192 J. Opt. Soc. Am. A/Vol. 1, No. 2/February 1984
Spectral and imaging properties of uniform diffusers
Marek Kowalczyk
Institute of Geophysics, Warsaw University, Pasteura 7, 02-093 Warsaw, Poland
Received January 31, 1983; accepted September 12, 1983
The use of thin phase diffusers in coherent imaging systems is analyzed. A condition is derived that must be ful-filled to ensure that no part of incident light is specularly transmitted. This condition is expressed in terms offirst-order statistics of the phase of light emerging from a diffuser. It is shown that random diffusers whose ampli-tude-transmittance argument (phase) is uniformly distributed in the (-7r, 7r) interval do not pass the specular lightwhile their rms phase corresponds to only 0.29 of a wavelength. Such diffusers will be called uniform ones. Amethod for forming a uniform diffuser is proposed. It is based on a recording of normal speckle patterns in phase-photosensitive materials with consideration of the exposure characteristics. Autocorrelation functions and powerspectra of diffuser transmittance are evaluated for two types of exposure characteristics. For both these cases theimage contrast of the uniformly illuminated diffuser is calculated.
1. INTRODUCTION
A number of optical systems exist for which it would be de-sirable to have a diffuser whose directional characteristic ofscattering could be shaped. Imaging with diffraction noisereductionl Fourier transform (FT) hologram recording,2 andincoherent systems in which the diffusers are used as viewscreens 3 or reflecting filters of infrared radiations are exam-ples. First, our analysis concerns the diffusers applicable inthe two coherent systems mentioned above.
A power spectrum of the diffuser transmittance suitable forsuch systems would have a forml"5' 6
(,o 2 + W y 2) 1/2 .pSt (O, y) circ = circ-,
Pd Pd(1)
where (, wy) are the spatial-frequency coordinates and Pdis the cutoff frequency of spectrum. A circularly symmetric,band-limited spectrum does not settle the question of diffuserusability as an element of a coherent imaging system. Ad-ditional knowledge about its phase and its first and secondspatial derivatives is needed since diffuser imaging propertiesdepend on these parameters.5 7 By imaging properties wemean statistical properties of irradiance variation in the imageof a uniformly illuminated diffuser, especially its averagecontrast. In general, small variations of phase and its deriv-atives are desirable.
The most significant papers dealing with the design of adiffuser whose power spectrum is uniform and band limitedare these of Kurtz et al.8 and Nakayama and Kato.9 Theyanalyze the one-dimensional (1-D) situation. However, thetwo-dimensional (2-D) generalization performed accordingto their suggestion does not yield a circularly symmetricspectrum. Moreover, the diffusers described in Ref. 8 weredesigned without taking their imaging properties into con-sideration. For example, large rms phase is the particularfeature of the diffusers. 10 An artificial diffuser with a circu-larly symmetric power spectrum with small variations of phaseand its spatial derivatives was proposed by wamoto.5 On theother hand, the power spectrum of Iwamoto's diffuser containswell-marked discrete components that correspond to the lowdiffracted orders. When a hologram of amplitude transpar-
ency is recorded in the FT scheme and such a diffuser is used,then the zero-order component usually saturates photosen-sitive material in the region of the lowest frequencies. A lackof these frequencies in the reconstructed beam causes a vari-ation of irradiance to appear in the reconstructed image of theuniformly illuminated diffuser. This is similar to the phasecontrast effect.7
We propose a random diffuser whose circularly symmetricpower spectrum does not contain discrete components andwhose rms phase is equal to only 1.81 rad.
2. FORMULATION OF THE PROBLEM
We assume that the diffusers in question are thin and non-absorbing; then we can describe their operation on incidentradiation by means of a complex amplitude transmittancetd(X, y) = exp[isp(x, y)]. Phase p(x, y) is related to the dif-fuser surface-relief height h (x, y) and to its refractive indexnd (X, y) by the formula
y(x, ) = [nd(x, y) - 1]h(x, y), (2)
where X is a wavelength. Equation (2) also describes thephase of the light that emerges from the diffuser, providedthat the incidence is normal and the refractive index of thediffuser environment is equal to unity. We also assume thatyo(x, y) is a continuous, smoothly modulated function ofcoordinates and that it is a sample function of an isotropic,wide-sense-stationary random field (2-D stochastic process).We use the same notation for the stochastic process and itsspecific sample. A proper interpretation of the notation isapparent from the context.
