Sun et al. Vol. 16, No. 6 /June 1999 /J. Opt. Soc. Am. A 1387

Statistical properties of triple-random-modulateddynamic speckles

Hong Sun

Department of Modern Applied Physics, Tsinghua University, Beijing 100084, China

Ling-Yun Zhang

CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China, and Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100080, China

Ying Liu and Shining Ma

Department of Applied Physics, Tianjin University, Tianjin 300072, China

Received January 26, 1999; accepted February 5, 1999

Statistical properties of triple-random-modulated dynamic speckles (TRMDS) are studied theoretically and ex-perimentally. The spatiotemporal correlation function of the intensity fluctuation of the speckles is obtained;it is a Gaussian distribution. The correlation time is inversely proportional to the velocity of the moving dif-fuser. The average radius of TRMDS is smaller than that of single-random-modulated dynamic speckles(SRMDS). The correlation time of SRMDS is approximately 70–80 times longer than that of TRMDS. Thetheoretical results are consistent with the experimental results. © 1999 Optical Society of America[S0740-3232(99)01306-X]

OCIS codes: 030.6140, 030.6600, 030.1640, 290.5880, 290.4210.

1. INTRODUCTIONThe statistical properties and applications of dynamicspeckles produced by a diffuser moving at constant veloc-ity have been investigated.1–5 The most general methodfor studying the statistical properties of dynamic specklesis to use the spatiotemporal correlation function of thespeckle intensity, because the dynamic speckle field var-ies randomly in space and time. Obviously, the spa-tiotemporal randomness of dynamic speckles is based ona stochastic process that is due to the random character-istics of the diffuser. The random light modulation re-sulting from a diffuser is described by various statisticalmethods. Generally, a diffuser can be regarded as a ran-dom phase screen, which retards the phase of the incidentlight in a spatially random fashion. So the illuminatinglight is randomly modulated when it is scattered by a dif-fuser. By studying the fundamental statistical proper-ties of dynamic speckles, one can extend the applicationsof dynamic speckles to various measurements such asscientific,6 industrial, and medical7–8 measurements,which have the advantage of being noncontact measure-ments. One of the most important applications of dy-namic speckles is to measure the velocity of a moving dif-fuser, because its statistical properties contain someinformation about the motion of the diffuser.9–14

However, in some circumstances it is not suitable to ap-ply the theory of single-random-modulated (normal) dy-namic speckles (SRMDS) to multiple-random-modulateddynamic speckles (MRMDS).15–19 MRMDS have re-ceived special attention because they are more usefulthan SRMDS formed by a single diffuser.20–23 A thor-

0740-3232/99/061387-08$15.00 ©

ough investigation has not been made until now, to ourknowledge, although there are some experimental resultswith speckled speckles, such as triple-random-modulateddynamic speckles (TRMDS).24

In this paper we investigate the statistical properties ofTRMDS theoretically and experimentally. TRMDS areformed by two diffusers, placed parallel and close to eachother, one moving at a constant velocity and the other sta-tionary. The illuminating laser light is random modu-lated first by the stationary transparent diffuser and thenby the moving diffuser, which is behind the stationarytransparent diffuser. The double-random-modulatedlight is reflected by the moving diffuser to the stationarytransparent diffuser and modulated by it again. Finally,the correlation function of the TRMDS is determined inthe observation plane. Using statistical optics theory, weobtain the spatiotemporal correlation function of theTRMDS intensity fluctuation, which is a Gaussian distri-bution. The correlation time is inversely proportional tothe velocity of the moving diffuser. The average radiusof TRMDS is smaller than that of SRMDS. The correla-tion time of SRMDS is approximately 70–80 times longerthan that of TRMDS. The theoretical results are in goodagreement with the experimental results.

2. COMPLEX AMPLITUDE OF TRIPLE-RANDOM-MODULATED DYNAMICSPECKLESFigure 1 shows schematically an arrangement for produc-ing TRMDS. Transparent diffuser D1 is located at plane

1999 Optical Society of America

1388 J. Opt. Soc. Am. A/Vol. 16, No. 6 /June 1999 Sun et al.

Fig. 1. Optical arrangement for the formation of TRMDS at the far-field diffraction plane by two diffusers under illumination by aGaussian beam.

