homogenize the single emitters. The common features of both types of semiconductor lasers in pulsed laser operation are the high required peak currents and the relatively poor conversion efficiency. Moreover, the achievement of introduction 5 short laser pulses in the ns range requires the use of complex driver cir- cuits, as well as good shielding of electromagnetic compatibility (EMC) sensitive components due to the high electromagnetic fields. A possible alternative to semiconductor lasers is the use of a solid- state laser. Especially for lidar applications with pulse energies above a few microjoules, passively Q-switched solid-state lasers offer enormous potential and provide certain advantages compared to semiconductor lasers. First of all, those lasers generally possess an almost diffraction-limited beam quality which allows high spatial resolutions to be achieved in lidar applications. Furthermore, compared to semiconductor lasers, the pulse energy does not scale with the emitter or array size, but only with the average power of the pump laser. Consequently, extremely com- pact systems with scalable pulse energies down to the double-digit mJ range are already state-of-the-art [21, 22]. In addition, shielding of EMC- sensitive components is not required due to the lack of high peak cur- rents and complex driver circuits. In particular, passively Q-switched microchip lasers possess a short resonator, which allows short pulse du- rations in the ns or sub-ns range [23, 24]. In general, this not only allows for a small package volume but also enables a small longitudinal res- olution of the lidar sensor due to short laser pulse durations. Another advantage of solid-state lasers is that in the case of rare-earth ion doped crystals they can have very small gain bandwidths. Therefore operation on a single longitudinal mode is possible with short resonators. Also, the atomic transition leads to temperature-dependent wavelength shifts in the range of a few picometers per degree Kelvin [25]. Likewise, semi- conductor lasers can also achieve relatively narrow linewidths <0.2 nm [26], but this typically translates into smaller output powers or signifi- cantly more complex device design and fabrication [27]. On top of that, the greater thermal wavelength shift of 0.3 nm K−1 for edge emitters and 0.07 nm K−1 for surface emitters, respectively, typically leads to lim- 6 introduction itations when considering the automotive temperature requirement for lidar sensors [28, 29]. However, for pulsed lidar sensors based on solid-state lasers, semicon- ductor lasers are still required. Diode lasers in continuous wave (CW) and QCW operation are extremely efficient with wall-plug conversion efficiencies >50 % and are therefore ideally suited as pump sources for solid-state lasers. Due to the relatively narrow-band emission, a good spectral overlap with the absorption bands of active ions of solid-state laser crystals can be generated. Furthermore, laser diodes in the longi- tudinal pump configuration allow a high spatial overlap with the solid- state laser mode and consequently an extremely efficient excitation. Overall, it appears that diode-pumped passively Q-switched solid-state lasers are well suited for lidar applications. However, there are certain challenges and points that need to be considered. Passively Q-switched lasers offer an extremely simplistic system design without active mod- ulators and large applied voltages, but the timing is determined by the laser dynamics itself. The resulting temporal jitter is roughly on the or- der of a few percent of the laser repetition rate [30] and consequently requires timing monitoring and synchronization of the system to the laser for use in the lidar sensor. Another important issue is market availability combined with wave- length availability for Q-switched solid-state lasers. While in semicon- ductor lasers the emitting wavelength can be shifted by design, solid- state lasers have fixed atomic transitions and are thus much more lim- ited. These solid-state laser transitions are often located in the near and far-infrared spectral range above one micrometer. Above one microme- ter, atmospheric water absorptions increase strongly and further inex- pensive state-of-the-art silicon-based detector systems can no longer be used. Conversely, there are comparatively few laser crystals that can be operated at emission wavelengths in the near-infrared spectrum (NIR) introduction 7 below 1000 nm. To achieve narrow-band spectral transitions, only rare-earth ions in com- bination with certain host crystals can be considered. Primarily, var- ious Nd3+-doped host materials operating as quasi-three-level lasers based on Nd3+:YAG at 946nm [31], Nd3+:YVO4 at 914 nm [32], or Nd3+:GdVO4 at 912 nm [33] have thus been realized in the past. Com- pared to the classical four-level system typically operating above one micrometer, these quasi-three-level systems are much more difficult to realize. First, for these quasi three-level lasers, the stimulated-emission cross-section is about one order of magnitude smaller, which is the rea- son why high resonator losses for the undesired four-level transition have to be realized by the cavity coatings. Apart from that, the coating- induced reflectivity for the desired emission wavelength has to be as high as possible, while on the pump facet a high transmission of the pump wavelength has to be ensured. This sets high demands on the optical coating to operate the laser on the desired wavelength as loss- free as possible and thus with maximum efficiency. Another limitation for quasi-three-level systems is reabsorption since the lower laser level also forms the upper ground-state manifold and therefore has a non- negligible thermal occupancy. Finally, for the use in a lidar sensor, these lasers require a compact and robust, if possible monolithic, cavity de- sign without active temperature control. Concerning the possible amplification materials, it turns out that Nd3+- doped YVO4 and GdVO4 offer several advantages compared to the pop- ular and commonly used YAG-based host material: i) the stimulated- emission cross-section and the absorption cross-section is larger [34, 35], ii) the isotropy of the YAG host crystal leads to random linear polariza- tion with additional depolarization losses in comparison to YVO4 and GdVO4 favoring a linear polarisation [36], and iii) the slightly shorter wavelength of the quasi-three-level transition leads to a better quantum efficiency in commercially available silicon detectors [37]. 8 introduction Compared to the CW operation of these quasi-three-level systems [32, 33, 38–40], passive Q-switching possesses further challenges, and hence only a few systems based on Nd3+-doped vanadate crystals have been investigated in the past [41–43]. Therefore, to fully exploit the potential of passively Q-switched solid- state lasers for use in automotive lidar sensors based on silicon detec- tors, further work and investigations are needed. The goal of this thesis is the investigation and realization of a passively Q-switched Nd3+:YVO4 microchip laser at 914 nm for the application in an automotive lidar sensor based on silicon detector technology. In particular, the case of a scanning or rotating lidar system is considered and thus a QCW operation mode with a low duty cycle is selected. In this context, the laser should preferably provide µJ pulse energies and pulse durations in the sub-10 ns range at repetition rates in the double- digit kHz regime. Additional simulations allow a comparison with the underlying theory and an outlook in case of variation of the different system parameters. As a final step, a passively Q-switched laser based on a monolithic resonator setup serves as a demonstrator for future li- dar systems and experiments. In addition to this introduction, the dissertation is divided into the fol- lowing chapters: Chapter 2 introduces the relevant fundamentals of laser operation and passive Q-switching of solid-state lasers. Chapter 3 presents the different simulation methods, which are used in the following chapter to match experimental investigations. introduction 9 Chapter 4 investigates different system relationships of a passively Q- switched laser considering the requirement for use in an automotive lidar sensor. Additional simulations were performed to provide a theo- retical alignment and give a possible outlook on how the system would behave if certain system parameters were varied. Chapter 5 presents a single-pulse operating passively Q-switched mi- crochip laser design and provides a possible outlook for its use in a lidar system. Chapter 6 summarizes the results of this thesis and gives an outlook on possible future experiments. 2B A S I C S This chapter summarizes the basics of laser crystals and their energy levels, light-matter interaction, as well as lasers and optical resonators. Furthermore, the fundamentals of passively Q-switched lasers and sat- urable absorbers for passive Q-switching are discussed. Unless otherwise stated, the following fundamentals are inspired by various books [44–47]. 