first-order statistics of displayed speckle patterns in electronic speckle pattern interferometry

474 J. Opt. Soc. Am./Vol. 71, No. 4/April 1981

First-order statistics of displayed speckle patterns inelectronic speckle pattern interferometry

Gudmunn A. Slettemoen

Department of Physics, Norwegian Institute of Technology, N-7034 Trondheim-NTII, Norway

Received September 3,1980

The first-order probability density functions of displayed speckle patterns in electronic speckle pattern interferom-etry are calculated. We show that in specular-reference-beam setups the brightness in the displayed pattern is al-ways (X)2 distributed. For speckle-reference-beam setups the probability density depends on the effective resolu-tion of the recorded speckle pattern. This function is calculated in the case of fully resolved patterns and shownto approach the (X)2 density function as the size of the aperture increases. The statistics obtained with specular-and speckle-reference-beam setups are compared. General expressions are derived for the average value and thestandard deviation of the monitor brightness. In specular-reference-beam setups the speckle contrast is found al-ways to be equal to I2, and in speckle-reference-beam setups it is found to be in the region from \/2 to \/g . Insetups in which speckle-reduction techniques have been used the speckle contrast decreases below these numbers.We also calculate this reduced speckle contrast.

i. INTRODUCTIONElectronic speckle pattern interferometry (ESPI) is a holo-graphic method by which a TV camera is used to recordimage-plane holograms. These holograms are reconstructedelectronically and displayed on a TV monitor. In comparisonwith the resolving power of holographic film, the resolvingpower of the TV camera is poor. Therefore the speckles onthe monitor display are large and clearly seen. The principlesof recording and reconstruction in ESPI are similar to theprinciples in other holographic techniques. The first-orderstatistics of the sum of a speckle pattern and a coherentbackground and the first-order statistics of the sum of twocoherent speckle patterns can be found in Ref. 1. In Refs. 2-4some of the first-order statistics in the reconstructed imagesare calculated. They consider conventional holographicsetups, and the recorded interference pattern in those setupsis the result of the sum of a speckle pattern and a coherentbackground (a specular-reference-beam setup). In otherholographic setups, in which the recorded interference patternis the result of the sum of two coherent speckle patterns (aspeckle-reference-beam setup), the statistics in the recon-structed images have not been calculated. We refer below toESPI with a specular reference beam as specular-referenceESPI. The other setup will be called speckle-referenceESPI.

In order to maximize the contrast of the displayed fringesin ESPI we calculate in Refs. 5 and 6 the ensemble average ofthe monitor brightness. The underlying speckle structurecauses difficulties when we want to determine the exactbrightness of the fringes. In this paper we shall calculate theother first-order properties of the brightness in the recon-structed images. The displayed patterns in speckle-referenceESPI seem to have larger speckle contrast than in specular-reference ESPI. 7 This effect will be studied in detail.

In an earlier paper 7 it was demonstrated that the speckle

noise can be reduced by simple means. The effectiveness ofthis speckle-reduction technique will also be discussed.

2. SIGNAL PROCESSING IN ELECTRONICSPECKLE PATTERN INTERFEROMETRY

The signal processing in specular-reference ESPI is describedin Ref. 6, and this processing in speckle-reference ESPI isdescribed in Ref. 7. Here we give a brief description only andrefer to Refs. 6 and 7 for more-detailed accounts.

In Fig. 1 the two types of ESPI setups are shown schemat-ically. The first setup, Fig. 1(a), is shown with a specularreference beam introduced by the beam splitter. The secondsetup, Fig. 1(b), is shown with a speckle reference beam in-troduced by the rough object 2 and the wedges inside the ap-erture. [This aperture is a multislit aperture, and the frontview is shown in Fig. 4(a).] The second setup is a compactversion of the speckle-reference ESPI that we presented inRef. 7. The imaging of the rough object 1 on the photosurfaceof the TV camera and the principles of the electronic pro-cessing are common to both setups. In the figures andthroughout this paper the beams are labeled by numbers,where number 1 denotes the object beam and number 2 de-notes the reference beam.