(a, b), reflecting diffuser D2 is located at plane (j, h), andthe observation plane is at plane (x, y). D1 , withrandom-phase-modulation function tD1

(a) for the roughsurface, is illuminated by the waist region of a Gaussianbeam. Then the single-modulated light propagates todiffuser D2 through distance H. The amplitude distribu-tion in front of diffuser D2 can be written as

E1~j ! 5 E2`

`

E~a!tD1~a!h1~j, a!d2a, (1)

where a is a vector notation for point (a, b) and

E~a! 5 exp@2~a 2/v2!# (2)

is the amplitude of the incident Gaussian beam with ra-dius v, and h1(a, j ) represents the propagation functionof the light from plane (a, b) to (j, h). Considering the la-ser beam incident on the plane (a, b) at a small angle, thelaser is distributed in the near field of the z axis on the(a, b) and (j, h) planes, and the Fresnel condition @(a2 j )/H#2 , 1 is satisfied,25 and therefore the (j, h) planeis in the Fresnel diffraction region. The propagationfunction is given by25

h1~j, a! 5exp~ikH !

ilHexpF i

k

2H~j 2 a!2G . (3)

Now the field E1(j ) is further modulated in amplitudeand is reflected by diffuser D2 , which is moving at a con-stant velocity v perpendicular to the z axis. Before thefield passes through diffuser D1 , the optical-field ampli-tude distribution at plane (a, b) is expressed as

E2~a8! 5 E2`

`

E1~j !t2~j, t !h2~a8, j !d2j, (4)

where tD2(j, t) is the modulation function of diffuser D2

and h2(a, j ), which is the propagation function fromplane (j, h) to plane (a, b), can be written as25

h2~a8, j ! 5exp~ikH !

ilHexpF i

k

2H~a8 2 j !2G . (5)

Finally, the field amplitude is modulated by diffuser D1again and propagates to observation plane (x, y) in thefar-field Fraunhofer diffraction region. The propagationfunction can be expressed as25

h3~x, a! 5exp~ikz !

ilzexpF iS k

2z Dx2G3 expF2iS 2p

lz D a8 • xG . (6)

The amplitude at observation plane (x, y) can be writtenas

E~x, t ! 5 E2`

`

E2~a8!tD1~a8!h3~x, a8!d2a8. (7)

Therefore, with Eqs. (1), (4), and (7), the complex ampli-tude of the TRMDS is obtained:

E~x, t ! 5 EEE2`

`

E~a!tD1~a!tD2

~j, t !tD1~a8!

3 h1~j, a!h2~a8, j !h3~x, a8!d2ad2jd2a8.

(8)

3. AMPLITUDE SPATIOTEMPORALCORRELATION FUNCTION OF TRIPLE-RANDOM-MODULATED DYNAMICSPECKLESWe now turn our attention to the spatiotemporal correla-tion properties of TRMDS. The stochastic behavior ofthe complex amplitude is conventionally described by thecorrelation function. The spatiotemporal correlationfunction of the complex amplitude E(x, t) is26

G~x1 , x2 ; t1 , t2! 5 ^E~x1 , t1!E* ~x2 , t2!&, (9)

where ^¯& refers to an ensemble average. Substitutingthe expression of the complex amplitude [Eq. (8)] into Eq.(9), we obtain

G~x1 , x2 ; t1 , t2!

5 K EEEEEE2`

`

E~a1!E0* ~a2!

3 tD1~a1!tD2

~j1 , t1!tD1~a18 !tD1

* ~a2!tD2* ~j2 , t2!

3 tD1* ~a28 !h1~j1 , a1!h2~a18 , j1!h3~x1 , a18 !h1* ~j2 , a2!