2.1 laser crystals and energy structure Solid-state lasers consist of an active medium into which laser-active ions are doped, which have the optical transitions at the desired laser wavelength. The host material, in which the active ions are incorporated to a very small percentage, consists of a crystal, glass, or ceramic. The electric field of the host material essentially influences the energy levels of the laser-active ions. In general, a distinction is made between two main groups of laser-active ions, the rare-earth ions, and the transition metal ions. 11 12 basics 2.1.1 Rare-earth ions The rare-earth ions include the atoms with an atomic number from Z = 58 to Z = 71 (Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu). For laser operation, these ions are doped at a low percentage into a suitable host material and are triply ionized. The electronic transitions are usually pure 4f-4f transitions (exception is Ce3+ with a 5d-4f tran- sition), which are spectrally very sharp [47]. Due to further outlying 5s and 5p electrons, which do not participate in the laser transition, the 4f electrons are strongly shielded from the electric fields of the host crys- tal. Here, due to the weak influence of the crystal field, the degenerate energy levels are only weakly split (Stark splitting) and spectrally only slightly shifted compared to the free ions. Thus, the energy levels of the ions in the crystal field are very similar to those of the free ions and the term symbol notation is still valid. Accordingly, the degenerate energy levels in the crystal field are split into 2J+1 sublevels. Here J is the total angular momentum which is the sum of orbital angular momentum L and spin angular momentum S (J = L + S). As a further consequence of the weak crystal field influence, rare-earth ions have a relatively narrow absorption and emission bandwidth. The narrow absorption bandwidth thus enables very efficient pumping with narrowband laser diodes, while the narrow emission bandwidth en- ables the realization of single-frequency lasers. For rare-earth ions, the pump and laser transitions are so-called weakly allowed transitions with a comparatively small oscillator strength. Weakly allowed transitions are transitions where dipole transitions are excluded by the selection rules. Due to the internal electric and mag- netic fields of the crystal, however, symmetry breaking can occur, which makes these transitions possible with a significantly higher probabil- 2.1 laser crystals and energy structure 13 ity [47]. While for allowed transitions the upper state lifetimes due to spontaneous emission are typically in the range of several nanoseconds, these can be comparatively long for weakly allowed transitions with life- times of micro to milliseconds. This, in turn, is an advantage for pulsed lasers, since large pulse energies are possible, as much energy can be stored in the upper level [47]. Furthermore, the lifetime of the upper energy levels is prolonged due to the narrow decay routes. Compared to transition metal ions, rare earth ions typically have differ- ent photon decay routes starting from the upper pump level. Thus, a crystal offers different wavelengths for laser operation, and the desired transition can be selected by losses in the mirror and crystal coatings [44, 45]. 2.1.2 Transition metal ions The transition metal ions include the atoms with an atomic number from Z = 21 to Z = 30 (Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn). When these atoms are doped into a host material, different valence states are formed, which, depending on the substitution site in the host material, occur in divalent, trivalent, or tetravalent form. In these atoms, the 3d orbital is the outermost orbital occupied by electrons and no 4f electrons are present. Consequently, the 3d electrons are not shielded from exter- nal electric fields by outer electrons, which leads to a strong shift and a strong splitting of the degenerate energy levels when doped into a host material. The strength of the shift and the splitting depends strongly on the electric field of the host material but also on the local site symmetry. Since the energy levels of the ions deviate significantly from those of the free ions due to the external electric field, a notation according to the term symbol is no longer permissible. The transition metals have ab- sorption and emission bandwidths which are very strongly broadened 14 basics due to the strong interaction of the electronic transitions with phonons. For this reason, lasers based on transition metal ions are also called vi- bronic lasers. Due to the broadband emission, these can be used very well for broadband tunable lasers. The tunability of wavelength is gen- erated by the energy of the laser transition can be continuously parti- tioned between photons and phonons [44]. On the other hand, broad- band absorption offers the use as an absorber for passive Q-switching. The broadband absorption in the visible spectral region gives rise to the observable color of transition metal ion-doped materials. Compared with rare-earth ions, transition metals have only one suitable transition for laser operation. The lower laser level, from which non- radiative decay processes occur due to phonons, is slightly above the ground state. Therefore, lasers based on transition metal ions are usu- ally quasi-three-level lasers. This means that when these laser crystals are heated, a wavelength shift into the infrared occurs with a narrowing emission bandwidth. Moreover, with increasing temperature, increas- ingly non-radiative processes, such as cross relaxation, take place in the upper laser levels, limiting the quantum efficiency. For these reasons, ef- ficient cooling is very important for transition metal lasers [45, 47, 48]. 2.1.3 Host materials and importance of phonons There are several requirements when choosing the right host crystal. Basically, the host material should have high transparency for both the pump and laser wavelengths. Good thermal and mechanical properties are also important. These include, of course, good thermal conductivity and a low thermo-optical refractive index. As shown in the previous two chapters, the electric field of the host material also plays a very important role in splitting and shifting the energy levels of the dopant ion. In addition, the emission and absorption cross-section, as well as 2.1 laser crystals and energy structure 15 the fluorescence lifetime are significantly influenced by the crystal field. Since a lattice atom of the host material is replaced by a laser ion dur- ing the doping process, both the size and the valence should be well matched. In addition, the host crystal should have lattice sites where the local electric field has the desired influence on the doped ion in terms of symmetry and strength so that the ion has the necessary spec- troscopic properties for the desired laser operation. In general, the ion should have a long radiative lifetime and emission cross-sections in the order of 10−20 cm2 in the host crystal [44]. In addition, a possible directional dependence of the crystal lattice in- fluences the local electric field and thus a preferred direction of light po- larization. Thus, isotropic crystals have a symmetric local crystal field and thus no fixed preferred direction for polarization. This does not mean that the light from a doped ion is unpolarized, only that there is no significant preferred direction for polarization. Microscopic crys- tal differences may nevertheless lead to a preferred polarization, which may not be stable against external influences. For example, thermally in- duced depolarization can occur in Nd3+:YAG with an isotropic crystal lattice. In comparison, yttrium orthovanadate (YVO4) is an anisotropic crystal and therefore the crystal field has a certain asymmetry that po- larizes the ion and its emission. The natural birefringence of the crystal leads to a preferential direction of polarization compared to Nd3+:YAG [47]. Phonons are the quantized vibration of the crystal lattice. In terms of host materials, they are primarily associated with the removal of heat but have a much more important role in laser operation. When degener- ate energy levels are split in the crystal field, phonons enable rapid (on the order of picoseconds) thermalization between sub-levels, forming a Boltzmann distribution within states. This rapid thermalization allows effective transition cross-sections to be used rather than each individ- ual sublevel being attributed its own cross-sections. Experimentally, the 16 basics determination of these individual cross-sections would be difficult to implement, since the degenerate levels have the same energy. Besides fast thermalization, multi-phonon transitions are essential for laser op- eration. These processes lead to the fact that after excitation of the ion with a spectrally broad pump light source, the electron is very quickly in the upper laser level, and thus stimulated emission can take place. The spectral broadness of the pump source excites the electron into one of the upper manifolds. Without phonons, it would be nearly impos- sible for the electron to relax into the upper pump state. With spon- taneous emission, the transition to a lower state with a larger energy difference would be much more likely. A similar problem would also occur after the laser transition, when the ion is in the lower state of the laser transition. Here, rapid thermalization by phonons prevents reab- sorption of the laser emission and allows rapid relaxation to the lower pumping level. This allows the ion to participate in further emission processes as quickly as possible again and efficient laser operation is possible [47]. 2.1.4 Transitions and selection rules For free ions, transitions between two energy states are only possi- ble if the selection rules are satisfied. These are summarized for light hydrogen-like atoms for the lowest order of multipole transitions in Ta- ble 2.1. The left and middle columns list the selection rules for electric and magnetic dipole transitions, respectively. The right column gives a short note on the respective selection rule. 2.1 laser crystals and energy structure 17 electric dipole magnetic dipole note ∆l = ±1 ∆l = 0 ∆S = 0 ∆S = 0 spin is preserved ∆L = 0,±1 ∆L = 0 L = 0 → 0 is forbidden ∆J = 0,±1 ∆J = 0,±1 J = 0 → 0 is forbidden Table 2.1: Selection rules for electric and magnetic dipole transitions of light hydrogen-like atoms. The probability for such a transition is calculated from the transition matrix element pi→ f = ⟨ψ f |µ|ψi⟩ , (2.1) where |ψi⟩ is the initial state, ⟨ψ f | is the final state and µ is the tran- sition moment operator. The transition moment operator distinguishes between electric dipole transition and magnetic dipole transition. Even if an electric dipole transition is forbidden by the selection rule, a mag- netic dipole transition can take place in the state. However, compared to electric dipole transitions, magnetic dipole transitions are about five orders of magnitude weaker [49]. If the selection rules are violated, the transition matrix element is 0 and this transition is called dipole forbidden or weakly allowed. This means that although the transition is not possible by dipole radiation, it may be allowed by other mechanisms such as multipole radiation (quadrupole or octupole). The electric and magnetic crystal fields can lead to sym- metry breaking so that a forbidden dipole transition becomes possible by mixing states of different parity. However, the transition probability is significantly lower compared to dipole transitions. Here, the parity describes the symmetry of the wave function concerning reflection and can take the values +1 and -1. It is important to mention that the selec- tion rules for multipole radiation differ from the rules shown in Table 2.1 [46, 47]. 