Optical and Quantum Electronics 8 (19 76) 5 2 3 - 5 2 9
Velocity measurement of a diffuse object
by using time-varying speckles
JUNJI
OHTSUBO,
TOSHtMITSU
ASAKURA
Research Institute of Applied Electricity, Hokkaido University, Sapporo, Hokkaido,
Japan
Received 12 April 1976, revised 11 June 1976
The relation between the velocity of a moving diffuse object and the mean number of integrated
speckles in the certain time interval at the far-field plane of the object is obtained. By using this relation
based on the first order statistics of integrated speckles, a new method of measuring the velocity of the
diffuse object is proposed and experimentally verified. The first order probability density function of
time-integrated speckles is also studied from which the validity of the present method is verified.
1. I n t r o d u c t i o n
Recently, the statistical properties of laser speckle patterns have attracted the special attention of many
investigators [1-8]. The recent book edited by Dainty [9] contains an extensive bibliography and can
be referred to for the past work on this subject. In parallel with this fundamental study, several applications for various scientific and industrial measurements using laser speckle patterns have also been contrived due to the superiority of their non-contacting measurements [8, 10-14]. One of those applications
is the velocity measurement of a diffuse object which is based mainly on the correlation or the power
spectrum of speckle intensity variations [12-14]. Velocity measurement of the particles undergoing
Brownian motion has been studied by Jakeman et al. [15] and Jakeman [16] on the basis of the moving
speckle phenomenon.
In this paper, a new simple method based on the first order statistics of speckle intensity variations is
proposed for measuring the velocity of a diffuse object. The method proposed here is based on the statistics of light intensity variations of time-varying laser speckles produced in the far-field region of an
illuminated area of the moving diffuse object and integrated over a fixed time interval. Among the
fundamental quantities characterizing the statistics of integrated intensity variations, the present method
uses only their mean and standard deviation in order to determine the velocity of the diffuse object.
However, the first order probability density function of time-integrated intensity variations of timevarying speckles is also examined in this paper since the validity of the present method is verified by
comparing the theoretical and experimental results of the probability density function. This probability
function is automatically produced and used in the present data processing of integrated speckle intensity variations. As has been generally assumed in velocity measurement, the diffuse object is again
assumed to have values of the rms (root mean square) surface roughness greater than the wavelength of
light used and the speckle field produced in the detecting region is also assumed to obey Gaussian statistics.
Before entering into the main subject of the present paper, the following two subjects which use the
statistics of the integrated light intensity and are similar to the present paper should be carefully pointed
out. The integrated intensity variations extensively used in this paper have been considerably studied in
the field of photoelectron counting in the determination of the spectral and statistical properties of
light beams. We can refer to a comprehensive review article [17] for the past work in this field. Among
the papers in this field, the probability density function of integrated intensity variations has been dis9 1976 Chapman and Hall Ltd. Printed in Great Britain.
523
J. Ohtsubo, T. Asakura
524
cussed in detail, for example, by Jakeman and Pike [18] and Mehta and Mehta [19]. In addition, the
statistics of laser speckles spatially integrated by using a finite-sized aperture has been studied by several
investigators [1-6], while that of moving speckles integrated in a finite time interval has been investigated by McKechnie [7]. These statistics of space-integrated speckles are quite similar to the theoretical
background of the present method.
2. The first order statistics of speckle field
Consider first the formation of speckle patterns in the far-field region of a transmitting (or reflecting)
diffuse object whose rms surface roughness is greater than the wavelength of illuminating light used.
Completely spatially coherent light is employed for illuminating the diffuse object moving with constant
velocity. The speckle pattern varying in time is then observed in the far-field region of an illuminated
area of the object and used for velocity measurement of the object. We now investigate the problem of
the statistics of speckles integrated over a fixed time interval after they are detected by a photodetector
having a sufficiently small aperture. This problem is equivalent to measuring speckles by the photodetector having a finite-size aperture [1-6] and to determining the statistical properties of the integrated
light intensity from photocounting measurements [17-19].
