256 J. Opt. Soc. Am./Vol. 73, No. 3/March 1983 Deschamps et al.

Gaussian Schell-model sources: an example and someperspectives

J. Deschamps, D. Courjon, and J. Bulabois

Laboratoire de Physique G6n6rale et Optique, Facult6 des Sciences et des Techniques, 25030 Besangon Cedex,France

Received May 26,1982; revised manuscript received September 30,1982In this paper we deal with the use of feedback systems to generate Gaussian Schell-model sources. Owing to theapproximation of quasi-homogeneity of the beam and assuming slow transmittance variations of diffracting ele-ments, the coherence-propagation equation can be simplifed and leads after several transits to a multiple-convolu-tion product. According to the central-limit theorem and whatever the pupil functions are, this product reducesin general to a Gaussian distribution. This result is applied to imaging and lensless feedback systems.

1. INTRODUCTION

Following recent work in the field of radiometry, new typesof noncoherent sources have been defined. These sources,often called quasi-homogeneous' or quasi-stationary 2 sources,are characterized by an intensity that remains essentiallyconstant over regions whose linear dimensions are of the sameorder as the correlation length of light. Moreover, the lineardimensions of the source are large compared with the corre-lation length across the source. Finally, the degree of co-herence is strictly stationary.

Among various physical models that satisfy the assumptionof quasi-homogeneity, one is particularly interesting, thequasi-homogeneous Gaussian Schell-model source,3 in whichboth optical intensity and degree of coherence are Gaussian.Such sources may theoretically generate highly directional andnoncoherent beams. Several experimental verifications havebeen carried out.4 -6 However, the results obtained do notpermit the practical construction of such a source with anenergetic efficiency large enough to permit its use as a primaryone. The common point of all proposed methods is passivefiltering, which is necessary for modifying both the intensityand the degree of coherence of the primary source.

It must be emphasized that these sources are of interest, forexample, in noncoherent processing, in which the light beammust be as directional as possible without necessarily beingcoherent.

In this paper, a method is described for the generation ofGaussian Schell-model sources. It is based on the use of afeedback device. Although the results obtained seem to becharacterized by the same defects as those of existing methods,it will be shown that the proposed method leads to interestingperspectives. For instance, a simple reflecting cavity (similarto a laser cavity) could permit the realization of a noncoherentenergetic and directive source.

2. PRINCIPLE

The proposed solution is based on the use of a feedback sys-tem. This system is described in Fig. 1. A primary quasi-

monochromatic and spatially noncoherent source Sp is imagedby means of the lens L limited by the pupil P in the outputplane of the setup. Two beam splitters, S, and S2 , and twomirrors, Ml and M2 , constitute the loop permitting the pas-sage of the beam through L and P several times.

The source is assumed to be quasi-homogeneous. Thishypothesis is in fact nonrestrictive because it has been shownelsewherel that a large number of physical noncoherentsources are part of the family of quasi-homogeneous ones.

A. Direct Transfer from Object to Image PlaneLet us consider first the direct passage from the object to theimage plane. The coherence-propagation laws lead to thepropagation equation

W(yl, Y2) J W(xI, x2 )K(xl, yl)K*(x 2 , y2 )d2 xld2x2,

(1)

where W(xl, x2) and W(yl, Y2) are the cross-spectral densityfunctions in the object plane and in the image plane, respec-tively, at a chosen frequency w, and K(x, y) is the impulseresponse of the optical system. Assuming quasi-homogeneity,the cross-spectral density takes the simple form

W(xl, x 2 ) - I[/ 2 (x + X 2)](xl -X2), (2)

where I corresponds to the optical intensity and ,u to thecomplex degree of spatial coherence. The impulse responsecan be written as7

K(x, y) = I exp [i 2 (y2 + X2)]

X P(Q)exp -i d (x - y) * . d2, (3)

where X is the wavelength and d is the distance between thepupil plane and the image plane.

