Three-dimensional speckle dynamics in paraxial optical systems

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Three-dimensional speckle dynamics in paraxial optical systems H. T. Yura Electronic Technology Center, The Aerospace Corporation, P.O. Box 92957, Los Angeles, California 90009

S. G. Hanson and R. S. Hansen Department of Optics and Fluid Dynamics, Risø National Laboratory, P.O. Box 49, DK-4000, Roskilde, Denmark

B. Rose IBSEN Micro Structures A/S, Gammelgaardsvej 65, DK-3520, Farum, Denmark Received October 29, 1998; revised manuscript received February 16, 1999; accepted February 18, 1999 Three-dimensional (3D) speckle dynamics are investigated within the paraxial approximation as represented by ABCD-matrix theory. Within the paraxial approximation, exact expressions are derived for the space– time-lagged intensity covariance that results from an in-plane translation, an out-of-plane rotation, or an inplane rotation of a diffuse scattering object that is illuminated by a Gaussian-shaped laser beam. As illustrative examples we consider the 3D dynamical nature of speckles that are formed in free space and in Fourier transform and imaging systems. The spatiotemporal characteristics of the observed 3D speckle patterns are interpreted in terms of boiling, decorrelation, rotation, translation, and tilting. Experimental results, which support the quantitative theory, are presented and discussed. © 1999 Optical Society of America [S0740-3232(99)02006-2] OCIS codes: 030.6140, 030.6600, 290.5880, 030.1640, 110.1220, 110.6150.

1. INTRODUCTION In this paper we are concerned with the space–time evolutionary nature of three-dimensional (3D) speckle that results from an in-plane translation, an out-of-plane rotation, or an in-plane rotation of a diffuse scattering object that is illuminated by a Gaussian-shaped laser beam. Coherent light scattered off a rough surface produces a speckle pattern that extends in space some distance from the object. The speckles originate from elementary interference from many independent (elementary) microscopic areas within the illuminated region of the object. The space–time evolution of speckles in a plane has been treated extensively in the literature, where many applications regarding measurements of surface roughness, object displacement, and velocity have been made.1 Many optically based systems rely on the use of speckle dynamics for determining translation, rotation, tilt, etc., for solid objects. The dynamic range of these systems is usually determined by speckle decorrelation, which gives the maximum allowable object deflection. Knowledge of the dynamic speckle behavior in three dimensions, for example, may facilitate placement of the detector (CCD array, holographic plate, etc.) in the proper orientation so as to preserve optimally the spatial correlation of the probed field. Yoshimura and Iwamoto2 investigated the dynamic properties of 3D speckles formed in free space that result from scattering off a planar diffuse object moving in its own plane. In many applications of interest, however, optical components are present between the object plane 0740-3232/99/061402-11$15.00

and the observation plane. No comprehensive analysis has been given in the literature for the dynamic properties of 3D speckles that is valid for arbitrary paraxial optical systems, as characterized here by ABCD ray-matrix techniques. Here we are concerned with the intrinsic 3D nature of speckles that are formed after the scattered light has propagated through an arbitrary optical system.3 Except for imaging systems, which are treated separately below, we consider, for simplicity in notation, only optical systems without limiting apertures (i.e., realvalued ABCD optical systems). This is because the corresponding analytic results for general complex systems are very long, cumbersome, and not readily amenable to physical interpretation. The speckles are affected by the motion of the diffuse object. In particular, we discuss the 3D behavior of the speckles for the case of an in-plane translation, an out of-plane rotation, or an in-plane rotation of a planar diffuse object that is illuminated, in a direction perpendicular to its surface, by a laser beam with a Gaussian-shaped intensity distribution. As indicated in Fig. 1, we consider a Cartesian coordinate system with the x – y axes lying in the plane of the object and the z (or optic) axis normal to its surface. We assume here that the object either moves in its own plane with constant velocity or rotates at a uniform angular velocity. In Section 2 a general expression, valid for an arbitrary ABCD paraxial optical system, is derived for the space– time-lagged covariance of intensity that characterizes the statistical nature of 3D speckles obtained from scattering or reflection off a diffuse object. In-plane translation and © 1999 Optical Society of America

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complex optical field U(p, t) in the observation planes is a circular complex Gaussian random variable with zero mean, it can be shown that4 C I ~ p1 , p2 ; t ! 5 Fig. 1. Propagation of reflected light from the object plane through an arbitrary ABCD optical system to an observation plane. The vector r is a 2D vector in a plane perpendicular to the optic axis at z.

out-of-plane rotation of the object are considered in Section 3, where general analytic results are derived for the intensity covariance. The 3D nature of the speckle pattern is interpreted in terms of decorrelation, translation, and tilting. We then derive the general conditions under which one obtains either pure translation or tilting of the 3D speckles. As special cases of interest, we discuss and compare the dynamical nature of 3D speckles obtained in free space, Fourier transform systems, and an imagelike configuration. In addition, we present corresponding results for an imaging system, for which a relatively simple analytic expression for the covariance is obtained. In Section 4, in-plane rotation of the object is considered, where the optic axis and axis of rotation are parallel, and the resulting nature of the 3D speckles is interpreted in terms of decorrelation, boiling, and rotation. In particular, it is shown, in contrast to the case of in-plane translation, that the resulting nature of the 3D speckles is independent of the wave-front radius of curvature of the illuminating beam. Experimental evidence, which supports the quantitative theory, is presented and discussed in Section 5. Finally, in Section 6 we discuss, compare, and summarize some of the novel features of 3D speckle obtained under various circumstances, followed by some concluding remarks.

2. GENERAL CONSIDERATIONS The most general description of the space–time evolution of speckle patterns is given by the spatiotemporal covariance of the optical intensity,1 C I (p1 , p2 ; t 1 , t 2 ), where, as indicated in Fig. 1, p1,2 5 $ x 1,2 , y 1,2 , z 1,2% denote the two spatial observation points of interest and t 1,2 are the corresponding observation times. In its normalized form this quantity is defined as

C I ~ p1 , p2 ; t 1 , t 2 ! 5

G ~ p1 , p1 ; 0 ! G ~ p2 , p2 ; 0 !