3 h2* ~a28 , j2!h3* ~x2 , a28 !d2a1d2j1d

2

3 a18d2a2d2j2d2a28L . (10)

In order to calculate the spatiotemporal correlationfunction, we will assume that (1) the single-modulated

Sun et al. Vol. 16, No. 6 /June 1999 /J. Opt. Soc. Am. A 1389

speckle from diffuser D1 obeys a circular Gaussian ran-dom process25 and (2) if a large number of speckles arepresent on diffusers D1 and D2 , triple-modulated ampli-tude statistics are a circular complex Gaussian process.16

An important property of such random variables is thatthey obey the complex Gaussian moment theorem, whichis given by

^u1* ¯ uk* uk11 ¯ u2k& 5 (p

^u1* up&^u2* uq& ¯ ^uk* ur&,

(11)

where (p denotes a summation over the k! possible per-mutations (p, q, ..., r) of (1, 2, ..., k).

Since the random feature of the speckle amplitude isproduced by the diffusers and characterized by the ran-dom modulation functions tD1

(a) and tD2(j, t), the en-

semble average in Eq. (10) is performed only over themodulation functions. So Eq. (10) becomes

G~x1 , x2 ; t1 , t2! 5 EEEEEE2`

`

E~a1!E0* ~a2!

3 ^tD1~a1!tD2

~j1 , t1!tD1~a8!tD1

* ~a2!

3 tD2* ~j2 , t2!tD1

* ~a28 !&h1~j1 , a1!

3 h2~a18 , j1!h3~x1 , a18 !h1* ~j2 , a2!

3 h2* ~a28 , j2!h3* ~x2 , a28 !d2a1d2j1

3 d2a18d2a2d2j2d2a28 . (12)

Because diffusers D1 and D2 are statistically indepen-dent, we have

^tD1~a!tD2

* ~j !& 5 ^tD2~j !tD1

* ~a!& 5 0. (13)

On the basis of the complex Gaussian moment theoremand the assumption that the direction of the incidentGaussian beam is approximately perpendicular to thesurface of diffuser D1 , the ensemble average reduces to

^tD1~a1!tD2

~j1 , t1!tD1~a18 !tD1

* ~a2!tD2* ~j2 , t2!tD1

* ~a28 !&

5 ^tD1~a1!tD1

* ~a2!&^tD2~j1 , t1!tD2

* ~j2 , t2!&

3 ^tD1~a18 !tD1

* ~a28 !& 1 ^tD1~a8!tD1

* ~a2!&

3 ^tD2~j1 , t1!tD2

* ~j2 , t2!&^tD1~a1!tD1

* ~a28 !&

' 2^tD1~a1!tD1

* ~a2!&^tD2~j1 , t1!tD2

* ~j2 , t2!&

3 ^tD1~a18 !tD1

* ~a28 !&. (14)

If it is assumed that the light field is modulated only inphase with the diffusers, the phase modulation functionsare

tD1~a! 5 exp@2if1~a!#,

tD2~j, t ! 5 exp@2if2~j, t !#. (15)

Generally the autocorrelation function of rough surfacesis of Gaussian form. Using the results of Beckmann,27

we can obtain

RD1~Da! 5 ^tD1

~a1!tD1* ~a2!&

5 exp@2p ~Da 2/a2!#,

RD1~Da8! 5 ^tD1

~a18 !tD1* ~a28 !&

5 exp@2p ~Da82/a2!#,

RD2~Dj 2 vt! 5 ^tD2

~j1 , t1!tD2* ~j2 , t2!&

5 exp@2p ~Dj 2 vt!2/b2#, (16)

where Da 5 a1 2 a2 , Da8 5 a18 2 a28 , Dj 5 j1 2 j2 , t5 t1 2 t2 , and a and b are the constants related to thesurface correlation distance. Substituting Eqs. (2), (3),(5), (6), and (16) into Eq. (12), and letting Dx 5 x1 2 x2the amplitude correlation is written as

G~x1 , Dx; t!

