Harthong et al. Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. A 405

Speckle phase averaging in high-resolutioncolor holography

Jacques Harthong, John Sadi, Marc Torzynski, and Dalibor Vukicevic

Laboratoire des Systemes Photoniques, Ecole Nationale Superieure de Physique de Strasbourg,Universite Louis Pasteur, Boulevard S. Brant, 67400 Illkirch, France

Received October 16, 1995; revised manuscript received August 2, 1996; accepted August 20, 1996

Interest in wavelength multiplexing in holography derives naturally from the need for realistic color renditionas well as from resolution requirements. Imaging in coherent illumination is compromised by speckle.Speckle is prejudicial to quality, sharpness, contrast—in a word, to the fidelity of reproduction or holographicreconstruction. In incoherent light these patterns are canceled by spatial phase averaging. Incoherent lightcan indeed be regarded as a superposition of a very large number of coherent components whose phase factorsare distributed at random. It is demonstrated that the averaging effect, ultimately caused by the law of largenumbers, is achieved by the superposition of only three components, thus allowing simultaneously a true colorrendition and an improvement in spatial resolution. The spatial statistical behavior of the amplitude of thesum of three intrinsically coherent waves, when they are incoherently superposed in an imaging system, isinvestigated. A random variable representing the amplitude of this sum is introduced. Then the cumulativeprobability function and the probability density function of the resulting amplitude are calculated. The white-light (infinite-wave illumination) case and the purely coherent (one-wave) case are analyzed. The results areinterpreted with a heuristic vector model. © 1997 Optical Society of America [S0740-3232(97)00502-4]

1. INTRODUCTIONThe information-carrying capacity of an optical system issubject to a diffraction limit of (Etendu)/l2. The Etendutakes into account not only the size of the source but alsothe angular aperture of the observing system. Theamount of information that can usefully be fed into thesource, therefore, depends on the mode of observation.1,2

Nevertheless, it is highly desirable not to feed into theinput anything approaching the theoretical maximum ofthe information capacity. This means that the systemhas redundancy; that is, it is carrying the input informa-tion several or many times over in parallel channels.This in turn makes it resistant to noise, i.e., increases thefidelity of transmission. It is well known that coherentsystems, which are frequently nonredundant, are easilyupset by the presence of dust or small disturbing par-ticles. There is only one way to ameliorate the image-forming fidelity: to increase the number of parallel chan-nels (number of degrees of freedom), i.e., to increase thesampling capacity of the image-forming system.Paraphrasing the famous parable of Eddington,3,4 An

ichthyologist should not be surprised, if exploring theocean by a fishing net with a mesh size of 2 inches, to findthat in the ocean there are no creatures smaller than 2inches, nor should he be surprised, if using several netssimultaneously, to find things smaller than the smallestopenings of all the nets used.Any calculation concerning the resolution limit of an

image is at best a guide to expected performance. Ratherthan attempting the task of defining resolution5 in anyimage-forming system, we shall adopt the principle thatthe reciprocal of the image spot size resulting from a pointsource is the resolution limit in lines per millimeter.In holographic precision imaging, i.e., microscopy6–9

0740-3232/97/020405-06$10.00 ©

and endoscopy,10 the spatial resolution is determined byspeckle. Speckle is the elementary information cell ofthe image-forming system11–13; therefore it determinesthe system’s information handling (sampling) capacity:It plays the role of the mesh sizes in Eddington’s fishingnets. Finally, the information-handling capacity of anoptical system is determined by the total number of thequanta of optical information (the total number of speck-les). To increase the image fidelity, one should not try toeliminate the speckles. It is an impossible task, in anycase, as the speckle-free beam is also information free,i.e., image free, and therefore useless. To increase the fi-delity, image resolution, and contrast, one should in-crease the number of parallel channels and thus increasethe density of speckle beyond the resolution of the detec-tion system (the human eye, perhaps) to achieve speckleaveraging.For us to be able to investigate the nuclei of individual

human tissue cells, details as small as 1 mm in size haveto be resolved. The best resolution achieved until now inholographic endoscopy was of the order of 2.5 mm.14

Spatial averaging of speckle patterns in imaging withcoherent light15 can improve resolution as well as white-light reconstruction of image-plane microholograms.14,16