, (2.2a)

where G ~ p1 , p2 ; t ! 5 ^ U ~ p1 , t 1 ! U * ~ p2 , t 2 ! &

(2.2b)

is the mutual coherence function of the optical field and the asterisk denotes the complex conjugate. Because of the circular complex Gaussian statistics of the optical fields, it can be shown that the corresponding joint probability-density function of I 1 @ 5I 1 (p1 , t 1 ) # and I 2 @ 5I 1 (p2 , t 2 ) # takes the general form4

F

exp 2 P~ I1 , I2! 5

I1 1 I2

^I&~ 1 2 CI!

^ I & 2~ 1 2 C I !

G

I0

F

2 AI 1 I 2 C I

^I&~ 1 2 CI!

G

,

(2.3)

where C I [ C I (p1 , p2 ; t ), I 0 ( • ) is the modified Bessel function of the first kind and zero order, angle brackets denote the ensemble average, and it has been assumed that the average intensity ^ I & 5 ^ I 1 & 5 ^ I 2 & . Examination of Eq. (2.3) shows that the joint probability density depends totally on C I . Note, in particular, that as C I → 1, P(I 1 , I 2 ) → P(I 1 ) d (I 1 2 I 2 ), where d (•) is the Dirac delta function and P(I 1 ) 5 exp(2I/^I&)/^I& is the well-known negative exponential probability-density function for intensity at a point. In this limit, the corresponding conditional probability distribution of I 2 , given that the value of intensity is known to be I 1 , is given by P(I 2 u I 1 ) 5 P(I 1 , I 2 )/P(I 1 ) 5 d (I 2 2 I 1 ). This means that as C I approaches unity, the most probable value of I 2 clusters about the known value of I 1 . That is, if we are given that at time t 1 a bright speckle is located at p1 , then this speckle will, with a high degree of certainty, be located at p2 at time t 2 . Thus information regarding the (mean) motion of a speckle as a whole in space and time can be obtained from knowledge of where the cross covariance approaches unity. On the other hand, information regarding the relative spatial intensity characteristics of the speckles at a given time (e.g., size, shape, and orientation of static 3D speck-

^ I ~ p1 , t 1 ! I ~ p2 , t 2 ! & 2 ^ I ~ p1 , t 1 ! &^ I ~ p2 , t 2 ! & $ @ ^ I ~ p1 , t 1 ! 2 & 2 ^ I ~ p1 , t 1 ! & 2 #@ ^ I ~ p2 , t 2 ! 2 & 2 ^ I ~ p2 , t 2 ! & 2 # % 1/2

where I(p, t) is the observed intensity at point p and time t and angle brackets denote the ensemble average over the stochastic realization of the surface-height fluctuations of the diffuse object. Because we assume quasimonochromatic conditions and statistical stationarity, first-order moments, such as ^ I(p, t) & , are independent of absolute time, whereas second-order moments, such as ^ I(p1 , t 1 )I(p2 , t 2 ) & are functions of the time difference, t (5t 2 2 t 1 ). For fully developed speckle, where the

u G ~ p1 , p2 ; t ! u 2

,

(2.1)

les) can be obtained directly from the corresponding spatial distribution of the normalized spatial autocorrelation function of intensity, C I (p1 , p2 ; 0). This is because C I (p1 , p2 ; 0), by definition, gives the extent in space where the intensity is mutually correlated. Thus, from knowledge of the 3D spatial distribution of the normalized spatial autocorrelation function of intensity, one can obtain quantitative estimates of the characteristic lateral and longitudinal speckle sizes and the corresponding ori-

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entation. Similarly, the relative temporal characteristics of a speckle located at the point p are obtained from C I (p, p; t ). The paraxial optical system intervening in the space between the diffuse object and the observation plane is represented by its corresponding ray-optics matrix. ABCD ray-matrix techniques combined with the paraxial approximation of the Huygens–Fresnel formulation of wave optics is a useful tool in the analysis of wave propagation through a general class of optical systems.5,6 Within the generalized ray-matrix formulation we derive exact analytic expressions for the space–time-lagged covariance of intensity; the results presented below are valid for arbitrary real ABCD optical systems in both the Fresnel and the Fraunhofer regions. In the following, we assume, for simplicity, cylindrically symmetric ABCD systems7 and that the object plane and the observation plane are both in the same media. The optical field in an observation plane can be expressed in terms of the corresponding field (e.g., the reflected or the scattered field) in the object plane by the use of the paraxial approximation of the Huygens–Fresnel principle, as5,6 U ~ p, t ! 5

E

d2rU 0 ~ r, t ! G ~ r, p! ,

(2.4)

where the integration indicated in Eq. (2.4) is over the entire object plane (i.e., the r x 2 r y plane at z 5 0), G(r, p) is the Green’s function for the ABCD optical system, given (to within an unimportant multiplicative phase factor) by

F

G

ik ik G ~ r, p! 5 2 exp 2 ~ Ar2 2 2r • r 1 D r2 ! , 2pB 2B (2.5) where the two-dimensional (2D) vector r 5 $ x, y % ; x and y are the components of the observation point p normal to the optic axis; k is the optical wave number; A, B, and D are the corresponding ray-matrix elements of the optical system; and we assume that the detector integration time is short compared with the characteristic speckle correlation time. The quantity U 0 (r, t) is the instantaneous reflected (or scattered) field in the object plane. Specifically, we model the reflected optical field as U 0 ~ r, t ! 5 U i ~ r, t ! c ~ r, t ! ,

(2.6)

where U i (r) is the incident field (assumed time independent and fixed in space) and c (r, t) is the stochastic timevarying (owing to the motion of the diffuse object) complex reflection coefficient. We assume that the microscopic structure of the object changes the phase of the incident light only, where for fully developed speckle we have

^ c ~ r1 , t ! c * ~ r2 , t ! & 5 const. 3 d ~ r1 2 r2 ! ,

(2.7)

where d (r) is the 2D Dirac delta function. For a diffuse object at any instant of time, the reflected fields from distinct points on the object are uncorrelated; that is, the scattering process is completely incoherent. This implies that for a scatterer within the illuminated region at time t and position r1 the mutual coherence function of the reflected field in the object plane is given, to within an unimportant constant, by

G 0 ~ r1 , r2 ; t ! 5 ^ U 0 ~ r1 , t ! U 0* ~ r2 , t 1 t ! & 5 U i ~ r1 ! U i* ~ r2 ! d ~ r2 2 r18 ! ,

(2.8)

where r18 is the position at time t 1 t of the scatterer that was located at position r1 at time t. Hence substituting Eqs. (2.4) and (2.6) into Eq. (2.2b), making use of Eq. (2.8), and simplifying yields G ~ p1 , p2 ; t ! 5

E

d2rU i ~ r! U i* ~ r8 ! G ~ r, p1 ! G * ~ r8 , p2 ! . (2.9)

For the case of an in-plane translation at a constant velocity v(5$ v x , v y , 0 % ) we have r8 5 r 1 vt .