We analyze the autocorrelation Rt (Ax, Ay) of td(x, y):
Rt(/\x, lAy) (td (Xl, Y1)td*(X2, Y2)), (3)
where Ax = X2 - x1, Ay = Y2 -y y, and ( ) denotes ensembleaveraging. In accordance with the Wiener-Khintchine the-orem, R (Ax, Ay) is the FT of the amplitude transmittance'spower spectrum S (w, wy), which is defined as follows":
Vol. 1, No. 2/February 1984/J. Opt. Soc. Am. A 193
St(w., oy)
£:IJ si td(x,y)exp27ri(xwX +ywy)]dxdy (4)
Because of a direct correspondence between the angularspectrum of plane waves and the spatial-frequency spec-trum,12 a simple relationship between the light-scatteringprofile and the power spectrum of the amplitude transmit-tance can be found.
In order to justify our concept of the diffuser and to esti-mate its imaging properties, we state briefly the elements ofthe phase imagery theory. For this purpose we utilize Iwa-moto's approximate results regarding the influence of low-passfiltering and defocusing on the quality of the phase objectimage. We also use Thompson's results dealing with theconventional imaging of complex objects,7 and we extend theirapplicability to holographic imaging.
If one follows the method developed by Iwamoto for the 1-Dcase exactly, one can show that the irradiance variation I(u,v) in the perfectly focused image of the uniformly illuminateddiffuser (Fig. 1) is approximated by
where p8 is the cutoff frequency of the rotationally symmetricimaging system and (p(u, v) denotes the phase distribution inthe diffuser plane formally depending on (u, v). It followsfrom Eq. (5) that variation I(u, v; Ps), which appears for anydiffuser, vanishes as ps becomes infinite. The above suggeststhat the power spectrum of any diffuser in fact extends toinfinity. Therefore, in general, the mean-square spread1 3
seems to be a more-adequate parameter of the diffuser spec-trum than the cutoff frequency is. Nevertheless, in someparticular cases, a cutofflike frequency is well marked. 6, 8 9
When an observation of the diffuser image is made at adistance b from the image plane and we put Ps equal to in-finity, the irradiance variation is approximated as
Xb [O2 p(u, V) a 2~0(U, V)I(u,v;b) 1-2 + I
2r AL 0u 2 av 2 I( Xb 12 82~0(U, V) a22 0(u, v)
+-I aU2 + I
+ f{ap(u, V)]2 + [( U, v)12 22
(6)
If the image focusing is perfect, then the noise pattern de-scribed by Eq. (6) vanishes. We may assume that the imageof a transparency and a diffuser image are well focused in thesame plane, i.e., light propagation between the diffuser andthe transparency is negligible for small rms phase only. It isobvious now that the specular component of scattered lightmay not be eliminated by phase standard deviation in-crease.
Fig. 1. Imaging system with diffraction noise reduction or holo-graphic imaging system with a smoothed-out object beam in thespatial-frequency plane. Recording and reconstructing systems arepresented jointly. Elements described in parentheses appear in theholographic system only. Diffraction noise reduction also takes placein the holographic system as an additional effect.
The approach presented here [Eqs. (5) and (6)] holds truewhen p, is large and b is small;14 therefore it may be consid-ered an alternative approach to the image-plane speckleproblem in the case when a small number of scatterers N (N< 1) contributes to the observed intensity.15
Equations (5) and (6) apply to both a holographic and aconventional imaging system. There is another factor. First,for holographic imaging in the FT scheme, which causes theimage of the diffuser to deteriorate, low frequencies areeliminated. If we consider a recording system with a commondiffuser, e.g., Gaussian or exponential, the power spectrumof which is .a sum of a discrete component ab(w,, co,) and acontinuous component St (w,, Wy),13 where a and : areconstant factors, then the average irradiance of object beamis given by (the derivation is given in Appendix A)
(If( , n)) K {1T5(W., w)J2 [ab(co., coy)
+ S~tc(W.,,, w)]1@ = vWy = 7/Af
[al T. (cox, WC)2 + AI T(, Y) 2
® St,(Wx, o, L., = v/ay = 17/Xf
(7)
where denotes the convolution operation, f is the focallength, and T,(w,, wy) denotes the FT of the object trans-mittance t, (x, y). The distinctive feature of the term ac| §T, 2is a sharp maximum at the origin. If we assume that the termd T 2 ® St, is linearly recorded, then the part of the hologramthat surrounds zero frequency is saturated by the aI TJ 2 term.A hologram recorded in such a manner may be regarded aslinearly recorded and put together with the ideal high-passfilter, the cutoff frequency of which corresponds to the firstzero in the FT of the window. In this position the irradiancevariation in the reconstructed image is given by7
I(u, v) t 2(u, v) + 1 - 2ts(u, v)cos k(u, v). (8)
The term 1 - 2t, cos (p describes the noise pattern that ap-pears because of the elimination of low spatial frequencies.The above is an approximation of the real situation since, infact, the filter equivalent of the saturation effect is not idealand there is an intermediate region in the hologram where therecord is nonlinear.
