1012 J. Opt. Soc. Am./Vol. 73, No. 8/August 1983

Effect of spatial incoherence of the laser inphoton-correlation spectroscopy

P. N. Pusey, J. M. Vaughan, and D. V. Willetts

Royal Signals and Radar Establishment, St. Andrews Road, Malvern, Worcestershire WR14 3PS, UK

Received February 11, 1983

The effects of spatial incoherence of the laser output in photon-correlation spectroscopy are examined both theo-retically and experimentally. It is shown that, when several transverse cavity modes oscillate simultaneously atdifferent frequencies, the amplitude of fluctuations in the intensity of scattered light can be reduced significantly.For simultaneous operation of TEMoo and TEMoi* (doughnut) modes the reduction of the zero-time-delay inter-cept g(2)(0) - 1 [where g(2 )(r) is the normalized intensity-correlation function] can amount to a factor of 3. Theeffects are shown to be particularly marked when high-gain ion lasers are used, and hitherto unexplained apparent-ly random changes of intercept have been found. Possible implications of these observations for speckle applica-tions, holography, and other forms of coherent optical processing are pointed out.

INTRODUCTION

Photon-correlation spectroscopy (PCS) measures the tem-poral correlation function of fluctuations in the intensity I, (t)of scattered laser light.1-3 If the laser light is fully coherentand the scattering medium contains a large number of inde-pendently fluctuating regions, the scattered-light field Es (t)is, by the central-limit theorem, a complex Gaussian randomvariable. Then the normalized intensity-correlation functiong(2)(,) can be written as

g (2)() = (I.(0)1 8(r)/(1r) 2 = 1 + [H(T)]2, (1)

where

Is(t) = IE8 (t) 12, (2)

the angle brackets indicate an ensemble average (over fluc-tuations assumed to be statistically stationary), and T is thecorrelation delay time. H(r) is a function that describesscatterer dynamics and decays from 1 at r = 0 to 0 at r = -.[In the ideal situation described above, H(r) can also beidentified as the modulus of the correlation function ofEs (t).]

In practice it is usually found that measured intensity-correlation functions are better described by

(I8(0)I,(r))/(IJ) 2 = 1 + fl[H('r)]2, (3)

where : is a constant for a given experimental arrangement.Several well-understood phenomena, 4 e.g., nonzero detectionaperture, dark count and stray light, and clipping in the signalprocessing, can cause f: to have a value significantly differentfrom (usually less than) its ideal value of 1 [Eq. (1)]. Althoughthis is not widely reported in the literature, it is a commonexperience of experimenters using Ar+- or Kr+-ion lasers that,even when the above phenomena are properly accounted for,the quantity fl can still be less than the theoretical value bya factor as large as 2 or more. In the practice of PCS this isan undesirable occurrence since it leads to a considerable re-duction of the signal-to-noise ratio. In this paper we suggestthat the effect is caused by spatial incoherence of the incident

laser light that arises when the laser oscillates simultaneouslyin several transverse cavity modes, each having a differentfrequency. Other properties of such laser operation werediscussed previously. 5 -7

In the next section we outline a theory of the effect anddiscuss some of its consequences. Some experimental con-firmation of the theory is then described briefly. Finally wegive a simple, largely qualitative, explanation of the effect anddiscuss some of its implications for PCS experiments. Wepoint out that a suitable small aperture placed inside the lasercavity should ensure operation in the TEMoo modes only andtherefore complete spatial coherence of the output.

Spatial incoherence of light from lasers in multimode op-eration was discussed previously by various authors, 8 thoughnot in a form directly adaptable to our present purpose. Ageneral discussion of source coherence effects in scattering;experiments, to which reference can be made for detailstreated superficially here, has been given elsewhere.9 "10

THEORYGeneralSeveral assumptions are made in this section that simplify thecalculation while retaining its important features.

First we assume that the scattering volume V, seen by thedetector, is near a focus of the laser beam and small enoughthat the wave fronts in V can be taken to be plane. The in-cident light field at point r in V at time t can then be writtenas

M r Z VEINC(r, t) = fE f(P)exp i~Wm .. til;

m=l I C

here we have adopted a cylindrical coordinate system r = (p,z) in which the laser beam propagates along z and the modeamplitudes /r depend only on the radius vector p in the x-yplane. In Eq. (4) M is the number of modes and m is a modeindex, c is the velocity of light, and the mode frequency conmlies within the Doppler-broadened line profile of width Aco andcentral frequency coo.

