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Laser speckle reduction by ultrasonic modulation
Volume 27, number 1
OPTICS COMMUNICATIONS
October 1978
LASER SPECKLE REDUCTION BY ULTRASONIC MODULATION Yoh IMAI and Yoshihiro OHTSUKA Department of Engineering Science, Faculty of Engineering, Hokkaido University, Sapporo, Hokkaido, Japan Received 27 June 1978
A new method is proposed for reducing speckle in the far field. The degree of coherence of an illuminating laser beam is phase-modulated via light-ultrasound interactions so that speckle may be partially removed. The averaged speckle-contrast is obtained from the autocorrelation measurement of the speckle intensity-distribution. For one of the diffusers used, the contrast is decreased down to 0.56 with increase in the ultrasonic power.
1. Introduction Laser speckle has been recognised to be unwanted noise for the holographic reconstruction of an image. It arises from the interference effect by scattered light; then the coherence condition of an illuminating light is in a close relation with the speckle formation [1,2]. From this point of view, several attempts have been made so far to remove speckle in an observation plane. Elbaum et al. [3] and Jeorge and Jain [4] studied its properties theoretically and experimentally with polychromatic or multi-mode laser light. Van Ligten [5] utilised a rotating prism for the purpose of degrading temporal coherence of light. Lowenthal et al. [6,7], SchrSder [8], and Ih [9] also demonstrated effects of speckle reduction using one or two moving diffusers. Dainty and Welford [10] and McKechnie [11] proposed ingenious methods by introducing a moving aperture to average the speckle intensity in holographic image-reconstruction. The present work aims to add a new method for speckle reduction by ultrasonic modulation. Theoretical aspects of the coherence condition of light dependent on the ultrasonic wave have already been described by Ohtsuka [12,13]. According to his results, the partial coherence condition can be controlled electronically via the light-ultrasound interaction, which suggests feasibility that the speckle formation 18
can also be controlled by the presence of the ultrasonic wave. Based on this concept, it is demonstrated experimentally that the speckle removal takes place partially in the far field when a laser beam for illuminating a diffuser object is modulated in phase by a progressive ultrasonic wave.
2. Principle Diffraction of light by an ultrasonic wave in a relatively low-frequency range has been well understood in terms of the so-called Raman-Nath's phase lattice theory [14]. This theory is based on the fact that the ultrasonic wave acts in a light-ultrasound interaction as a moving phase lattice on an incident light beam. This postulation also makes it easy to formulate the expression of the mutual intensity [15] at Pl(Xl) and P2(x2) across the exit plane of the ultrasonic column (see eqs. (4) and (9) with r = 0 in ref. [10] ) P12(0) = P'12(0)J0 [2vlsin (K/2) (x 2 - x 1)1].
(1)
Here, P]2(0) is the mutual intensity of the light beam coming to the entrance plane of the ultrasonic column, J0 the Bessel function of the first kind and zeroth order, v the Raman-Nath parameter dependent on the ultrasonic power, K the wavenumber of the ultrasonic wave. Note that the x-coordinate is taken along the propagation direction of the ultrasonic wave.