We assume polarized laser light so that the reference waveis coherent with the object wave. These two waves will add,and at position x,y the exposure of the TV camera's photo-surface will be proportional to 6

I(x,y) = Ai(x,y)Al(x,y) + A2(x,y)A2(x,y)

+ 2 Re[Al(x,y)A 2 (x,y)M(x,y)]= Ii(x,y) + I2(X,y) + Ic(x,y), (1)

where M(x,y) is the fringe functions and Al and A2 are com-plex amplitudes. II = A1Al is the intensity in the image ofobject 1, i.e., of beam 1 alone. For both setups this image is

0030-3941/81/040474-09$00.50 © 1981 Optical Society of America


Vol. 71, No. 4/April 1981/J. Opt. Soc. Am. 475

Fig. 1. Schematic drawing of two ESPI setups. (a) Specular-ref-erence ESPI, (b) speckle-reference ESPI.

a speckle pattern. The intensity I2 = A2A' of the referencebeam may be uniform or a speckle pattern, depending onwhether the setup is a specular-reference ESPI or a speckle-reference ESPI. The last term, Ic = 2 Re(A1A2M), is theuseful cross-interference term that, through the fringe func-tion M, carries information about the object movement. Thespatial intensity variations that are due to this last term willbe referred to either as the cross-interference speckle patternor simply as the cross-interference pattern.

The terms I, and I2, the optical self-interference terms, areunwanted, and they should be removed in the electronicprocessing. In ordinary off-axis holography 9 these terms areremoved by spatial filtering. The same terms in ESPI maybe removed by electronic filtering of the video signal, a processthat is equivalent to the off-axis filtering method in conven-tional holography. The unwanted terms may be removed byother means, and in Ref. 7 those methods are discussedbriefly.

The resolution properties of the TV camera and the transfercharacteristics of the electronic filter are described togetherby one impulse-response function h(Ax,Ay). 6 The result willbe a linear space-invariant operation on the cross-interferenceterm. After the electronic filter the video signal will be thesmoothed version IO(x,y) of the cross-interference term

I'(x,y) = jfh(x - x1,y - y)Ic(x 1,yi)dxidyi. (2)

The coordinates refer to the corresponding positions on thephotosurface of the TV camera or on the TV monitor. To

demodulate the fringe signal we square-law rectify the filteredvideo signal. The carrier wave is the cross-interferencespeckle pattern. The resulting monitor brightness Im maybe written

Im(r) = [I'c(r)]2

= Jh(r - r')h(r - r")I (r')I,(r")dr'dr". (3)

We have introduced the short notation r to mean one of thecoordinates x or y. The extension of the final results to thetwo-dimensional case is straightforward. The brightnessIm (r) gives a speckle pattern. If the object has moved duringthe exposure of the photosurface, this speckle pattern ismodulated by fringes.

The video signal may also be demodulated with charac-teristics other than the square law. We choose square-lawrectification here because it corresponds to an intensity de-tection of the reconstructed image as in conventional holog-raphy. Hence the results in this paper can be compared withthe results for other holographic setups.

3. FIRST-ORDER PROBABILITY DENSITYFUNCTION OF DISPLAYED SPECKLEPATTERN: CROSS-INTERFERENCE SPECKLEPATTERN FULLY RESOLVED

We assume in this section that the cross-interference patternis fully resolved by the camera and filter systems. Then thedisplayed monitor brightness is given by

Im = 4[Re(A'A 2M)] 2 = 4M2 [A rAr + A1A ]2. (4)

Here Ar denotes the real part and Ai denotes the imaginarypart of the corresponding complex amplitude A, and an ir-relevant proportionality constant is left out. With the elec-tronic readout from the TV camera we do not discriminatebetween the real and virtual image, and the complete cross-interference pattern 2 Re(A*A2M) contributes to the monitordisplay. This is in contrast to optical reconstruction in con-ventional holography. There, A1A2M* and A1A2M give thevirtual and real images, and these images are usually sepa-rated. Therefore we expect that the statistics in the recon-structed image in ESPI will be different from the statistics inconventional holography. We assume in Eq. (4) that thefringe function is real (a constant phase is unimportant), andwe assume that it varies slowly with position r, i.e., the fringesare assumed to be broader than the average speckle size.

The statistics of the brightness Im depend on the statisticsof the complex amplitudes A1 and A2. The probability den-sity of complex amplitude A1 is known to be, in most practicalcases, a circular Gaussian function. 10 The complex amplitudeA2 in a speckle-reference ESPI is also a circular Gaussianvariable, whereas it is a constant (deterministic) in the spec-ular-reference ESPI. The joint probability density functionof the real and imaginary parts of a circular Gaussian randomvariable is10

i(Ar Ai) = (A)i121 p (A r)p (Ai),

(5)

where (I) is the ensemble average of the intensity I = AA*



and the amplitudes Ar and Ai are seen to be independentvariables.

Since the statistics of complex amplitude A2 depend onwhether a specular or a speckle reference beam is used in thesetup, we divide the further calculation of the brightnessstatistics into two parts.

