Analysis of Mechanical Oscillations by Speckling
H. J. Tiziani
The application of the speckle phenomenon to the analysis of in-plane translations and oscillations is
first reviewed briefly. Then a practical method of investigating out-of-plane rotations (tilts) even in the
presence of in-plane movements is studied. This goal can be achieved by recording the speckle patterns
in the Fourier transform plane before and after the tilt or as a time-average exposure for an oscillating
object. Young's fringes related to the tilts are observed in the Fraunhofer diffraction pattern when the
developed photographic plate is illuminated with coherent light. This leads to a very simple engineer-
ing tool for the analysis of movements. Theoretical and experimental results will be shown.
I. Review of Previous Work
An optically rough surface illuminated with highly
coherent light, such as from a laser, leads to a speckled
appearance everywhere in space. The main features of
the speckle patterns in the Fourier transform plane as
well as in the image were investigated recently by a
number of authors. 1-4
The application of the speckle phenomenon for the
measurement of in-plane translations and rotations was
investigated lately.'- 2 In addition, this technique
was found convenient for the investigation of in-plane
oscillations.10-12 Vibratory movements perpendicular
to the object surface were studied by the observation of
the contrast reduction in the speckle patterns."3-" An
object surface with a given roughness leads to a particular
speckle pattern in the Fourier transform plane as well as
in the image plane. The smallest dimensions of the
individual speckle in the image and in the Fourier
transform plane are inversely proportional to the
lateral dimensions of the lens aperture and the illumi-
nated object field, respectively.1-4 Let us consider the
surface itself to be characterized by the coherent speckle
pattern generated by illuminating it with laser light.
To simplify the argument this speckle pattern may be
considered to be attached to the optically rough object.
Photographing the speckle patterns before and after the
movement of the surface has taken place will lead to
two practically identical, but shifted speckle patterns
from which the in-plane translation or rotation can be
derived by analyzing its optical diffraction effect.
It was shown 6-" that by recording two identical, but
The author is with the Institute of Applied Physics, Swiss
Federal Institute of Technology (ETH), 8049 Zurich, Switzer-
land.
Received 29 February 1972.
shifted speckle patterns, by double exposure in the
image plane, for instance, Young's fringes are obtained
in its Fraunhofer diffraction pattern when illuminated
with coherent light. The focal plane of the lens where
the fringes are studied is referred to as the Fraunhofer
diffraction plane (FDP). The fringe spacing is in-
versely proportional to the translation of the speckle
pattern. In hologram interferometry, the amplitude of
vibration of a surface (usually in the direction of the
perpendicular) can be measured by applying the well-
known time-average technique, 6 in which a single
exposure is made while the surface is harmonically
oscillating. A corresponding technique can be used for
measuring vibratory motions by speckle photography.
A single exposure is recorded in the image plane for in-
plane oscillations even in the presence of small out-of-
plane movements.
The intensity distribution obtained in the FDP of the
image plane, where the speckle patterns before and
after the movements or as a time-average exposure
(exposure over a number of oscillations) were recorded, is
given by the square modulus of the autocorrelation
function of the exit pupil of the lens forming the image
of the object multiplied by
(1) a (cosine) 2 function for a double exposure with an in-plane
translation between the exposures,
(2) a (zero-order Bessel function)2 for a time-average re-
cording of a harmonically oscillating object or image,
(3) a (sinx/x)' for a time-average recording for a translation
with continuous velocity.
From the fringe spacing p we obtain the in-plane trans-
lation A/ and rotation as well as the amplitude of oscil-
lation 0o:
A = XfI/Mp, o = 0.76(Xfu/Mp,),
where f, is the focal length of the lens or the distance
between the developed photographic plate and the focal
December 1972 / Vol. 11, No. 12 / APPLIED OPTICS 2911
/Fig. 1. Photograph of the tuning fork.
plane depending on the particular arrangement for ob-
taining the Fraunhofer diffraction pattern, 11 is the
lateral magnification, and X is the wavelength of the
light used in vacuum. The separation from the origin
to the first minimum of the Bessel function of order zero
(and first kind) is denoted by p1/2 .
Since the interference fringes are perpendicular to the
direction of the displacement, the orientation (azimuth),
but not the direction of the translation, can be deter-
mined. If the movements for the different regions of
the object are different in amplitude and phase, the
entire object can be recorded before and after the in-
plane movements or as a time-average exposure, and the
developed photographic plate can be illuminated after-
wards, point by point, with an appropriately narrow
laser beam to select the region of constant movement.