0030-3941/83/081012-06$01.00

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We now assume that the scattering volume contains a largenumber N of noninteracting particles at (fluctuating) posi-tions [ri(t)]." Provided that the incident light is temporallycoherent over V, i.e., that its coherence length 27rc/Aw is muchgreater than V113, the typical linear dimension of V, the lightfield scattered by the particles in direction R (modulus R) canbe writtenl 0 as

ES(R, t) K R-1 exp [iwo (R - t)] E exp[i(wo - wm)t]

NX _ fm [pi(t)]exp[i K * ri(t)];

i=l

and

M N= E; E J(f2(pi))(f2(pj))

m,n=l ij=l

+ (fm(Pi)fn(Pi)) (fm(Pj)fn(pj)) [H(r)]2}

. M 2= N2 E (fm 2(p) )

Vm I

(5) Here

here pi (t) is the radial position of particle i and K (modulusK) is the scattering vector associated with the central fre-quency "o and direction R. Next we omit quantities thatultimately disappear through normalization and obtain fromexpressions (2) and (5) the instantaneous scattered inten-sity

MI,(R,t) = F exp[i(wmw- w.)t]

m,n=1

NX E fm[pi(t)]fn[pj(t)]expfiK- [ri(t) -rj(t)].

i~j=1

(6)

Consider the case in which the mode frequencies are dif-ferent, i.e., wm FD Wn. Typical intermode beat periods are lessthan 10-8 sec, whereas typical sample times (over which I, isintegrated) used in PCS are greater than 10-7 sec. Thus inEq. (6) only terms for which m = n will contribute to thetime-integrated intensity IsI(t), giving

M NI'I(t) = Z Z fm[Pi(t)]fm[pj(t)]

m=1 ij=l

X expliK- [ri(t) - rj(t)]1 (7)

and

I8'(O)I8'(T)

M NE E fm[PiZ(O)]m[Pi(O)]fn[Pk(r)]fn[Pl(T)]

m,n=l ij,k,l=l

X expliK- [ri(0) - rj(0) + rk(r) - rl(r)]1. (8)

We now require the averages of Eqs. (7) and (8) over allpossible positions of the particles in V. As will generally bethe case, we assume that the reciprocal K- 1 of the scatteringvector is much smaller than the laser-beam radius o. Twoconsequences of this assumption are discussed elsewhere1 0' 12:First, it is a good approximation to perform separately aver-ages over phase factors texp[iK - (... .)]} and amplitude factors/m (pi)fn (pj) .. .]; second, whenever a (slowly decaying)

time-correlation function of amplitude factors multiplies a(rapidly decaying) time-correlation function of the phasefactors the former can be evaluated at zero delay time. Fi-nally, we note, as has also been discussed elsewhere,10 that,for a large number of noninteracting particles, only i = j termscontribute to the average of Eq. (7); significant contributionsto the average of Eq. (8) arise only for i = j $ k = I and i = I

j=k. Thus we get

M N M(Is') = F (fm2 (Pi)) =N Z (fm

2(p)) (9)

m=1 i=1 m=1

M+ N' E (fm(p)fn(p))2[H(T)]2.m,n=l

H(r) = (exptiK- [r(O) - r(r)]})

(10)

(11)

which, for many independent identical particles in Brownianmotion, takes the well-known form

H(r) = exp(-DK2r), (12)

D being the particle-diffusion constant. In Eqs. (9) and (10)the averages over mode amplitudes reduce simply to integralsover the cross-sectional area A of the beam that defines thescattering volume; for example,

(13)(fm (P)Wn(P)) = fA dp fm (p)fn (p).

Comparison of Eqs. (9) and (10) with Eq. (3) gives finallyME (fn(P)fn(P)) 2

0 m,n=l

IM 2E ( f 2(p) )1m=l

(14)

Specific ExamplesConsider first the case in which the laser beam illuminatingthe scattering medium is unrestricted by apertures, for ex-ample, so that area A in Eq. (13) is infinite. It then followsimmediately from the orthogonality of the transverse modesthat

(fm(p)fn(p)) = (fm 2(P) ) bmn; (15)

furthermore, it is easily verified from Eq. (4) that (fm 2 (p)) is

simply the power of mode m. Use of Eq. (15) in Eq. (14) thengives

MZ (fm2(p))2

m=l=.M 2F, (fm 2(p))Im=lI

(16)

Thus the reduction i of the amplitude of intensity fluctua-tions of light scattered from a laser operating in several spatialmodes of different frequencies is the ratio of the sum ofsquares of the mode powers to the square of the sum. Thisis a remarkably simple result, though perhaps not a surprisingone when viewed with hindsight.