On substituting into the right-hand side of Eq. (1) fromexpressions (2) and (3), the cross-spectral density in the imageplane becomes

0030-3941/83/030256-06$01.00 © 1983 Optical Society of America

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Fig. 1. Schematic diagram of the imaging feedback system.

W(y 1, Y2) 1 ( 2 exp Jj 2 (y12-Y22)J(X2d2)2 2d

XJf f i I[1/2(xl + X2)].(Xl - X2)PQ1)P*(6)

X exp k (x12 - X22) exp (-ij [(xi - y 1) * .1

- (X2 - Y2) * 2]} d2 xid2x2d2t 1d2%2 - (4)

The integral of expression (4) can be simplified if we use theproperties of quasi-homogeneous sources. For such sources,the complex degree of coherence is a narrow function, beingsomewhat different from zero only within the domains whoselinear dimensions are much smaller than those of the source.In this case, the integral has appreciable values only for x1 -x2 , and so expUj(k/2d)(x1

2 - x2 2)] can be approximated byunity.

The validity of this approximation has been demonstratedin Refs. 8 and 9 for coherent fields scattered by rough surfaces.Because of the stochastic nature of partially coherent fields,the transposition is straightforward and implies that

d > 2S,7- Xwhere S, is the coherence area in the object plane and S, is thearea of the object.

Let us assume, moreover, that the pupil amplitude P(s) ispurely real and varies slowly across the pupil so that it isperceptibly constant over regions whose linear dimensions areof the order of the field-correlation distance, i.e., the effectiverange of the complex degree of spatial coherence in the pupilplane. This condition is easily verified in practice because,according to the generalized Van Cittert-Zernike theorem,the complex degree of coherence is, apart from a phase factor,proportional to the normalized Fourier transform of the dis-tribution of the optical intensity across the source. When thesource is sufficiently extended, the coherence area in the pupilplane is much smaller than the pupil extension. With thisassumption,

Q1 + 2P(Q1)P* (6,) - 1p <2,

and W(y1 , Y2) can be written as

W(y1, Y2) 1 I - exp [j 2 (y12 - Y22)J(X2d2)2 2d

x Jf 33 I['/2(Xl + x2)]A(xl - x2)AP['A2(l + U2X12

X exp {- d [(xi - Yl) * t1- (X2 - Y2) *2]}

X d2x1 d2x2d2 t1 d2

2 . (5)

By setting 4, + 62/2 = ql\d and noting that d2%2/d2 ,q =

4X2d2,

W(Y 1 , Y2) - X2

d2 4 f I[h/2(Xl + X2)]9(Xl - X2 )

X f exp f-j d (xl + x2 - Y1 - Y2) * 6 d21

X f [P(Xq1d)] 2 expl-j47r(y 2 - X2) - ?71d 2nd 2 xid 2x2 - (6)

After partial integration,

W(y 1, Y2) - 4 exp Ii (y2 - Y22)}

X I[1/2 (xl + x2)]M(xl - x 2 )h[2(y2 - X2)]

X 62(X1 + x2 - Y1 - y2 )d 2 x1 d2 x 2 , (7)

where 62 (x1 + X2 - Y1 - Y2) is a bidimensional Dirac distri-bution and

h[2(Y2 - X2)] = [P(X71d)]2 exp[-j27r(y2 -X2)-n7d2n.

(8)

Finally, by putting a = 2x1 - (Y1 + Y2), and, after a straight-forward calculation, the cross-spectral density is reduced to

W(y1, Y2 ) - exp 2 (y12 -Y22)] I['/2(Yl + Y2)]

X (a)h [a- (Y1 - Y2 )]d2a, (9)

the integral on the right-hand side is the convolution of thecomplex degree of coherence with the function h, which be-haves as a peculiar impulse response of the imaging system.