The fringes are then to be studied in the FDP. The
stress of a material can be derived by measuring the
change of width of the fringes from neighboring portions
of the object.
The tuning fork of an electronic watch is introduced
in Fig. 1. Examples of the fringes displayed in the
FDP of the speckle patterns recorded in the image plane
are shown in Fig. 2. The corresponding regions of the
tuning fork are marked in Fig. 1. The experimental
details will be given in Sec. III.
Small in-plane translations (amplitude and azimuth)
of an oscillating object can also be measured. The
time-average speckle pattern before and after the
translation can be recorded in the image plane. In the
FDP two fringe systems are obtained. The (cosine)2
and the Jo2 fringes are superimposed, leading to the
analysis of the two movements. A typical example is
demonstrated in Fig. 3, where the (Bessel function of
order zero) 2 was modulated by the (cosine) 2 fringes (two
dark fringes symmetrical to the center).
Small movements orthogonal to the object surface
and tilts lead to a reduction in the contrast of the
fringes obtained. It should further be noted that the
absolute phase of the light illuminating the object is not
important in the above application, and the require-
ments on the temporal and spatial coherence of the light
used are rather relaxed compared with those of holo-
graphic techniques. However, the in-plane movements
to be measured should be larger than the smallest
speckle size in the object space (given by the aperture of
the image-forming system) and not larger than, say,
one-tenth of the lateral dimension of the object. For
the analysis of very small movements the holographic
techniques are the more sensitive than the speckle
methods. A moderately high aperture lens is needed
for recording the speckle patterns to obtain a number of
fringes with good contrast in the FDP.
I1. Study of Diffraction of Speckle Patterns
Recorded in Fourier Transform Plane of Object
Holographic techniques suitable for the study of
three-dimensional movements were analyzed recently
by a number of authors. The interpretation of the
fringes in hologram interferometry is usually rather
difficult for a nonexpert, especially when larger amounts
of tilts (rotations with the axis of rotation orthogonal to
the line of sight) occur in the presence of considerable
in-plane translations. Some of these problems are
overcome in the present study where the speckle pat-
terns are recorded in the Fourier transform plane of the
object before and after a tilt or as a time-average expo-
sure for an oscillating object.
The novel feature of the investigation reported here is
that the shift of the speckle pattern in the Fourier
(a) (b)
Fig. 2. Photographs of the fringes obtained in the FDP of a time-average exposure of a speckle pattern of the tuning fork recorded in
the image plane (a) at the center of the symmetrical tuniing fork where pol = 9.5 jim, (b) at position 2 whcre P02 = 5.4 Am.
2912 APPLIED OPTICS / Vol. 11, No. 12 / December 1972
Fig. 3. Fringes displayed in the FDP when the central portion
of the image with speckle patterns was illuminated. The image
was recorded in the time average before and after a lateral dis-
placement of 5 ,4m (ol = 9.5 jim).
transform plane is due to a tilt in another plane, the
object plane. A tilt of an angle y introduced into the
object rotates the speckle pattern, leaving the object as
a whole by an angle, which is 2 y for coherent light
incident and scattered approximately along the surface
normal, i.e., the surface acts like a mirror. An oblique
illumination, however, is usually more convenient for
practical applications. A rotation of the original
speckle pattern through a different angle depending on
the angle of incidence of the light was found.
We assume that the shift of the speckle pattern result-
ing from a tilt of the object is larger than the smallest
speckle dimension. Let the optically rough object
surface described by A(t) be illuminated by a plane
wave incident at an angle as shown in Fig. 4. The
amplitude on the object may be written as
AoQ) = A(t) exp[i(2r/X)t sinf]. (1)
A tilt of an angle y introduces a phase change into the
original wavefront as will be shown. The one-dimen-
sional notation is used to simplify the writing. The axis
of rotation at H is orthogonal to the line of sight (opti-
cal axis) as shown in Fig. 4. The point P moves to
P2 after a tilt y resulting in an optical path change of
[QP2]. Assuming the refractive index in air to be 1, we
have
[QP2J = 2( - ) sin(-y/2) cos[3 - (/2)]
; -(tto)7 cosfl (2)
for small y's. In our model it is assumed that, at least
for small angles of tilt, the relative phase variations due
to the roughness of the surface remain practically
unchanged, but rotated by an angle ny, so that the
complex amplitude just in front of the tilted object is
A (t) exp[i2rt(sing/X)] exp[i27r(y/X)tI
X exp[i2r - o)(Qy/X) cosfl].