For example, if the laser is operating in a doughnut modecomposed of two equal-power TEM01 modes,5-7 f = 1/2. It issometimes found that ion lasers operate in a combination ofa doughnut mode and a TEMOO mode. If the ratio of thepower of the latter to that of one of the TEM01 modes is x, itfollows immediately from Eq. (16) that

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(Is I (OV)Is I(T) )

1014 J. Opt. Soc. Am./Vol. 73, No. 8/August 1983

3 = (2 + x2)/(2 + x)2. (17)

Thus when x = o (pure TEMOO), /3 = 1; when x = 1 (threemodes of equal power), :3 = 1/3; when x = 0 (pure doughnutmode), / = 1/2, as above. In this not-uncommon three-modecase it is therefore possible for the fluctuation amplitude tobe reduced by a factor of 3.

A second situation, which we consider in the next section,is one in which a pure doughnut mode illuminates the scat-tering medium through a small aperture. In this case theorthogonality condition [Eq. (15)] does not apply since thearea A in Eq. (13) is restricted, and the general result [Eq. (14)]must be used. The two TEM01 modes comprising thedoughnut can be written as

fl(p) = P exp(-p 2 /2o'2 )sin 0, (18a)

f2(P) = P exp(-p 2 /2o-2)cos 0, (18b)

where we have adopted circular coordinates p - (p, 0). Sincethe p dependence of each mode is the same, only angular av-erages contribute in Eq. (14). Thus

(sin20)2 + 2(sin 0 cos 0)2 + (cos 20)2

/3= ( 19)((sin26) + (cos 20))2

where, for example,

(sin 2 6) dO sin 2 O,

and the range of integration 0 is determined by the illumi-nating aperture. Two cases are of interest. First we take thesmall aperture to be centered on the beam axis. Then H en-compasses the whole range 0 < 0 < 2ir, (sin20) = (cos20) = 7r,(sin 0 cos 0) = 0, and :3 = 0.5 as in the case of the unrestricteddoughnut mode. In the second case the small aperture isplaced well off axis and 0 is essentially constant at 0 = 0 0, say.Then (sin20) = sin2O0, (sin 0 cos 0) = sin Oo cos Oo, etc., and/ = 1, implying the full fluctuation amplitude in the scatteredlight.

EXPERIMENTAL INVESTIGATIONS

For these studies the 90° scattering arrangement shown in Fig.1 was used with a Kr+-laser source. This permitted good

E=-E3 N.D. FILTERS

'"='LENS Li

A1 A 2

Fig. 1. The experimental arrangement. The scattered light is im-aged on the aperture at Al, which with aperture A2 at distance I de-fines the degree of coherence in the light detected at the photomul-tiplier PM.

flexibility in the choice of experimental parameters. Bysuitable selection of apertures Al and A2 and adjustment oftheir separation 1, the degree of coherence over the collectionarea at A2 could be varied. This was usually chosen to providea geometrical factor /3geom between 0.8 and 0.95. The size ofthe laser beam at the scattering cell could be altered bychanging the focal length f of lens Ll. In addition, apertureAL could be used to select the portion of the laser beam thatentered the cell. In all cases the diameter of the beam in thecell was chosen so that its image at Al matched the size of theaperture at Al.

The scattering cell contained a suspension of monodispersepolystyrene spheres in water; their concentration was chosento provide an adequate scattering rate without appreciablemultiple scattering. The intensity-correlation functions wereformed with a 96-channel correlator operated in the scalingmode.",2 Initial experimental checks included (1) confir-mation of proper normalization with g(2)(r) - 1 at long timedelay; (2) minimization of stray light; (3) investigation of ex-perimental intercepts at Tr - 0 with a low-power laser ofknown stability, checking that /geom agreed with theoreticalpredictions for selected values of Al, A2, and 1; (4) evaluationof correlation functions from static scattering and white-lightsources as a check on photomultiplier and correlator perfor-mance.