If the integral has appreciable values only for Yi - Y2, thequadratic phase term can be neglected. If this requirementis not verified, the quadratic term can be directly dropped bychoosing a focal-plane to focal-plane geometry or by replacingthe analysis plane with a hemispherical surface. In any case,it does not influence the coherence area or the intensity dis-tribution in the plane y in any way. For the sake of simplicity,let us thus assume a hemispherical observation surface; thecorresponding cross-spectral density W, is then

Ws(y 1, Y2 ) - I[1/2 (Yl + Y2 )]g(Yl - Y2)A (10)

with g = , * h, * being the convolution symbol. After nor-malizing and introducing the complex degree of spatial co-herence, 's (Y - Y2 ), W. becomes

W s(Y 1, Y 2) -g(0)I['/2 (Yl + Y2 )]A's(YI - Y2), (11)

that is,

W.(y1, Y2) -I'[ 1 /2(Y1 + Y2)]A2',s(Y1 -Y2) (12)

where I' is the intensity distribution in the image plane. Fi-nally, according to our assumptions concerning I, ,i, and P,the image of the quasi-homogeneous source is also quasi-homogeneous.

B. Images After n TransitsLet Wi be the cross-spectral density after i transits, and letus consider the first one without taking into account thebeam-splitter coefficients1 0 :

Deschamps et al.

258 J. Opt. Soc. Am./Vol. 73, No. 3/March 1983 Deschamps et al.

WI k 1y1Yi, Y2 (X2d2)2 exp [j 2d (yi2 - y22)I

Xff ff I[/2(xl+ x2)]g(xl -X2

X exp [k 2 (X12 - X22)l 1P[P/2 (Q1 + 62)]12~j2dI

X exp -i [(xl - YI) - (X2 - Y2) *2]1

X d2xid2x2d2 td 2 42. (13)

As before, the quadratic term inside the integral can be ne-glected, and W1(yI, Y2) reduces to

W1(yI, Y2) - exp [i -d (y12 - Y22)]

X I[1/2(Y1 + y2 )]g'(yl - Y2), (14)

where

gl = g * h = *h * h. (15)

By iteration we deduce the cross-spectral density after ntransits:

Wn(yl, Y2) ( X~d2)2 exp i - (y12 - 2)(2d2)2 [2d Y2)

XJJ ffsr- I[1/2(xl + X2 )]gnl(Xl - X2)

X exp [ 2d (X12-X22)] IP[P/2\Q1 + 62)]12

X exp {-i d [(x, - Y1) *i - (X2 - Y2) * 2iJX d 2

X d 2x 2d 2

td 22 . (16)

Finally, after n transits through the loop, the cross-spectraldensity W8 will take the form

W" n(yl, Y2) I[1/2(Yl + Y2)] (A * h * h ... h (17)

n + 1 timeswithin the limit of the validity of previous approximations ofquasi-homogeneity.

C. Consequences of Expression (14)The image after n transits retains the properties of quasi-homogeneity of the primary source, and the complex degreeof spatial coherence can be easily deduced owing to the cen-tral-limit theorem. According to the latter, under certaingeneral conditions, if

f W) = h ) *f2() A . ), (18)

f(x) approaches a normal distribution."This theorem is satisfied even if the convolved functions

are identical. Finally, after a certain number of transits, thedegree of coherence in expression (14) may approach aGaussian distribution. This behavior is numerically andexperimentally verified in following paragraphs, but first weconsider a particular case of quasi-homogeneous sources.

D. Gaussian Schell-Model SourcesFor this type of source, the intensity distribution and thedegree of spatial coherence are both Gaussian, i.e., they may

be represented in the form

I(r) = A exp(-r 2 /2a,2 ),AWr') = exp(-r' 2 /2Ur 2 ), (19)

with A, uj, and A, being positive constants and r' = r, - r2.These sources generate a field whose radiant intensity J, atfrequency c and in a direction specified by unit vector smaking an angle 0 with the normal to the source plane, can bewritten as

J(0) = J(0) cos 20 exp(-sin20/2A2 ), (20)where

A = (1/koy)[l + (oJ2aj)2 ] 1/2

and

k a > 1/2A, ka, > 1/A.