(2.10)

On the other hand, assuming an in-plane rotation at a constant angular velocity v about the optic axis, we have

S D S

D

x8 x cos v t 2 y sin v t 5 . y8 x sin v t 1 y cos v t

(2.11)

Note that the time lag always appears in the dimensionless form given by u 5 v t , where u is the corresponding angular lag. Thus the results presented below are valid for the corresponding covariance of intensity for an inplane angular displacement u of the object. To be specific, we assume that the plane perpendicular to the z axis that contains the observation point p2 is displaced by a distance Dz with respect to the corresponding plane that contains p1 . We denote by A, B, C, and D (where AD 2 BC 5 1) the ray-matrix elements corresponding to propagation from the object plane and the plane that contains p1 . Then the corresponding raymatrix elements for the displaced plane that contains p2 are given by

F

Ad

Bd

Cc

Dd

G F 5

1 0

GF G

Dz A 1

C

B

D

.

(2.12)

That is, A d 5 A 1 DzC, B d 5 B 1 DzD, C d 5 C, D d 5 D, and the corresponding displaced Green’s function, G d (r8 , p2 ), is given by Eq. (2.5) with A, B, and D replaced by A d , B d , and D d , respectively. To proceed, we assume Gaussian illumination, where the incident field is given, to within an unimportant multiplicative factor, by

F S

U i ~ r! 5 exp 2r 2

1 w

2

1

ik 2R

DG

,

(2.13)

where w and R are the 1/e 2 intensity radius and the wavefront radius of curvature, respectively, of the illuminating beam. Strictly speaking, Eq. (2.13) is valid for planar objects. However, it also applies to an object rotating about an axis of radius R 0 , with its axis perpendicular to direction of propagation of the illuminating beam (i.e., out-ofplane rotation), where the quantity 1/R in Eq. (2.13) is replaced by 1/R 1 2/R 0 (Ref. 8), and v t is replaced by u R 0 , where u is the out of-plane angular displacement. Note that the sign convention used here for the wave-front radius of curvature is such that a converging illuminating beam corresponds to R , 0, whereas a diverging illuminating beam corresponds to R . 0. For simplicity in notation, in the following we denote the observation plane

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that is perpendicular to the z axis and contains the point p1 as the p plane, with coordinates given by $ r, z % , where z is the free-space distance between the object and the observation plane. The corresponding plane displaced by a distance Dz along the z axis contains the point p2 , whose coordinates are denoted by $ r 1 D r, z 1 Dz % .

Substituting Eqs. (2.5), (2.13), and (2.10) into Eq. (2.9), performing the resulting (Gaussian) integrations, and simplifying yields9 C I ~ r, D r, Dz; t ! 5

@ 1 1 ~ Dz/l z ! 2 #

F

1

˜ vt exp 2 2 u D r 2 A r0

˜ vt ! /B u 2 2 DDz ~ r 1 A

G

3 exp@ 2~ v t /w ! 2 # ,

(3.1)

where ˜ 5 A 1 B/R, A lz 5

r0 5 >

(3.2)

4B ~ B 1 DDz ! kw 2 2 ~ B 1 DDz ! kw 2B kw

>

4B 2 kw

, 2

(3.3)

@ 1 1 ~ Dz/l z ! 2 # 1/2

@ 1 1 ~ Dz/l z ! 2 # 1/2 .

(3.4)

The first and second terms on the right-hand side of Eq. (3.1) relate to the 3D spatiotemporal properties of the speckle pattern, whereas the third term is the temporal decorrelation factor, which is identical to that of 2D speckles. The approximation indicated in both Eqs. (3.3) and (3.4) is consistent with the paraxial approximation in that Dz cannot be large compared with B/D. From approximations (3.3) and (3.4) we find that l z / r 0 5 2B/w @ 1 for most practical cases of interest. That is, the length of a speckle along the optic axis is much larger than the corresponding transverse length. See Appendix A, where the corresponding case of pure longitudinal displacement of the diffuse object along the z axis is considered. For the special case of Dz 5 0 (i.e., 2D speckles in a plane normal to the optic axis) Eq. (3.1) becomes

F

C I ~ r, D r, Dz; t ! 5

1 @ 1 1 ~ Dz/l z ! 2 #

exp@ 2~ v t /w ! 2 #

F

G

1 3 exp 2 2 u D r 2 A 0 vt 2 Dz ~ r 1 A 0 vt ! /z u 2 , r0

(3.6)

where

3. IN-PLANE TRANSLATION

1

1405

G

˜ vt u 2 uDr 2 A C I ~ r , D r ; t ! 5 exp@ 2~ v t /w ! 2 # exp 2 , ~ 2B/kw ! 2

(3.5)

which is identical to that obtained previously for realvalued ABCD systems.8 Specifically, w/v and 2B/kw give the characteristic temporal deformation time and the mean transverse size, respectively, of two-dimensional speckles in a plane perpendicular to the z axis. For free-space propagation we have that A 5 1 1 z/R, B 5 z, D 5 1, and Eq. (3.1) becomes

A 0 5 1 1 z/R,

l z 5 4z 2 /kw 2 ,

r 0 5 2z @ 1 1 ~ Dz/l z ! 2 # 1/2/kw. On comparing Eq. (3.6) with the corresponding results given by Eq. (8) of Ref. 2, we find that the only difference is that instead of having the (correct) 2D vector quantities Dr and r in the exponent, Yoshimura and Iwamoto’s result has the 3D vectors Dp and p. For real-valued ABCD systems, examination of Eq. (3.1) reveals that for a given observation point p, the space–time covariance of intensity will have an absolute maximum of exp@2(vt/w)2# for the 2D vector Dr given by ˜ vt , Dr 5 A

(3.7a)

Dz 5 0.