For the experiments described below, the laser power en-tering the cell, and hence the scattering rate, was adjusted byneutral-density filters so that dead time and after-pulsingeffects in the photomultiplier were negligible. Intercepts atr = 0 were determined by extrapolation of the measuredcorrelation functions to zero time delay. The lie decay timeof g(2)(r) - 1 was found to be about 110 ,sec at 476-nmwavelength. Typically, correlation sample times of 8-12 ,usecwere used.

Measurements at Different Laser CurrentIn some preliminary investigations, changeable interceptswere found for several of the higher-gain Kr+-laser lines,particularly at larger current. The first investigation thusconsisted of a series of measurements at different currentlevels made using the red line at 647 nm and the blue line at476 nm (which was typical of the other lines, 468, 483, 521, and568 nm). These results are shown in Fig. 2. For the red line,only at the highest current is there any reduction of interceptand that only a few per cent. However, for the blue line theonset of the reduction is quite sharp and amounts eventuallyto a factor of about 2. For these measurements a prism lineselector was used but not a longitudinal-mode-selecting eta-lon. It must be emphasized that throughout these initial in-vestigations the laser beam had to all appearances a Gaussianprofile with maximum intensity at the center. Only fromcareful observation was an increase of beam diameter ap-parent for the blue line, starting at -22 A. With the suppo-sition that the two TEM0 1 modes are of equal power, the in-tercept at high current for 476 nm would suggest that thepower in the Gaussian TEMoo mode was about four timesgreater [from Eq. (17) for / = 0.5, x = 4]; this of course con-firms that the change of laser profile would not be at all ob-vious.

These initial experiments were extended by inserting amode-selecting 6talon and measuring correlation functions

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Vol. 73, No. 8/August 1983/J. Opt. Soc. Am. 1015

under a wide variety of tuning conditions. These may besummarized as follows:

(1) The observed intercepts were changeable with tuningand alignment, confirming that as the relative power in thedifferent modes varies the factor :3 can change rapidly.

(2) At the highest current, even with the most carefultuning, it was extremely difficult to obtain an observed in-tercept IBobs greater than c_0.8, compared with a /3geom of -0-9-This confirms that additional discrimination againsthigher-order modes, e.g. an intracavity aperture, is requiredat the highest current if the full intercept is to be obtained.

(3) By suitable detuning to produce an obviously enlargedlaser beam, observed intercepts as low as /obs _ 0.3 could beobtained. This of course would correspond to the case of:=

1 /3 for x = 1 in Eq. (17).

At this stage it became clear that the TEMo1* puredoughnut mode had interesting properties, and, after we in-vestigated the interference phenomena described in Ref. 7,the following correlation studies were made.

Measurements on the TEMa1 * Doughnut ModeWith the laser tuned to a symmetrical doughnut mode so thatthe TEM 01 components were of approximately equal power,the observed intercept was close to half of the geometricalfactor, i.e., Sobs = 0.5/geom. It was noticeable that this rela-tionship did not change dramatically when the componentswere obviously unequal in power. This is explained by Eq.(16) since if the ratio of the mode power is y, then for y = 2,a = 5/9 and for y = 3,3 = 5/8, both still reasonably close to0.5.

The second set of observations was to constrain the beamentering the cell with different circular apertures at AL.Provided that the aperture was precisely centered on the axisof the beam, it was immediately apparent that the interceptwas independent of the size of the aperture and remained atBobs = 0.50geom. This was still true even when the aperturewas quite small and relatively little light near the core of thebeam entered the cell. This initially surprising result is now

(2Z9'Z(

1.01 | I . . * I- .

-0--O a \647nm0.8-

0) -1

0.6

OA X476nm

0.2k

018 22 26 Amps 30

Discharge currentFig. 2. The experimental interceptg( 2)(0) - 1 of the intensity-cor-relation function for two lines from a Kr+ laser, at different dischargecurrents. Note that the low-current intercepts are different becauseflgeom is wavelength dependent.