One possibility for reducing the speckle noise in animage-plane hologram, regarded simply as a spatial car-rier image, is to extend the recording source size,17 i.e., toreduce the spatial coherence. Another possibility for in-creasing the resolution by speckle averaging is to recordholograms with several wavelengths (thus in effect in-creasing the number of Eddington’s fishing nets). Theresolution of microscopic observation of a coherently illu-minated object is indeed ameliorated by wavelengthmultiplexing.18–21 The mere addition of the speckle pat-terns caused by different laser wavelengths22,23 makes

1997 Optical Society of America

406 J. Opt. Soc. Am. A/Vol. 14, No. 2 /February 1997 Harthong et al.

their averaging possible. This means that it would bepossible to obtain both a higher resolution and a naturalcolor reproduction24 if the hologram were recorded with,for example, three laser lines of adequate wavelength(color recording).Interest in wavelength multiplexing comes naturally

from the need for realistic color rendition and directlyfrom resolution requirements, as well. Wavelength mul-tiplexing in color holography has already been studiedfrom colorimetric, intensity, and diffraction-efficiencyviewpoints. Colorimetric analysis has shown that a spec-tral sampling effect resulting from the use of a discretenumber of laser lines that are spectrally narrow owingto coherence requirements has to be taken into accountfor matching true colors of an object. Peercy andHesselink25,26 discussed wavelength selection by investi-gating the sampling nature of the holographic process.During the recording of a color hologram, the chosenwavelengths point sample the surface-reflectance func-tions of the object. This sampling of color perception canbe investigated by the tristimulus values of points in thereconstructed hologram, which is mathematically equiva-lent to integral approximations for the tristimulus inte-grals. According to Peercy and Hesselink, the samplingapproach indicates that three monochromatic sources arealmost always insufficient to preserve all of the object’sspectral information accurately. Four, five, or more laserwavelengths may be required. Only further experimentsfor each specific case can determine the necessary numberand the optimal combination of wavelengths. The otherdecisive factor that influences the choice of the recordingwavelengths is their availability.Intensity studies confirm that speckle averaging arises

from wavelength variety, thus enhancing resolution.Diffraction-efficiency analysis27–29 confirm that the prob-lems of reciprocity law failures are managed best by si-multaneous rather than sequential recording of all wave-lengths. In any case, sequential exposure is impossiblein pulsed recordings.Here we shall show that the behavior of luminous radi-

ance composed of three light waves of different wave-length is statistically closer to that of an incoherent thanto that of a coherent light beam. We shall focus our at-tention on three geometrically identical waves, existingsimultaneously but with mutually random phases. Weshall study the behavior of the sum of these three compo-nents to see whether this sum takes certain particularvalues more frequently.

2. MODELThe fact of interest is that the averaging of speckle noiseis achieved by the superposition of only three waves. Toexplain it, a simplified one-dimensional analysis of the in-coherent superposition of a finite number of coherentwaves is presented. If the number of waves is very large(infinite), their superposition is tending toward its nonco-herent limit.Intrinsically (in themselves) coherent waves are desig-

nated here by index ( j), that is, (f j), for example. Thewave f(x) can be written in complex notation as

f~x ! 5 a~x !exp@iu~x !#, (1)

or in a compact form,

f [ a exp~iu!, (2)

f j [ aj exp~iu j!. (3)

Mutual noncoherence between intrinsically coherentwaves f1 , f 2 , f 3 , . . . f j , etc., is treated by assumingthat the corresponding u1 , u2 , u3 , . . . u j are stochasticallyindependent.If certain illumination is composed of N mutually non-

coherent waves f j( j 5 1, 2, 3, . . . N), then the imagewave f(x) of a single object point x is given as a superpo-sition:

f~x ! 5 (j51

N

f j~x ! 5 (j51

N

aj exp~iu j!. (4)

The illumination intensity of the image point is then

I~x ! 5 uf~x !u2 5 U(j51

N

f j~x !U2, (5)

and, according to Eq. (3),

I~x ! 5 U(j51

N

aj exp~iu j!U2. (6)

A mere expansion of this gives

I~x ! 5 (j51

N

uaju2 1 2RH (j51

N

(k51j,k

N

ajak* exp@i~u j 2 uk!#J ,(7)

where aj and ak are complex and ak* stands for the com-plex conjugate of ak .In a shorter form,

I~x ! 5 (j51

N

uaju2 1 R, (8)