(3.7b)

That is, for a time lag that is short compared with the speckle decorrelation time w/v, a given speckle that at time t 1 is located at p1 5 $ r , z % will most likely be located ˜ vt , z % . Thus, for in-plane translation and at p2 5 $ r 1 A out-of plane rotation of the diffuse object, distinct speckles at some distance from the object move in a corresponding (parallel) plane with no apparent motion along the z ˜ vt , axis. Note that the change in transverse position, A is identical to the result obtained for the corresponding motion of speckles in a plane.8 In particular, no temporal change of position of speckles is obtained for optical ˜ 5 A 1 B/R 5 0 [e.g., at a focus in systems that satisfy A free space or for Fourier transform systems for collimated beams (i.e., R → `)]. The corresponding 2D speckle characteristics in a plane perpendicular to the z axis are referred to in the literature as being in a boiling motion (i.e., the speckles change, in a time period of the order w/v, only in structure without translation). As discussed in Section 2, the spatial characteristics of 3D speckles can be obtained directly from C I (p1 , p2 ; 0), which from Eq. (3.1) takes the form C I ~ r, D r, Dz; 0 ! 5

F U

1

1 exp 2 2 D r r0 @ 1 1 ~ Dz/l z ! # 2

2 DDz r /B

UG 2

.

(3.8)

Examination of Eq. (3.8) reveals that for a given value of r and Dz(Þ0) the spatial intensity covariance attains a relative maximum for Dr 5

D B

rDz.

(3.9)

That is, the direction of relative maximum correlation of intensity lies along the straight line defined by Eq. (3.9). Following the authors of Ref. 2, we define this line as the

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major axis of a 3D speckle centered at point p. Examination of Eq. (3.9) reveals that, in general, the inclination (or slope) of a static 3D speckle is a linear function of absolute lateral position r and displacement along the z axis Dz, and it is independent of the wave-front radius of curvature of the illuminating beam. For optical systems that satisfy D 5 0 (e.g., Fourier transform system) Eq. (3.9) indicates that the speckles are parallel to the z axis. On the other hand, for free space, where B 5 z, and D 5 1, the speckles are oriented toward the origin, in agreement with the results obtained in Ref. 2 for static speckles. Similar results apply near a focal plane. Another quantity of interest is the mean (static) speckle length, which denotes the mean length of speckles along the major axis. Because the spatial covariance of intensity both along the major speckle axis [i.e., for transverse separations satisfying Eq. (3.9)] and along the optic axis (i.e., r 5 0, and D r 5 0) are identical, it follows that all speckles located in a given plane have the same projection along the z axis. From Eq. (3.1) we have C I ~ Dz, 0 ! 5

1 @ 1 1 ~ Dz/l z ! 2 #

.

4B 2 kw 2

.

(3.11)

The mean length of the speckle, L, is given by L 5 L z /cos a, where a is the angle between the major axis and the z axis. From Eq. (3.9) it can be shown that tan a 5 Dr/B, where r 5 u ru , and hence L5

4B kw 2

AB 2 1 D 2 r 2 .

u ~ t ! [ D r /Dz 5

(3.12)

Equations (3.11) and (3.12) are the generalizations of the corresponding free-space results, given by Eqs. (15) and (16) of Ref. 2, to ABCD systems. Examination of Eq. (3.12) shows that, in general, the speckle length increases in proportion to (B/w) 2 . Although the projection of the speckle length on the z axis is independent of transverse position, the speckle length increases with transverse position from the z axis, as indicated by Eq. (3.12). However, for optical systems where D 5 0 (e.g., Fourier transform systems) the speckle length is also independent of transverse position. In these systems, the 3D static speckles are aligned parallel to the z axis, with a length L 5 4B 2 /kw 2 , which is the generalization to ABCD systems of the depth of focus. In summary, 3D speckles are such that at any instant in time they are elongated and inclined with respect to the z axis. Both the slope and the characteristic longitudinal length depend on transverse position only, as indicated by Eqs. (3.9) and (3.12), respectively. Furthermore, in a plane transverse to the z axis (i.e., Dz 5 0), the characteristic speckle size is given by Eq. (3.3), which is identical to that obtained previously.10 Next we investigate the 3D properties of dynamic speckles. Consider a speckle that at a given time is lo-

D B

˜ vt ! . ~r 1 A

(3.13)

Note that the orientation of the major axis of the speckle depends only on the transverse position of the speckle at any given instant and is independent of the specific details of the motion of the object. That is, at any given transverse position, the slope of the speckle is independent of time. Equation (3.13) indicates that speckles obtained from optical systems that satisfy D 5 0 (e.g., Fourier transform systems) will have their slopes always oriented parallel to the z axis. As another illustrative example, consider free space where A 5 1, B 5 z, and D 5 1, and Eq. (3.13) becomes

(3.10)

Following Yoshimura and Iwamoto,2 we define the mean speckle length along the optic axis, L z , from the condition that the intensity covariance equals 1/2. Hence combining Eqs. (3.3) and (3.10) yields LZ 5

cated at position $ r, z % with a slope given by Eq. (3.9). After a time period t, which is small in comparison with w/v, this speckle most likely will be located at ˜ vt , z % , with the slope of the major axis of the $r 1 A speckle, u (t), given by the new position in space, i.e., from Eq. (3.9):

u ~t! 5

˜ vt r1A z

.