Case 1, Specular-Reference Electronic Speckle PatternInterferometryIn this case the complex reference amplitude A2 is deter-ministic, and in Eq. (4) we set A' = 0 without lack of gener-ality. The monitor brightness is then given by

Im = 4(I 2)(AD)2 . (6)

Here we leave out the constant fringe function squared sinceit, in our discussions, will be an irrelevant proportionalityconstant. (12) is the uniform reference intensity, and am-plitude A' is Gaussian with a density function given by

p(Al) = exp[-(A') 2 /(I 1)]. (7)(7r (I,) 11/2

To find the probability density of Im we square-law proba-bility transform the density function in Eq. (7). A generalsquare-law probability transformation of a symmetric func-tion px (x) gives the probability density py (y), where y = x 2,i.e.,11

Py (Y) = IP (X;)

Py(Y) = 0

for y > 0,

for y < 0.

With the use of Eqs. (6)-(8) and a simple change of variables,the probability density of the monitor brightness will be

P ) = 2(7r(I1) (2)Im)2 exp(-I./4(I1) (12))

forlm °0,

veloped speckle pattern."' In the reconstructed image inconventional holography the first-order statistics are givenby the statistics of the term A A,*, i.e., the probability densityis also a negative-exponential density function .3 12 Forcomparison these two functions are drawn in Fig. 2, and wesee in particular that the speckle contrast in the specular-reference ESPI is higher than the contrast in a fully developedspeckle pattern.

Case 2, Speckle-Reference Electronic Speckle PatternInterferometryIn this case the calculation of the first-order probabilitydensity of brightness Im is more elaborate, and we perform thecalculation in three steps. We first find the probabilitydensity of the product of two independent Gaussian variables,i.e., the probability density pr(Ir = AjA2) of one of the ad-dends of Eq. (4). Thereafter, we find the probability densityPc (I, = 2AA' + 2A lA 2) of the cross-interference term. Sincethe real and imaginary parts of the complex amplitudes areindependent, this cross-interference term is the sum of twoindependent variables. Finally, we derive the probabilitydensity p 1 (Im = 1) of monitor brightness Jin. This brightnessis equal to the cross-interference term squared.

For independent A' and A 2 the probability that Ir = A1Ais less than a certain value ir is given by the probability dis-tribution function (see, e.g., Ref. 11 for details)

+ f+ p(/A2)p(A2)dAidAd+ r r p(Ar)p (A rWA rdA r0 }-(

= 2 Jo fA p(A )p(A2)dAdA2. (10)

The probability density is found by derivation with respectto I. on both sides. This gives

Po(I,.) = 0

This function is the so-called (X)2 density function [the (x2 )density function for 10 of freedom]." It is different from theprobability density of the object intensity I, = A,1A 1 = (A 1r)2+ (A')2 . The last function is a negative-exponential densityfunction, and the speckle pattern is often called a fully de-

,.0-

pr(Ir) = 2 Jp (Ir/Ar)p(Ar) AdAr . (11)

The real parts of complex amplitudes Al and A2 are bothGaussian distributed. Insertion from Eq. (5) into Eq. (11)gives

Pr((r) = 2

1r A)2 (Ir)2 1

X 'f 2 -exp - (I2- dA2.A 2 Y ( U )(AD2

(12)

We can write this integral in terms of the modified Hankelfunction of order zero, Ko(z). By use of the integral repre-sentation in Eq. (6,447) of Ref. 13, the result is

p2(Jr) = 2 Ko 2 Ir1p(Ir - K0W/ [(( (2) )1/2]

I -

(13)

We now derive the probability density of the cross-inter-ference term Ic, This term is the sum of two independentvariables, and the probability density is given by the convo-lution between the probability densities of the respectiveaddends." We Fourier transform these functions sinceconvolution gives the simple product in the transform domain,i.e.,

Fc(f) = Fr(f)Fi(f)27r = F2(f)2r. 14)

Fig. 2. po(Ini) is the probability density function of the monitorbrightness for a specular-reference ESPI. The broken line shows theprobability density for a fully developed speckle pattern.


for I,, < °- (9)


Here, properly indexed, F(f) denotes the Fourier transformof the corresponding probability density function p (t). Theprobability densities for 2A'A' and 2A'A' are equal, and F(f),which is the characteristic function, is defined by

F(f) = - eif tp(t)dt.27r -E (15)

The Fourier transform of a modified Hankel function of orderzero is derived from the transform pair p,(t) - F(f), Eq.(5,116) of Ref. 13, where

Ps, = I t1), r (,(/2) )K,(qltl),p~)=(T!) rS P(I'/2)

2(f2 + q2)s+1/2 (16)

In Eq. (16) q is a positive constant and F( ) denotes thegamma function. The index s is the order number of themodified Hankel function. From Eqs. (13)-(16) the charac-teristic function of the cross-interference term is

FJ(f) = 1 (17)

27r(II)(1 2 )(f2 + (I)(j))

From Eq. (16) we also see that the inverse Fourier transformthat gives the probability density of the cross-interferenceterm may be written

P(I ) =1 I IC 1/2

X(2V2/2(vIc2) (18)We now continue on the last step in our derivation. Since

Im =2J the square-law probability transformation in Eq. (8)gives by insertion

Pi(Im) = r(1/2 )V2(I1 ) (I2)

X K1/2 ( V' ('i I |)for Im > 0,

pi(Im) = 0 for Im < 0. (19)

We have thus found the probability density pI(Im) of themonitor brightness. In this case the setup is a speckle-ref-erence ESPI with a fully resolved cross-interference pat-tern.