R being

R 5 2RH (j51

N

(k51j,k

N

ajak* exp@i~u j 2 uk!#J , (9)

where R stands for the real part of the complex expres-sion within the braces.In noncoherent illumination imaging, R is neglected, as

the summation is taken over a very large number(N → `) of incoherent waves. In the particular case ofinterest, N is always small (N 5 3, 4, or 5, for example),and consequently this term also has to be taken into ac-count.Instantaneous phases u j , u j8 ( j, j8 5 1, 2, 3, . . . N)

of the coherent light waves are characterized by a strongmutual correlation approaching 1 in the ideal coherentcase. In that case they are simply proportional (u j5 Ku j8) to each other.In contrast, in the noncoherent case, the instantaneous

phases u j , u j8( j Þ j8) would have uncorrelated randomvalues.

Harthong et al. Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. A 407

3. THREE-WAVE MULTIPLEXINGIn color holography, wavelength multiplexing takes placeat the reconstruction. Usually three spectrally narrow(although broadened with respect to the laser lines usedfor recordings) light waves are emerging from each recon-structed image point.Therefore N 5 3, and the total wave amplitude f(x)

reads as

f~x ! 5 (j51

3

f j 5 a1 exp~iu1! 1 a2 exp~iu2!

1 a3 exp~iu3!. (10)

Of course, only the real part of this complex amplitudehas a physical meaning and, therefore it has to be ana-lyzed further. If it is denoted by S,

S 5 R$f~y !% 5 a1 cos u1 1 a2 cos u2 1 a3 cos u3 .(11)

As u1 , u2 , and u3 are random variables, so is the sum S.For a noncoherent superposition the phases u j are sta-

tistically independent, and they can embrace any value(0 < u j < 2p) at random and with no preferences. Inother words, it is very likely that the probability P(a< u j < b) of encountering (finding) certain u j dependsonly on (b 2 a). Owing to the normalization require-ment,

P~a < u , b ! 5b 2 a2p

, 0 < a , b , 2p. (12)

Each of amplitudes a1 , a2 , and a3 is determined as aproduct of three factors: (1) the spectral reflectance func-tion of the object that is holographed, (2) the relative in-tensities of three chromatic laser light components usedfor recording the hologram, and (3) the relative intensitiesof the same spectral components of the white-light recon-structing beam.

4. CUMULATIVE PROBABILITY FUNCTIONOF THE SUM OF THREE WAVESSince the random variables u1 , u2 , and u3 are stochasti-cally independent, the cumulative probability functionFs(x) of the random variable S,

FS~x ! 5 P~S , x !

5 P~a1 cos u1 1 a2 cos u2 1 a3 cos u3 , x !, (13)

can be expressed in the form of the integral

P~a1 cos u1 1 a2 cos u2 1 a3 cos u3 , x !

5 E E Ea1cosu11a2cosu21a3cosu3,x

du1du2du3 . (14)

The integration domain of this integral, Fig. 1, is a vol-ume in three-dimensional space with orthogonal refer-ence frame (u1 , u2 , u3).

The volume of integration is bounded by a surface ex-pressed by the equation

a1 cos u1 1 a2 cos u2 1 a3 cos u3 5 x. (15)

If we change variables adequately,

uj 5 cos u j , and hence 21 < uj < 1, (16)

the integration domain is then transformed into thesimple form of a truncated cube (Fig. 2) centered at theorigin and with one corner cut off by the plane:

a1u1 1 a2u2 1 a3u3 5 x. (17)

Rewritten, the cumulative probability function is now

F~x ! 5 P~a1u1 1 a2u2 1 a3u3 , x !; (18)

that is,

F~x ! 5 E E Ea1u11a2u21a3u3,x

du1

A1 2 u12

du2

A1 2 u22

du3

A1 2 u32

since du j 5duj

A1 2 uj2. (19)

Fig. 1. Three-dimensional integration domain.

Fig. 2. Integration volume after a change of coordinates.