18 basics 2.2 lasers and optical resonators A laser generally consists of a gain medium and an optical resonator. Depending on the laser medium used, or the interaction between the gain medium and the resonator, different numbers of energy levels of the gain crystal are involved in the laser operation. A basic distinction is made between four-level and three- or quasi-three-level systems. In this section, the basics of the quasi-three-level system, optical resonators and modes, and thermal effects will be discussed. 2.2.1 Three-level lasers In addition to four-level lasers, quasi-three-level lasers are widely used as bulk solid-state or fiber lasers. Pure three-level lasers are used more rarely and have considerable disadvantages compared to four-level and quasi-three-level lasers. Figure 2.1 shows the idealized energy levels of a three-level laser on the left, a four-level laser in the middle, and a quasi-three-level laser on the right. The first laser was built in 1960 and was a three-level laser based on a Ruby crystal (Cr3+:Al2O3). The left side of Figure 2.1 shows a pure three-level laser. Here the lower laser level (1) is also the ground state. The laser thresh- old is comparatively high since a population inversion can only be achieved when more than half of the ions are in an excited state (2), or state (3), respectively. In this case, the transition τ23 is very fast (on the order of picoseconds) and prevents excited ions from being de-excited by stimulated emission due to pump photons. 2.2 lasers and optical resonators 19 Three-level system Four-level system Quasi-three-level system 1 1 1{ {3 22 2 33 4 τ31 wp wp wp τ34 τ23 τ41 τ23 τ23 τ31 τ31 ϕ ϕ ϕ Figure 2.1: Simplified view of the energy levels of different laser systems. The pump transition is shown in blue, while non-radiative phonon transitions are shown in green. In addition, orange and red arrows correspond to transitions by spontaneous emission and stimulated emission, respectively. Compared to the three-level laser, four-level lasers have a lower thresh- old because the lower laser level (2) is not the ground state (1) and is depopulated very quickly by multi-phonon transitions. As a result, the achievable population inversion is larger and there is also negligi- ble reabsorption of the laser light. The best known solid-state laser is a four-level Nd3+:YAG laser operating at 1064 nm. The right schematic in Figure 2.1 is a quasi-three level system. An ex- ample of such a system is the 4F3/2 → 4I9/2 operation of a Nd3+:YAG laser emitting at 946 nm. In comparison to a four-level laser, the lower laser level is the upper Stark multiplet of the ground state manifold and therefore only a few 100 cm−1 above the ground state multiplet. Due to this, the lower laser level has a high thermal occupancy in thermal equi- librium at room temperature (kT ∼ 207 cm−1 at 300 K). Comparatively, in a typical four-level system the lower laser level is thermally nearly unoccupied because it is about 10 · kT above the ground state at room temperature (300 K). Due to the thermal occupation of the Stark manifolds, the population in- version becomes smaller and reabsorption processes take place, which indirectly reduces the laser efficiency and increases the laser threshold. However, reabsorption is not equivalent to an actual energy loss, since the absorbed photon is not lost, but re-excites the ion, which can then 20 basics contribute to the laser operation again by stimulated emission. How- ever, a higher excitation level is required due to reabsorption and the associated increased threshold, so additional losses are incurred due to spontaneous emission. In addition, for longitudinally pumped lasers, a trade-off must be made between pump absorption efficiency and re- absorption losses due to the process of reabsorption. If the complete pump power shall be absorbed, the laser crystal has to be chosen ac- cordingly long or has to be highly doped. The longer or higher doped laser crystal increases the reabsorption while at the weakly pumped end of the crystal comparatively little pump light is absorbed. Therefore, to optimize the system for maximum conversion efficiency, a trade-off be- tween pump efficiency and reabsorption has to be made. Another con- sequence of the higher excitation level is that quasi-three level lasers require more pump power, which in turn can lead to further thermal effects. Due to the temperature-dependent refractive index, the thermal lensing is increased and the additional heat input causes a reduction of the effective laser cross-sections. While pump saturation is usually not an issue for four-level lasers, it should not be disregarded especially for passively Q-switched quasi three-level lasers with small pump spots. The Q-switching initially increases the threshold value and the gain medium is pumped significantly longer compared to CW operation. For a quasi-three level laser, the laser threshold in CW mode is Pth = πhνp 4σgτ( fa + fb) (ω2 p + ω2 l )[2 faσg NgLg + ln(R)], (2.2) where hνp is the pump photon energy, σg is the stimulated emission ef- fective cross-section, τ is the fluorescence lifetime, and fa and fb are the relative occupation densities in the lower and upper Stark manifolds, re- spectively [50]. Furthermore, ωp and ωl are the radii of the pump and resonator modes, Ng is the doping of the gain medium, Lg is the length 2.2 lasers and optical resonators 21 of the gain medium, and R is the resonator internal losses including the output coupler mirror (OCM) and other parasitic losses. The equation consists of two parts that contribute descriptively to the threshold value. The first part ∝ 2 faσg NgLg is the pump power required to achieve a population inversion. This corresponds to the round-trip absorption of the laser wavelength and is therefore also called the reab- sorption loss. The second part ∝ ln(R) is the pump power required to compensate for the cavity round-trip losses due to the OCM and other parasitic losses. To minimize the threshold, both a small pump and resonator mode should be chosen with the length of the gain medium being as short as possible. These two conditions speak for the use of laser diodes as pump lasers since they can be focused on a much smaller pump volume (∼ pump mode × length of the gain medium) compared to flash lamps. The saturation fluence of the laser crystal is an important parameter, especially for passive Q-switching (see the following subsection). It has a significant influence on the achievable pulse energy and the repetition rate of the laser. If a pulse with the fluence Fpulse (unit energy density) propagates through the laser crystal, it is amplified and reduces the gain in the crystal. The saturation fluence of the gain material is defined as Fsat = hν σg + σreab , (2.3) and reduces the gain by 1/e (∼ 37 %). Hereby, σreab is the effective reab- sorption cross-sections of the gain medium. From the saturation fluence along with the mode area A, the saturation energy can be calculated as Esat = A · Fsat = A · hν σg + σreab . (2.4) 22 basics This is often more practical for design and optimization since the laser mode area is known or can be measured. In the three-level system, the effective reabsorption cross-section σreab of the laser wavelength is taken into account, which is omitted in the case of a four-level system. 2.2.2 Optical resonators and modes Optical resonators are an essential part of lasers since they provide opti- cal feedback to the gain medium and thus amplify the stimulated emis- sion. The resonator determines the mode structure in the transverse and longitudinal direction as eigensolutions of the electric field. Transversal modes, as the name implies, have a field distribution per- pendicular to the direction of propagation in the resonator. The trans- verse modes are so-called eigenmodes of the resonator, which means that the field distribution, apart from diffraction losses, must be in the initial state after one resonator round trip. The transverse modes have a major influence on the beam divergence and the beam diameter of the laser. Mathematically, the transverse modes can be described either by Her- mite polynomials (in cartesian coordinates) or Laguerre polynomials (in cylindrical coordinates), dependent on the choice of coordinate system. The overlap between resonator mode and the mode diameter of the pump laser has significant influence on the transverse operation modes of the laser. By choosing a pump spot that is too large, higher trans- verse modes can resonate, while with a pump spot that is too small, the conversion efficiency of the laser may deteriorate. A laser operating in the fundamental transverse mode has a beam with a Gaussian intensity profile, which is also called a Gaussian beam. The 2.2 lasers and optical resonators 23 name follows from the shape of the transverse intensity profile, which can be written mathematically as I(r, z) = P πw(z)2/2 exp ( −2 r2 w(z)2 ) , (2.5) where r is the distance from the optical axis, z is the position of the prop- agation direction of the beam, P is the optical power, and w(z) is the beam radius. Although the intensity distribution maintains a Gaussian profile for all propagation positions z, the beam radius w(z) changes as a function of the axis position z and thus the width of the intensity distribution. The beam radius w(z) of a Gaussian beam as a function of propagation z is described by a hyperbolic form w(z) = w0 √( 1 + z zR )2 (2.6) with the minimum beam diameter 2 · w0 and the Rayleigh length zR = πw2 0 λ , (2.7) where λ is the wavelength of the laser. The Rayleigh length is defined as the distance from the minimum beam diameter at which the beam area has doubled (w(zR) = √ 2w0). Alternatively, the confocal param- eter b is often defined as b = 2zR. In this range, the beam is assumed to be collimated. Figure 2.2 shows schematically the hyperbolic shape of a Gaussian beam around the position of the focus. The Gaussian in- tensity profile extends to infinity, but the intensity for diameter 2w0 has already dropped to 1/e2 (13.5 %), which is indicated by the red enve- lope. In the far-field, the hyperbolic shape can be approximated by a linear divergence angle, resulting in θ = lim z→∞ an 2w(z) z = 2λ πw0 . (2.8) 24 basics w(z) b w0 θ/2√2 w0 z Figure 2.2: Spatial propagation of a Gaussian beam along the z-axis. Different parameters characterizing the beam are indicated by the black arrows. This means that the beam radius w(z) and the divergence angle θ in- crease linearly in the far-field. By rearranging Equation 2.8, it can be seen that the product of beam radius w0 and divergence angle θ is con- stant and equals λ/π. The constant product implies that as the beam radius becomes smaller, the divergence angle increases and vice versa. The product of the two quantities is defined as beam parameter product (BPP) and used as a measure to determine beam quality. A Gaussian beam has the smallest possible BPP and therefore the high- est possible beam quality, causing it to be called diffraction-limited. Beams with a larger BPP can also be described by Gaussian beams considering the beam quality factor M2. While the fundamental mode of the resonator has an M2 = 1, higher transverse modes have an M2 > 1. The larger the M2, the more higher modes are present, and the larger the minimum focusable diameter becomes. The factor M2 re- sults from the quotient of the measured BPP and the minimum possible diffraction-limited BPP and is a measure of how well the laser can be 2.2 lasers and optical resonators 25 focused or collimated. Taking into account the beam quality factor M2 and the Rayleigh length the beam radius w(z) finally changes to w(z) = w0 √√√√(1 + M2λz πw2 0 )2 . (2.9) When propagating through lenses and crystals, Gaussian beams main- tain their Gaussian intensity profile and beam quality, whereas only the associated beam parameters w0 and θ change. However, this assump- tion is valid only under consideration of the paraxial approximation, which loses validity for large divergence angles. In a linear resonator, a standing wave is formed analogously to a Fabry- Pérot etalon. The constructively interfering waves with frequency ν are called longitudinal modes and follow the resonance condition ν = q c 2Lopt . (2.10) Here q is the order of the longitudinal mode, c is the speed of light, and Lopt is the optical resonator length. From this condition follows an equidistant frequency spacing ∆ν = c/(2Lopt) between adjacent fre- quencies which in the wavelength space becomes ∆λ = λ2 2Lopt , (2.11) where λ is the wavelength of the propagating light. For an exemplary Nd3+:YAG laser at 1064 nm with a compact resonator having an optical length of 20 cm, this results in a longitudinal mode spacing of about 28 pm. Of course, the number of laser modes does not only depend on the resonator but is also determined by the gain medium, since only modes which are located within the gain bandwidth are amplified. For a rough estimate, the number of longitudinal modes can be calculated 26 basics from the ratio of the gain bandwidth and the mode spacing. For the above resonator, in the case of Nd3+:YAG with a gain bandwidth of 0.6 nm, the laser operates on an estimated number of 21 modes. Simply put, if the resonator is shortened, the number of longitudinal modes is reduced, and theoretically, a single-frequency laser can be generated which operates on a single longitudinal mode. 2.2.3 Thermal effects The pumping process and operation of a solid-state laser leads to a heat contribution in the gain medium, which has a great influence on the performance and stability of the laser. The heat input and asso- ciated thermal effects reduce the maximum achievable output power, as well as the beam quality, and in the worst-case lead to breakage of the laser crystal. There are various causes and effects of heat in- put. These are partly based on the intrinsic properties of the materials used but can be minimized by system optimization. The quantum de- fect is the energy difference of the pump photon to that of the emitted laser photon and has a significant contribution to the heat due to non- radiative phonon transitions. Depending on the crystal, laser transition, and pump laser used, this value varies. As an example, the quantum de- fect for an 808 nm pumped Nd3+:YVO4 laser is 11.6 % when the laser is operated at 914 nm. This means that at 10 W of continuous pump power, more than one watt of the heat already accumulates in the crys- tal and has to be dissipated. When pumping at 880 nm, the quantum de- fect is only 3.7 %, but the absorption cross-section is also much smaller so that a longer crystal or higher doping must be selected, which can lead to further effects. Moreover, the pump spot diameter should be chosen to match the resonator mode, otherwise, regions in the gain crystal will be pumped, leading either to higher transverse modes or 2.2 lasers and optical resonators 27 an additional heat contribution by fluorescence. With a good overlap between pump laser mode and resonator mode, the contribution of flu- orescence during laser operation in saturation can be neglected, since the contribution by stimulated emission is usually much larger than the spontaneous emission and thus relatively few ions are excited by fluo- rescence. Concerning deposited heat, the use of diode-pumped lasers is also advantageous compared to flash-lamp pumped lasers. Due to the relatively narrow spectral width of pump diodes, the pump transition of the crystal can be well-targeted. In contrast, the wide pump spectrum of flash lamps leads to background absorption by the host material and impurity atoms. In longitudinally pumped solid-state lasers, the heat introduced by the pumping process leads to a temperature gradient in the laser crystal. The crystal is typically actively cooled via the side faces, where ther- mal bonding can be improved by inserting an indium foil between the crystal and the crystal holder. The end faces of the crystal, on the other hand, are passively cooled by ambient air, resulting in an inhomoge- neous and non-uniform temperature profile in the crystal. The result- ing temperature gradient leads in the crystal in radial direction and along the optical axis finally to an optical path difference (OPD) and consequently to a thermal lens with a focal length of f = KA Pheat [ 1 2 dn dT + αthCrn3 + αthr0(n − 1) l ]−1 (2.12) with Pheat = η · Ppump. (2.13) Here, K is the thermal conductivity, A is the cross-sectional area of the pump spot, Pheat is the heat dissipated in the crystal, η is the pump absorption efficiency, Ppump is the applied pump power, dn/dT is the temperature-dependent refractive index change, ath is the thermal ex- pansion coefficient, Cr is the photoelastic coefficient, n is the refractive 28 basics index, r0 is the radius of the crystal, and l is the crystal length [44]. Equation 2.12 consists of three different parts which are related to the temperature and stress-dependent change of the refractive index and distortion and the end-face curvature of the crystal. Due to the differ- ent temperature gradients in the radial and axial direction, a refractive index difference dn/dT arises which leads to a thermal lens. Depen- dent on the gain crystal and the associated refractive index variation, the modification can have a positive or negative sign and the resulting lens is either convex or concave. In addition to this, the thermal gradient leads to non-uniform crystal expansion, which creates mechanical stress in the crystal lattice and thus also contributes to an inhomogeneous re- fractive index change dn/dr. Furthermore, the thermal expansion of the crystal can lead to bulging of the planar end faces of the crystal and thus a lensing effect. At the pumped end face of the crystal, the temperature gradient and the resulting thermal lensing are most pronounced. The exponential absorption of pump light along the optical axis results in an exponentially decaying temperature profile. In the radial direction, the cooled side surfaces lead to strong heat dissipation and a strong temperature gradient. In the radial direction, the cooled side surfaces lead to a strong logarithmic decrease in the temperature. Overall, the contribution of the temperature-dependent refractive index variation is dominant concerning the induced thermal lensing. For instance, ex- periments with a Nd3+:YAG laser showed at higher pump power levels that a comparatively small contribution of 20 % for the stress-dependent variation of the refractive index, while the temperature-dependent ex- pansion had hardly any influence at less than 6 % [51]. For this rea- son, especially for low pump powers, the bulging effect of the crystal end faces, the stress-induced influence of the refractive index, and the 2.2 lasers and optical resonators 29 temperature-dependent crystal expansion can be neglected, reducing Equation 2.12 to f = Kπω2 p ηPpump(dn/dT) ( 1 1 − exp(−αthl) ) . (2.14) In general, the thermal lens deteriorates the beam quality and nega- tively affects the resonator stability as a function of pump power. In the case of plane-parallel an resonators, however, some thermal lensing is quite desirable and essential to stabilize the resonator, which is other- wise operated at its stability limit. In principle, even a strong thermal lens can be compensated by adjust- ing the laser design for a specific operating point. Especially for high output powers and correspondingly high intracavity powers, a slight deviation of the power can have a large impact on the stability of the laser and stop the laser operation or cause crystal fracture. Further, a strong thermal lens inevitably leads to strong aberrations, making high beam quality difficult to achieve. As an example, Figure 2.3 shows the resulting thermal lens for a Nd3+:YVO4 crystal according to Equation 2.14 as a function of the applied pump power and the assumption that 24 % of the pump power is converted to heat [52]. In addition, a crys- 0 2 4 6 8 10 12 14 16 0 40 80 120 160 200 f = Kπω2 p ηPpump ( 1 1−αth ·l ) applied pump power (W) fo ca ll en gt h (m m ) calculated thermal lens Figure 2.3: Calculated thermal lens of a Nd3+:YVO4 crystal as a function of applied pump power. 