The modified Hankel function of order 1/2 can be expressedin terms of the exponential function, and from Eq. (6,479) ofRef. 13 the probability density of the monitor brightness is

=2((I) (=2)Im)1/2 exp[-(Im/(I1) (12))1/2]

for Im > 0,P1(Im) = 0 for lm < 0. (20)

In Fig. 3 this probability density is compared with the (X)2density function. We see that they have similar shape. Butwe also see that the speckle-reference ESPI gives highercontrast than the specular-reference ESPI. [As was pointedout by Pedersen,' 4 Eqs. (9) and (20) can be derived in a moredirect way by using the 6-function formalism of Ref. 15.However, our calculations can easily be extended to cases in

3

Fig. 3. pi(Im ) is the probability density function of the monitorbrightness for a speckle-reference ESPI (the cross-interferencespeckle pattern fully resolved). This function is compared with theprobability density po(Im ) for a specular-reference ESPI.

which fully resolved patterns are incoherently added at therecording stage.]

4. FIRST-ORDER PROBABILITY DENSITYFUNCTION OF DISPLAYED SPECKLEPATTERN: CROSS-INTERFERENCE SPECKLEPATTERN PARTIALLY RESOLVED

The cross-interference pattern is usually not fully resolvedby the combination of a TV camera and an electronic filter.We must consider the statistics of spatially averaged signals.In the general case those calculations become difficult.However, for specular-reference ESPI we can find all thefirst-order properties of the displayed monitor brightness.The probability density that the speckle-reference ESPI givescan be calculated only in certain special cases. We shall cal-culate the limiting case in which the resolution of the cross-interference pattern goes to zero because of increasing aper-ture size.

Case 1, Specular-Reference Electronic Speckle PatternInterferometryThe video signal now contains a smoothed version of thecross-interference term, and the displayed monitor brightnessis given by Eq. (3). For specular-reference ESPI this reducesto

(21)Im (r) = 4 (I2) f h(r - r')A (r')dr'I.

We recognize the integral as a linear superposition of Gauss-ian-distributed amplitudes. Any linear combination ofGaussian-distributed variables gives a new Gaussian vari-able."1 Therefore the probability density of the monitorbrightness will always be the (X)2 density function discussedearlier. We return to an explicit expression in the next sec-tion.

Case 2, Speckle-Reference Electronic Speckle PatternInterferometryIn the speckle-reference ESPI a multislit aperture (a largeaperture) is usually used.7 We show one aperture in Fig. 4(a).



N-I= 2 [Ar.(Arn_ 1 + Arn+,) + A',(A,-, 1 + A',+ 1))

1= 1+ A2NA2N-1 + A2NA2N-1

N-1= 2 X 2 + A2nA~n-)

ti= 1

N-1+ 2 E (Ar.Ar.+1 + A'nA'.+i)

n=1

+ 2 (A2NA2N-1 + A2NA2N -). (23)

TV-camera Aojc-' j /, { object

Fig. 4. (a) Multislit aperture that is used in speckle-reference ESPI.The effective aperture used in the object 1 imaging system is denotedby 1, and the effective aperture used in the object 2 imaging systemis denoted by 2. (b) Modification of the aperture in (a). (c) Sche-matic of Young's experiment. The maximum distance d that givesresolvable interference fringes at the TV camera is denote by L in (a)and (b).

In the figure, distance L, corresponding to the limit of TV-camera resolution, is indicated. The aperture is the same asthe one in Fig. 1(b). To explain the electronic filtering processand the significance of distance L we need not, in the presentpaper, employ the whole mathematical formulation. 6' 7 Webase our simple explanation on Young's experiment. In Fig.4(c) the formation of Young's fringes is illustrated. Thespatial frequency of the fringes at the TV camera is inverselyproportional to the distance d between the two small holes.If the two holes are slid in front of the multislit aperture, onlythe waves that pass through these holes at distance d less thana certain distance, distance L, contribute to the resolved partof the interference pattern. The part of the interference pat-tern that is not resolved by the TV camera gives a dc term inthe video signal. The electronic filter removes this dc term.The setup is arranged so that, in addition, the electronic filterremoves the two unwanted terms I, and 12.