408 J. Opt. Soc. Am. A/Vol. 14, No. 2 /February 1997 Harthong et al.

5. CALCULATIONS AND RESULTSIntegral (19) can be calculated by three successive one-dimensional integrations, but only the first one can beperformed analytically by its primitive arcsine function.After this first integration is done, the cumulative prob-ability function comes out in the form of an elliptical sur-face integral:

F~x ! 5 E E arcsin~h ! 1 p/2

A1 2 u12A1 2 u2

2du1du2 , (20)

where h is the distance of the cutting plane Py (varyingwith the sum x) from the plane u3 5 21.

h 5x 2 a1u1 2 a2u2

a3. (21)

The remaining surface integral (20) can be calculatedonly by numerical procedures. If we apply the Gaussquadrature method,30 for example, we obtain the cumula-tive probability density function F(x) for each value (x),ranging from xmin 5 (2a1 2 a2 2 a3) to xmax 5 (a1 1 a21 a3). During this integration the cutting plane is suc-cessively truncating the cube of integration starting at itslower corner (21, 21, 21) and ending up at its upper cor-ner located at (1, 1, 1). This can be imagined as a planemoving parallel to itself through the cube from the start-ing point xmin to the final position xmax .

Fig. 3. Probability functions for $a1 5 a2 5 a3 5 1.0%.

Fig. 4. Probability functions for $a1 5 1.5, a2 5 2.0,a3 5 2.5%.

After the cumulative probability function is obtained,the corresponding probability density function is calcu-lated by deriving the cumulative probability function. Inpractice this derivation is done by a stepwise finite-difference approximation. Obviously, probability func-tions are strongly dependent on the choice and combina-tion of the initial parameters (a1 , a2 , a3). Three plotsof the probability functions for three different sets of am-plitudes (a1 , a2 , a3) are presented in Figs. 3, 4, and 5.Finally, the range of the most probable values of the

wave amplitude (11) of the non-coherent superposition ofthree waves emanating from each holographically recon-structed object point, could be estimated.

6. ANALYSIS AND INTERPRETATIONS1. Let us assume that at first the three amplitudes(a1 , a2 , a3) are mutually not very different from eachother, for example, as in a scene with a large-band spec-tral reflectance function. In that case the most probablevalue for the resulting amplitude (x 5 a1 cos u11 a2 cos u2 1 a3 cos u3) [Eq. (15)] of the noncoherentsuperposition of three (chromatically) different waves iszero (x 5 0) (see Figs. 3 and 4). This represents the bestpossible three-wave averaging of the spatial phase varia-tions (speckle). To reduce the speckle variations evenmore, (N) should be chosen larger than three:(N . 3). The interesting question is, What is the mini-mum number of statistically independent (noncoherent,different color) waves that should be used for obtaining agiven level of speckle noise reduction and consequentlythe spatial resolution improvement that approaches thedesirable white-light limit?2. In the case in which the condition (a1 ' a2 ' a3) of

equal amplitudes is not satisfied, for example, (a1. a2 1 a3) (Fig. 5) the zero value is no longer the mostprobable one. In other words, the averaging over therandom spatial phases is no longer so good.3. To elucidate the two previous points, we can imag-

ine the complex amplitude (aj) of each wave as a vector ina complex space. Different vectors corresponding to dif-ferent noncoherent waves are pointing toward randomdirections (to account for random phases), each havingdifferent (but comparable) constant lengths (correspond-

Fig. 5. Probability functions for $a1 5 3.0, a2 5 1.0,a3 5 1.0%.

Harthong et al. Vol. 14, No. 2 /February 1997 /J. Opt. Soc. Am. A 409

ing to wave amplitudes aj). Obviously, the sum of sev-eral (three) vectors with random phases [each turningwith constant angular speed (2pn j) in a mutually inde-pendent, noncorrelated way] will become closer to zero asthe vectors’ lengths converge. The probability of a maxi-mum constructive interference is not excluded completelybut is very small.

7. CONCLUSIONSImaging in coherent illumination is compromised by theformation of diffraction patterns in a variety of forms andintensity distributions from Airy disks to generalspeckles.31 Such patterns are prejudicial to the quality,sharpness, and contrast, that is, to the fidelity of the re-production or holographic reconstruction. In incoherentlight these patterns are canceled by spatial phase averag-ing. Incoherent light can indeed be regarded as a super-position of a very large number of coherent componentswhose phase factors are distributed at random. This av-eraging effect, ultimately caused by the law of large num-bers, can be represented mathematically by the meanvalue of the complex sum

S 51N(

j51

N

exp~ia j!, (22)

where the a j’s are independent and uniformly distributedrandom variables. In this case the probability density ofuSu is approximately Gaussian32 and zero centered, with astandard deviation of order 1/AN. In other words, S willseldom deviate from zero for more than a few 1/AN.Consequently, when N is very large (N → `), the prob-

ability of uRu being greater than a given b/AN is very low:

probS uRu .b

AN D } 102f~b!, (23)

where f(b) is an integer-valued function depending on theb chosen. In this case Eq. (8) becomes