Volume 27, number 1
OPTICS COMMUNICATIONS
October 1978
-t-~
x
I0') =
y
f
q-oo
f
P'12(O)Jo[2Vlsin(K/2)(x2-Xl)[]
dion Plane
X O(x 1)O*(x2) exp [-i(k]f) (Xz-X 1)y] dx l dx 2 ,
,J Slit
Fig. 1. Schematic diagram of the optical system. It should be emphasised that the Bessel function in eq. (1) is space-invariant since it depends on only the difference between P1 and P2. On the assumption of F]2(0 ) to be also space-invariant, the complex degree of coherence [15] follows directly from eq. (1):
(3) where O(x) refers to the complex amplitude transmittance across the object plane, the asterisk the notation of the complex conjugate, and k the wavenumber of light. This formula tells us that the speckle intensity obtained is equivalent to what would result if the same object were illuminated with the light having the mutual intensity given by eq. (1). From the fact that F]2(0) can be regarded constant across the entrance plane of the ultrasonic column in the experiment made, we have for the ensemble-averaged speckle intensity +oo
(/(y)) = f _
~(x) = ~o(X)Jo [2ulsin (KX/2 )II
(2)
with x = x 2 - Xl, where/a0(x ) is the normalised form of Fl 2 (0). Accordingly, the degree of coherence [ 15 ] is given by the absolute value of eq. (2). As will be stated later on, the laser beam used for the experiment has been found to be spatially, completely coherent. This implies that/a(x) depends on only the ultrasonic modulation factor J0 [2olsin (KX/2)I] and then is simply determined electronically by controlling the ultrasonic power for given K and x. We shall be concerned with speckle pattern formed in the far field, as shown in fig. 1. A slit aperture to limit the light beam in width is set between the exit plane of the ultrasonic column and the diffuser, which aims to examine how the slit width is associated with the speckle formation. As a result, some combined effects due to three components (ultrasonic wave, slit aperture, and diffuser) will appear on speckle pattern. It should be noted that the diffuser is made to move slowly for the purpose of taking the ensemble average over the speckle intensity-distribution. Let x and y be the one-dimensional coordinates of the object and observation planes, respectively. Speckle is formed via the Fourier transforming lens F with the focal length f. The speckle intensity-distribution is thus written in the far field in the form
Jo[2Vlsin(KX/2)[] R(x) oo
× exp [-i(k/f)xy]
dx
(4)
with
R(x) =f
(OOCl)O*(x1 -- x)) dXl,
(5)
where the sharp brackets stand for taking the ensemble average. Here, F]2(0 ) is deleted for the sake of simplicity. It is understood that R(x) represents the ensemble-averaged autocorrelation of O(x). Let us consider next the average contrast of the speckle intensity, which is a measure of speckle reduction. The averaged contrast is normally defined as follows: C = [(i2) _ (i)2]
1/2/(i)"
(6)
Also, it is known that a valuable knowledge concerning the statistical property of speckle pattern is inferred from the autocorrelation of the speckle intensity: +~
(/@1)1@2))=
if if
Jo[Z°lsin(K/Z)(x2 - Xl)[]
× J0 [2vlsin (K/Z)(x 4 - x3)l ] X (O(x l)O*(x2)O(x3)O*(x4)) exp [-i(k/f) X {(x 1 - x2)Y l + (x 3 - x4)Y2) ] dx 1 dx 2 dx 3 dx 4.
(7) 19
Volume 27, number 1
OPTICS COMMUNICATIONS
So, the ensemble-averaged square intensity is simply derived by settingy 1 =Y2 as a special case. In view of eq. (4), we therefore have [(eq. (7)}yl=y2 - {eq. (4)} 2] 1/2 C= eq. (4)
Once the statistics of O(x) is given, more quantitative discussions on the speckle property dependent on the ultrasonic wave will be made with the help of eqs. (4), (7), and (8). Such investigations are now in progress. From an experimental point of view, however, let us look for a more convenient form of the average contrast. Assuming that the speckle intensity-distribution is also space-invariant and its statistics obeys ergodic process, we obtain for the intensity autocorrelation
D(Ay) = (I(y)I00 + Ay)) = (AI00)AI(y + Ay)) + (100))2 ,
(9)
where the ensemble average equals the time average, Ay is the distance between two points QI(Y) and Q2(Y + Ay), and A/00) is the intensity deviation from its mean value (I(y)). In general, the correlation property leads to lim (AI(y)AI(y + Ay)) = 0.
Ay ~ o o
(10)
Then, D(oo) = (i(y))2. Also it is evident that
D(0) = (/2 00)).
(11)
(13)
This expression illustrates that the average contrast can be determined from the correlation measurement of the speckle intensity.