In the present discussion we number both the slits in themultislit aperture [Fig. 4(a)] and the corresponding ampli-tudes sequentially. The total cross-interference term is thengenerally written as

N NI, = 2 (A r2. A- + A',, A'.-,) - (22)

n'=1 n=1

Here, even index numbers denote the amplitudes that add upto give the total complex amplitude Al, and odd index num-bers denote the amplitudes that give the total complex am-plitude A2 . As is shown in Eq. (2), this cross-interferencepattern is smoothed by the combination of the TV camera andthe electronic filter.

In the previous section we treated the limiting case in whichthe cross-interference pattern was fully resolved by the TVcamera. We now discuss the limiting case in which the di-mensions of the aperture in Fig. 4(a) are much larger than theTV-camera resolution limit L. We treat the usual case7 inwhich the resolved part of the cross-interference pattern is dueonly to interference between waves transmitted throughneighboring slits. The resolved part of the total cross-inter-ference term can then be written as

The size of the speckles incident upon the aperture is ex-tremely small,'0 and it follows that the amplitudes with dif-ferent indices are statistically independent. I, in Eq. (23) isthen expressed as the sum of three terms, where the two firstterms are again written as the sum of statistically independentterms. If we invoke the central-limit theorem on the two firstterms separately, then they will be Gaussian distributed asthe number N goes to infinity. The sum of two Gaussianvariables is again a Gaussian variable.'1 As N goes to infinity,the probability density of the last term will, compared withthe probability densities of the two first terms, approximatea Dirac delta function. Therefore in the limit we end up withan equation similar to Eq. (21). In Eq. (22) the smoothedversion of the Gaussian variable I, is square-law rectified togive the (x)2 density function of the monitor brightness. Inthe general case in which the aperture dimensions increase farbeyond the TV-camera resolution limit L, our argument showsthat the first-order statistics of the monitor brightness forspeckle-reference ESPI approach the first-order statistics ofthe monitor brightness for specular-reference ESPI.

5. CONTRAST OF DISPLAYED SPECKLEPATTERNS IN ELECTRONIC SPECKLEPATTERN INTERFEROMETRY

In Ref. 16 the speckle noise of image displays is discussed. In

order to determine the minimum detectable contrast level foran object in the presence of speckle noise, it is demonstratedthat the speckle contrast, the ratio ,m/ (I, ), is an importantparameter. U,, is the standard deviation of the monitorbrightness, and (Im ) is the average value of this brightness.It is demonstrated that the minimum detectable contrast levelincreases with the speckle contrast. In our case, the objectthat we want to detect is the fringes that are due to objectmovement. We now study this speckle contrast in ESPImonitor displays.

Case 1, Specular-Reference Electronic Speckle PatternInterferometryIn this case we know from the previous section that theprobability density is always the (x)2 density function

PO ) = (21( )Im)/2 exp(-Im/2 (in)) for Im > 0,

Po(Im) = 0 for Im < 0. (24)

The ensemble average (In) has been calculated in Ref. 6 andis equal to

(25)

where


,y2(I.) = 2 (II) U2) 12,


S IP1(v)J 2JH(v)I 2dv712 = (26)

f IPI(v) I 2dv

712 is the degree of resolution of the fringe carrier that is ourcross-interference speckle pattern. Pl(v) is the pupil functionof the aperture used in the imaging of object 1, and v is a shortnotation for spatial frequencies that are referred to the TV-camera photosurface. H(v) is the transfer function of thecombination of the TV camera and the electronic filter andis the Fourier transform of the previously considered im-pulse-response function h(Ar).

The standard deviation a,, for the (X)2 density function isgiven by

amn = (Irn) = 2VJ2(I1) (I2) 12. (27)

Therefore the speckle contrast in a specular-reference ESPIis

C = -Y.I ) =X*(28)

This contrast is independent of other details in the setup.

Case 2, Speckle-Reference Electronic Speckle PatternInterferometryAlthough in this general case we do not know the probabilitydensity of the monitor brightness, it is possible to find explicitexpressions for both the standard deviation and the averagevalue of this brightness. The average value was given in Ref.7. For completeness we include the derivation here.