It has been shown that such sources can generate highly di-rectional beams. From previous paragraphs, it can be con-cluded that the cross-spectral density in the output plane ofa feedback system is given by

Wn(yl, Y2) = I'['/2(Y1 + y2)Iexp[-(y, - y2)2/2 o-A2 1. (21)

If

I'f'/ 2(Y1 + Y2)] = A exp [(Y + Y2) / 20-2] (22)

the resulting secondary source at the output of the systempossesses the characteristics of a Gaussian Schell source. TheGaussian profile of intensity across the source can easily beobtained when a suitable spatial filter is used or merely be-cause of the vignetting effect introduced by multiple transitsthrough the system.

3. NUMERICAL VERIFICATION OF THECENTRAL-LIMIT THEOREM

The problem now is to estimate the number of transits nec-essary to obtain a Gaussian profile. A numerical verificationis then carried out. The results are shown in Figs. 2 and 3.

Two cases are considered:

1. Theoretical case (function characterized by strongand fast variations). The input function is defined by

f(x) = rect x/a(1 + cos x2) (23)(see Fig. 2).

2. Practical case (function characterized by slow localvariations. The input function corresponds to the impulseresponse of an imaging system in which the pupil is composedof a photographic recording of the diffraction pattern of auniform circular aperture (Fig. 3). This pupil function obeysthe assumption

P(W1 )P*(62) -PP[1/ 2 (-i + 62)]2. (24)Two observations can be made:On one hand, the convolution products rapidly tend toward

a Gaussian distribution since, even in the case of Fig. 2 withonly three convolutions, the differences of the resulting curveand the corresponding Gaussian curve are no longer visible.

On the other hand, the extension (LP) of the curve, wherethe function has perceptible values, increases more slowly than

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s F.F.F

X

F.F.F.F

I . LP > ILT

Fig. 2. Numerical example of multiple convolutions. In spite of the strong variations of the input function, the convolution product rapidlytends toward a Gaussian distribution.

F.F.F

F. F

F.F*F.F

-LT I

Fig. 3. Numerical example of multiple convolutions. In this case,no longer perceptible after the first product.

the theoretical extension (LT) corresponding to the definitiondomain of the function.

In considering expression (14) and observing Figs. 2 and 3,we can expect to obtain in a feedback device a degree of spatialcoherence with a Gaussian profile after only two or threetransits.

This result is important because the previous assumptionsare available only in the case of weak interactions between thepupils of the system and the light beam. After a limitednumber of transits, the diffraction effects remain sufficientlysmall that our assumptions are justified.

4. EXPERIMENTAL VERIFICATION

A laser beam illuminates a rotating diffuser, the image ofwhich is projected in the output plane of the setup. Twobeam splitters, S1 and S2, and two mirrors, M1 and M2, con-stitute the loop allowing the passage of the beam through Land P several times. z is a rotary-shearing interferometeryielding the determination of the degree of coherence (Fig.4).

the difference between the Gaussian curve and the convolution is already

The experimental verification is carried out by measuringthe degree of coherence and the intensity in the image afterone, two, or three transits in the loop. In this case, the pupiltransparency P is constructed by the photographic recordingof the Fraunhofer diffraction pattern of a circular aperture.It allows one to obtain an impulse response K, the form ofwhich is similar to that used in the numerical example in Fig.3.

The results given in Fig. 5 represent in Fig. 5(a) the intensityacross the secondary source and in Fig. 5(b) the degree of co-herence determined from the visibility of fringes generatedby the rotary-shearing interferometer, without the loop (case0) and after one, two, or three transits through the loop (cases1, 2, and 3). In the third case, the output signal is so weak thatthe measure of the degree of coherence cannot be achieved bythe proposed interferometric technique.

It can be noted that the intensity distribution does not re-main constant during the multiple transits through the loopbut approaches (more slowly than the degree of coherence)a Gaussian distribution. This point is in disagreement withour theoretical-development expression (12). In fact, the

J"-" . - -

Deschamps et al.