The autocorrelation function R v(x, x; t, t') of the speckle amplitude varying in time at a certain point
in the far-field region of the diffuse object is written as
nv(x, x; t, t') = (V(x, t)V*(x, t'))
(1)
where V(x, t) is the amplitude of speckles produced at a point with co-ordinate vector, x in the far-field
region at time t and the symbol (...) indicates ensemble averages. In general, the stationarity [7] is
assumed for the time autocorrelation function Rv(x, x; t, t') of speckle patterns so that it becomes
Rv(xl x; t, t') = Rv(x; t -- t') = Rv(t -- t')
(2)
The intensity of speckle patterns produced at the far-field plane by the diffuse object moving with
constant velocity is detected by a photodetector and the detected signal is next integrated over a timed
time interval [0, T]. Then the total intensity integrated within the time interval, T, is given by
I=
l fro lV(x, t)12dt.
(3)
The random complex amplitude V(x, t) of speckle patterns is expressed by the Karhunen-Lo~ve expansion using an orthonormal set of the functions, ffk (t):
V(x,t) = V(t)= ~ akqJk(t).
(4)
k=O
The function ffk(t) in this equation satisfies the following homogeneous Fredholm integral equation
Xkfk(t) = f ro Rv(t-- t') ~0k(t')dt'
(5)
where Xk is a set of the eigenvalues. The orthonormal set of the functions ~Ok(t) is given by a solution of
the above homogeneous Fredholm equation (5) with the kernel Rv(t -- t'). By substituting Equation 4
into Equation 3, the integrated intensity of speckles within the time interval, T, can be written by
I = ~1 ~ lat~12.
(6)
It is now assumed that the surface area of the diffuse object contributing to the formation of speckles
at any point of the far-field plane consists of a very large number of independent scattering points. With
this assumption, the varying amplitude of speckles follows a complex Gaussian process and the coefficients ak of the orthonormal functions ek (t) become independent complex Gaussian variables with
Velocity measurement of a diffuse object by using time-varying speckles
525
zero mean and variance kk. By using these properties, the probability density function of variations of
the integrated speckle intensity in the time interval T at the far-field plane is obtained as the inverse
Fourier transform of the characteristic function (exp (izl)) of the integrated speckle intensity, I, and
given by
=
P(I)
f_o
**(exp (izI)) exp (-- izI)dz
1 fexp (-- izI)
- 2-rr ~k~o [1-iz(X~lT)] d z
(7)
If we assume that independent speckles of the mean number, N, are detected in the time interval T and
that the intensity of each of these speckles holds the same weight in a statistical sense, i.e. Xa = Xo
(k = 0, 1 . . . . . N - 1) and kk = 0 (k ~>N), then Equation 7 in a normalized form reduces to
N N I N-1 exp
P(I/(1)) =
(--NI)
(8)
F(N)
where the mean number is given by
(1) 2
N
-
(12) _ (1)2
(9)
and F(N) is a gamma function. Equation 9 indicates that the mean number, N, of integrated speckles
within the fixed time interval T is obtained by finding the mean value (corresponding to the numerator)
and the square of the standard deviation (corresponding to the denominator) of integrated intensity variations of time-varying speckles. The discussion given here for the integrated speckle intensity has been
conducted in a similar way by Parry [20] in the statistics of polychromatic speckle patterns.
3. Velocity of the diffuse object
We next consider the average size of speckles varying in time due to the motion of a diffuse object and
produced at the far-field plane of the object. The experimental geometry for the formation of speckles
due to a moving transmitting object is shown in Fig. 1. The light amplitude of speckle patterns at time t
at the far-field plane, a distance R away from the diffuse object, is given by
V(•
=f_ exp - - - ~
ir
exp t - - ~ - u . x ] d u
(10)
where w is the radius of the illuminating beam at the object plane, r
is a random phase of the diffuse
object at the co-ordinate vector u, v is the velocity vector of the moving diffuse object, and X is the
wavelength of light used. In Equation 10, the laser beam of a single Gaussian mode, having the radius w,
is employed for the illumination.
T
] _i-f--. _ . . R
~ .
~o
-cell
6,,
Diffusing
object
Diffraction
plane
~A~_m_m~7-e~-~ g--r~e~'
[ '~[,'.:.'~' "l----tComputerl
,
C
o
signal I
generator
L__
nerotor-]
n
t
Figure I Optical arrangement for producing
speckle patterns and the signal-analysing system of
detected speckles.