52

l6H4z2 expF S ip

lz D ~2x1 2 Dx!DxG3 EEEEEE

2`

`

expF22a 1

2 2 2a1Da 1 Da 2

v2 G3 RD1

~Da!RD2~Dj1 2 vt!RD1

~Da8!

3 expF S 2ip

lH D ~Da 2 1 Dj 2 2 2DaDj 1 2Daj1

2 2Djj1 1 2Dja1 2 2Daa1!GexpF S 2ip

lH D3 ~Da82 1 Dj 2 2 2Da8Dj 1 2Da8j1 2 2Djj1

1 2Dja18 2 2Da8a18 !GexpF S 2i2p

lz D3 ~a18Dx 1 Da8x 2 Da8Dx!Gd2a1d2j1

3 d2a18d2Dad2Djd2Da8. (17)

Using the definition of d function and its properties

d ~x ! 51

2pE

2`

`

exp~2ikx !dk 51

2pE

2`

`

exp~ikx !dk,

d ~at ! 51

uaud~t !, (18)

together with the Fourier transform formula of theGaussian function,

g ~x, y ! 5 A exp@2a~x2 1 y2!#,

F ( g ~x, y !) 5 E E2`

`

g ~x, y !exp@2j2p ~ fx x 1 fy y !#dxdy

5 AAp

aexpF2

p2~ f x2 1 f y

2!

aG , (19)

after some complicated integration we obtain the follow-ing results:

1390 J. Opt. Soc. Am. A/Vol. 16, No. 6 /June 1999 Sun et al.

G~x1 , Dx; t!

5A2/m3pv

l4H3z2 expF S 2ip

lz D ~2x1Dx 2 Dx2!S 2H

z2 1 D

1 S i2p

lz D S x1 2Dx

2 D 1

m3S 2pH

a2zDx

1pH

b2zDx 2

p

b2 vt D G3 expF2

p2v2Dx2

2l2z2 2p2~x1 2 Dx/2!2

m3l2z2

24pH2

a2z2 Dx2 2p

b2 S H

zDx 2 vt D 2

11

m3S 2pH

a2zDx 1

pH

b2zDx 2

p

b2 vt D 2G , (20)

where

m3 51

2v2 12p

a2 1p

b2 . (21)

This is the final expression of the spatiotemporal correla-tion function of TRMDS. Equation (20) indicates thatG(x1 , Dx; t) is related to spatial coordinate x1 as well asto space difference Dx and time difference t. Therefore itrepresents a spatial nonstationary process.

4. NORMALIZED SPATIOTEMPORALCORRELATION FUNCTION OFTHE INTENSITY FLUCTUATIONSince the physical magnitude actually detected by thephotodetector at the far-field diffraction plane is thespeckle intensity, the most important factor characteriz-ing TRMDS is the correlation function of the speckle in-tensity fluctuation. The circular complex Gaussian pro-cess with zero mean is further assumed. Thespatiotemporal correlation function of the intensity fluc-tuation of the speckles is defined as

GI~x1 , x2 ; t1 , t2! 5 ^I~x1 , t1!&^I~x2 , t2!&

1 uGE~x1 , x2 ; t1 , t2!u2. (22)

The definition of the intensity fluctuation is

DI~x, t ! 5 I~x, t ! 2 ^I~x, t !&, (23)

and the spatiotemporal correlation function of the inten-sity fluctuation is expressed as

GDI~x1 , x2 ; t1 , t2! 5 uGE~x1 , x2 ; t1 , t2!u2. (24)

Thus the normalized spatiotemporal correlation functionof the intensity fluctuation can be given as

gDI~x1 , x2 ; t1 , t2! 5uGDI~x1 , x2 ; t1 , t2!u

^I1&^I2&

5uGE~x1 , x2 ; t1 , t2!u2

^I1&^I2&, (25)

where

I1~x1 , t1! 5 uE~x1 , t1!u2,

^I1& 5 ^E~x1 , t1!E* ~x1 , t1!&,

I2~x2 , t2! 5 uE~x2 , t2!u2,

^I2& 5 ^E~x2 , t2!E* ~x2 , t2!&. (26)

Substituting the expression of the amplitude spatiotem-poral correlation function into Eq. (25), we find that

gDI~Dx, t! 5 expF2p2v2Dx2

l2z2 1p2Dx2

2m3l2z2

28pH2

a2z2 Dx2 22p

b2 S H

zDx 2 vt D 2

12

m3S 2pH

a2zDx 1

pH

b2zDx 2

p

b2 vt D 2G .