It is seen from Eqs. (5), (6), and (8) that the irradiancevariation in the image of the uniformly illuminated diffuser[t, (x, y) = 1] depends on so(x, y) and its first and second de-rivatives. If we use the diffuser whose spectrum does not
Marek Kowalcyzk
194 J. Opt. Soc. Am. A/Vol. 1, No. 2/February 1984
contain the zero-order term and if we focus the reconstructedimage well, then we can neglect the dependence of I(u, v) onp, a2p/aU2, and d2cp/8v2, provided that the variation of the firstand second derivatives of the phase is limited and that p islarge enough so we can neglect the last term in Eq. (5). Toproceed further, we assume that this is the case. A quanti-tative relation that tells us how large p, should be is given inSection 7. Now the irradiance distribution in the image planemay be expressed by
I (a, v) 1- f[p9 P u~ v]+ R ou, 0 ) 1 _ Do (, ),
do 2P r au + [a'p UV)j2J D2 (U, V) 9
where D is the magnitude of the 2-D gradient of the phase.We see that the diffuser image quality is determined by thecutoff frequency p and the statistics of the squared magni-tude of the phase gradient.
We proceed as follows: First we show that the existence ofa diffuser that does not pass the specular light and whose rmsphase is smaller than 2 rad is theoretically possible.Therefore it is possible to avoid the generation of noise de-scribed by Eq. (8) and to obtain good focusing of the recon-structed image (b X/2) simultaneously. We propose twosimilar ways of producing such a diffuser, and we calculateautocorrelation R (Ax, Ay) for each case. Then the calcu-lation of conditional density functions W(aso/dxls) will permitan estimation of noise-pattern contrast in the image plane forboth types of diffuser.
3. INFLUENCE OF FIRST-ORDER STATISTICSOF PHASE ON DIFFUSER STRENGTH
In order to show that the power spectrum St(w, y) of adiffuser whose rms phase is smaller than 2 rad must notcontain a zero-order term or, equivalently, that the autocor-relation R (Ax, Ay) must not contain a constant component,we note that the value of this component is equal tolim,-CRt (r),6 where ' = [(AX) 2 + (y) 2 ]1/2 . Since we assumean isotropic random field td (Xy), then R (Ax, Ay) and St (w,Wy) are circularly symmetric. Let us calculate the limitRt (f):
Rt ( ) lim J exp [i (° - PAT-.- CD E <
where cT(v) is the characteristic function of so. Thus St (wa,coy) does not contain a zero-order term if and only if
11(1)12 = 0. (12)
Equation (12) is exactly fulfilled for W(w) such that 4(v) 2
= 0 for a finite v. If, however, I I(V) 2 decreases to zero as vbecomes infinite in such a way that for any v we have I 1(v)l 2> 0, then Eq. (12) is approximately fulfilled for large rms ,oonly.16 It may be noted that, for an oscillating functionI(v)I 2, there are regions of anomalous behavior of the specularcomponent intensity versus rms s, i.e., this intensity increasesas rms s° increases. Equation (12) holds for an arbitrary phasedensity, including the discrete one, characterizing pseudo-random phase diffusers.6 9
In order to determine the W('p) that minimizes the rmsphase and whose characteristic function satisfies Eq. (12), weimpose on W(p) additional conditions that follow from theproposed production process of the diffuser and from its ap-propriation. We assume that the diffuser is made by meansof recording, on phase-photosensitive material, a random lightfield (Fig. 2) resulting in a negative exponential or Gaussianprobability density function of exposure E(x', y') = OI(x', y'),where is the exposure time. We also assume that the ma-terial characteristic 'P(E) is linear for small E and saturateswhen E becomes large (e.g., for photoresists1 7 ), or it has athreshold of sensitivity, an approximately linear region, anda saturation region (e.g., for a bleached silver halide emulsion).These additional conditions we formulate as follows:
(1) W('P) is symmetric about ( ) and does not increasewhen I - ()I increases. If condition (1) is fulfilled, or, inother words, phase = () is the more likely value than phaseid (qp), then rapid phase variations and consequently a highlevel of noise as described by Eq. (9) are not expected. It isconvenient to put () = 0.
(2) W(p) > 0 when |H < , and W(P) 0 0 otherwise.This condition follows from saturation of 9(E).
(3) WW) is a continuous function on an interval (o, 'Po).Condition (3) follows from the continuity of W(E) and,p(e).