3. Experiments and discussions Returning to fig. 1, the ultrasonic wave of 10 MHz propagates in pure water and its wavelength is'calculated to be approximately 0.15 ram. The diffuser made of ground glass moves with a constant speed of 1.0 cm/min towards the same direction as that of the ultrasonic wave. A photomultiplier having an aperture window much smaller than the average size of a speckle receives the scattered light via the lens F. The photocurrent is fed into an electronic correlator followed by a preamplifier, from which the timeaveraged intensity-autocorrelation is obtained. The experiment was conducted using two kind of ground glass, named (A) and (B). Three photographs of speckle pattern formed by (A) are shown in fig. 2. The photograph (a) is taken with no ultrasonic wave i.e., o = 0, whereas (b) and (c) with the presence of the ultrasonic waves. It is evident that speckle is partially removed due to the modulation effect by the ultrasonic wave. From only these pictures, however, the difference between (b)
Fig. 2. Speckle patterns in the far field.
20
(12)
With the reference to eq. (6), we finally obtain
C= [O(O)/D(oo)_ 1] 1/2 (8)
October 1978
Volume 27, number 1
OPTICS COMMUNICATIONS
October 1978
<1 (A)
(A)
I 1.0
O:
ix~, ~
v=1.87 v = 1.23
0.5 v=0
5',0
lO'.O
x : v = 1.23
,,^,x/~
T
v=0
iX: v = l . B 7
15.o' (m~) • Ay
1.0 <3
2.0
3.0
4.0 I
(B)
1.o v = 1.87 v = 1.23 v=0
......
0.5
--!•'
L
(B)
O:
i
51o
x : v = 1.23 A : v = 1.87
x'~. ~
i
lOO
v=O
(mr'n) • Ay
Fig. 3. Autocorrelation functions of the speckle intensities for the two kinds of diffuser, (A) and (B). and (c) is not clear. The intensity autocorrelation functions obtained with the same conditions of o's as in fig. 2 are denoted in fig. 3. As may be seen from these figures, the correlation curves are monotoneously decreasing with increase in Ay between the two points across the observation plane and the average intensity, given by [D(oo)] 1/2 [see eq. (1 I)], becomes large as the u-value increases. This is why sharp variations in the speckle intensity are made smooth by the ultrasonic modulation effect. More quantitative-
l.0
i
(B)
~5
0
0
I
I
I
0.5
1.0
1.5
I
2.0 I1 V
Fig. 4. Average contrasts as a function of u for (A) and (B).
0
1.0
2.0
3.0
4.0 >
I
Fig. 5. Probability densities of the speckle intensity-variations for (A) and (B). ly is represented its behavior in fig. 4, as the average contrast obtained from these correlation curves [see eq. (13)]. As is expected, the contrast decreases when the u-value becomes large. This is because the ultrasonic wave makes the degree of coherence of the illuminating laser beam degrade. The roughness properties of the two kinds of ground glass used are likely to give discrepancy between the curves (A) and (B). In the experiments, measurements were made with and without the slit aperture of (A/2) in width between the ultrasonic cell and the diffuser, where A is the wavelength of the ultrasonic wave. Note that the degree of coherence takes the minimum at x = (A/2) (see ref. [10] ). However, no clear effect by 21
Volume 27, number 1
OPTICS COMMUNICATIONS
this aperture could be observed. Although the aperture effect has not been fully understood, the reason why speckle pattern was not influenced by the aperture is that the correlation length of the diffuser used might be shorter than (A/2). Fig. 5 shows the probability density functions of the speckle intensity dependent on the o-values. It can be seen that the peak position shifts towards the larger value of I/(I) when the o-value increases. The degree of spatial coherence for the laser beam was measured by the Young's double slit experiment, from which it was found to take almost unity across the cross section of the laser beam. From all the experimental results it can be concluded that speckle formed in the far field is partially removed with the presence of the ultrasonic wave and its property depends on the ultrasonic power as well as the statistical property of the diffuser used. Much more speckle reduction will be expected by passing the laser beam successively two or three similar ultrasonic waves. The experiment was done for only the one.dimensional situation but extension to the two-dimensional problem will be possible with the use of two-crossed ultrasonic waves. Such arrangement might also cause much more effective specklereduction.