The monitor brightness I, is given by Eq. (3), and the en-semble average will be given by

(I) = Jh(r - r')h(r - r")(Ic(r')Ic (r))dr'dr'. (29)

Using Eq. (1), with M left out as in Eq. (6), the intensity cor-relation function in the integrand can be written in terms ofthe complex amplitudes

(I (r')Ic(r")) = 4(Re[A '(r')A 2(r')]Re[A 1(r")A2(r")])

= 2Re[(A1(r')Ai(r')) (A2(r')A2 (r"))]. (30)

Equation (30) is valid for the case in which complex ampli-tudes with index 1 (light scattered from object 1) are inde-pendent of complex amplitudes with index 2 (light scatteredfrom object 2) and terms such as (AI(r')A 1 (r')) and(At(r')Ai(r")) are zero for circular Gaussian-distributedvariables. The complex amplitudes result from coherentimaging processes and are all of the form

Al(r') = f hi(r'- r1)AX(ri)dri. (31)

Here hi(Ar) is the appropriate coherent impulse-responsefunction, and Al(rl) is the complex amplitude at object 1. Byinsertion of equations such as Eq. (31) into Eq. (30) we get

(I. (r')I,(r'))

= 2Re rr hl(r' - rli)h(r' - r2)(A1(rI)AI(r2))dridr 2

X jfJ'h2(r' - rl)h2(r" - r2)(Aq2(rl)q2(r2 ))drldr2] . (32)

We assume that the microstructure of the object surface isso fine as to be unresolvable by our lens systems. The corre-lation properties of the complex object amplitudes can thenbe written 17

(AX(r1 )Ai(r 2 )) = 716(ri - r2),

(A2(r1)Ai2 (r2 )) = 726(ri - r2). (33)

We assume stationary conditions at the objects and that theobject intensities 71 and 12 will be constant. Using Eq. (33)and representing the impulse-response functions in terms ofthe corresponding transfer functions, where

h(r) = fP(vI)e27Tirv'dvI, (34)

we get from Eq. (32)

(IC(r')IC(r"/)) = 2I7 2 Re1 J, I P1(V')J21P2(v")J2

X e2rr v~ e7i'^~"d' ](35)

With proper indexes the transfer functions P( ) are the pupilfunctions of the imaging systems.18 v'y"... are spatialfrequencies that are referred to the TV camera.

To find the ensemble average of the monitor brightness wefinally insert Eq. (35) into Eq. (29), which gives

('in) = 27172 ff JP1(VI)I21P 2 (VI)I2

X I H(v' - V",) I2 dv'dv". (36)

We use the symbol * for the cross-correlation operation, andwe rewrite Eq. (36) in terms of the average intensities (I,) and(12) at the TV camera. The result is

rf [jP 1(V)j2 * IP 2 (V)12 11H(v)I 2 dv

S [IPl(v)1 2 * JP 2 (v)12 ]dv

= 2 (I1)(I2) 12. (37)

Here 712 is the degree of resolution of the fringe carrier. Fora fully resolved cross-interference pattern (a fully resolvedcarrier), 712 = 1. We have thus rederived the ensemble av-erage of the monitor brightness given in Ref. 7.

To find the standard deviation a,, of the monitor bright-ness, the ensemble average of the squared monitor brightnessmust be calculated because

(38)

By insertion from Eq. (3), the average of the squared monitorbrightness is

(IM') = 5 h(r - r')h(r - r")h(r - r"')h(r -riv)

X (Ic(n')Ic (r/)Ic (n/)Ic(riv))dr'dr/Idr/fdriv. (39)

We see that in the general case we need a four-point intensitycorrelation function. This requires eight-point correlationfunctions of the complex amplitudes. Written out, the cor-relation function in Eq. (39) is


(I. ) = 2 (I, ) (I2))

a 2 = (J2 ) - (I. ) 2.M M


(I, (r')I,, (r")I, (r ..)1, (riv))= (QAi(r')A2(r') + Aj(r')A*,(r')]

X [A (r")A 2(r") + Al(r")A;(r")]X [A;(r"')A2 (r"') + Al(r"')A2(r"'X [Ai(riv)A 2 (riv) + Ai(riv)A >(riv)]). (40)

Multiplied out, there are sixteen terms in the parenthesesinside the brackets, of which eight are the complex conjugatesof the other eight; i.e., we have the real part of eight terms.

ssss

The complex amplitudes result from coherent-imaging pro-cesses. As in the calculation of the ensemble average (In ),we invoke the Dirac delta correlation properties of the com-plex object amplitudes. 17 We also replace all impulse-re-sponse functions with the corresponding transfer functions.The result is

rn, = (Im) 2(2 + 3T),

where the parameter T is given by

Pi(v')J 2JPi(v")2PP2 (v 2p 9(viv)J 2 H(v' - v')H(viv - v")H("' - v"')H(v'- viv)dv'dv"dv/..dviV

(45)

- (46)

Because of the symmetry in Eq. (39), the coordinates can beinterchanged to show that the remaining eight terms aregrouped together to give

(I, Wr)I, (r')I, (r"')I, (riv) )= 2 Re[(A ;(r')At(r'')A;(r/')A (riv))