Among the basic densities used in probability theory andstatistics' 8 that satisfy conditions (1)-(3) and Eq. (12), thesmallest rms has the density
(10)
where sp = O(xI, l, 'P2 = O(x2, Y2), and W(p1, 'P2; ) is thesecond-order density function of the phase. If we assume thatchanging the order of integration and the limit calculation ispermissible, we have
Rt( M= s: exp[i((p - P2)1
X lim W('PI, 'P2; r)d'pod'P2T-:
=3 exp(if l) W(si)df l
x f' exp(-iP 2)W(P 2)dP 2 =I(f(1)1 2, (11)
strongdiffuser
laser !beam
paserecording
z ; materialFig. 2. System for forming uniform diffusers. Speckle pattern thatappears in the (x', y') plane is recorded in the phase-photosensitivematerial. In order to ensure stationarity of the uniform diffusertransmittance td(x, y), the central part of the speckle pattern, wherethe average irradiance distribution (le (x,' y')) is nearly constant,should be utilized only. Circularly symmetric illumination of thestrong diffuser (dotted area) is required to ensure isotropy of therandom field Ie(x,' y') and consequently the isotropy of td(x, y).
Marek Kowalcyzk
X W((Pl, IP2; -Od�oA02,
Vol. 1, No. 2/February 1984/J. Opt. Soc. Am. A 195
= I (2ir)-1 1J < w0 otherwise
(13)
The above uniform density has the characteristic function'I (v; spo = r) = (sin 7rv)/7rv. From Eq. (13) we see that rmsso equals 1.81 rad. This corresponds to a rms wave-frontmodulation depth equal to 0.29 A. Since the density given byEq. (13) does not follow from the rigorous optimizing proce-dure, it may be possible to find a density that satisfies con-ditions (1)-(3) and Eq. (12) whose rms o is even smaller than1.81 rad.
4. AUTOCORRELATION OF td(x, y FOR TWOTYPES OF UNIFORM DIFFUSER
In order to obtain a diffuser with the density W(f ) given byEq. (13), we can record the normal speckle pattern 1 9 on thephase-photosensitive material that has the following char-acteristic p(E) (Fig. 3):
Pe(Ie)= 7r 1-2 exp ((I )] (14)
where unity exposure time is assumed. The subscript e marksquantities associated with the single exposition of the normalspeckle pattern irradiance Ie (x', y'), which obeys a negativeexponential law, in order to distinguish them from those as-sociated with Gaussian statistics of exposure, which aremarked g. If a random light field leading to the Gaussiandensity of exposure is used, then the dependence of phase shiftversus exposure should be given by (Fig. 3):
pg (E) )= rerf(g- (Eg)) (15a).\-2OEg
where
erfs = 2 JS exp(-t 2 )dto17r
(15b)
a
a
.CC
1,/0e
Fig. 3. Characteristics of phase-recording materials suitable forforming uniform diffusers.
amplitude light field that appears at a distance z from a strongdiffuser illuminated with a coherent beam (Fig. 2). For ourpurpose Me (T) can be calculated from the equation
(18)Ae (T) =
f |P(1)121dl
where Jo(... .) is a Bessel function of the first kind, zero order,and IP(1)12 is the circularly symmetric irradiance distributionincident upon the scattering spot. Such a symmetry is re-quired to ensure isotropy of tde (X, y). If we make use of theidentity 2 3
-s exp t_ + _ 2) [2(ttnl O Ln(t,)Ln(t2)s",
(19)
and aEg is the standard deviation of exposure. A Gaussiandensity of exposure can be obtained by means of a multipleexposition (e.g., 15 times2 0) of a normal speckle pattern,provided that successive patterns are not correlated. Char-acteristics 'Pe(Je) and Pg(Eg) are the properly scaled andshifted distribution functions of Ie and Eg, respectively. Herewe just utilize the fact that a function of a random variable isuniformly distributed in the (0, 1) interval if it has a form ofthe distribution function of this variable.2 '
Now we calculate the autocorrelation Rte (r) of the randomfield tde (X, Y) expli'pe [Ie (x, y)]1:
Rte(T) = f f expUi[pe(Iei) - Pe(Ie2)]1
X W(Ie, Ie2; )dIeldIe 2.