526
J. Ohtsubo, T. Asakura
If the amplitude V(x, t) of speckles varying in time obeys a complex Gaussian process, the time autocorrelation of variations of the speckle intensity at the far-field plane is characterized by a normalized
absolute square of the second order correlation of V(x, t):
Rt(x;t--t'
) =
[(g(x, t)V*(x, t')) lz
i(I V(x, t)t2)12
(11)
By substituting Equation 10 into Equation 11 and knowing that the phase ~(u) of the diffuse object
varies greatly within a certain range, the resultant integrand of the ensemble average of the phase term
in Equation 11 may be approximately expressed by
( exp {i~b(u1 -- vt) -- iq~(u2 -- vt')} ) = 8 (u 1 -- u2 - vr)
where r = t -- t' and 8(u) is a delta function. Then the normalized absolute square RI(X;
correlation function becomes
RI(x;7-) = exp
w2-----]
(12)
7-) of the
auto(13)
where the replacement of v = Iv[is taken. The time correlation length of speckle intensity variations at
the far-field plane due to the moving diffuse object with constant velocity v may be defined by
W
r = -.
v
(14)
By means of this correlation length of speckles varying in time, the mean number, N, of speckles detected within the time interval, T, is given by
vT
N = -(15)
2w
from which the velocity of the moving object is determined using the relation
2w
v = --N
T
-
2w
(132
T (12) -- (1)2.
(16)
In the experiment, the beam radius, w, of the illuminating laser light at the object plane is measured
beforehand and the integrating time interval, T, of moving speckles at the far-field plane of the diffuse
object is made fixed. Then, if we obtain the mean and square-mean values of time-integrated speckle
intensity variations, the velocity of the object can be determined by Equation 16. In other words, the
velocity of the moving diffuse object is measured experimentally by the first order statistics of integrated speckle intensity variations over the fixed time interval.
4. Experiments and results
The optical arrangement for producing speckle patterns at the far-field plane of the moving diffuse
object and the signal-analysing system for the detected speckles are schematically shown in Fig. 1. A
single mode He-Ne gas laser with a wavelength of 6328 A having a Gaussian form, is focused on a transmitting diffuse object moving with constant velocity, v. The radius, w, of the Gaussian beam focused on
the diffuse object is given by
w =
xf
rrw i
(17)
where w i is the radius of the Gaussian beam incident directly on the lens and f i s its focal length. The
radius w of the Gaussian beam is chosen as 43/am in the present experiment. The rotating ground glass
having an rms roughness of about 3.5/am is used as the moving diffuse object. By changing the distance
of the illuminationg spot from the rotating centre of the ground glass, various velocities of the diffuse
object are prepared. The diffuse object rotates across the illuminating Gaussian beam with constant
Velocity measurement o f a diffuse object by using time-varying speckles
527
angular velocity and this angular velocity is theoretically calculated and measured beforehand for comparison with the velocity to be experimentally obtained from speckles.
A speckle pattern is produced at the far-field plane of the illuminated diffuse object and its intensity
distribution varies randomly in time and space due to the rotation of the ground glass. In this experiment,
the distance R from the object plane to the detecting plane is taken as 50 cm which satisfies the far-field
condition. The speckle intensity variations at the far-field plane is detected by a photo-cell together with
a pinhole which is made much smaller in size than the average spatial grain size of speckles (therefore,
the photo-cell faithfully records the speckle intensity variations).
The signal current, i(t), from the photo-cell is firstly amplified and passes through the gate 1 which
repeats on- and off-states with period T by means of the clock. It is brought into the integrator and
integrated within the time interval T by an order of the control generator. Next the integrated value, I,
proportional to the total speckle intensity in the time interval T, passes through the gate 2 and is stored
in the pulse-height analyser. At this stage, the probability density distribution P(I/<I)) of integrated
speckle intensity variations I, which is given by Equation 8, is obtained as a function of the mean number, N, of speckles detected in the time interval T. From the obtained probability density distribution,
the mean and square-mean values of integrated speckle intensity variations are calculated by a computer
and then the mean number, N, of speckles detected in the time interval T is calculated from Equation 9.
The gate time T is chosen as 8 ms in this experiment. Therefore, the velocity of the moving diffuse
object is obtained from Equation 16 as a function of the mean number N of detected speckles in the
time interval T:
(I) z
v = 10.8N = 10.8 •
2.
(18)
As an example and for comparison of experimental and theoretical results, Fig. 2 shows the probability density functions obtained from the experiment and calculated theoretically from Equation 8. For
calculating the theoretical curves, the values of the mean number N obtained experimentally are used.