(3.14)

Equation (3.14) indicates that dynamic free-space speckles are always oriented toward the origin. This is in contrast to the results of Ref. 2, where it was concluded that the major axis of dynamic free-space speckles is directed toward the point vt /2 in the object plane. The reason for this discrepancy is related to the fact that the authors of Ref. 2 obtained the orientation of the speckles from the correlation of the space–time-lagged intensity. This is physically incorrect. As indicated in Section 2, this function gives information regarding only the movement in space–time of the speckles as an entity. Information regarding the orientation of the speckles can only be obtained from the autocorrelation properties of the intensity that exists at the same time [i.e., from C I (p1 , p2 ; 0)]. The mean change in slope in a time period t is given by Du ~t! 5

D B

˜ vt . A

(3.15)

Examination of Eq. (3.15) reveals that speckles obtained ˜ 5 0 or D 5 0 do in optical systems that satisfy either A not change their slopes as a function of time. Speckles ˜ 5 A 1 B/R obtained from optical systems that satisfy A 5 0 (e.g., in free space or at a focal plane) neither translate nor change their tilts with respect to the z axis as a function of time; they are in a pure boiling state. However, speckles obtained from optical systems that satisfy ˜ Þ 0, and D 5 0 are in a state of pure translation. A Let us now consider the velocity of the moving speckles. The change in transverse position of a speckle centered in some transverse plane over a time period t is given by ˜ vt . Because the speckle changes its slope during this A time period, the corresponding change in transverse position of this speckle obtained in the plane displaced by an ˜ vt 1 DzD u ( t ). Hence the veamount Dz is given by A locity of the speckle, which is the rate of change of position with respect to time, is given by

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˜v1 VS ~ Dz ! 5 A

D B

˜ v. DzA

where m 5 f 2 /f 1 is the system magnification and (3.16)

The first term on the right-hand side of Eq. (3.16), which is independent of Dz, is the component of velocity that is due to in-plane speckle translation (i.e., it represents a shift of the speckle as a whole). On the other hand, the corresponding second term, because it is linearly dependent on Dz, is the component of the velocity that is due to the tilting motion of the speckle. Because the speckle length along the z axis is of the order L Z [see Eq. (3.11)], an estimate of the maximum velocity difference of a speckle is given by ˜ u vu /B 5 8A ˜ BD u vu /kw 2 . V S ~ L Z ! 2 V S ~ 2L Z ! 5 2DL Z A Imaging systems without limiting apertures have B 5 0 in the image plane. Hence to include such systems we must generalize the analysis to complex-valued ABCD ray-matrix elements (i.e., systems containing limiting apertures). Although analytic results can be obtained for an arbitrary complex ABCD system, the resulting complex expressions are very long and physically uninteresting. Therefore, to obtain the 3D characteristics of speckle obtained near an image plane, we consider the special case of clean imaging discussed by Yura and Hanson.10 This system consists of a large positive thin lens of focal length f 1 that is placed a distance f 1 to the right of a diffuse source. A distance f 1 to the right of this lens is a Gaussian limiting aperture of radius s. A further distance f 2 to the right of this aperture is a large positive lens of focal length f 2 . Finally, the image plane is located a further distance f 2 to the right of this lens. An imaging system of this type is convenient, because by varying the limiting-aperture radius s, one can match the target resolution to ensure a fully resolved speckle pattern across the detector array. Furthermore, since the limiting aperture is placed in the Fourier plane, the clean imaging system is relatively insensitive to vignetting effects and quadratic phase factors, which are present in direct imaging systems. The absence of such factors leads to mathematical simplification of the complex field in the image plane. We analyze the clean imaging configuration to illustrate the characteristics of 3D speckle obtained in the image plane. Here we have A 5 2f 2 /f 1 , B 5 22if 1 f 2 /k s 2 , and D 5 2f 1 /f 2 (see Ref. 10). We also assume, for theoretical simplicity, that s @ 2 f 1 /kw (i.e., the illuminated spot on the diffuse object is well resolved by the optical system). In this case the speckle characteristics near the image plane are dominated by the effects of the limiting aperture in the Fourier plane (i.e., speckle effects due to the finite size of the diffuse object are negligible).11 Proceeding in a manner similar to that which led to Eq. (3.1) yields C I ~ D r, Dz; t ! 5

exp@ 2~ v t /w ! 2 # @ 1 1 ~ Dz/L ZI ! 2 #

F

1407

u D r 2 mvt u 2 3 exp 2 ~ 2 f 2 /k s ! 2 @ 1 1 ~ Dz/L ZI ! 2 #

G

(3.17)

L ZI 5

4f 2 2

(3.18)

ks 2

is the depth of focus. Examination of Eq. (3.16) reveals that in an imaging system the 3D dynamical nature of speckles is such that their statistical characteristics are independent of absolute lateral position. The speckles remain directed parallel to the z axis and have a characteristic length along the optic axis of the order of the depth of focus, with a decorrelation time of the order w/v. That is, in-plane target translation gives rise to a corresponding (magnified) translation of the speckles without tilt.

4. IN-PLANE ROTATION First, for simplicity we consider the case in which the optic axis and the axis of rotation coincide. Substituting Eqs. (2.5), (2.13), and (2.11) into Eq. (2.9) and performing the resulting (Gaussian) integrations yields9 C I ~ r, D r, Dz; t ! 5

1

X HS

1 exp 2 2 r0 @ 1 1 ~ Dz/l z ! # 2

S

12 11

D Dz B

DF

Dr 2

R5

F

B

2 r • ~ r 1 D r! sin2

1 D r • R • r sin v t where

D

0

1

21

0

Dz r

D

2

vt 2 (4.1)

G

(4.2)

and the other quantities have been defined above. Examination of Eq. (4.1) reveals that, in contrast to the case of in-plane translation, in general in the case of in-plane rotation, the spacetime properties of both 2D and 3D speckles are independent of the wave-front radius of curvature. This is because the respective contribution to the phase that is due to wave-front curvature by the scatterer at time t is equal to that at time t 1 t . As indicated by Eqs. (2.9) and (2.13), the contribution to the mutual coherence function from the elementary point radiator at transverse position r is proportional to U i (r)U i* (r8 ) } exp@2ik(r2 2 r82)/2R # 5 1, because for in-plane rotation the magnitude of r equals the magnitude of r8 [see Eq. (2.11)]. For Dz 5 0, Eq. (4.1) reduces to the equation obtained by Churnside for the corresponding 2D case.11 The inplane-rotation temporal autocorrelation function at lateral position r is obtained by setting D r 5 0 in Eq. (4.1). In many cases of interest we have that r / r 0 5 r /(2B/kw) @ 1. Under these conditions the autocorrelation function consists of a series of Gaussian-like peaks located at integral multiples of v t 5 2 p . 12 Near one of the peaks the autocorrelation function becomes C I ~ r, 0; t ! 5 exp@ 2~ t / t R ! 2 # ,

(4.3)

where the characteristic rotational deformation time is given by

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t R 5 r 0 / rv .