Joint density W(Iel, Ie2; T) is given by22
W(Ie, 1 e2; T)
(Ie 1 + e 2)exp [(I)(1 -I Me (T)1 2)
(Ie ) 2(1 - I e (T)l 2)
x Io ((T)| ))
(16)
where Ln (t) is a Laguerre polynomial of order n defined by
Ln(t) = p() dn [tn exp(-t)],n! dtn
(20)
then we can easily expand W(IeI Ie2; T) into a series of La-guerre polynomials Ln (Ie/(Ie)). By replacing W(Ie1, Ie2; r)in Eq. (16) with its expanded form and 'pe(Ie) with its explicitform given by Eq. (14), we obtain the autocorrelation func-tion
Rte(T) = T ICni 2 I ,e (r)I 2n,n=O
(21)
where
LCn = |o Ln (-In t)exp(-27rit)dt. (22)
The integral in Eq. (22) gives c0 = 0. This means that Rte (r)does not contain a constant component. It can be seen fromEq. (21) that Rte (i-) D 0; thus the power spectrum Ste (wx, w,)cannot be similar to that of Eq. (1). The correlation coeffi-cient r, (-r) of a random field Ie (x', y') is equal to I Me (r)l 2,22
(17) so we can rewrite Eq. (21) as
where Io(- . .) is a modified Bessel function of the first kind,zero order, and Me (r) is a coherence factor of a complex-
(23)Rte(T) =Z ICnI2rIen(r).n=O
Marek Kowalcyzk
(E -(Eg))/\(-2o-9
196 J. Opt. Soc. Am. A/Vol. 1, No. 2/February 1984
The above procedure when applied to the random fieldtdg(x, y) = expfi gEgf(x, y)]J gives
Rtg (T) = I 1 2(rE, () (24)
m-O m 27where
dm =-f4 Hrn(t)exp(-t2 + i-r erf t)dt, (25)-V -where the Hermite polynomial of order m, Hm(t), is definedby
Hm(T) = (1)m exp(t2 ) dM [exp(-t 2 )]. (26)dtm
In calculating Eq. (24) we have assumed that Eg(xi, yi) andEg(x 2, Y2) are jointly Gaussian with correlation coefficientrEg(r), so their joint density is as follows2 4:
in the (0, 2r) interval, and random fields y(x, y) and D(x, y)are statistically independent.
In this section we calculate the conditional densityW(do/dxl p) of dy~/Ox assuming yp when o and dy/dx aremeasured at the same point (x, y). Also, the second-orderdensity W(Pl, y 2; r) of phase yp and the joint density W(yp,acp/Ox) between yp and dy/Ox are given as intermediate re-sults.
In order to calculate the joint density W(yo, dy/ox) for thediffuser based on a single exposition of a normal specklepattern, we introduce two auxiliary random variables he and#e, which are defined by
he (Oe el, e2; AX) y_ e2 ± %Pel Pe(x + AX, Y) +( (x, y)2 2
(31a)
W(Egi, Eg2; z-) [1 -rE- 2 (T)]"l/2I (Egi - (Eg))2 - 2rEx(T)(Egl - (Eg))(Eg2 - (Eg)) + (Egi - (Eg))212 T7r eE x 2 pi 2O-Eg
2[1 - rEg2 (T)] I
In order to expand W(Egl, Eg2; r) into an infinite series ofHermite polynomials Hm [(Eg - (Eg))/V7Eg], the followingidentity is used 2 5 :
1 2tlt2 , - (t 2 + t22)s21
7=exp1 _ s2 2(1 - 2)
= (X) (X) k=0 (.2
(28)
It follows from Eq. (25) that, for m = 0, we have the coeffi-cient
d= =-f' exp[-t2 + iHr erf tldtVrW _co
= - ' d[sin(zr erf t)] = 0.7r
(29)
Thus Rtg(r) also does not contain a constant component. Ifthe method of multiple exposition of a normal speckle patternis applied, then rE () = rIe (r) (see Appendix B), and conse-quently rEg (r) > 0. Therefore it follows from Eq. (24) that,in this latter case also, the autocorrelation of the diffusertransmittance is a nonnegative function.