(27)

Since the expression indicates that the spatiotemporalcorrelation function of the intensity fluctuation is depen-dent only on space difference Dx and time difference t, theintensity fluctuation is stationary in the spatial and tem-poral domains. If we let Dx 5 0, the temporal correla-tion function of the intensity fluctuation is obtained as

gDI~0, t! 5 exp~2t 2 /t c32 !, (28)

where

tc3 5 F2p

b2 S 1 2p

m3b2D G21/2

uvu21. (29)

The spatial correlation is of Gaussian form, and its tem-poral correlation length is inversely proportional to thevelocity v of the moving diffuser, because m3 5 (2v)21

1 2pa22 1 pb22 is not related to the distances H and zif a and b are constants.

If we let t 5 0 and a 5 b, the spatial correlation func-tion of the intensity fluctuation is obtained as

gDI~Dx, 0 ! 5 exp~2Dx2/rs2!, (30)

where

rs 5 rs0S 1 110H2l2

pv2a2 21

2m3v2 218H2l2

m3v2a4D 21/2

(31)

is the average radius of TRMDS and rs0 5 (lz)/(pv) isthe average radius of SRMDS (see Appendix A).

5. DISCUSSION AND EXPERIMENTALRESULTSTo confirm the theoretical results described in Section 4,we conducted an experimental study, using the arrange-ment shown in Fig. 2. A ground glass is employed as astationary transmitting diffuser, and a piece of abrasivepaper acts as a reflecting diffuser moving perpendicularto the optical axis at a constant velocity. The paper dif-fuser is located 10 mm behind the ground glass. The sur-face roughness of the two diffusers is so great that they

Sun et al. Vol. 16, No. 6 /June 1999 /J. Opt. Soc. Am. A 1391

Fig. 2. Block diagram for detection of the correlation function of the TRMDS intensity fluctuation. A/D, analog to digital; PMT, photo-multiplier.

may be regarded as deep random-phase screens with cor-relation length 10 mm. The Gaussian beam from a 5-mWsingle-mode He–Ne laser has a 0.7-mm waist radius onthe glass surface at 633-nm wavelength. The distancefrom the ground glass to the detector is 1 m. The inten-sity signal is coupled into a photomultiplier by an opticalfiber of 50-mm core radius. After being amplified, thephotocurrent passes through a low-pass filter and is inputinto a computer.

After transmission and reflection, the laser light re-ceived by the photomultiplier is very weak. As the pho-tocurrent is only nanoamperes, a high-resistance ampli-fier is connected to the photomultipler. The TRMDSchanges very slowly because diffuser D2 is moving slowly(several tens of micrometers per second). Since the sig-nal frequency is under 500 Hz, we can use a low-pass fil-ter with 500-Hz cutoff frequency. Diffuser D2 is put on amoving stage driven by an electrical device.