U(j51

N

f jU2 > (j51

N

uaju2. (24)

A noncoherent superposition of a large number of in-trinsically coherent waves is intensity based, behavinglike a thermal source having a Gaussian intensity distri-bution.Now the question is, as has already been pointed out,

whether such an averaging effect occurs if a rather smallnumber of coherent light components are superposed. Incolor holography seldom more than three different coher-ent components will be applied, and in the reconstructionthey will overlap incoherently. In Fig. 6 a one-dimensional simulation of the spatial phase averaging ef-fect is presented.The first line represents an irregularly oscillating func-tion that can be regarded as the intensity modulation in atypical speckle pattern. This speckle function is first ar-bitrarily shifted and then rescaled by a factor propor-tional to the wavelength ratios of the three-chromatic

components. The mean value of the superpositions of theoriginal speckle function with the first shifted and re-scaled speckle function is shown as the second line of theFig. 6. The third line represents the mean value of theoriginal signal superposed on its first two shifted and res-caled versions, and so on. The law of large numbers en-sures that the limit (far beyond the eighth line) will beflat: This would be the perfect state of averaging. Nev-ertheless, here the only aim is to demonstrate that thethird line can be considered to have significantly lessspeckle than the first one.The amplitudes of the original speckle function at given

abscissas can be considered to be distributed at random iftheir mutual distances are larger than a few periods(wavelengths). This holds for any particular speckle dis-tribution in one, two, or in three dimensions. If the threeof these structures are superposed, the statistical fre-quencies of the deviations will be described by the prob-ability density function [Eq. (11)] of the random variableS,

S 513

~cos u1 1 cos u2 1 cos u3!, (25)

where u1 , u2 , u3 are uniformly distributed in the interval[0, 2p].The resulting probability density functions were pre-

sented in Figs. 3–5. One should observe that most of thedeviations are confined within one half of the maximum,and this complies with the simulation curves (see Fig. 6).Fluctuations in the third line are half as large as the fluc-tuations of the original signal. But one should not forgetthat both the statistical model and the simulations areonly heuristic, and they do not represent the real diffrac-tion patterns in any sense. Moreover, even if they did,the mean values represented by the lower lines of Fig. 6could eventually present only the structures (intensitypatterns) recorded within the hologram (refractive-indexmodulation). But there is only an indirect and not at allsimple relationship between the fidelity of the interfer-ence intensity pattern recordings and the fidelity of theholographic reconstruction. To find this relationship,one should perform a complete rigorous analysis of thereconstruction-beam diffraction at the complex three-dimensional modulation, and this is not an easy task.The case of three multiplexed waves treated above con-

sequently appears to lie between the following two ex-treme cases:

Fig. 6. Superposition of N random phase waves.

410 J. Opt. Soc. Am. A/Vol. 14, No. 2 /February 1997 Harthong et al.

2N 5 1, illumination by a single, coherent beam

~e.g., a laser),

2N → `, illumination by a noncoherent source

~e.g., a white light).

In conclusion, the averaging of the diffraction struc-tures achieved by the incoherent superposition of onlythree chromatic components of the color hologram recon-struction, is explained through an analysis that wasbased on a very simple model. In reality, the phases(u1 , u2 , u3 , . . . u j) are not random variables but randomfunctions of coordinates. However, this does not affectthe qualitative results.

REFERENCES AND NOTES1. G. L. Rogers, Noncoherent Optical Processing (Wiley, New

York, 1977).2. F. T. S. Yu, Optics and Information Theory (Wiley, New

York, 1976).3. A. Eddington, The Philosophy of Physical Sciences (Cam-

bridge U. Press, Cambridge, 1939), pp. 16, 62.4. D. Gabor, ‘‘Light and information,’’ in Progress in Optics,

E. Wolf, ed. (North-Holland, Amsterdam, 1965), Vol. I, pp.111–152.

5. ‘‘The resolution of two points is impossible, except if onea-priori doesn’t dispose of an infinite amount of informationon the object that is observed,’’ G. Toraldo di Francia,‘‘Super-gain antennas and optical resolving power,’’ NuovoCimento, Suppl. 9, 427 (1952).