.gv ((p~~ (P ; E)°2 e ge ( + AX A - e (X, Y) 'Pe (~ei, ~ePeA ) 'PeA -X
(x Ax
(31b)
It follows from Eq. (31) that
lim !e(yeib 4Pe2; AX) = 'PesAx -0
lim l/e(/eyei, e2; Ax) = ayPe/OX.Ax-0
(32a)
(32b)
Also, the following limit relation holds26 :
lim W(re,# e; Ax) = W (e,-e)Ax p0 A5x-
(33)
Therefore at first we determine the joint density W(te, l/e;Ax), and then we calculate its limit as Ax approaches zero.For determination of W(Ge, 'e; Ax) we must at first determineW(yPel, yPe2; Ax) from Eqs. (14) and (17) by performing theusual transformation technique of random variables.2 7 FromEq. (14) we have
5. STATISTICAL PROPERTIES OF 0y 0 /dxAND dipgIOx
Statistical properties of the spatial, partial derivative of phasedy/Ox are of interest to us since dy/Ox and the phase gradientmagnitude D, the second power of which determines the dif-fuser imaging properties (see Eq. (9)], are simply related toeach other; namely,
dyo(x,Y) = D(x,y)cos y(x,y), (30)
ax
where y is the angle formed by the vector [adp/ax, dy/dy] andthe x axis. Since we assume an isotropic random field yp(x,
y), it is reasonable to assume that y is uniformly distributed
*°ei = r [1-2 exp I '\ (le I],
i = 1, 2. (34)
Equations (34) have the unique joint solution of the form
Ie = -Ie) n _ ei (35)
and its Jacobian function, calculated according to Eq. (35) asa function of 5°el and ye2, is as follows:
d(yOe1 , yPe2) 4w2 (1 _ Oelf( 1 y 2 (36)
(Iel, Ie2) (1 ) 2 k2 2X rJ2 27!
(27)
Marek Kowalcyzk
Vol. 1, No. 2/February 1984/J. Opt. Soc. Am. A 197
From Eqs. (17), (35), and (36) we obtain
(P1 Pei 1 YPe2l IiAe(r)12
I _i Il- IIe(T)12
[W2 2w7M2 271 JW(cpe, Y°e2; r) 47 2(1 - lie(r)J2)
'Pei\ (1 Pe2\]1/2
Io ( (2 2w / (2 2 )]X 10 2'~~~~g (r~
IPeil < . (37)
If we repeat the preceding procedure starting from the sec-ond-order density given by Eq. (37) and the transformationgiven by Eqs. (31), then we obtain
To calculate the limit in Eq. (33) we assume that Ie (x, y) isdifferentiable in the mean-square sense28 and, consequently,that
It is difficult to integrate Eq. (39) over p°e in a closed form.Therefore we determine the conditional density W(asoe/axi Ye):
_ __ (i~ A \ 21 I e )IIlA xi 1 1 1 i - t ( X~ 2 ' [( 2
4w2[1 - 1,4e(AX)12]
2 I + in (-- -A )] 1 e(Ax)l
xI 2 2wr 4wr - 2 2wr 4 wj /0 o - Ite (AX)l 2 I I < , II'eAXI < 27r. (38)
To derive Eq. (38) we substitute Ax for r in Eq. (37). Theabove substitution does not destroy the generality of ourformulation. Since 'Pe (x, y) is an isotropic field, it is alwayspermissible to orient the axes of the coordinate system in sucha manner as to obtain Ay = 0. Since te and te are randomvariables but not random fields, Ax in W(te, 1,Ve; Ax) no longerhas the meaning of the distance between the two points, wheree and Ie are measured or evaluated. It is just a numerical
parameter. Taking into account Eqs. (32), (33), and (38) andalso the fact that I ,e (T-)( 2 = re (r), we finally conclude that
exp ax
*~( { (epa 2 (r- Y0e) [Ie(0) n (2 )]1 /12}
'a~~e W(Y,,e ax / P" W (e)
I~~e ax ) ( dsI _Ido) exp
= \2 {(7r - Ye) Ie(0) n 2- ) 1/212
-\(2-i Or- 'pe) [Ie() n (1- 2-e)]11 2
(42)
This conditional density and Eqs. (9) and (30) are sufficientto calculate the variance of I(u, v). It is easily seen thatW(OPe/axl pe) is Gaussian with zero mean and variance:
This variance takes on its maximum at 'Pe = -0.2 wr (not at Pe= 0) and approaches zero as YPe approaches w. Its maximumvalue is equal to 7.26 Pie(O). When the above procedure isapplied to the diffuser that is produced by means of a multipleexposition of a normal speckle pattern, the following is theresult:
198 J. Opt. Soc. Am. A/Vol. 1, No. 2/February 1984
W ' I pg) = 2W W(pg, $ (46)
where W(sog, dopg/dx) is given by Eq. (45) and O(.. .) is theinverse of erf(.. .). The conditional density W(dsog/dxl 'Pg)is Gaussian with zero mean and the following variance:
~g 2((pg) = -2PI:e(0) exp [202 (g)J. (47) 0.5Co0
This variance takes on its maximum at pg = 0 and approacheszero as Sg approaches ±sr. Its maximum value is equal to- 6.28 Pi ()-
6. CONTRAST OF THE NOISE PATTERN INTHE IMAGE PLANE
We define contrast C in the image plane of a uniformly illu-minated diffuser by
=[I(U, V) - (I(u, v)) 2)l/2 (48)(I(u, v))
The reciprocal of C may be considered the signal-to-noiseratio. Equations (9) and (30), with regard to the statisticalindependence of random fields y(x, y) and D(x, y) yield
C
2 ~ (( 8 U ((dii2-a ( (aX )211= U
a(u
In order to determine contrast C we have to calculate thesecond and fourth moments of d'p/Bu. For this purpose weuse the formula2 9
<<Xl Y>> = (X), (50)