Figure 3 shows the experimental results of the tempo-ral correlation function of the intensity fluctuation of theTRMDS, in which diffuser D2 is moving at different ve-locities. The correlation curves from top to bottom corre-spond to v1 5 21 mm/s, v2 5 38 mm/s, v3 5 60 mm/s, v45 81 mm/s, and v5 5 100 mm/s. Corresponding to thesevelocities, the correlation lengths are tc1 5 229 ms, tc25 120 ms, tc3 5 78 ms, tc4 5 60 ms, and tc5 5 48 ms.The different tc related to different velocities representthe time it takes for the normalized temporal correlationfunction to fall to 1/e. From the experimental results wecan see that the temporal correlation function of the in-tensity fluctuation of TRMDS is in Gaussian form andthat its temporal correlation length tc is inversely propor-tional to the velocity of the moving diffuser: The greaterthe velocity of the diffuser, the shorter the temporal cor-relation length. Figure 4 shows the experimental resultsof the reciprocal of the correlation time as a function ofthe diffuser velocity. Figure 5 shows how the calculatedspatiotemporal correlation function changes with timeand space differences, with parameters a 5 b 5 10 mm,H 5 10 mm, v 5 0.7 mm, and v 5 21 mm/s. The calcu-lated value is tc 5 228 ms. The calculated temporal andspatial correlation functions of TRMDS are both Gauss-ian in form. When we change the distance H from 10mm to 12 mm and keep the velocity constant (v5 38 mm/s), the temporal correlation length (tc

5 120 ms) is the same as that of H 5 10 mm, which up-holds the conclusion of Eq. (29). The experimental re-sults are shown in Fig. 6. The theoretical and experi-mental results are in a good agreement.

Our theoretical model can also be used to explain someprevious experimental results.27 First, in our theoreticalresults, Eq. (29) indicates that the proportional coefficientof tc3 /(1/v) 5 a is independent of distance z. This resultis the same as the experimental results in Ref. 24, whichshow that the dynamic properties of TRMDS intensityfluctuations are scarcely affected by any misalignment ofthe imaging system used. That is, the relationship be-tween the correlation time and the velocity of the movingdiffuser does not change when the distance from the sta-tionary diffuser to the observation plane changes slightly.

Second, we are able to compare the properties ofTRMDS and SRMDS intensity fluctuations. The rela-

Fig. 3. Correlation curves (from top to bottom) corresponding tov1 5 21 mm/s, v2 5 38 mm/s, v3 5 60 mm/s, v4 5 81 mm/s, andv5 5 100 mm/s.

Fig. 4. Experimental results of the reciprocal of the correlationtime as a function of the diffuser velocity.

1392 J. Opt. Soc. Am. A/Vol. 16, No. 6 /June 1999 Sun et al.

Fig. 5. Calculated spatiotemporal correlation function of the TRMDS intensity fluctuation.

tionship between the correlation time and the velocity ofthe moving diffusers with SRMDS can be written as (seeAppendix A)

1

tc15 F2p

a2 S 1 2p

m1a2D G1/2

v, m1 51

2v2 1p

a2

(32)

and that of TRMDS as

1

tc35 F2p

b2 S 1 2p

m3b2D G1/2

v,

m3 51

2v2 12p

a2 1p

b2 . (33)

If we let a 5 b, a1 5 tc1 /(1/v), and a3 5 tc3 /(1/v) wecan obtain the expression

a1

a35

tc1

tc3. (34)

Using Eqs. (32)–(34) we get

tc1

tc35 Fm1~m3a2 2 p!

m3~m1a2 2 p!G1/2

5 F1 14pv2

~a4/2v2! 1 3pa2G1/2

. (35)

In our experiments, a 5 10 mm and v 5 0.7 mm, a! v, and a4/v2 is negligible, and Eq. (35) is reduced to

a1

a35

tc1

tc3'

2v

A3a. (36)

According to this result, the ratio of the proportionalitycoefficient of SRMDS and TRMDS depends on the beam-waist radius v and the surface correlation length a.When a ; 1025 m and v ; 1023 m, the ratio a1 /a3 is ap-

proximately 81. This means that the intensity fluctua-tion of TRMDS has larger temporal stochastic propertiesthan that of SRMDS. The experimental value of the ra-tio in Ref. 24 is 72, which is of approximately the samemagnitude as our theoretical value: approximately70–80 times. Finally we can compare the spatial corre-lation of TRMDS with that of SRMDS, which is shown bythe speckle average radius. The expression of the aver-age radius of TRMDS is given by Eq. (31). The param-eters are H ; 1022 m, l ; 1027 m, a ; 1025 m, and v; 1023 m. The radius can be reduced approximately to

rs ' rs0S 1 14H2l2

pv2a2 D 21/2

. (37)

Since (4H2l2)/(pv2a2) . 0, the average radius ofTRMDS is smaller than that of SRMDS, which is inagreement with the experimental results.24

To sum up, our theoretical model can be used to explainthe experiments and is in good agreement with the ex-periments.