(4.4)

Speckles close to the z axis persist longer than those that are further away. This is expected physically, because for observation points near the axis of rotation, less apparent motion of the object is perceived than for observation points further away. Note that although both the static characteristic speckle length L and speckle size r 0 obtained here are identical to the properties obtained for the case of in-plane translation, the corresponding 3D dynamical properties are, however, quite different. The space–time properties of the motion of the speckles as a whole are obtained from the location of the absolute maximum of C I ( r, D r, Dz; t ). Differentiating Eq. (4.1) with respect to Dx, Dy, and Dz and setting the resulting equations equal to zero yields x ~ t ! 5 ~ x cos v t 2 y sin v t ! , y ~ t ! 5 ~ y cos v t 1 x sin v t ! ,

(4.5a)

Dz 5 0,

(4.5b)

and where x( t ) and y ( t ) are, respectively, the x and y coordinates of the position of the center of the speckle, as a function of time lag, of the speckle that at t 5 0 was centered at r 5 $ x, y % . 13 Examination of Eq. (4.5a) shows that for a general ABCD system the 3D speckles rotate about the z axis with the same angular velocity as the object. That is, in a coordinate system rotating with the target, the speckle pattern will remain fixed in time, as expected physically. As discussed in Section 2, the spatial characteristics of 3D speckles can be obtained directly from the location of a relative maximum of C I ( r, D r, Dz; 0). Examination of Eq. (4.1) for t 5 0 reveals that for a given value of r and Dz Þ 0 the spatial intensity covariance attains a relative maximum for D r 5 D rDz/B, which is, as expected, identical to that obtained for in-plane translation.14 Thus, for example, in a Fourier transform system, where D 5 0, the direction of the speckles is parallel to the z axis, independent of axial position. For free space, however, where B 5 z and D 5 1, the major axis of the speckles is directed toward the origin [i.e., (0, 0, 0)]. Next we consider off-axis illumination, where the illuminating spot is displaced a radial distance a from the axis of rotation. Because of practical considerations (e.g., power budget), the optic axis is assumed centered about the illuminating spot rather than about the axis of rotation.15 A microscopic scattering center located initially at $ x, y, 0% will after a time t be located at $ x 8 , y 8 , 0% , where

S D S

D

x 8~ t ! x cos v t 2 ~ y 1 a ! sin v t 5 , y 8~ t ! x sin v t 1 ~ y 1 a ! cos v t 2 a

static speckles with respect to the optic axis is independent of the motion of the diffuse object and is given in Eq. (3.9). Thus, for example, in free space the major axis of the speckles will always be directed toward the center of the illuminated spot rather than toward the origin. Finally, as discussed in Ref. 15, the temporal correlation properties at any point in space consist of a series of Gaussian-like peaks and consist of two different terms that describe the decorrelation process of speckle boiling and rotation. Speckle boiling has the decorrelation time t B 5 a/ v w, while the corresponding decorrelation time ˜ u 21 . that is due to speckle rotation t R 5 ( r 0 / v ) u r 1 aA

5. EXPERIMENTAL OBSERVATIONS In this section the 3D properties of speckles are investigated experimentally and compared with the theoretical results given above. Only static measurements have been performed to measure the mean speckle length and slopes for various optical configurations. A. Measurement Setup A schematic of the experimental setup that has been used for the measurements is shown in Fig. 2. A collimated He–Ne laser beam with a wavelength l of 632.8 nm and a spot size w of 1 mm illuminated the target, and a 2D image sensor collected the scattered light. The CCD image sensor had 512 3 512 pixels with a pitch of 11.7 mm in the x direction and 8.8 mm in the y direction. An optional converging lens f had a focal length of 60 mm and a diameter of 18 mm. A matt silicon wafer giving rise to fully developed speckle was used as target. The CCD image sensor was placed on a translation stage that provided displacement of the image sensor along the optic axis. The readout from the image sensor after displacement Dz was cross correlated with the reference readout taken before (i.e., when the image sensor was placed at z 2 ). Note that in the measurements performed here, the speckle size is larger than the size of the individual pixels. Hence there is no appreciable speckle averaging, and the

(4.6)

where, for definiteness, we assume that the illuminated spot is displaced along the y axis (see Fig. 1 of Ref. 15). That is, in a coordinate system whose transverse positions are given by Eq. (4.6) the speckle pattern will remain fixed in time. This means that a speckle that was originally centered at $ x, y, z % will after time t be centered at $ x 8 ( t ), y 8 ( t ), z % , where x 8 ( t ) and y 8 ( t ) are given by Eq. (4.6). As discussed above, the (mean) slope of the

Fig. 2. Optical diagram for the measurement setup. The first image is acquired at the position z 2 , whereafter additional measurements are taken for displacements of Dz. For free-space measurements, the lens is omitted and the distance z 2 1 Dz denotes the distance between the target and the image sensor.

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Fig. 3. Division of the captured image (speckle pattern) into subimages.

Vol. 16, No. 6 / June 1999 / J. Opt. Soc. Am. A

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The measurements are clearly Lorentzian, which is in good agreement with Eq. (3.10). As indicated by Eq. (3.11), the expected projection of the speckle length along the z axis, L Z , is 2 mm, which should be compared with the measured result of 1.1 mm. The discrepancy between theory and measurements is most likely caused by electronic decorrelation that occurs when the speckle patterns are digitized and by uncertainties in measuring the exact spot size at the target. In the free-space configuration we now measure the speckle slopes at different positions in space. Note that we measure the slope at five different positions for various values of longitudinal position z 2 . The resulting slopes in the x and y directions are illustrated in Figs. 5(a) and 5(b), respectively. Note that for both the x and y directions the speckles are oriented toward the origin (i.e., they are oriented toward z 2 5 0). The reason the origin is not located at $ x,y % 5 $ 0,0% is that the image sensor is not displaced parallel to the optics axis but slightly inclined. This gives the same bias to all the measured slopes, and thus it is important to note that the speckles are oriented toward z 2 5 0, as depicted in Eq. (3.14) Consider now the imagelike configuration where the distance between the target and the lens z 1 is 2 f (5120 mm). We measure the speckle slopes at different positions in space immediately after the lens. The resulting

Fig. 4. Lorentzian fit to the measured peak values of the cross covariance for free-space propagation. The reference image is captured at z 2 5 70 mm.