where X and Y are statistically dependent random variables.In other words, the conditional mean (XI Y) is regarded as arandom variable, being a function of Y. Since W(d'P/dulp)is the zero-mean Gaussian density and W('p) is given by Eq.(13), we have
__ = f 'P1 W(p)dp = y j' c d2(f)d9,
(51)
4f aX 4f 3X
-\OuJ 3 = r _ f W ()d'P= f(')d .(52)
Performing the integration in Eqs. (51) and (52) for -e2('pe)
given by Eq. (43) and then for pg2('og) given by Eq. (47), weconclude finally that
Ce =
Cg =
8 1 1gl/2 (-il 1 (O))'405) \1 Ps 2
40 (rIe ()5--I 9 Ps2
4 (~ - 1)1/2(-Iie(0))
4 -( .4 -u (°)
- P'
We see that the contrast is determined completely by the ratioof pIe(0) to ps 2 . Curves for C-ie(0)/ps 2 ] and
Fig. 4. Average contrast of the noise pattern in the image plane.
Cg [-ie (0)/p8 2] are shown in Fig. 4. It is seen that the diffuserassociated with Gaussian statistics of exposure possessesslightly better imaging properties for any -PIe (O)/Ps2
value.
7. DISCUSSION AND CONCLUSIONS
The main purpose of this theoretical study has been to showthat it is possible to design a random diffuser whose powerspectrum does not contain a zero-order term and whose rmsphase is quite small. In our particular case the rms phasecorresponds to 0.29 X only. This indicates that common cri-teria of diffuser strength based on the ratio of the rms phaseto 27r rad (Ref. 1) do not always lead to correct results.
It follows from Eqs. (2) and (12) that a diffuser that doesnot pass specular light should be designed for a given A, say,A0. If we assume that n (x, y) = constant, then we can satisfyEq. (12) for A < So by putting an immersion layer with aproperly chosen refractive index between the diffuser and theobject. For A > AO we have no possibility of compensation forthe difference X - A0. Note however that, for a uniform dif-fuser, a small deviation of X with respect to A0 and conse-quently a small deviation of the interval length 2 'po with re-spect to 27r does not lead to a rapid increase of the specularcomponent because [ 4% (1; 'P0)1 2/8Po],o=O = 0. This stabilityis a desirable feature of uniform diffusers from the experi-mental point of view.
Autocorrelations of diffuser transmittance in Eqs. (23) and(24) are derived as infinite series. However, some conclusionsmay be drawn about the transmittance power spectrumwithout computing the coefficients c and dn. If the Fou-rier-Bessel transform of the series in Eqs. (23) and (24) iscalculated by term, then the nth term of the power-spectrumseries is a product of [IP(1)12102n (self-convolution iterated 2n- 1 times) and I CI 2 or IdI 2/m!2-. It follows from the abovethat the power spectra of both diffusers extend to infinity. Adiffuser that is based on a single exposition of the normalspeckle pattern generates noise because of the diffraction limitPS, and this total noise power is larger than that generated bya diffuser based on multiple exposition (see Fig. 4). Thereforewe conclude that the mean-square spread of its power spec-trum is larger.
Two similar methods of producing a uniform diffuser areproposed. Their essential point is to find a random field ofexposure E(x, y) and such a characteristic 'p(E) of a phase-photosensitive material that
Marek Kowalcyzk
Vol. 1, No. 2/February 1984/J. Opt. Soc. Am. A 199
s(E) = 27r[Z(E) - 0.5], (55)
where Z(E) is the distribution function of E. Thus our pro-posal to use an exponential or Gaussian field of the resultantexposure that is generated in the free-propagation geometryis not a unique way of realizing our method.
As was presented in Section 2, elements of phase imagerytheory are of a deterministic character, i.e., they enable us tocalculate I(u, v) in the case when phase distribution in theobject plane is known. This distribution may be known onlyfor deterministic and pseudorandom diffusers. Neverthelessit is shown that Eqs. (5), (6), and (8) are also useful when noisein the image of a really random diffuser is analyzed, as theyenable us to consider the noise caused by the diffraction limitps, defocusing b, and the saturation of holographic recordingmaterial separately. On the other hand, for such consider-ations, statistical characteristics of phase and its first andsecond spatial derivatives should be known. It is seen, how-ever, that under our assumptions knowledge of conditionaldensities W(soye/xls y) and W(0yg/Oxl g) is sufficient todetermine contrast Ce and Cg. These conditional densitiesare Gaussian. Thus, contrary to our assumptions, arbitrarylarge values of dy/Ox are probable. In practice, By/Ox will belimited by the finite resolution of the phase-recording mate-rial.