_0 100 P m 200Radial position

Fig. 3. The experimental intercept g(2)(0)-1 with scattered lighttaken at different radial distances from the axis of a TEMo1*doughnut mode. Note that flobs = 0.45 at the core is exactly half offlobs = 0.90 well off center.

obviously explicable from Eq. (19) and discussions. It maybe noted that this observation is quite different if a TEMoomode is present. In this case a small aperture at the centertends preferentially to exclude light from the higher-ordermodes, and in consequence the intercept is increased andapproaches #geom-

In the third set of observations an aperture of 100 -ptm di-ameter at AL was accurately positioned at successive radialdistances from the axis of the TEMo1* beam, which was itselfadjusted to about 300-Mm diameter. The experimental pointsare shown in Fig. 3 and illustrate clearly the factor-of-2 changefrom Fobs = 0.45, on axis, to #obs = 0.90 flgeom, well offaxis.

Finally, the 100-,um aperture was moved around the cir-cumference of the TEMo 1* beam at 250 ,um off axis and ameasurement made in eight different positions. As expected,the intercept was unchanged at its full value of 0.90.

Measurements on a TEM01 Two-Spot ModeA pure two-spot mode was obtained by positioning a needleprecisely in the beam within the optical cavity. As would beexpected for single-mode operation, successive measurementsof the intercept showed the full value of 0.90 for light takenfrom the whole beam, from portions of either lobe, and fromparts of both lobes together.

DISCUSSION

We have outlined a theory of the effects of spatial incoherenceof the illuminating laser light in PCS; we have also describedsome experimental evidence in support of this theory.

We now attempt a simple explanation of the effect. Bysubstitution of Eqs. (18) in Eq. (4) we calculate the intensity(at plane z = 0) in the beam of a laser operating in a doughnutmode:

IINC(P, 0, t) = 2 exp(-p 2 /U2 )[1 + sin 20 cos(w, -2)t].(22

(20)

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, . . . . . . . . . . . . .

1016 J. Opt. Soc. Am./Vol. 73, No. 8/August 1983

ca) Cb)

CW1- u)t =2 n r (wi-w.) t -(2n+l1/2) r

Cw1-W2)t=C2n+1)r Cwf-w%)t=(2n+312)ir

Fig. 4. Sketches showing the combination of two orthogonal TEMo1modes of frequencies w, and W2 at different relative phase. Thecontours are at approximately half of the peak intensity of thedoughnut profile and one quarter of the peak of the two-spot.

Provided that wl FD W2, the intensity is a function of p, 0, andt and goes through a cycle in time with period 27r/(w, - °2)-Constant-intensity profiles as a function of time for one cycleare sketched in Fig. 4. We see that in Figs. 4(a) and 4(c) thelaser is essentially illuminating two different sets of particles.Our simple picture for this situation is that the intensity, in-tegrated over many intermode beat periods, can be writtenas the sum of two intensities, each derived from independent(since the two sets of particles constitute independent en-sembles) complex Gaussian fields:

IsI (O = Il (t) + I2(t), (21)

for which (I12) = 2 (I,) 2, etc. and UJ12) = U01 (I2) - If wetake (II) = (I2), it is simple to show that

((I J)2)/(IsJ)2 = 1.5 (22)

so that ,B=1/2, the same result as that obtained above.Equation (21) can be generalized for the case of many inde-pendent intensities, leading to a result similar to Eq. (16).

A virtue of this picture is that it is immediately apparentthat the time-varying illuminating intensity profile (in otherwords, spatial incoherence) results from interference betweendifferent transverse spatial modes having different frequen-cies (see also Ref. 8). If these modes were to operate at thesame frequency, the beam would have full spatial coherenceso that:f = 1. Formally this can be shown by setting all w°m=wco in expression (5) and proceeding with the analysis as in

the section on theory, above.For simplicity in the theory we assumed that each

transverse mode operated at only one frequency, i.e., on onlyone longitudinal mode. It is a simple matter to generalize thetheory to include more than one longitudinal mode for eachtransverse mode,13 and the results, e.g., Eqs. (14) and (16), areunchanged if the labels m apply only to spatial indices.Physically this is because the operation of a number of lon-gitudinal modes with the same spatial indices affects the

temporal coherence of the laser but not its spatial coherence.In terms of the simple picture given above, the output of sucha laser would be temporally modulated with time constant27r/Aw, but its spatial profile (its p and 0 dependence) wouldbe constant. For delay times T >> 27rlAw such a laser operatesin PCS in the same way as a single-mode laser, provided thatthe temporal-coherence criterion 27rc/Aw >> V1/3 is still sat-isfied (see Refs. 9, 10, and 13 for further discussion).