X (A2(r')A2(r')A2(r"')A2(riVM)

+ 8 Re[(A;(r')A;(r'')At(r '')Ai(riv))X (A2(r')A2(r')A2(r"')A2*(riVM)

± 6 Re[(A;(r')A;(r')AI(r''')AI(riv))X (A2(r')A2(r'')A2(r ')A.*>(riV))]. (41)

Here complex amplitudes with index 1 (light scattered fromobject 1) have been assumed to be independent of amplitudeswith index 2 (light scattered from object 2). To simplify fur-ther, we use a moment relation1 9 for circular Gaussian vari-ables:

(XX 2X,3X4 ) = (X 1X 2)(X3X 4 ) + (X1X3)(X2 X 41 )

+ (X1 X4 ) (X2X3 ) -2 (xl) (X 2 ) (X3) (X 4 ). (42)

For circular complex Gaussian variables, terms such as (x),(x1x2),and (x1x2) arezero. Therefore,byuseofthemomentrelation in Eq. (42), the first and second terms in Eq. (41) arezero and only the last term remains. By use of the same mo-ment relation, the last term in Eq. (41) is written in terms ofpair correlation functions as

=12 Re[(A*I(r')Al(r'')) (A*(r'')AI(riv))X (A2(r')A2*(r"'))(A2(r')A,*,(riv))I

+ 12 Re[(A;(r')Ai(riv))(At(r'')A1(r'''))X (A2(r')A2(r''))(A2(r'')A.*,(riv))]. (43)

We compare the first term in this equation with the square ofthe average value (Im,) in Eq. (30) and find that it is equal to3 (I,,l )2. By inserting Eq. (43) into Eq. (38), we find that thestandard deviation of the monitor brightness is then

For the speckle contrast in a speckle-reference ESPI we finallyobtain

C = ,,m/(Im) = (2 + 3T)l/2. (47)

Because of the complexity of the integral in the numeratorin Eq. (46), this speckle contrast is not easily evaluated. Tosee how the contrast depends on the resolution of the cross-interference pattern, we consider a simple example.

We mask off some of the original aperture in Fig. 4(a), andwe get the aperture in Fig. 4(b). This aperture consists ofmany equal and distant elementary pairs of apertures. Aspeckle-reference ESPI with such an aperture will functioninterferometrically like the original one.7 Apertures such asthe one in Fig. 4(b) can be described by the sums

NP1(v) = O P1 (v-p )

p=O

NP2(v) = Y P2(v -k vo0 ).

k=O(48)

Here the pupil functions P'1 (v) and P'2 (v) together describethe first elementary pair of apertures, and for simplicity weuse a one-dimensional description (in this case, a two-di-mensional description gives the same result). The setup isarranged such that only the elementary cross-interferencepatterns contribute to the final display, i.e., the distance be-tween any elementary pairs is larger than the TV-cameraresolution limit L. Insertion of Eq. (48) into the general ex-pression for r, Eq. (46), gives

1T1,NT

(49)

where the parameters r and r' are defined by the same equa-tion. To find r' from Eq. (46) we have only to substitute Pl(v)and P2(v) for Pl(v) and P2(v). If each of the elementarycross-interference patterns is fully resolved, r' = 1, and we getthe simple result

a' = (Im)2{2 + 3

14 ffff (A;(r')A1(riv)) (A1(r')A1(r'w))

X (A,)(r')A' (r"')) (A,)(r')A,*,(riV))

X h(r - r')h(r - r")h(r -r')

X h(r - riv)dr'dr'dr/'driv]/ (I.)22

r = 1/N. (50)

Therefore, as N goes from 1 to infinity, the speckle contrastgoes from N/5 to N/-. This effect is demonstrated in Fig. 5,in which four photographs of the monitor display are shown.Instead of the elementary double-slit apertures in Fig. 4(b),we used pairs of small circular holes. The average monitorbrightness was the same in all photographs, and the pictures

(44) were processed alike. As the number of pairs increases wemay recognize that the total sum of the completely black and

-

If [Ipl(V)12. 12]IH(v)12dv 21 P2(V) I



N=2 appears as the result of an addition of T statistically inde-pendent variables.1 ' These variables have the same proba-bility density functions. Therefore the speckle contrast inthe speckle reduced image is

Theoretically, T may go to infinity, and we can obtain zerospeckle noise in the display. The corresponding value in aspecular-reference ESPI would be

N=11

Fig. 5. Photographs of the monitor display. The setup is aspeckle-reference ESPI, and N is the number of elementary pairs ofsmall-circular apertures. The distance between any elementary pairsis larger than the TV-camera resolution limit L.

white (saturated) areas becomes smaller. We may alterna-tively look for the fine vertical gray lines; the total area occu-pied by them becomes larger as N increases. Another visualeffect is due to the apparent change of the speckle size. Todetermine this size we must calculate higher-order statistics.In this paper we limit ourselves to the calculation of thefirst-order statistics, and we are unable to explain the lasteffect.