30 basics tal of 3x3x5 mm3 and a Gaussian pumping distribution with a diam- eter of 400 µm was assumed. For the material-dependent constants, K = 5.1 W m−1 K−1, dn/dt = 8.5 × 10−6 K−1 and ath = 5.32 cm−1 were as- sumed [53]. 2.3 passive q-switching Passive Q-switching is a technique to obtaining laser pulses in the nano- and sub-nanosecond regime. The principle of Q-switching is that high cavity losses are introduced while continuous pumping enables high storage of energy without laser operation. Sudden reduction of inter- nal losses results in an intense laser pulse as a significant amount of stored energy is extracted. For passive Q-switching, the temporary cav- ity losses are attributed to a passive element like a saturable absorber. 2.3.1 General operation principle The dynamics of a passively Q-switched operation are shown in Figure 2.4. Hereby, the evolution of the gain coefficient is shown in blue, the saturable absorber losses in green, and the generated photon density in the form of a laser pulse in red. Initially, the population inversion, which is proportional to the gain coefficient, is increased via the ex- ternal pumping process and the laser operation is suppressed due to the high losses of the saturable absorber. Consequently, the population inversion reaches a level that is far above the natural laser threshold without the saturable absorber. The maximum energy stored in the gain crystal is limited by the process of spontaneous emission or by 2.3 passive q-switching 31 too low losses of the saturable absorber. If now suddenly the losses generated by the saturable absorber are reduced, an electric field in the resonator builds up exponentially from spontaneous emission. The gain decreases simultaneously and gets saturated once the saturation fluence of the gain medium is reached. Further, when the gain reaches the remaining resonator losses (equal to the threshold inversion) the maximum photon density is reached. As the change of inversion is neg- ative and further the intracavity power is high, the population inversion still decreases until a final population density is reached. The passive Q-switching element is increasing intracavity losses once the intracav- ity photon density decreases. Usually, the Q-switching element should recovery quicker than the gain material to obtain stable pulse operation. For continuously pump lasers, the population inversion increases again above threshold inversion, and the whole process repeats. In comparison to active Q-switching, passive Q-switching [44, 47] re- quires no active elements such as an electro-optic or acusto-optic driver as well as a RF signal generator. Therefore a simple and cost-effective system is possible which can be additionally quite compact and robust. In addition, high voltages can lead to electromagnetic compatibility 0.0 0.5 1.0 time (µs) ga in / lo ss es (a rb .u ni ts ) gain 0.0 0.5 1.0 ph ot on de ns it y (n or m al iz ed ) gain coefficient absorber losses generated pulse Figure 2.4: Dynamics of a passively Q-switched laser showing the time evolu- tion of the gain coefficient (blue), the saturable absorber losses (green) and the generated photon density in the form of a laser pulse (red). 32 basics problems with other components and are also potentially dangerous from a work point of view. Another advantage of passively Q-switched lasers is the possibility of generating very high repetition rates whereas actively Q-switching is somehow more limited due to a maximum re- quired switching time. In contrast, passively Q-switched lasers show temporal pulse-to-pulse fluctuations as the laser dynamics determine the time when the pulse is triggered. For example, even small varia- tions in pump energy or ambient conditions can lead to significant tem- poral jitter. Further, the lower laser output is achieved in general due to residual absorption in form of non-saturable losses of the Q-switching element. 2.3.2 Features and properties of passively Q-switched laser In the following, important features and properties will now be intro- duced and discussed in the context of passively Q-switched lasers con- cerning the later experiments. For passively Q-switched lasers with short resonators the pulse build- up time is in the regime of 1 µs as the number of round-trips in the res- onator is approximately 1000 once the saturable absorber is switched. In contrast for active Q-switching, pulse formation takes place within sev- eral tens of round trips and therefore the pulse build-up time is an order of magnitude smaller. In passively Q-switched lasers with saturable ab- sorbers, the switching process is triggered by the first built-up longitu- dinal mode. Compared to other modes starting later, this mode already consists of a certain photon density. As soon as the second and other modes reach the threshold, the first mode has already grown signifi- cantly due to the highest gain. As a result, the first longitudinal mode usually differs from other modes by more than one order of magnitude, 2.3 passive q-switching 33 and single-frequency operation is easier [54, 55]. Furthermore, the lon- gitudinal mode spacing can be increased by shortening the resonator. Due to the relatively narrow gain-bandwidth of rare-earth-doped crys- tals, modes can be selected by placing them outside the gain. Another possibility to achieve single-frequency operation is the insertion of a frequency-selective element, e.g. a thin Fabri-Pérot etalon in the form of an undoped crystal. Due to the long pulse build-up time, many res- onator cycles are covered and frequencies that are not resonant at the etalon are suppressed [56]. Passively Q-switched lasers possess pulse duration in the nanosecond and sub-nanosecond regime dependent on the utilized gain material, saturable absorber, output coupler reflectivity and internal cavity losses. Usually, it takes several round trips until the exponentially growing photon flux extracts the stored energy. A reasonable estimate for the pulse duration in the case of a Q-switching element with strong satura- tion is τp ∼ 4.6 Tround q0 , (2.15) where q0 (∼ 1 - T0) is the saturable loss and T0 the initial transmission of the Q-switching element. Further, Tround is the resonator round trip time which is again linearly dependent on the resonator length. Thus, the easiest way to change pulse duration is done by adjustment of the resonator length whereas a change of the saturable losses typically af- fects the general laser dynamics [47]. The pulse energy of passively Q-switched lasers typically ranges from sub-micro joule up to a few milli-joules dependent on the pulse repe- tition rate and design specification. Analytically, the pulse energy of a passively Q-switched laser can be derived as Ep = hνA 2σγ ln ( 1 R ) ln ( ni n f ) , (2.16) 34 basics with the laser photon energy hν, effective beam cross-section area A, stimulated-emission cross-section σ, inversion reduction factor γ, and output coupler reflectivity R [57]. Large pulse energies are obtained for a big mode-area which can be realized by a large pump spot but this usually results in higher-order transversal modes and an increase of laser threshold. Further, a small stimulated-emission cross-section and an output coupler mirror with low reflection are favorable. Moreover, the ratio ni/n f is directly related to the saturable losses of the element. This means that large pulse energies can be achieved more easily if the saturable losses are low which again increases the pump laser threshold. It should be noted that high pulse energies are always a trade-off and typically reduced the repetition rate. For pump power levels well above threshold pump power, the repetition rate is approximated by frep ∼ [ τ ln ( Pth/P Pth/P − 1 )]−1 , (2.17) with gain medium fluorescence lifetime τ, pump power threshold Pth, and applied pump power P [58]. In the case of a low-power system us- ing a saturable absorber for passive Q-switching the maximum achiev- able repetition rate is limited by the recovery time of the absorber. For high repetition rates, the time between adjacent pulses approaches a regime where the saturable absorber is not able to fully recover after a pulse is generated. In that case, unstable laser dynamics dominate and disturb steady laser operation. In contrast, for high-power lasers, the maximum achievable repetition rate is typically restricted due to ther- mal effects at higher pump powers [56]. With satellite pulses, the actual laser pulse is followed by a second pulse which is delayed in time concerning the main pulse and which has a smaller pulse amplitude. The interval between the two pulses is of the order of nanoseconds and is thus significantly smaller than the 2.3 passive q-switching 35 actual pulse interval at repetition rates in the kilohertz range. In gen- eral, the occurrence of satellite pulses can have two reasons: 1 When the first longitudinal mode has built up in the form of a laser pulse, the remaining gain can be sufficiently high that a second adjacent longitudinal mode and thus a second laser pulse builds up. However, this requires that the remaining gain is above the threshold and the dif- ference in build-up time between the two modes is not too large. Since the gain is already significantly reduced by the build-up of the first pulse, the second pulse has considerably smaller pulse energy and thus a longer pulse duration. When the satellite pulses are generated by this effect, the laser runs on two different longitudinal modes. In this case, satellite pulses can be suppressed if there is a large gain difference be- tween the main mode and competing mode. For this purpose, the mode spacing should be as large as possible (short resonator length) and the main mode should be as close as possible to the maximum of the gain curve (tuning via temperature or pump power). A good rule of thumb is: The closer the main mode is tuned to the maximum of the gain curve, the greater the delay between main and satellite pulse and the smaller the amplitude of the satellite pulse [59]. 