Several experimenters have found that the fluctuationamplitude 13 can be increased by use of an intracavity 6talon.Although this, of course, limits the number of longitudinalmodes operating, it is not expected to affect significantlytransverse-mode operation. (Indeed, several of the experi-ments discussed in the previous section and elsewhere5 7 wereperformed using an intracavity 6talon.) The implied im-provement in spatial coherence observed by these workersprobably resulted from a decreased overall cavity gain, whichfavors the operation of TEMOO modes.

Fortunately, however, the gain of off-axis spatial modes canbe reduced below threshold simply by decreasing the diameterof the intracavity aperture fitted to many commercial ion la-sers. As this diameter is decreased the laser spot, magnifiedand displayed on a screen, can frequently be seen to jump froma rather broad (TEMoo + doughnut) profile to a cleanerGaussian (TEMoo) profile.

In conclusion, we note that the comments made and thetheory outlined in this paper apply equally well to laserspeckle experiments (indeed, PCS can be regarded as a studyof moving speckle 9" 0). In such experiments, the expected fullspeckle contrast may not be observed unless careful attentionis paid to the laser-mode structure.'4 Furthermore, one mightexpect spatial incoherence of the illuminating laser to be aproblem in all forms of coherent optical processing, e.g., ho-lography, that depend on interference phenomena. By con-trast, it might be possible to exploit the effect to suppressunwanted speckle noise in such applications as laser radars,in which the factor-of-3 reduction achieved by using theTEMoo + doughnut mode configuration could prove distinctlybeneficial.

REFERENCES

1. H. Z. Cummins and E. R. Pike, eds., Photon Correlation andLight Beating Spectroscopy (Plenum, New York, 1974).

2. H. Z. Cummins and E. R. Pike, eds., Photon Correlation Spec-troscopy and Velocimetry (Plenum, New York, 1977).

3. B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley,New York, 1976).

4. C. J. Oliver, "Correlation techniques," in Photon Correlation andLight Beating Spectroscopy (Plenum, New York, 1974), pp.151-224.

5. J. M. Vaughan and D. V. Willetts, "Interference properties of alight beam having a helical wave surface," Opt. Commun. 30,263-267 (1979).

6. D. V. Willetts and J. M. Vaughan, "Properties of a laser mode witha helical cophasal surface," in Laser Advances and Applications,Proceedings of the Fourth National Quantum Electronics Con-ference, Edinburgh, Scotland, 1979, B. S. Wherrett, ed. (Wiley,New York, 1980), pp. 51-56.

7. J. M. Vaughan and D. V. Willetts, "Temporal and interferencefringe analysis of TEMo 1 * laser modes," J. Opt. Soc. Am. 73,1018-1021 (1983).

8. D. C. W. Morley, D. G. Schofield, L. Allen, and D. G. C. Jones,"Spatial coherence and mode structure in the He-Ne laser," Brit.

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J. Appl. Phys. 18, 1419-1422 (1967); B. E. A. Saleh and J. M.Minkowski, "On the spatial coherence of laser beams," J. Phys.A 8, 120-125 (1975).

9. E. Jakeman, P. N. Pusey, and J. M. Vaughan, "Intensity fluctu-ation light scattering spectroscopy using a conventional lightsource," Opt. Commun. 17, 305-308 (1976).

10. P. N. Pusey, "Statistical properties of scattered radiation," inPhoton Correlation Spectroscopy and Velocimetry (Plenum,New York, 1977), pp. 45-141.

11. In fact, the conclusions reached in this article apply also tomore-complicated scattering media, provided that the scattering

volume is much larger than the volume over which spatial corre-lations persist in the medium (see, e.g., Ref. 10).

12. M. Drewel and P. N. Pusey, "Number fluctuation spectroscopyin standing-wave fringes," Opt. Acta (to be published).

13. L. Mandel, "Correlation properties of light scattered from fluids,"Phys. Rev. 181, 75-84 (1969).

14. If an ion laser illuminates ground glass and the speckle is projectedon a screen, as the intracavity aperture is closed down and off-axismodes extinguished, the increase in contrast between brightspeckles and dark regions is frequently quite obvious to the un-aided eye.

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