22
October 1978
References [1] T.S. McKechnie, Speckle reduction, in: Laser speckle and related phenomena, Topics in Applied Physics, Vol. 9, ed. J.C. Dainty (Springer-Verlag, Heidelberg, 1975). [2} J .C. Dainty, The statistics of speckle patterns, in: Progress in optics, Vol. 14, ed. E. Wolf (North-Holland, Amsterdam, 1976). [3] M. Elbaum, 1. Greenbaum and M. King, Opt. Commun. 5 (1972) 171. [4] N. George and A. Jain, Appl. Optics 12 (1973) 1202. [5] R.F. van Ligten, Appl. Optics 12 (1973) 255. [6] S. Lowenthal and D. Joyeux, J. Opt. Soc. Am. 61 (1971) 847. [7] H. Arsenault and S. Lowenthal, Opt. Commun. 1 (1970) 451. [8] E. Schrder, Opt. Commun. 3 (1971) 68. [9] C.S. Ih, Appl. Optics 16 (1977) 1473. [10] J.C. Dainty and W.T. Welford, Opt. Commun. 3 (1971) 289. [11] T.S. McKechnie, Optik 44 (1974) 34. [12] Y. Ohtsuka, Opt. Commun. 17 (1976) 234. [13] Y. Ohtsuka, Opt. Commun. 17 (1976) 238. [14] C.V. Raman and N.S. Nath, Proc. Ind. Acad. Sci. 2A (1935) 406, 2A (1935) 413. [15] M. Born and E. Wolf, Principles of optics, 2nd Ed. (Pergamon Press, New York, 1964).
OPTICS COMMUNICATIONS
October 1978
LASER SPECKLE REDUCTION BY ULTRASONIC MODULATION Yoh IMAI and Yoshihiro OHTSUKA Department of Engineering Science, Faculty of Engineering, Hokkaido University, Sapporo, Hokkaido, Japan Received 27 June 1978
A new method is proposed for reducing speckle in the far field. The degree of coherence of an illuminating laser beam is phase-modulated via light-ultrasound interactions so that speckle may be partially removed. The averaged speckle-contrast is obtained from the autocorrelation measurement of the speckle intensity-distribution. For one of the diffusers used, the contrast is decreased down to 0.56 with increase in the ultrasonic power.
1. Introduction Laser speckle has been recognised to be unwanted noise for the holographic reconstruction of an image. It arises from the interference effect by scattered light; then the coherence condition of an illuminating light is in a close relation with the speckle formation [1,2]. From this point of view, several attempts have been made so far to remove speckle in an observation plane. Elbaum et al. [3] and Jeorge and Jain [4] studied its properties theoretically and experimentally with polychromatic or multi-mode laser light. Van Ligten [5] utilised a rotating prism for the purpose of degrading temporal coherence of light. Lowenthal et al. [6,7], SchrSder [8], and Ih [9] also demonstrated effects of speckle reduction using one or two moving diffusers. Dainty and Welford [10] and McKechnie [11] proposed ingenious methods by introducing a moving aperture to average the speckle intensity in holographic image-reconstruction. The present work aims to add a new method for speckle reduction by ultrasonic modulation. Theoretical aspects of the coherence condition of light dependent on the ultrasonic wave have already been described by Ohtsuka [12,13]. According to his results, the partial coherence condition can be controlled electronically via the light-ultrasound interaction, which suggests feasibility that the speckle formation 18
can also be controlled by the presence of the ultrasonic wave. Based on this concept, it is demonstrated experimentally that the speckle removal takes place partially in the far field when a laser beam for illuminating a diffuser object is modulated in phase by a progressive ultrasonic wave.