-X

J-X

-x

260 J. Opt. Soc. Am./Vol. 73, No. 3/March 1983 Deschamps et al.

5. TRANSPOSITION IN LENSLESS FEEDBACKSYSTEMS

The system chosen is a laser cavity. Calculations are basedon the use of the central-limit theorem and on the model ofFox and Li,'12 which is a series of identical circular aperturesof diameter a with spaces d (Fig. 6).

Let us consider the transfer in the cavity through the vari-cus apertures. By applying Fresnel diffraction equations, wecan write the amplitude in plane y that is due to a point sourceS placed in plane x as

K(x, y) = 1 exp - k ( + y2)X2d2 [_ 2d

Fig. 4. Experimental setup. The spatially incoherent source isobtained by destroying the coherence of a laser beam by means of arotating diffuser. In order to avoid the overlapping of images after1 ... n transits, the optical system is slightly off axis. Diameter ofobject, -5 mm; diameter of Airy disk of the pupil P, -10 mm; distanceof object to lens, -600 mm.

(a)

I

0x

X

I

,I

1 Jl l,

2

(b)

'Y'

'TI

/" y'.

3 I

Fig. 5. Experimental result: (a) cross section of image intensity,(b) degree of spatial coherence after 0, 1, 2, and 3 transits.

vignetting effect cannot be neglected in the process, becauseit is responsible for the evolution of the intensity toward aGaussian distribution.

A quantitative evaluation of the characteristics of both thedegree of coherence and the intensity in the output plane in-dicates that the resulting noncoherent source is typically aGaussian Schell-model source. The divergence of the beamis about 10 mrad, corresponding to that of a multimode gaslaser. The energy losses introduced by filtering are so greatthat there is no practical use for this beam.

x exp fj d [42 - (x - y)t]} rect t/ad2 4.

After putting

P(s) = exp j d2). rect s/a,

(25)

(26)

we again find relation (3) of Section 2.A.As before, if the degree of spatial coherence is sufficiently

small in the pupil plane, expL(k/d)42] is almost constant inregions whose dimensions are of the order of the field corre-lation.

Various approximations suppose that the number of tran-sits in the cavity is sufficiently small (unlike the case of thelaser effect) that the final degree of coherence has an extentin agreement with the approximation (see Section 3).

In comparison with the preceding development, the cross-spectral density W, becomes

W. (Y1, Y2 ) - I[1/2(Y1 + Y2 )]g(Y1-Y2), (27)

where

g(yl - Y2) = y(a)h[a - (Y1 - Y 2 )]d 2 a. (28)

Such a result is similar to that of feedback systems based onimagery. After a number of transits n, the cross-spectraldensity takes the form

W.n(y1, Y2) IE'/2(Y1 + Y2)] (A * h * h * h *. . . h) (29)

n timesFinally, after a small number of transits, a cavity generates

Gaussian-correlated quasi-homogeneous sources. TheGaussian intensity distribution can be obtained either withsuitable spatial filters or by the inherent vignetting effect.

6. EXPERIMENTAL CONFIRMATION

The possibility of generating highly directional noncoherentbeams in cavities seems to be verified by the amplified spon-taneous-emission effect (ASE). This effect appears in lasercavities before the threshold of laser emission and until nowhas been considered a parasitic effect.13"14

___V ad d

Fig. 6. Lensless feedback system (model of Fox and Li).

Vol. 73, No. 3/March 1983/J. Opt. Soc. Am. 261

(a) (b)Fig. 7. (a) Amplified spontaneous emission and (b) laser emissionin dye laser after 5 m of propagation. The area of the spot, distortedby aberrations, is almost the same in both cases.

We have observed and roughly measured the directivity andcoherence properties of the ASE in a laser cavity (dye laser).In spite of the broadband spectrum (about 500 A) of the flu-orescence emission and its low spatial coherence, the resultingbeam has a very good directivity (about 10 mrad), which is ofthe same order as that of a multimode laser (see Fig. 7). Thiseffect could then be used for generating highly directionalnoncoherent beams. The problem is to choose the fluorescentmedium in order to obtain a good energetic ratio.