The laterally shifted speckle pattern in the Fourier
transform plane can therefore be written as
a[x - (sin,3/X) - (y/X)(1 + cos,6)J exp[i2(ro/X)-y cosfl], (3)
where x = X/Xf, with X = coordinate in Fourier trans-
form plane (Fig. 4) and f = focal length of the lens used
to perform the Fourier transform of the object, a(x) =
Fourier transform of A ().
The intensity i(x) recorded in the Fourier transform
plane of the doubly exposed speckle patterns before and
after a tilt is therefore
i1(x) = a[x - (si#/X)112
+ a[x - (sinj#/X) - (y/X)(1 + cosl)]12. (4)
For a harmonically oscillating object illuminated by a
single plane wave incident at an angle , the time-aver-
age intensity photographed in the Fourier transform
plane can be written as
(i2)) = (a[x - (sin#/X) - (1 + cosf)(y0/X) cos2rptJJ2)t, (5)
where v is the frequency of oscillation and yo the maxi-
mum angular tilt (rotation with the axis of rotation
perpendicular to the line of sight). For an intui-
tive argumentation it should be noted that the laterally
oscillating speckle pattern in the Fourier transform
plane remains almost stationary for a short time near
the extreme positions. After photographic development
of the plate, which is either doubly exposed or recorded
as a time-average exposure, another Fourier transform
in coherent light is performed to obtain the fringes in the
FDP:
A'(u') = f T(x)(expi27ru'x)dx, (6)
where T(x) is the amplitude of transmission of the
developed photographic plate and u' = i'f/f,, with i' =
coordinate in the FDP and f, = focal length of the
second lens or distance between the plate and focal
plane, as pointed out in the first section.
By considering the amplitude transmittance of the
photographic material to be proportional to the re-
corded intensity, 7 the intensity in the FDP can be
written for the doubly exposed plate (before and after a
tilt Y),
1' (u') = 2A'(u') A'(-uI)12
X {1 + cos[2ru'(y/X)(1 + cosi3)]},
-o
I0
- a)
0 0
(7)
a)
Cq
0L
a)
Fig. 4. Arrangement to record the speckle patterns in the
Fourier transform plane.
December 1972 / Vol. 11, No. 12 / APPLIED OPTICS 2913
and for the time-average recording of the harmonically
oscillating object, for an exposure time to >> 1/v
I2'(ul) = A'(u') ® A*(-U)2 JO2[27r(y0/X)u'(1 + cosi3)]. (8)
Since A'(u') is the Fourier transform of a(x) and 0
indicates convolution and Jo is the Bessel function of
order zero and first kind, we obtain in the FDP an
intensity proportional.to the square modulus of the
autocorrelation of the illuminated object field, multi-
plied by (cosine) 2 for a tilt or the (zero-order Bessel func-
tion) 2 for an oscillating tilt or (sinu'/u') 2 for a continuous
tilt.
From the width of the cosine fringes A' [in expres-
sion (7)] the tilt angle in the object plane is found to be
= Vfi/f(l + cosW)zt' (9)
and the maximum amplitude yo of the angular tilt [from
expression (8) ], by denoting the separation of the first
minimum from the origin by al/2, as
,yo = .76Xf1/f(l + cos3)al. (10)
An interesting observation can be made when the time-
average speckle pattern is first recorded and then super-
imposed on the speckle pattern of the stationary object.
This yields a modified fringe pattern in the FDP,
namely,
Ia(u') = IA'(u') X A*'(-u')I2
X K + J[2ir( 70,/X)u'(1 + cosji)]J1, (11)
where K is a constant depending on the illumination of
the stationary and time-average speckle patterns. If
K > 1Kol (Ko = amplitude of the first minimum of Jo)
and Jo' > 0, the minima for I3'(u') appear for Jo' = 0.
This result may sometimes be useful when larger move-
ments are analyzed with the same experimental con-
figuration; examples are shown in Fig. 7(d) and (e).
111. Experimental Arrangement and Procedure
In our applications an He-Ne laser of 18-mW power
with a wavelength of 632.8 nm was used as a light
source. For convenience, Agfa-Gevaert Scientia 10E75
plates and films were chosen for registration of the
speckle patterns. They have the necessary fine grain
and give minimum grain noise scatter. The exposure
times were of the order of a few seconds. No special
care was needed during the processing of the plates and
films to obtain fringes of good contrast. No additional
treatment of the surface such as painting was needed.