2. Principle Diffraction of light by an ultrasonic wave in a relatively low-frequency range has been well understood in terms of the so-called Raman-Nath's phase lattice theory [14]. This theory is based on the fact that the ultrasonic wave acts in a light-ultrasound interaction as a moving phase lattice on an incident light beam. This postulation also makes it easy to formulate the expression of the mutual intensity [15] at Pl(Xl) and P2(x2) across the exit plane of the ultrasonic column (see eqs. (4) and (9) with r = 0 in ref. [10] ) P12(0) = P'12(0)J0 [2vlsin (K/2) (x 2 - x 1)1].
(1)
Here, P]2(0) is the mutual intensity of the light beam coming to the entrance plane of the ultrasonic column, J0 the Bessel function of the first kind and zeroth order, v the Raman-Nath parameter dependent on the ultrasonic power, K the wavenumber of the ultrasonic wave. Note that the x-coordinate is taken along the propagation direction of the ultrasonic wave.
Volume 27, number 1
OPTICS COMMUNICATIONS
October 1978
-t-~
x
I0') =
y
f
q-oo
f
P'12(O)Jo[2Vlsin(K/2)(x2-Xl)[]
dion Plane
X O(x 1)O*(x2) exp [-i(k]f) (Xz-X 1)y] dx l dx 2 ,
,J Slit
Fig. 1. Schematic diagram of the optical system. It should be emphasised that the Bessel function in eq. (1) is space-invariant since it depends on only the difference between P1 and P2. On the assumption of F]2(0 ) to be also space-invariant, the complex degree of coherence [15] follows directly from eq. (1):
(3) where O(x) refers to the complex amplitude transmittance across the object plane, the asterisk the notation of the complex conjugate, and k the wavenumber of light. This formula tells us that the speckle intensity obtained is equivalent to what would result if the same object were illuminated with the light having the mutual intensity given by eq. (1). From the fact that F]2(0) can be regarded constant across the entrance plane of the ultrasonic column in the experiment made, we have for the ensemble-averaged speckle intensity +oo
(/(y)) = f _
~(x) = ~o(X)Jo [2ulsin (KX/2 )II
(2)
with x = x 2 - Xl, where/a0(x ) is the normalised form of Fl 2 (0). Accordingly, the degree of coherence [ 15 ] is given by the absolute value of eq. (2). As will be stated later on, the laser beam used for the experiment has been found to be spatially, completely coherent. This implies that/a(x) depends on only the ultrasonic modulation factor J0 [2olsin (KX/2)I] and then is simply determined electronically by controlling the ultrasonic power for given K and x. We shall be concerned with speckle pattern formed in the far field, as shown in fig. 1. A slit aperture to limit the light beam in width is set between the exit plane of the ultrasonic column and the diffuser, which aims to examine how the slit width is associated with the speckle formation. As a result, some combined effects due to three components (ultrasonic wave, slit aperture, and diffuser) will appear on speckle pattern. It should be noted that the diffuser is made to move slowly for the purpose of taking the ensemble average over the speckle intensity-distribution. Let x and y be the one-dimensional coordinates of the object and observation planes, respectively. Speckle is formed via the Fourier transforming lens F with the focal length f. The speckle intensity-distribution is thus written in the far field in the form
Jo[2Vlsin(KX/2)[] R(x) oo
× exp [-i(k/f)xy]
dx
(4)
with
R(x) =f
(OOCl)O*(x1 -- x)) dXl,
(5)
where the sharp brackets stand for taking the ensemble average. Here, F]2(0 ) is deleted for the sake of simplicity. It is understood that R(x) represents the ensemble-averaged autocorrelation of O(x). Let us consider next the average contrast of the speckle intensity, which is a measure of speckle reduction. The averaged contrast is normally defined as follows: C = [(i2) _ (i)2]
1/2/(i)"
(6)
Also, it is known that a valuable knowledge concerning the statistical property of speckle pattern is inferred from the autocorrelation of the speckle intensity: +~
(/@1)1@2))=
if if
Jo[Z°lsin(K/Z)(x2 - Xl)[]
× J0 [2vlsin (K/Z)(x 4 - x3)l ] X (O(x l)O*(x2)O(x3)O*(x4)) exp [-i(k/f) X {(x 1 - x2)Y l + (x 3 - x4)Y2) ] dx 1 dx 2 dx 3 dx 4.