7. CONCLUSIONThe narrow connection between high directivity of beams andcomplete coherence has for many years been considered afundamental principle. The laser effect, the optical proper-ties of which have been described by the (coherent) theory ofFox and Li, seemed to be proof of this rule.

However, the role of the cavity in coherence propagationwas laid down some years later by Wolf,15 who pointed outthat a completely space-coherent beam could be generatedfrom noncoherent sources after a certain number of transits.Coherence (and directivity) can then be introduced by dif-fraction of a noncoherent beam through suitable transpar-encies. Unfortunately, the filtering process introduces atremendous loss of energy, and, in the case of complete co-herence, the energy tends toward zero. A more suitable so-lution is based on the recent work dealing with the Gauss-ian-correlated sources that lead to the important result thatdirectional beams can be generated without reaching completecoherence of the emerging field. If the directional beam isgenerated from a primary incoherent source, it is obvious thatpartial coherence of the resulting field needs more-moderatefiltering than that which must be achieved to obtain completecoherence.

In this paper, we have tried to verify, in the case of a feed-back system with the help of the central-limit theorem, that

the directivity of beams can be ensured with a low number oftransits; in other words, with a low coherence of the emergingbeam.

Although the passive spatial filtering introduced is not toosevere, the problem of energy losses remains. It could besolved by using a suitable active medium. Amplified spon-taneous emission seems to offer an interesting solution.

The authors wish to thank W. H. Carter, A. Lacourt, andJ. Monneret for helpful remarks.

REFERENCES

1. W. H. Carter and E. Wolf, "Coherence and radiometry withquasi-homogeneous planar sources," J. Opt. Soc. Am. 67, 785-796(1977).

2. H. P. Baltes, J. Geist, and A. Walther, Radiometry and Coher-ence, Vol. 9 of Topics in Current Physics (Springer-Verlag, Berlin,1978), pp. 135-139.

3. E. Collett and E. Wolf, "Is complete spatial coherence necessaryfor the generation of highly directional light beams?" Opt. Lett.2, 27 (1978).

4. J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highlydirectional beams from a globally incoherent source," Opt.Commun. 32, 203-207 (1980).

5. P. de Santis, F. Gori, G. Guattari, and C. Palma, "An example ofa Collett-Wolf source," Opt. Commun. 29, 256-260 (1979).

6. F. Gori, "Collett-Wolf sources and multimode lasers," Opt.Commun. 34, 301-305 (1978).

7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1968).

8. J. W. Goodman, "Some effects of target-induced scintillation onoptical radar performance," Proc. IEEE 53, 1688-1700 (1965).

9. J. W. Goodman, Laser Speckle and Related Phenomena, Vol.9 of Topics in Applied Physics (Springer-Verlag, Berlin, 1975),pp. 61-63.

10. Although the reflection and transmission coefficients stronglylimit the available energy in the output plane, they have beendropped in the mathematical development since this paper dealsmore with the qualitative study of the propagation of a partiallycoherent field in the system than with energy considerations.

11. See, for instance, R. Bracewell, The Fourier Transform and ItsApplications (McGraw-Hill, New York, 1965), pp. 168-173; A.Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), pp. 227-239.

12. A. G. Fox and T. Li, "Resonant modes in a maser interferometer,"Bell Syst. Tech. J. 40, 453-588 (1961).

13. G. Dujardin and P. Flamant, "Conversion d'6nergie dans lesamplificateurs A colorants en presence de superfluorescence," Opt.Acta 25, 273-283 (1978).

14. V. Ganiel, A. Hardy, G. Neumann, and D. Treves, "Amplifiedspontaneous emission and signal amplification in dye-laser sys-tems," IEEE J. Quantum Electron. QE-li, 881-891 (1975).

15. E. Wolf, "Recent researches on coherence properties of light,"presented at Third International Conference on Quantum Elec-tronics, Paris, 1964.

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