resulting intensity covariance characterized by Eq. (3.8) is valid for the measurements described in this section. To measure the speckle slope, we divided the captured speckle pattern into five subimages, as is illustrated in Fig. 3. As indicated in this figure, each subimage consists of 40 3 40 pixels, which are distributed across the CCD frame. We performed measurements for three different optical configurations: free-space, imagelike, and a Fourier transform system. In the free-space configuration the lens is omitted and z 2 1 Dz denotes the distance between the target and the image sensor. In the imagelike configuration the distance between the target and the lens z 1 is 2 f, and for the Fourier transform system, the distance z 1 is f. Because the aperture defined by the converging lens 2 f is large compared with the diffraction spot size (i.e., s @ lz 1 /w), the corresponding ABCD matrix elements can be assumed to be real. B. Measurements First, consider the free-space setup where z 2 5 70 mm. After the reference image is acquired, the image sensor is displaced along the optics axis Dz. For each displacement we measure the peak value of the corresponding cross covariance (here performed on the entire speckle pattern). The results are illustrated in Fig. 4, where we have fitted a Lorentzian curve to the measurements.

Fig. 5. (a) Speckle slope in the x direction for various transverse positions for a free-space configuration. Note that for positions 2, 4, and 5, the measurements are overlapping. (b) Speckle slope in the y direction for various transverse positions for a freespace configuration. Note that for positions 1, 2, and 3, the measurements are overlapping.

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6. DISCUSSION AND CONCLUSION A thorough theoretical investigation of the static and dynamic properties of 3D speckles that are due to in-plane translation, out-of-plane rotation, and in-plane rotation of a diffuse scattering object has been conducted within the framework of real-valued ABCD-matrix optical systems (i.e., systems without inherent limiting apertures). Specifically, three commonly used optical systems have been considered for illustration: free-space propagation, Fourier transformation, and imagelike transformation. For each of these systems the static properties have been examined extensively, and analytical expressions for the length and slope of the speckle grains with respect to the optic axis have been presented. With use of the formulas, it can be shown that the slope of the speckle grains in the x – z plane (or in the y – z plane) will be parallel to the interference lines that arise if the object is replaced by two point sources arranged at the two opposite edges of the illuminated spot of the target. The speckle grains thus point toward the center of the illuminated area for free-space propagation irrespective of the radius of curvature of the incident beam, whereas the speckle grains will be aligned parallel to the optic axis in a Fourier transformation configuration. In an imagelike configuration the speckle grains will be pointing toward the corresponding image point. For the clean-image configuration, we have

Fig. 6. (a) Speckle slope in the x direction at various transverse positions for an imagelike configuration. Note that for positions 2, 4, and 5, the measurements are overlapping. (b) Speckle slope in the y direction at various transverse positions for an imagelike configuration. Note that for positions 1, 2, and 3, the measurements are overlapping.

slopes in the x and y directions are illustrated in Figs. 6(a) and 6(b), respectively. Note that for both the x and y directions the speckles are oriented toward the position where the image plane intersects the z axis. This is in accordance with Eq. (3.9), where D 5 21 and B 5 2 f 2 z 2 [ L, and hence u 5 D r /B 5 2r /L, which indicates that because L is the distance between the observation plane and the image plane, the speckles point toward the intersection of the image plane and the z axis. The speckles are not oriented exactly toward the image plane because of experimental difficulties in obtaining the conjugate image planes (i.e., z 1 5 2 f ). Note again that the displacement of the image sensor is not exactly parallel to the optical axis. Finally, consider a Fourier transform system in which the distance between the target and the lens z 1 is f (560 mm). We measure the speckle slopes at different positions in space after the lens. The resulting slopes in the x and y directions are illustrated in Figs. 7(a) and 7(b), respectively. Note that for both the x and y directions the speckle slopes are parallel to the z axis, which is in accordance with theory [see the discussion following Eq. (3.13)]. The slight change in directions is caused by experimental difficulties in placing the target exactly in the Fourier plane.

Fig. 7. (a) Speckle slope in the x direction at various transverse positions for a Fourier transform system. Note that for positions 2, 4, and 5, the measurements are overlapping. (b) Speckle slope in the y direction at various transverse positions for a Fourier transform system. Note that for positions 1, 2, and 3, the measurements are overlapping.

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Vol. 16, No. 6 / June 1999 / J. Opt. Soc. Am. A

included the effect of a Gaussian-shaped aperture in the Fourier plane in order to show that the speckle grains in the exact image plane will be directed parallel to the optic axis. Experiments have been performed that substantiate the theoretical findings for static speckles for all three configurations. Dynamic speckles have been analyzed theoretically for in-plane displacement, for out-of-plane rotation of the object, and for in-plane rotation with the axis of rotation parallel to the optical axis. The relation between the cross covariance of the field before and after displacement has been established, showing that a mere linear displacement will, for any paraxial ABCD optical system, result in a linear and pure lateral displacement of the 3Dspeckle pattern combined with a tilting of the speckle grains. The final tilt of the speckle grain under observation will be identical to the tilt of the speckle grain that was present at this location before the object displacement took place. The present work has relevance for the design of speckle-based optical systems in which knowledge of speckle displacement and speckle decorrelation is of vital importance. Specifically, detection schemes are usually based on examination of the interception of a spatial distribution of speckle grains with a plane defined by a series of detectors, a CCD array, or photographic film. Speckle boiling can thus be caused in two ways, either by a comprehensive decorrelation of the granular structure or by the shifting of the granular speckle pattern through the plane. In the latter case, a change in the optical configuration may reduce the susceptibility of the system with respect to speckle boiling. Furthermore, because the longitudinal behavior of the speckle pattern reflects both spatial and temporal coherence of the source, the corresponding axial behavior of the speckle pattern will contain information on both aspects (e.g., Fourier transform interferometry). Future and more extensive work on the 3D behavior of speckle grains may disclose new speckle-based systems.

C 1 ~ r, D r, Dz; t !