(7) 19
Volume 27, number 1
OPTICS COMMUNICATIONS
So, the ensemble-averaged square intensity is simply derived by settingy 1 =Y2 as a special case. In view of eq. (4), we therefore have [(eq. (7)}yl=y2 - {eq. (4)} 2] 1/2 C= eq. (4)
Once the statistics of O(x) is given, more quantitative discussions on the speckle property dependent on the ultrasonic wave will be made with the help of eqs. (4), (7), and (8). Such investigations are now in progress. From an experimental point of view, however, let us look for a more convenient form of the average contrast. Assuming that the speckle intensity-distribution is also space-invariant and its statistics obeys ergodic process, we obtain for the intensity autocorrelation
D(Ay) = (I(y)I00 + Ay)) = (AI00)AI(y + Ay)) + (100))2 ,
(9)
where the ensemble average equals the time average, Ay is the distance between two points QI(Y) and Q2(Y + Ay), and A/00) is the intensity deviation from its mean value (I(y)). In general, the correlation property leads to lim (AI(y)AI(y + Ay)) = 0.
Ay ~ o o
(10)
Then, D(oo) = (i(y))2. Also it is evident that
D(0) = (/2 00)).
(11)
(13)
This expression illustrates that the average contrast can be determined from the correlation measurement of the speckle intensity.
3. Experiments and discussions Returning to fig. 1, the ultrasonic wave of 10 MHz propagates in pure water and its wavelength is'calculated to be approximately 0.15 ram. The diffuser made of ground glass moves with a constant speed of 1.0 cm/min towards the same direction as that of the ultrasonic wave. A photomultiplier having an aperture window much smaller than the average size of a speckle receives the scattered light via the lens F. The photocurrent is fed into an electronic correlator followed by a preamplifier, from which the timeaveraged intensity-autocorrelation is obtained. The experiment was conducted using two kind of ground glass, named (A) and (B). Three photographs of speckle pattern formed by (A) are shown in fig. 2. The photograph (a) is taken with no ultrasonic wave i.e., o = 0, whereas (b) and (c) with the presence of the ultrasonic waves. It is evident that speckle is partially removed due to the modulation effect by the ultrasonic wave. From only these pictures, however, the difference between (b)
Fig. 2. Speckle patterns in the far field.
20
(12)
With the reference to eq. (6), we finally obtain
C= [O(O)/D(oo)_ 1] 1/2 (8)
October 1978
Volume 27, number 1
OPTICS COMMUNICATIONS
October 1978
<1 (A)
(A)
I 1.0
O:
ix~, ~
v=1.87 v = 1.23
0.5 v=0
5',0
lO'.O
x : v = 1.23
,,^,x/~
T
v=0
iX: v = l . B 7
15.o' (m~) • Ay
1.0 <3
2.0
3.0
4.0 I
(B)
1.o v = 1.87 v = 1.23 v=0
......