For the analysis of the in-plane movements the speckle
patterns were recorded in the image plane with a lateral
magnification = -3 and f ratio f/2.8. Most of the
work presented was carried out to study the mechanical
oscillations (in-plane and tilts) of the tuning fork of a
new design of an electronic watch (Fig. 1). The reso-
nant frequency of oscillation was 1050 Hz. The total
length of the tuning fork was 20 mm. The fringes
obtained in the FDF from the lateral in-plane move-
ments are displayed in Fig. 2 for the appropriate posi-
tion of the tuning fork. They result from the time-
(b)
As
14  

It 
!
I.
/ i 
( p
J , I' j f 
lj
' I ! 

 J I
(c)
Fig. 5. Photographs of the fringes obtained in the FDP from
the speckle patterns recorded in the Fourier transform plane of an
Al plate, illuminated by a plane wave ( = 45°). (a) Doubly ex-
posed speckle patterns with a tilt -y = 60 see of arc introduced
between the two exposures (-yt - in Fig. 4), (b) two doubly
exposed speckle patterns with a tilt e = 60 see of arc between the
first and second exposure and a tilt -y,7 = 60 see of arc perpendicular
to yt between the second and third exposure, (c) microdensitom-
eter trace of (a) slightly below the center.
2914 APPLIED OPTICS / Vol. 11, No. 12 / December 1972
(a)
IS [ mm-i
Fig. 6. The inverse of the fringe spacings obtained in the FDP
is plotted against the tilt angle for tilts y~(Qy = y in Fig. 4) and
tilts y,, perpendicular to y marked by 0. The results for tilts -Yr
with an additional translation of 40 m are marked by .
(, = 450).
average recording of the speckle patterns in the image
plane.
The experimental arrangement for the study of tilts
is shown in Fig. 4. In addition a jig was designed to
introduce known tilts into the object illuminated by a
plane wave. For comparison, the tilts introduced were
measured very accurately by autocollimation. It
should be noted that the axis of rotation must not coin-
cide with the object surface under test, as pointed out in
Ref. 18. The developed photographic plates and films
were then illuminated by a converging spherical wave
from the He-Ne laser, and the fringes were studied in the
rear focal plane (FDP).
Figure 5 shows the fringes obtained in the FDP from a
doubly exposed small Al plate (lateral width 6 mm),
where a tilt y = 60 see of arc was introduced between
the two exposures. A plane wave was used to light the
object at an angle 3 =45'. The tilt fringes in the FDP
for two doubly exposed speckle patterns of a larger
metallic surface are displayed in Fig. 5(b), where a tilt Yt
of 60 see of arc was introduced between the first and the
second exposure ( = y in Fig. 4) and a tilt , of 60 see
of arc perpendicular to ye was added before the third
exposure. The two fringe systems are equidistant and
perpendicular to each other. In addition, the fringes
that would occur by introducing y and y, in turn,
before and after a double exposure, are also visible. In
Fig. 5(c) a microdensitometer trace recorded slightly
outside the central portion of Fig. 5(a) is shown. The
results of a number of experiments where known tilts
Pyt and y,, were introduced are presented in Fig. 6, where
the inverse of the fringe spacings 1/As, is plotted,
against the angle of tilt. In-plane translations of 40
Am were added to the tilt in some experiments. They
had, however, no consequences on the results obtained,
as shown by the points marked ®9 in Fig. 6. Therefore,
tilts can be measured accurately even in the presence of
small in-plane translations. The results for different
(a) (b) (c)
Cd) (e)
Fig. 7. Photographs of the fringes displayed in the FDP for the time-average speckle patterns recorded in the Fourier transform plan
of region 2 in Fig. 1. The resonant frequency was about 1050 Hz, and the voltages applied to the coils were U = 1.0 V, 1.5 V, and
2.5 V for Fig. 7(a), (b), and (c) yielding tilts in the azimuth perpendicular to the fringes; (d) and (e) show fringes obtained in the FDP
of the doubly exposed speckle patterns (stationary and time-average) for U = 2.5 V and 3.5 V.
December 1972 / Vol. 11, No. 12 / APPLIED OPTICS 2915
[m]
(a)
U Volt]
1 1.5 2 2.5 3 3.5 4
Y0[10-41
(b)
the time-average speckle pattern in the image plane. A
fringe pattern obtained this way is shown in Fig. 9.