F U S DS

exp 2 5

11

1

r 02

Dr 2

kw 2

D B

rDz 1

B

UG 2

˜ 1 CDz ! ~A

˜ 2v t Dz 2 A Z

2

4B 2

v Zt r

1411

˜ /B ! v t 1 1 ~ D/B ! Dz 2 ~ A Z

D

2

,

(A2)

˜ D 2 BC where, in obtaining Eq. (A2) we have used A 5 1 and, to lowest order in Dz and v Z t , r 0 5 2B/kw. Examination of Eq. (A2) reveals that a speckle that was originally centered at $ r , z % will, after a time delay t, most likely be located at $ r 1 D r ( t ), z 1 Dz( t ) % , where ˜ 2v t Dz ~ t ! 5 A Z

(A3)

and, to terms of first order in v Z t , D r~ t ! 5

5

D rDz B

2

˜ v Zt r A B

v Zt r B

˜ 1 CDz ! ~A

˜D 2 1 2 A ˜ Cv t ! > v t rA ˜ C. ~A Z Z

(A4)

Equations (A3) and (A4) indicate that for small displacements parallel to the optic axis, speckles as a whole will move, in general, to different longitudinal and transverse positions.16 This is in contrast to in-plane displacement, for example, where no corresponding motion along the optic axis is obtained. In particular, for optical systems ˜ 5 0 (e.g., Fourier transform systems for that satisfy A collimated beams), no motion either along or perpendicular to the optic axis is obtained, and for systems satisfying C 5 0 (e.g., free-space) no transverse motion is obtained. The inclination of the speckles is independent of the motion of the object and is given by Eq. (3.9), and the component of the length of the speckles along the z axis is identical to that given by Eq. (3.3).

ACKNOWLEDGMENT APPENDIX A In this appendix we consider the case of (pure) longitudinal displacement of the diffuse object, where v 5 $ 0, 0, 6v Z % and the plus-and-minus sign refers, respectively, to a displacement toward and away from the observation planes. For pure longitudinal displacement we have that r8 5 r. Then the corresponding ray-matrix elements for the displaced plane that contains p2 is given by (for definiteness we assume a displacement toward the observation plane in the following)

F

Ad

Bd

Cc

Dd

G F 5

1 0

GF GF

Dz A 1

C

B

1

D 0

2v Z t 1

G

.

(A1)

That is, A d 5 A 1 DzC, B d 5 B 1 DzD 2 v Z t (A 1 CDz), C d 5 C, D d 5 D 2 Cv Z t , and the corresponding displaced Green’s function, G d (r, p2 ), is given by Eq. (5) with A, B, and D replaced by A d , B d , and D d , respectively. Proceeding in a manner similar to that which led to Eq. (3.1) yields

The research of H. T. Yura was performed while he was a guest scientist at Risø National Laboratory, Roskilde, Denmark, August–November 1998, through a grant from the Danish Research Academy, and general support was received from the Danish Technical Science Foundation under the LIS program.

REFERENCES AND NOTES 1.

See, for example, J. C. Dainty, ‘‘The statistics of speckle patterns,’’ in Progress in Optics, E. Wolf, ed. (NorthHolland, Amsterdam, 1976), Vol. XIV, Chap. 1; S. M. Kozel and G. R. Lokshin, ‘‘Longitudinal correlation properties of coherent radiation scattered from a rough surface,’’ Opt. Spectrosc. 33, 89–90 (1972); T. Asakura and N. Takai, ‘‘Dynamic laser speckles and their application to velocity measurements of the diffuse object,’’ Appl. Phys. 25, 179–194 (1981); T. Yoshimura, ‘‘Statistical properties of dynamic speckles,’’ J. Opt. Soc. Am. A 3, 1032–1054 (1986); B. Rose, H. Imam, S. G. Hanson, H. T. Yura, and R. S. Hansen, ‘‘Laser-speckle angular-displacement sensor: theoretical and experimental study,’’ Appl. Opt. 37, 2119–2129 (1998); T. Okamoto and T. Asakura, ‘‘The statistics of dynamic speckles,’’ in Progress in Optics, E. Wolf, ed. (Elsevier, Am-

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2. 3. 4. 5. 6. 7. 8. 9.

J. Opt. Soc. Am. A / Vol. 16, No. 6 / June 1999 sterdam, 1995), Vol. XXXIV, Chap. 3; H. T. Yura, S. G. Hanson, and T. P. Grum, ‘‘Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,’’ J. Opt. Soc. Am. A 10, 316–323 (1993). T. Yoshimura and S. Iwamoto, ‘‘Dynamic properties of three-dimensional speckles,’’ J. Opt. Soc. Am. A 10, 324– 328 (1993). H. T. Yura and S. G. Hanson, ‘‘Optical beam wave propagation through complex optical systems,’’ J. Opt. Soc. Am. A 4, 1931–1948 (1987). J. W. Goodman, ‘‘Statistical properties of laser speckle patterns,’’ in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), Chap. 2. A. S. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 20. H. T. Yura and S. G. Hanson, ‘‘Optical beam wave propagation through complex optical systems,’’ J. Opt. Soc. Am. A 4, 1931–1948 (1987). The generalization of the results derived here to systems that are not cylindrically symmetric is straightforward. H. T. Yura, B. Rose, and S. G. Hanson, ‘‘Dynamic laser speckle in complex ABCD systems,’’ J. Opt. Soc. Am. A 15, 1160–1166 (1998). S. Wolfram, M ATHEMATICA : A System for Doing Math-

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10.

11. 12.

13. 14. 15. 16.

ematics by Computer (Addison-Wesley, Redwood City, Calif., 1991). H. T. Yura, S. G. Hanson, and T. P. Grum, ‘‘Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems,’’ J. Opt. Soc. Am. A 10, 316–323 (1993). J. H. Churnside, ‘‘Speckle from a rotating diffuse object,’’ J. Opt. Soc. Am. 72, 1464–1469 (1982). Because each scatterer appears in the same position in the illuminated region once with each revolution of the object, the speckle pattern exhibits a periodicity with a frequency equal to the frequency of revolution. Substituting Eqs. (4.6a) and (4.6b) into Eq. (4.1) yields that the maximum value of the space–time cross covariance equals unity. This because the orientation of the (static) speckles is independent of the motion of the object. H. T. Yura, B. Rose, and S. G. Hanson, ‘‘Speckle dynamics from in-plane rotating diffuse objects in complex ABCD optical systems,’’ J. Opt. Soc. Am. A 15, 1167–1171 (1998). Note that the complex cross covariance is proportional to exp@ikJ 2 ik(J 2 v Z t ) # 5 exp@ivDt#, where J is the path length along the optic axis and v D 5 kv Z is the Doppler shift caused by the longitudinal motion of the object.