The values of the mean number N are determined experimentally from the variance of integrated speckle
intensity variations with the assumption that the approximate probability density function of Equation
8 holds. Fig. 2a shows the experimental curves of the probability density function for four different velocities of v = 86, 198,440 and 845 mm s-1 . The mean numbers corresponding to these velocities are
N = 8.3, 20.4, 39.3 and 80.0. Fig. 2b shows the theoretical curves of the probability density function
4.0
3.0
D
N
8,4
20.4
39,3
V(mm s~')
86
198
440
4,0
A
B
C
fl
D
80.0
845
3.0
~2.0
N
A
B
C
D
8
20
39
80
v(mm s~)
90
220
422
860
~2.0
B
1.0
1,0
1.0
[/<I>
(a)
2.0
310
00
1.0 i/<I> 2.0
(b)
3~0
Figure 2 Probability density distribution P(I/(I)) of integrated speckle intensity variations for four different velocities
of the diffuse object. (a) Experimental probability density distributions and (b) theoretical ones corresponding to (a).
528
J. Ohtsubo, T. Asakura
xl0~
1C
y
/
.y-
6
..//"
E
'
/Z."
/
./
Figure 3 Measured velocities of the diffuse object as a
/~
2'o
I
40
N
t
60
I
80
function of the mean number N of integrated speckles in
the fixed time interval. Black circles indicate the experimental values obtained from speckles while the straight
line stands for the theoretical curve.
corresponding to Fig. 2a. Comparison of Fig. 2a and b indicates that the experimental curves agree very
well with the theoretical ones. For the small mean number N of integrated speckles the experimental
results of the probability density function are slightly different from the theoretical ones (see curve A
of Fig. 2a and b). This deviation of the experimentally obtained probability density function from the
theoretical one may be due to the statistical uncertainty associated with the experiments. For the small
mean number of integrated speckles, the fluctuation of the moving ground glass plate becomes significant and its effect cannot be neglected. Accordingly, the fluctuation of the object motion is not statistically averaged out in the integration time interval. Another cause for the deviation between the experimental and theoretical results may be due to the approximation of the probability density function by
means of Equation 8. Namely, the exact eigenvalues of the eigenfunction in Equation 5 do not always
take the same value [5] for any values of N and, therefore, for the small mean number N of integrated
speckles the deviation from the gamma distribution occurs due to the replacement of eigenvalues by a
step function.
Fig. 3 shows the velocity of the diffuse object as a function of the mean number N of integrated
speckles. The black circle indicates the experimental values of velocity, against the mean number of
integrated speckles, obtained from the probability density functions shown as in Fig. 2a. The straight line
stands for the theoretical curve obtained from Equation 18. The deviation of experimental values of the
velocity from the theoretical ones is within about + 10% in the present experiment. The experimental
values of the velocity of the diffuse object are in good agreement with the theoretical curve of the velocity.
The mean number N of integrated speckles in the time interval, T, required to measure precisely the
velocity of the diffuse object is roughly about 10-100. For the small mean number N of integrated
speckles, the deviation of the experimental probability density function from the theoretical one occurs
as was already stated in Fig. 2, while for the large number N the width of the probability density function becomes narrow like a delta-like function and, consequently, the noise of the signal-analysing system cannot be negligible.
5. Conclusions
The relation between the velocity of a moving diffuse object and the mean number of integrated
speckles in the time interval T at the far-field plane of the object is clarified and the velocity of the diffuse object is determined using this relation based on the first order statistics of time-integrated speckle
intensity variations. The optical system for measuring the velocity of the diffuse object is very simple
Velocity measurement of a diffuse object by using time-varying speckles
529
in comparison with other correlation methods. Furthermore, the signal-analysing system is also simplified since the present method is experimentally based on the first order statistics of speckles produced
in the far-field region. On the other hand, this technique of velocity measurement has several disadvantages. Namely, the method can not be applied to the measurement of the varying velocity during the
observation and is also limited for the diffuse object having a sufficiently rough surface in comparison
with the wavelength of light used. For the weak diffuse object whose rms surface roughness is less than
the wavelength of light, the variance of integrated speckle intensity variations in the certain time interval will be inversely proportional to the velocity of the object. Another disadvantage of the present
method is that the direction of the velocity cannot be measured.
Since the present study is a preliminary experimental investigation for verifying the proposed method
of measuring the velocity of the diffuse object, a further precise measurement of the velocity will be
done by adjusting the optical system more rigidly together with the constants of the radius w of the
illuminating beam and of the integrating time T. The present technique may be very useful for noncontacting measurement of the velocity of a diffuse object.
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