0.5
--!•'
L
(B)
O:
i
51o
x : v = 1.23 A : v = 1.87
x'~. ~
i
lOO
v=O
(mr'n) • Ay
Fig. 3. Autocorrelation functions of the speckle intensities for the two kinds of diffuser, (A) and (B). and (c) is not clear. The intensity autocorrelation functions obtained with the same conditions of o's as in fig. 2 are denoted in fig. 3. As may be seen from these figures, the correlation curves are monotoneously decreasing with increase in Ay between the two points across the observation plane and the average intensity, given by [D(oo)] 1/2 [see eq. (1 I)], becomes large as the u-value increases. This is why sharp variations in the speckle intensity are made smooth by the ultrasonic modulation effect. More quantitative-
l.0
i
(B)
~5
0
0
I
I
I
0.5
1.0
1.5
I
2.0 I1 V
Fig. 4. Average contrasts as a function of u for (A) and (B).
0
1.0
2.0
3.0
4.0 >
I
Fig. 5. Probability densities of the speckle intensity-variations for (A) and (B). ly is represented its behavior in fig. 4, as the average contrast obtained from these correlation curves [see eq. (13)]. As is expected, the contrast decreases when the u-value becomes large. This is because the ultrasonic wave makes the degree of coherence of the illuminating laser beam degrade. The roughness properties of the two kinds of ground glass used are likely to give discrepancy between the curves (A) and (B). In the experiments, measurements were made with and without the slit aperture of (A/2) in width between the ultrasonic cell and the diffuser, where A is the wavelength of the ultrasonic wave. Note that the degree of coherence takes the minimum at x = (A/2) (see ref. [10] ). However, no clear effect by 21
Volume 27, number 1
OPTICS COMMUNICATIONS
this aperture could be observed. Although the aperture effect has not been fully understood, the reason why speckle pattern was not influenced by the aperture is that the correlation length of the diffuser used might be shorter than (A/2). Fig. 5 shows the probability density functions of the speckle intensity dependent on the o-values. It can be seen that the peak position shifts towards the larger value of I/(I) when the o-value increases. The degree of spatial coherence for the laser beam was measured by the Young's double slit experiment, from which it was found to take almost unity across the cross section of the laser beam. From all the experimental results it can be concluded that speckle formed in the far field is partially removed with the presence of the ultrasonic wave and its property depends on the ultrasonic power as well as the statistical property of the diffuser used. Much more speckle reduction will be expected by passing the laser beam successively two or three similar ultrasonic waves. The experiment was done for only the one.dimensional situation but extension to the two-dimensional problem will be possible with the use of two-crossed ultrasonic waves. Such arrangement might also cause much more effective specklereduction.
22
October 1978
References [1] T.S. McKechnie, Speckle reduction, in: Laser speckle and related phenomena, Topics in Applied Physics, Vol. 9, ed. J.C. Dainty (Springer-Verlag, Heidelberg, 1975). [2} J .C. Dainty, The statistics of speckle patterns, in: Progress in optics, Vol. 14, ed. E. Wolf (North-Holland, Amsterdam, 1976). [3] M. Elbaum, 1. Greenbaum and M. King, Opt. Commun. 5 (1972) 171. [4] N. George and A. Jain, Appl. Optics 12 (1973) 1202. [5] R.F. van Ligten, Appl. Optics 12 (1973) 255. [6] S. Lowenthal and D. Joyeux, J. Opt. Soc. Am. 61 (1971) 847. [7] H. Arsenault and S. Lowenthal, Opt. Commun. 1 (1970) 451. [8] E. Schrder, Opt. Commun. 3 (1971) 68. [9] C.S. Ih, Appl. Optics 16 (1977) 1473. [10] J.C. Dainty and W.T. Welford, Opt. Commun. 3 (1971) 289. [11] T.S. McKechnie, Optik 44 (1974) 34. [12] Y. Ohtsuka, Opt. Commun. 17 (1976) 234. [13] Y. Ohtsuka, Opt. Commun. 17 (1976) 238. [14] C.V. Raman and N.S. Nath, Proc. Ind. Acad. Sci. 2A (1935) 406, 2A (1935) 413. [15] M. Born and E. Wolf, Principles of optics, 2nd Ed. (Pergamon Press, New York, 1964).