Fig. 6. Experimental results for the temporal correlation func-tion of the TRMDS intensity fluctuation when H 5 10 mm (solidcurve) and H 5 12 mm (dotted–dashed curve).

Sun et al. Vol. 16, No. 6 /June 1999 /J. Opt. Soc. Am. A 1393

6. CONCLUSIONIn this paper the properties of TRMDS were investigatedtheoretically and experimentally. The complex-ampli-tude spatiotemporal correlation function and the spa-tiotemporal correlation function of the TRMDS intensityfluctuation were obtained. The former is not stationaryin space, whereas the latter is stationary in the spatialand temporal domains, and the temporal correlationlength is inversely proportional to the velocity of the mov-ing diffuser. By comparing the temporal change rate ofthe intensity fluctuation of the TRMDS with that ofSRMDS, we can conclude that it is approximately 70–80times that of SRMDS, which is consistent with the experi-mental results. Although our theoretical results are gen-eral, they can be used extensively to analyze some experi-ments, such as noncontact measurement by laser TRMDSand measurement of skin blood flow by using TRMDS, inwhich the front stationary diffuser is regarded as the skinsurface and the near moving diffuser corresponds to bloodflow in a capillary network.

APPENDIX AIn Fig. 1, if D2 is absent and D1 is moving with constantvelocity v perpendicular to the z axis, the amplitude atthe observation plane (x, y) can be written as

E~x, t ! 5 E2`

`

E~a!tD~a, t !h~a, x!d2a, (A1)

where

E~a! 5 exp@2~a 2/v2!#,

h~x, a! 5exp~ikz !

ilzexpF iS k

2z Dx2GexpF2iS 2p

lz D a • xG(A2)

and tD(a, t) is the random-phase modulation function ofthe diffuser, which satisfies

RD~Da 2 vt! 5 ^tD~a1 , t1!tD* ~a 2 , t2!&

5 exp@2p ~Da 2 vt!2/a2#. (A3)

According to the definition of the amplitude correlationgiven by Eq. (9), we can obtain the amplitude correlationfunction of SRMDS after complicated integration, usingthe same method as in Section 3. The expression is

G~x1 , Dx; t! 5A2m1pv

2l2z2 expF iS p

lz D ~2x1Dx 2 Dx2!

2 S i2p

lz D S x1 2Dx

2 D 1

m1S p

a2 vt D G3 expF2

p2v2Dx2

2l2z2 2p2~x1 2 Dx/2!2

m1l2z2

2p

a2 S 1 2p

m1a2Dv2t 2 G , (A4)

where

m1 51

2v2 1p

a2 . (A5)

Using Eq. (25), we obtain the normalized spatiotempo-ral correlation function of the SRMDS intensity fluctua-tion as

gDI~Dx, t! 5 expF2p2v2Dx2

l2z2 1p2Dx2

2m1l2z2G3 expF2

2p

a2 S 1 2p

m1a2Dv2t 2 G . (A6)

If we let Dx 5 0, we have the temporal correlation func-tion

gDI~0, t! 5 exp~2t 2 /t c12 !, (A7)

where

tc1 5 F2p

a2 S 1 2p

m1a2D G21/2

uvu21. (A8)

Then if we let t 5 0, the spatial correlation function ofthe intensity fluctuation becomes

gDI~Dx, 0 ! 5 exp~2Dx2/rs02 !, (A9)

where

rs0 5lz

pvS 1 2

1

2v2m1D 21/2

'lz

pv, ~a ! v!

(A10)

is the average radius of SRMDS.

ACKNOWLEDGMENTSWe are grateful for financial support from the NationalNatural Science Foundation of China and the China Post-doctoral Science Foundation.

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