6. E. N. Leith and J. Upatnieks, ‘‘Microscopy by wavefront re-construction,’’ J. Opt. Soc. Am. 55, 569–570 (1965).

7. R. F. van Ligten, ‘‘Contribution a l’holographie de hauteresolution, application a la microscopie holographique,’’Ph.D. dissertation (Universite de Paris Sud-Orsay, 1972).

8. R. A. Briones, L. O. Heflinger, and R. F. Wuerker, ‘‘Holo-graphic microscopy,’’ Appl. Opt. 17, 944–950 (1978).

9. R. W. Smith, ‘‘Holographic methods in microscopy,’’ J.Microsc. 129, 29–47 (1983).

10. M. Friedman, H. I. Bjelkhagen, and M. Epstein, ‘‘Endohol-ography in medicine,’’ J. Laser Appl. 1, 40–44 (1988).

11. D. Gabor, ‘‘Laser speckle and its elimination,’’ IBM J. Res.Dev. 509–514 (1970).

12. J. W. Goodman, ‘‘Some fundamental properties of speckle,’’J. Opt. Soc. Am. 66, 1145–1150 (1976).

13. N. Abramson, The Making and Evaluation of Holograms(Academic, New York, 1981), pp. 50–53.

14. H. I. Bjelkhagen, J. Chang, and K. Moneke, ‘‘High resolu-tion contact Denisyuk holography,’’ Appl. Opt. 31, 1041–1047 (1992).

15. P. S. Considine, ‘‘Effects of coherence on imaging systems,’’J. Opt. Soc. Am. 56, 1001–1009 (1966).

16. M. E. Cox and K. J. Vahala, ‘‘Image plane holograms for ho-lographic microscopy,’’ Appl. Opt. 17, 1455–1457 (1978).

17. E. N. Leith and G. C. Yang, ‘‘Interferometric spatial carrierimage formation with an extended source,’’ Appl. Opt. 20,3819–3821 (1981).

18. N. George and A. Jain, ‘‘Speckle in microscopy,’’ Opt. Com-mun. 6, 253–257 (1972).

19. N. George and A. Jain, ‘‘Speckle reduction using multipletones of illumination,’’ Appl. Opt. 12, 1145–1150 (1973).

20. N. George and A. Jain, ‘‘Space and wavelength dependenceof speckle intensity,’’ Appl. Phys. 4, 201–212 (1974).

21. M. Elbaum, M. Greenebaum, and M. King, ‘‘A wavelengthdiversity technique for reduction of speckle size,’’ Opt. Com-mun. 5, 171–174 (1972).

22. C. Saloma, S. Kawata, and S. Minami, ‘‘Laser-diode micro-scope that generates weakly speckled images,’’ Opt. Lett.15, 203–205 (1990).

23. B. Dingel and S. Kawata, ‘‘Speckle-free image in a laser-diode microscope by using the optical feedback effect,’’ Opt.Lett. 18, 549–551 (1993).

24. H. Bjelkhagen, T. H. Jeong, and D. Vukicevic, ‘‘Color reflec-tion holograms recorded in a panchromatic ultra-high-resolution single-layer silver-halide emulsion,’’ J. ImagingSci. Technol. 40, 134–146 (1996).

25. M. S. Peercy and L. Hesselink, ‘‘Wavelength selection forcolor holography,’’ in Practical Holography VII, S. Benton,ed. Proc. SPIE 2176, 108–118 (1994).

26. M. S. Peercy and L. Hesselink, ‘‘Wavelength selection fortrue-color holography,’’ Appl. Opt. 33, 6811–6817 (1994).

27. X. Y. Chen, ‘‘The beam ratio in volume holography,’’ J. Mod.Opt. 35, 1393–1398 (1988).

28. K. Bazargan, X. Y. Chen, S. Hart, G. Mendes, and S. Xu,‘‘Beam ratio in multiple-exposure volume holograms,’’ J.Phys. D 21, S160–S163 (1988).

29. K. M. Johnson, L. Hesselink, and J. W. Goodman, ‘‘Holo-graphic reciprocity law failure,’’ Appl. Opt. 23, 218–227(1984).

30. F. B. Hildebrand, Introduction to Numerical Analysis(McGraw-Hill, New York, 1956).

31. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).32. According to the central limit theorem, when N is very

large (N → `), the probability density function of the sumof a large number of random variables is Gaussian, no mat-ter what the individual law of each random variable is.