2 It has been shown that satellite pulses can occur if the photon life- time in the resonator is significantly smaller than the thermalization time in the degenerate Stark submanifolds of the lower laser level. Once a laser pulse is generated, the remaining population in the upper laser level can lead to a new population inversion if a delayed depopulation by thermalization occurs in the Stark sublevels of the lower laser level. If the new population inversion is larger than the threshold value of another longitudinal mode, a satellite pulse is generated. Since the gain has already been significantly reduced by the main pulse, the satellite pulse has a significantly smaller amplitude [59]. In this case, the satellite 36 basics pulse is generated similar to the gain-switching of pulsed lasers. Com- pared to the upper possibility of satellite pulse generation, both pulses are usually in the same longitudinal mode. In addition, the two pulses can have a very stable delay and amplitudes. Depending on which gain material is used, the thermalization time varies in the Stark multiplets and satellite pulses may occur. For ex- ample, while Nd3+:YAG has a very short lifetime of about 225 ps in the lower Stark multiplet, Nd3+:GdVO4 or Nd3+:YVO4 has around 20 ns. For resonator lengths of a few millimeters, the photon cavity decay time is on the order of nanoseconds and satellite pulses have been observed at Nd3+:GdVO4 [60]. Figure 2.5 shows an exemplary satellite pulse based on rate equation simulations. Here, the origin of the satellite pulse is based on possibil- ity 2 , where the thermalization time in the Stark submanifolds was reduced in the model to be shorter than the photon lifetime in the resonator. The generated main pulse with a pulse duration of 18.9 ns (FWHM) is followed by a satellite pulse with a pulse duration of 108.2 ns separated by 219 ns. For passively Q-switched lasers, the exact timing for laser pulses is determined by the laser system dynamics. By pump- 0 50 100 150 200 250 300 350 400 0.0 0.5 1.0 main pulse: 18.9 ns satellite pulse: 108.2 ns time (ns) si gn al (n or m al iz ed ) Figure 2.5: Exemplary satellite pulse based on rate equation simulations. The generated main pulse with a pulse duration of 18.9 ns (FWHM) is followed by a satellite pulse with a pulse duration of 108.2 ns separated by 219 ns. 2.3 passive q-switching 37 ing, a population inversion builds up in the laser crystal and a pulse is triggered as soon as a longitudinal mode builds up from spontaneous emission which is strong enough to switch through the saturable ab- sorber. Due to the dynamic process, various factors can influence the exact switching time of the saturable absorber and thus generate a tempo- ral jitter between successive laser pulses. Typical temporal jitter is on the order of the pulse duration or even larger. Starting with the pump laser, small fluctuations in the pump power, a slight drift of the pump wavelength but also temperature-induced changes in the polarization can disturb the dynamics of the laser. Furthermore, temperature influ- ences at the laser crystal and saturable absorber but also mechanical vibrations are added as disturbance of the laser dynamics. While the exact switching time of passively Q-switched lasers is influenced by these disturbances, the pulse energy, and pulse duration remain largely unaffected [61]. In principle, temporal jitter is an inherent property of passively Q- switched lasers, but there are several ways to reduce it. At low rep- etition rates, the pump power can be modulated and pulsed. In this case, the solid-state laser can be pumped continuously just below the laser threshold and pulses are generated by briefly raising the pump power until a single pulse is formed. The maximum switching speed of the pump laser but also the overall efficiency is limited with this method [62]. Furthermore, injection seeding can be used to reduce the pulse build-up time and to synchronize the wavelength of the seed laser and the passively Q-switched laser. However, this method requires very good temperature stability of the whole system and makes the laser setup larger and more complex [63]. In addition to this, there is also the possibility of bleaching the saturable absorber with another pulsed laser. In this case, however, the laser that bleaches the absorber must have a significantly better timing jitter than the laser that is passively 38 basics Q-switched. In addition, the setup is of course also significantly more complex and expensive than the single passively Q-switched laser [64]. 2.4 saturable absorbers for passive q-switching Saturable absorbers have a nonlinear absorption behavior, which can be used for a sudden reduction of the resonator losses and thus for the generation of short laser pulses. The nonlinear absorption behavior of the absorber can be generated here by saturating a spectral transition for high intensities. There are different types of saturable absorbers for passive Q-switching which are predominantly based on transition metals ion-doped crystals or glasses. Probably the most commonly used saturable absorber mate- rial is a chromium-doped yttrium aluminum garnet crystal (Cr4+:YAG). These absorbers have broadband absorption, as well as good chemical, thermal, and mechanical properties. Furthermore, semiconductor saturable absorber mirrors (SESAMs) are interesting for passive Q-switching, but due to a very low damage threshold, they are preferably used for passive mode-locking to gen- erate ultrashort pulses. A major advantage of SESAMs is that they can be specially designed and optimized for specific wavelengths. In ad- dition, compared to crystalline absorbers doped with transition metal ions, they increase the resonator length only insignificantly, which in principle can also lead to shorter pulses. However, due to the relatively low damage threshold and a short recovery time, SESAMs are preferred for applications with low pulse energy and high repetition rates. An- other important point is that SESAMs act as end mirrors by their design, therefore an additional output coupler mirror must be used which often makes the system design larger and more complex. Much rarer are ex- 2.4 saturable absorbers for passive q-switching 39 10−4 10−2 100 102 104 90 95 100 Fsat T0 Tsat ∆md ∆ns fluence (J/cm2) tr an sm it ta nc e (% ) Fsat T0 Tsat Figure 2.6: Characteristic switching behavior of the transmission of a saturable absorber for increasing fluences on the absorber. otic saturable absorbers based on quantum dots, organic dye solutions, color centers, or even graphene layers. Since the absorbers used in this work are based on transition metal ion-doped crystals, these absorbers will be the main focus of the following subsection. Figure 2.6 shows the characteristic switching behavior of the transmis- sion as a function of the fluence of a saturable absorber. The initial transmission T0 is the transmission of the absorber for small signals or fluence. For an absorber of thickness l with absorption coefficient, this results in T0 = exp(−αl) = exp(−NgsσGSAl), (2.18) where the absorption coefficient for small signals without the contri- bution of ESA can be calculated to α = NgsσGSA. Here, Ngs is the oc- cupation density in the ground state of the absorber and σGSA is the ground-state absorption (GSA) cross-section. The initial transmittance of an absorber can be determined relatively easily and, along with the thickness, is an important parameter in the design of a laser. The saturation fluence Fsat is the energy per area needed to increase 40 basics the initial transmission value by 1/e (∼ 37 %). It is expressed as Fsat = hν σGSA , (2.19) and thus the ratio of photon energy and GSA. Here, non-saturable excited-state processes are neglected. It is important to note that the saturation fluence usually does not depend on the thickness of the sat- urable absorber if the absorber is not too thick to change the fluence as a function of penetration depth. Likewise, the saturation fluence does not take into account the mode area on the absorber. The mode area is taken into account by the saturation energy which is the product of satura- tion fluence and mode area. The maximum achievable transmission of a saturable absorber is the saturation transmission Tsat. The maximum transmission is only 100 % in the ideal case and is reduced by the non- saturable losses ∆ns. These losses can be generated in ion-doped insula- tor saturable absorbers by ESA. In this process, the already excited ion is further excited to an even higher electronic state by the absorption of light. The strength of the ESA is characterized by the excited-state absorption cross-section σESA. In passive Q-switching, non-saturable losses are undesirable because they do not contribute to the switch- ing behavior, but reduce the achievable pulse energy and conversion efficiency of the laser while increasing the system threshold. ESA is a problem especially for broadband absorbing media such as transition metal ion-doped absorbers and does not play a major role for rare-earth- doped materials. While GSA leads to a higher absorber transmission with increasing fluence, ESA behaves in exactly the opposite way. While GSA leads to higher absorber transmission with increasing fluence, ESA behaves in exactly the opposite way. Absorption by ESA rises with in- creasing fluence as the excited state becomes more populated. For sat- urable absorbers which are used for passive Q-switching, the influence of GSA must be dominant. More precisely, σGSA > σESA must hold. Ab- 2.4 saturable absorbers for passive q-switching 41 sorbers, where the ESA is dominant compared to the GSA, are called reverse saturable absorbers [65]. These absorbers cannot be used for passive Q-switching but can be used, for example, for self-stabilization of passively mode-locked lasers since the total transmission decreases with increasing absorber fluence [66]. A good measure to judge the absorber quality is the ratio of both cross-sections. This ratio is called figure of merit (FOM) and is defined as FOM = σGSA σESA , (2.20) [67]. In general, however, the higher the FOM, the better the saturable absorber is suited for passive Q-switching. Similar to the stimulated- emission cross-section of the gain material, the GSA and ESA cross- sections depend on the wavelength. The modulation depth ∆md of a saturable absorber is the difference be- tween the maximum and minimum transmittance. This value is largely determined by the initial transmission and non-saturable losses of the absorber. In passive Q-switching, the modulation depth has a great in- fluence on the pulse energy and thus also on the pulse duration. To gen- erate high-energy pulses at a lower repetition rate, a large modulation depth is required. In addition, it is also important that the saturation energy in the gain medium, i.e. the maximum storable energy, is large. Similar to the saturable absorber, the saturation energy is the product of saturation fluence and mode area. Therefore, for large pulse ener- gies, a small stimulated-emission cross-section and the largest possible mode area in the gain medium are important. Conversely, high repe- tition rates with smaller pulse energies can be achieved more easily if the stimulated-emission cross-section is large and the mode area in the gain crystal is rather low. The saturation intensity Isat (power per area) becomes important for saturable absorbers only if the recovery time is in the order of mag- nitude of the passively Q-switched pulse duration since then the sat- 42 basics urable absorber already recovers during the pulse build-up time. The saturation intensity is defined as Isat = hν σGSAτ = Fsat τ (2.21) where Fsat is the saturation fluence and τ is the recovery time of the absorber. The recovery time τ of a saturable absorber is the time until the ab- sorber has recovered by 1/e (∼ 37 %) after complete bleaching. In sat- urable absorbers based on a doped insulator, the excited state decays back to the ground state by spontaneous emission. This period should ideally be significantly longer than the pulse duration of the generated laser pulse, to avoid that the saturable absorber already recovers during the pulse build-up time and thus attenuates the building-up pulse by re-absorption. On the other hand, the recovery time should not be too long, otherwise a satellite pulse can be formed by reaching population inversion in the gain medium again while pumping the system contin- uously. In addition, with passively Q-switched lasers at high repetition rates, a long recovery time can lead to a situation where the saturable ab- sorber is not fully recovered when the subsequent pulse is formed. If this is the case, the subsequent pulse has a reduced modulation depth and thus smaller pulse energy, which leads to unwanted dynamics and instability in the laser. As a rough estimate, the recovery time should be between the pulse duration and the upper-state lifetime of the gain medium being used. As an example, Figure 2.7 shows an idealized ex- ponential decay with a fictitious excitation by an ultrashort pulse at time τ0. The subsequent decay of the excited states with a recovery time of 4 µs (1/e) takes 18.5 µs until the absorber is almost completely recovered (>99 %). In this case, at repetition rates of about 54 kHz (≈ 1/18µs−1), the absorber would not recover completely between succes- 2.4 saturable absorbers for passive q-switching 43 0 10 20 30 0 0.5 1.0 1/e (∼ 37%) τ1% 1% τ37%τ0 time (µs) si gn al de ca y (n or m al iz ed ) Figure 2.7: Idealized exponential decay of excited states of a saturable absorber after it has been excited by a fictitious ultrashort pulse at time τ0. sive pulses and thus develop unwanted dynamics (e.g. satellite pulses) and instabilities. Compared to the idealized decay rate, the decay pro- cesses of saturable absorbers can be more complicated in reality and thus deviate from a simple single-exponential decay e.g. exhibiting a double-exponential decay. 3S I M U L AT I O N A N D S Y S T E M D E S I G N In this Chapter different simulation methods are introduced, which will be used in the following chapters to improve the understanding of the system and to reconcile theory and experimental investigations. In this context, a rate equation model is used to calculate the temporal occupation dynamics of the gain medium and the saturable absorber by coupled rate equations. This allows subsequently to calculate dif- ferent systems output quantities such as pulse energy, repetition rate, or pulse duration for a passively Q-switched system. Furthermore, the second threshold criterion for passively Q-switched lasers is considered and applied to a quasi-three-level system based on Nd3+:YVO4 as gain medium and Cr4+:YAG as saturable absorber. In addition, the resonator stability is investigated using a matrix formalism approach. Here, in ad- dition to the fundamental mode, higher-order transverse modes are also considered and the influence of the pump-induced thermal lensing on the resonator stability is examined. Finally, the gain crystal is thermally analyzed using finite element analysis (FEA), and the influence of the pump spot diameter, pump wavelength, and applied pump power on the peak temperature in the crystal is investigated. 45 46 simulation and system design 3.1 rate-equation model According to previous work on the theory of passively Q-switched lasers [44, 45, 68–70], the coupled rate equations taking into account ESA of the saturable absorber and reabsorption losses of the quasi-three level system are given by dϕ dt = ϕ tr [2σg Nglg − 2σreablg − 2σGSANGSlSA −2σESANESlSA − L + ln(R)] (3.1) dNg dt = Wp − Ng τg − γσgcϕNg (3.2) dNGS dt = NSA − NGS τSA − σGSAcϕNGS Ag ASA (3.3) with NSA = NGS + NES. (3.4) Where Φ is the photon density in the resonator, tr is the resonator round-trip time, σg is the stimulated-emission cross-section of the laser crystal, Ng is the population inversion density in the gain medium, lg is the length of the laser crystal, σreab is the reabsorption cross-section in the quasi-three level system, σGSA and σESA are the GSA and ESA cross-sections of the saturable absorber, respectively. Further, lSA is the length, NGS and NES are the occupation densities of the ground and excited absorber states, respectively, L is the non-saturable resonator losses, R is the reflectivity of the OCM, τg is the upper laser level life- time, γ is the inversion reduction factor, tSA is the excited lifetime of the saturable absorber, and Ag and ASA are the beam area in the gain 3.1 rate-equation model 47 medium and saturable absorber, respectively. In addition to this, the volumetric pump rate Wp = ηqPpump 1 − exp(−2σabslg) hνAglg (3.5) can be expressed by the pump efficiency nq, the applied pump power Ppump, the absorption cross-section of the gain medium σabs, the Planck’s action quantum h, and the pump frequency ν. Again, the resonator round-trip time is given by tr = lc/c (3.6) from the resonator length lc and the speed of light c. In addition, the initial population inversion in the gain medium for low intensities in the resonator can be expressed by Ngi = L − ln(R)− ln(T2 0 ) 2σglg (3.7) while the threshold occupation inversion assuming a fully bleached ab- sorber can be expressed as [71] Ngth ∼= 2σg NSAlSA + ln ( 1 R ) + L 2σglg . (3.8) Besides, under the assumption that the photon density is small after the Q-switched pulse is decoupled, the following transcendental equation Ngi − Ng f − Ngth ln ( Ngi Ng f ) = 0 (3.9) is obtained which couples the final occupation inversion of the gain medium with the initial and the threshold occupation inversion. 48 simulation and system design The numerical solution of this transcendental Equation 3.9 [72] finally leads to the values for Ngi , Ngth , and Ng f , which can be used to obtain the pulse energy [57] Ep = hνAg 2σgγ ln ( 1 R ) ln ( Ngi Ng f ) (3.10) and peak power Pp = hνAglg γtr ln ( 1 R )[ Ngi − Ngth − Ngth ln ( Ngi Ng f )] . (3.11) By the pulse energy Ep and peak power Pp finally the pulse duration tp ≈ Ep Pp (3.12) can be estimated. If the system is pumped continuously, the repetition rate frep of succes- sive pulses can be calculated by the upper laser level lifetime τg, the volumetric pumping rate Wp and the initial, as well as final occupation inversion Ngi and Ng f to [73] frep = 1 τg ln ( Wptg − Ngi Wptg − Ng f ) . (3.13) Figure 3.1 shows an exemplary numerical solution of the coupled rate equations 3.1 - 3.3 for a period of 40 µs in the case of a continuously pumped system. The equations were solved in Matlab using a stan- dard ordinary differential equation solver (ODE45) based on the Runge- Kutta method [72]. The time evolution of the occupation inversion of the gain medium and the saturable absorber is shown as blue and green line, respectively, while the resonator-internal photon density is shown as red line. The continuous pumping process causes the occupation in- 3.1 rate-equation model 49 20 25 30 35 40 45 50 55 60 0.0 0.5 1.0 time (µs)po pu la tio n de ns ity (a rb .u ni ts ) 0.0 0.5 1.0 ph ot on de ns ity (n or m al iz ed ) gain medium Ng saturable absorber NSA generated pulse φ Figure 3.1: Solution of rate equations 3.1 - 3.3 for repeated passive Q-switching based on simulation parameters of Table 3.1. The evolution of the occupation density of the gain medium (blue line) and saturable absorber (green line) are part of the left axis, while the resulting photon density is shown as a red line and belongs to the right axis. version in the gain medium to increase, while the spontaneous emission increases due to the increasing occupation inversion, which also leads to a limited increase of occupation inversion in the saturable absorber. At a certain point in time, the absorber suddenly switches through, causing the photon density in the resonator to increase abruptly and the energy stored in the gain medium to become apparent in a laser pulse. This reduces the occupation inversion in the gain medium and the process starts all over again due to continuous pumping. For a con- tinuously pumped system like this, the pump power and the resulting volumetric pump rate are used to set the duration of the build-up of the population inversion, which has a direct influence on the repetition rate of the system. For the simulation shown in Figure 3.1 and all subse- quent calculations in Chapter 4, the constants shown in Table 3.1 were used unless otherwise specified. 