N is the number of fully resolved elementary cross-inter-ference patterns. In a speckle-reference ESPI, in which thetotal cross-interference pattern is fully resolved, N = 1, andthe speckle contrast is found from Eqs. (47) and (50) to beequal to V5. As the number N goes to infinity, the specklecontrast in the speckle-reference ESPI approaches the con-trast in a specular-reference ESPI.

With the ordinary multislit aperture [shown in Fig. 4(a)]in the setup, the parameter i- can also be calculated. Sincethose calculations are more involved and since the results willnot differ much from the results above, we do not repeat themhere.

We began this section by referring to Ref. 16. In that paperspeckle noise in displays was investigated. If we assume acertain speckle size, it is obvious from the previous discussionthat the specular-reference ESPI gives the smallest specklenoise. However, the difference is small if, as is usually thecase, the speckle-reference ESPI has aperture dimensionsmuch larger than the TV-camera resolution limit L. Thespeckle contrast can be effectively reduced if speckle-reduc-tion techniques are applied. A simple speckle-reductiontechnique for speckle-reference ESPI was presented in Ref.7. By that technique, essentially uncorrelated speckle pat-terns were subsequently added at the display stage. If Tuncorrelated speckle patterns are added, the speckle contrastwill decrease proportionally with 1/VT. The value V'7

N=5

N=1

Cred = [(2 + 3T)/T]1 1 2 .

However, no speckle-reduction techniques have been devel-oped for specular-reference ESPI.

6. DISCUSSION AND CONCLUSION

We have assumed a square-law rectification of the video sig-nal, and we have calculated most of the first-order statisticsof the monitor brightness. From the results we conclude thatthe brightness distribution in speckle-reference ESPI isslightly different from the case of specular-reference ESPI.The probability density of the monitor brightness in a spec-ular-reference ESPI is always the (X)2 density function. Theprobability density in the speckle-reference ESPI has a similarshape, and it will approach the (x)2 density function as the sizeof the aperture increases (we assume a certain TV-cameraresolving power).

The speckle contrast, i.e., the ratio between the standarddeviation of the monitor brightness and the average monitorbrightness, is an important parameter used in the character-ization of speckle noise in displays. 16 The speckle contrastin specular-reference ESPI is always equal to \, whereasit is equal to (2 + 3r)1/2 in speckle-reference ESPI. T is aparameter that generally is a complicated function of thetransfer functions in the setup. If the cross-interferencespeckle pattern is fully resolved, r = 1. As the size of theaperture increases, r approaches zero, and the contrast willbe the same in speckle- and specular-reference ESPI. If weassume the same speckle size in the two kinds of setup, weexpect that the speckle-reference ESPI gives a slightly noisierdisplay. However, in Ref. 7 we developed a speckle-reductiontechnique for speckle-reference ESPI. Uncorrelated specklepatterns were subsequently added at the display stage. Inthe present paper we show that the speckle contrast decreasesproportionally with 1/V , where T is the number of uncor-related speckle patterns. By this means a very small specklecontrast can be obtained.

In addition to the first-order statistics treated in this paper,the speckle size (i.e., the second-order statistics) in comparisonwith the size of the displayed fringes is important for imagequality. An investigation of the second-order statistics seemsto be a natural extension of the present work.

Finally, we comment on the statistics of the displayedbrightness in ESPI images compared with the statistics inother holographic images. A main difference is due to the factthat in ESPI the real and virtual images are not separated.This causes, for example, the contrast of JV in specular-ref-erence ESPI compared with the contrast of 1 (Refs. 1 and 2)in conventional holography. In ESPI the displayed specklesare clearly seen (resolved), and the first-order statistics aresufficient to describe some of the quality properties of theimages. In conventional holography the displayed (recon-structed) speckle pattern may not be resolved, and the re-

(51)

Cred = V'WT. (52)



suiting speckle contrast may be less than the maximal contrastof 1 (see Refs. 2 and 3). In holography and speckle interfer-ometry, in which the real and virtual images are not separatedand in which the displayed speckle patterns are resolved,however, our calculations should give useful relations.

I thank C. T. Stansberg for stimulating discussions and P.R. Neiswander and H. M. Pedersen for helpful criticism of themanuscript. I also thank R. Bolso Moen for preparing thefigures.

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first-order statistics of displayed speckle patterns in electronic speckle pattern interferometry

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