The two bright fringes symmetrical to the central fringe
indicate an out-of-plane movement in the presence of an
in-plane oscillation of the tuning fork near 2 in Fig. 1.
A detailed analysis of the three-dimensional movements
with two-beam illumination will be given in a separate
publication. By contrast with one-beam illumination
the fringes of Fig. 2(b) were obtained.
The fringes in the FDP of a time-average speckle
pattern recorded in the Fourier transform plane of an
oscillating quartz ( = 8000 Hz) are shown in Fig. 10.
IV. Conclusions
The method described may become a very useful
engineering tool and has many potential applications.
No reference wave is needed, and the practical im-
plementations of the analysis are not very demanding.
No special processing and no sophisticated equipment is
necessary to perform the investigations. A conven-
tional camera can be used to photograph the speckle
patterns. In-plane oscillations can be studied in the
presence of small tilts by recording the speckle patterns
in the image plane. By contrast, tilts can be measured
with high accuracy, even in the presence of small in-
plane translations, by a double-exposure, or time-
U t Volt ]
1 1.5 2 2.5 3 3.5 4
Fig. 8. Amplitudes of the mechanical oscillations as function
of the voltages applied for (a) in-plane movements of the central
portion 1 and (b) tilts yo of position 2.
incident angles and for one- and two-beam illumination
are given elsewhere. 8 Very good agreement was
obtained between the theory based on the proposed
model and the experiments.
For the measurements of the rotational oscillations
(tilts) even in the presence of a lateral oscillation, the
time-average speckle pattern was recorded in the
Fourier transform plane. The corresponding fringes in
the FDP are shown in Fig. 7(a), (b), and (c) for different
voltages applied to the coils of the electromagnetic head.
By contrast, the fringes obtained in the FDP from the
superposition of the speckle patterns of the stationary
object on that of the time-average recording are shown
in Fig. 7(d) and (e) for higher voltages (larger tilts).
The exposure for the stationary object was chosen to
be half that of the oscillating one, namely, 2 see (f =
50 mm, f/4, and f, = 120 mm).
The amplitudes of the lateral in-plane mechanical
oscillations and tilts of the tuning fork (Fig. 1) are
plotted in Fig. S. As a consequence of the results a
further study of the shape of the tuning fork may be
carried out to reduce the tilts. Results obtained by
holographic techniques of the tuning forks are presented
in Ref. 11.
For some of our special applications it was found
convenient to apply two-beam illumination and record
Fig. 9. Fringes in the FDP of the speckling recorded in the
image plane for two incident light beams (position 2 in Fig. 1)
for U = 2.5 V ( - -2).
Fig. 10. Example of the fringes in the FDP obtained from a
time-average speckle pattern recorded in the Fourier transform
plane, of an oscillating quartz.
2916 APPLIED OPTICS / Vol. 11, No. 12 / December 1972
Is W
1514
14-
13
12
11
10
a -
7-
6-
5 -
4-
3-
l l
average, recording of the speckle patterns in the Fourier
transform plane. For some applications it may be
useful to reimage the recorded speckle patterns by a lens
with an aperture stop offset from the axis placed in its
rear focal plane to act as a filter. In this way it is pos-
sible to select the appropriate part of the motion. The
method proved to be very accurate (better than 3%) for
translation as well as tilt measurements. For tilt
measurements the angle of incidence of the illumination
has to be known. The 3% limit was mostly imposed by
the precision of our experimental equipment.
The new speckle method was found to be very
convenient and accurate for our applications. The use
of the speckle phenomenon for three-dimensional dis-
placement and vibration analysis can complement
holographic methods in many engineering applications.
They operate best at sensitivities slightly lower than
that of holographic interferometry. The requirement
for our application of the speckle phenomenon is that
the displacement of the speckle pattern in the appro-
priate plane due to a tilt of the object or a translation be
greater than the smallest speckle width in the corre-
sponding plane.
The author would like to thank the Director of the
Institute of Applied Physics and the Department of
Industrial Research at the ETH Zurich, E. Baumann,
for support of the work. Thanks are further due to A.
Greuter for providing the tuning fork and A. Aemmer
for helping with the preparation of some of the experi-
ments.
References
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Duncan Macdonald who was director of the Boston University Optical Research Laboratory when this photograph
was taken in 1946.
December 1972 / Vol. 11, No. 12 / APPLIED OPTICS 2917