50 simulation and system design symbol parameter value unit σg stimulated-emission cross-section 2.1 × 10−20 cm2 lg length gain medium 5 mm σr reabsorption coefficient 0.2 × 10−20 cm2 τg relaxation time gain medium 90 µs T0 saturable absorber initial transmission 96 % R output coupler reflectivity 92 % L non-saturable loses 3 % lSA length saturable absorber 0.7 mm σGSA GSA cross-section 4.4 × 10−18 cm2 σESA ESA cross-section 2 × 10−18 cm2 τSA relaxation time saturable absorber 4 µs ωgain waist radius gain medium 100 µm ωSA waist radius saturable absorber 100 µm γ inversion reduction factor 2 ηq pump efficiency 45 % Ppump applied pump power 25 W σabs pump absorption coefficient 8 cm−1 ν pump frequency 371 THz lc resonator length 15 mm Table 3.1: Overview of the simulation parameters used for the rate equation analysis for a passively Q-switched laser based on Nd3+:YVO4 and Cr4+:YAG. 3.2 second threshold condition The laser threshold of Equation 2.2 was defined as the point where the gain exceeds unsaturated cavity losses and the intensity inside the cavity can grow at all. For passively Q-switched operation another ex- pression called the second threshold condition is of great importance [74, 75]. The second threshold condition defines whether the temporal derivative of the intracavity photon density is positive or negative. A positive derivative is linked to an increase in photon density due to 3.2 second threshold condition 51 dominating laser gain whereas a negative sign means the density de- creases as the saturable absorber loses are dominating. Practically this means that the saturable absorber losses have to saturate before satura- tion of the gain medium takes place to obtain Q-switched operation. Starting from the first formulation in 1978 [76], the second threshold condition was continuously evolved [45, 77] and finally extended by Bai et al. [78] to include the influence of ESA. In the case that the inequation of the second threshold condition is not fulfilled the laser tends to operate in CW operation [45, 56]. The second threshold condition is defined as ln(1/T2 0 ) ln(1/T2 0 ) + ln(1/R) + L σGSA σgain Again Asa > γ 1 − β , (3.14) where T0 is the saturable absorber initial transmission, R the reflectiv- ity of the output coupler, L the non-saturable dissipative cavity losses, σGSA is the stimulated-emission cross-section of the gain crystal, and Again/Asa the ratio of laser mode area and mode area in the saturable absorber. Further, γ is the population inversion reduction factor (one for a four-level system and two for a three-level laser) and β = σESA/σGSA the ratio of the ESA and GSA cross-section. For β < 1 due to predominating saturable absorption, it is obvious that the right side of Inequation 3.14 is always greater than two for a three-level laser. However, the first term on the right side is always sig- nificantly smaller than one due to a certain initial transmission, output coupler reflectivity, and non-negligible dissipative cavity losses. Conse- quently, both the ratio of cross-sections σGSA/σgain as well as the ratio of laser mode areas Again/Asa have to be noticeably greater than one, to fulfill the inequation. It is obvious that this is not naturally fulfilled and puts serious conditions on the design of a Q-switched laser system. For this reason, the second threshold condition (Inequation 3.15) in the case of a quasi-three-level laser system based on Nd3+:YVO4 as gain 52 simulation and system design medium and Cr4+:YAG as saturable absorber is calculated and investi- gated subsequently for typical system parameters. Therefore, the initial transmittance T0 of the saturable absorber and the reflectivity R of the OCM were varied in a certain range for the later laser design. In ad- dition, the ratio of the beam area of the laser mode and the area of the saturable absorber was assumed to be one, since the plane-parallel resonator should be rather short to obtain short pulse durations and consequently the spot diameter in the gain medium and saturable ab- sorber does not differ significantly. Additionally, an illustrative value of 3 % was estimated for the nonsaturable losses, while literature val- ues were used for the cross-sections of the saturable absorber (σGSA = 3.9 × 10−18 cm2 and σGSA = 1.4 × 10−18 cm2 [79]) and the stimulated- emission cross-section of Nd3+:YVO4 at 914 nm (σgain = 4.8 × 10−20 cm2 [34]). The constants used for the calculation are summarized in Tab. 3.2. To perform the simulations, Equation 3.14 was adapted to ln(1/T2 0 ) ln(1/T2 0 ) + ln(1/R) + L σGSA σgain Again Asa 1 − β γ > 1, (3.15) symbol parameter value unit T0 saturable absorber initial transmission 80-100 % R output coupler reflectivity 80-100 % L non-saturable loses 3 % σGSA GSA cross-section 3.9 × 10−18 cm2 σESA ESA cross-section 1.4 × 10−18 cm2 σgain stimulated-emission cross-section 4.8 × 10−20 cm2 ωgain waist radius gain medium 100 µm ωSA waist radius saturable absorber 100 µm γ inversion reduction factor 2 Table 3.2: Overview of the simulation parameters used to calculate the second threshold criterion. 3.2 second threshold condition 53 and then a parameter sweep was performed on the initial transmission, as well as the reflectivity of the OCM in Matlab using loops. The result of the simulation is shown in Figure 3.2 for different initial transmis- sions and output coupler reflectivities between 80-100 %. Due to the rearranged second threshold condition (Inequation 3.15), the passive Q- switched operation requires that the left part of the inequation is greater than one. Overall, the calculated values are for all combinations of initial trans- mission and output coupler reflectivities well above a value of two and consequently, the passively Q-switched operation should work in prin- ciple. The reason that in quasi-three level operation the second threshold con- dition is satisfied quite easily is due to the relatively small stimulated- emission cross-section at 914 nm (σgain = 4.8 × 10−20 cm2) compared to the ground state absorption cross-section of the saturable absorber (3.9 × 10−20 cm2). In comparison to that, for the four-level system, the literature value of the stimulated-emission cross-section at 1064 nm (1.2 × 10−18 cm2 [80]) is larger by a factor of 25, and consequently, care must be taken in the choice of design parameters to satisfy the second threshold condition. A possible solution, in this case, is to reduce the spot size on the saturable absorber compared to the spot size in the gain medium. This can be done for example by using a curved resonator mirror. The minimum in Figure 3.2 was reached with 0.99 for a purely hypo- thetical initial transmission of 100 % and an output coupler reflectivity of 80 % (upper left corner). Like the maximum value of 24.2, which was reached for an initial transmission theoretical output coupler re- flectivity of 100 %, the minimum value for a purely hypothetical ini- tial transmission of 100 % makes no physical sense. For an initial trans- mission of 100 %, the saturable absorber would be vividly completely transparent and the laser would not be passively Q-switched but would 54 simulation and system design 80 84 88 92 96 100 80 84 88 92 96 100 outcoupling reflectivity (%) in it ia lt ra ns m is si on (% ) 0 5 10 15 20 25 va lu e of se co nd th re sh ol d co nd it io n Figure 3.2: Calculated parameter sweep of the left side of the second thresh- old criterion over the initial transmittance of the saturable absorber and the re- flectance of the output coupler mirror according to the left side of Inequation 3.15). Hereby, simulation parameters of Table 3.2 were utilized. run in CW mode. On the other hand, an output coupler reflectivity of 0 % would mean that the output coupler is completely transparent and consequently the laser would not work because the optical feedback through the resonator is missing. 3.3 matrix formalism - cavity stability 55 3.3 matrix formalism - cavity stability The following derivation is adapted from the work of Kogelnik et al. [81] and Hodgson et al. [82]. Accordingly, in a paraxial approximation, the beam propagation of an optical system can be described by ray transfer matrices [83]. Assuming a uniform heat input, a gain crystal in the resonator can be approximated consequently as a thick lens, with the distance h of the principal planes from the end are given by h = l 2n0 , (3.16) where n0 is the refractive index of the laser crystal and l is the length of the gain medium [84–88]. Without a thermal lens, the system can be described via the stability parameter gi = 1 − (d1 + d2)/ρi i, j = 1, 2; i ̸= j (3.17) and the resonator length L = d1 + d2. (3.18) Here, d1 and d2 are the respective mirror spacing and ρi is the radius of curvature of the mirror i. For the propagation of a Gaussian beam, the concept of the stability parameters gi can be adapted, according to which the system with a thermal lens of refractive power D = 1/ f can be described via g∗i = gi − Ddj(1 − di/ρi) i, j = 1, 2; i ̸= j (3.19) L∗ = d1 + d2 − Dd1d2. (3.20) 56 simulation and system design d1 d2 θ1 θ2 ω2ω1 hh Figure 3.3: Schematic overview of the components and parameters of the ray transfer matrix model for calculating resonator stability as a function of thermal dioptric power. As shown in Figure 3.3, d1 and d2 correspond to the mirror distance from the principal planes of the crystal while L∗ is the effective res- onator length considering a thick lens. Analogous to a resonator without a thermal lens, a stability diagram can be defined via the g∗ parameters, where the resonator stability is now a function of the refractive power D. Such a diagram is exemplar- ily shown in Figure 3.4 for a 5 mm long Nd3+:YVO4 crystal (n = 1.9647 [89]), where the in-coupling resonator mirror is dielectrically deposited on the crystal (d1 = h, r1 = ∞) and the planar external output coupler mirror (r2 = ∞) has a distance of d2 =10 mm to the principal plane of the equivalent thick lens principal plane. These and other parameters used below are summarized in Tab. 3.3. Without a thermal lens, the plane-parallel resonator is located at