It is seen from Fig. 4 that the contrast still increases when-ie (0)/p,2 exceeds about 0.5. But a contrast value greaterthan 1.0 has no physical interpretation; thus our approach isnot correct if p 2 < - 2 t e (0). The accuracy of our formulasfor contrast can be improved by making use of Eq. (5) insteadof Eq. (9) to express I(u, v). The contrast that is evaluatedon the basis of Eq. (5) should be smaller than that evaluatedon the basis of Eq. (9), as the noise term of Eq. (5) that is ne-glected in Eq. (9) is of the sign opposite that which is used.The accuracy of such a solution will be still better if, on de-riving the formula for I(u, v; pS), phase derivatives of an orderhigher than second are considered.' 4 Such a procedure is,however, inexpedient since it needs evaluation of densityfunctions for phase derivatives of high orders, and this israther difficult for non-Gaussian diffusers. Unfortunately,Gaussian diffusers are not of interest to us since they transmitthe specular light for arbitrarily large rms phase. We con-clude that our approach is valid when p equals at least-2tie(0). Experimental verification of Eqs. (53) and (54) mayshow that p, 2 should be even larger.
The existence of the inapplicable region of the theory pre-sented results from the fact that it uses Eqs. (5), (6), (8), and(9) to evaluate the irradiance distribution at the image plane.These equations express I(u, v) in terms of local values ofphase and its derivatives at the object plane, whereas, for smallPs 2, an effective number of scatterers N defined as the ratioof the point-spread-function area divided by the transmit-tance autocorrelation area contribute to the observed irra-diance. Indeed, trIe (0)/ps2 = 0.5 corresponds to N 1.
Finally, it may be noted that Eq. (Al), from which Eq. (7)is derived, may serve as more direct justification of thepower-spectrum shape in Eq. (1) than a similar convolutionformula for the amplitude domain [Eq. (1) in Ref. 5], as itcontains St(co, coy) explicitly. Equation (Al) may also beused to evaluate the average irradiance distribution in the farzone of any stationary diffuser. T, is then considered the FTof the complex amplitude of the illuminating wave in thediffuser plane. Equation (Al) differs somewhat from anal-
ogous formulas derived by Miller et al. [Eq. (26) in Ref. 16]and Goodman [Eq. (26) in Ref. 30]. Equation (Al) seems tobe more general since it is derived under a unique, essentialassumption (beside the assumptions of a scalar diffractiontheory), i.e., stationarity of td(x, y). For example, an arbitraryform of autocorrelation Rt (Ax, Ay) is allowable.
APPENDIX A
If the FT hologram of a 2-D transparency is recorded and aphase diffuser is used (Fig. 1), then the average irradiance inthe spatial-frequency plane is given by (without the referencebeam)
where t (x, y) is, in general, complex and Y is the FT operator,Eq. (Al) may be rewritten as
(5t, (X,Y) td(X, y)]I 2) = I T.,o (W,, y)2 @ St (W., WY). (A3)
Let us introduce radius vectors in the object and spatial-frequency planes: (x, y) = x, (W,, Wy) = a, and dxdy = dx.Then it is assumed that t, (x) is a deterministic function andtd(x) is a random stationary field so its autocorrelation de-pends on xI - x2 only. In this situation we have
S [t,(X, y)td(x, y)1l2 )
= (: rt(Xl)ts(X2)td(X1)td*(X2)
X exp[2riw- (x1 - x2)]dxidx2)
= r P ts(XI~ts*(X2)Rt(xi - X2)
X exp[2ric * (xI - x2)]dxdx 2
= I t(x)dxl | t8*(x)Rt(xi - x2)
X exp[2iriw- (x1 - x2)]dx 2
= t .r tXl)t* (x, - 2)Rt(X2)
X exp[2ricw. x2]dxdx 2
= - Rt(x2)exp(2wriw- X2)dx 2
Xf t8 (x1)ts*(x, - X2 )dxl
= f Rt(x2)[t(x 2 ) * t, x2)]exp(2iriW. X2)dX2
= S t (W) ® | T ( ')I2, (A4)
where * denotes the correlation operation and * denotes a
Marek Kowalcyzk
112),- (J9[t,(XY)td(XY)
200 J. Opt. Soc. Am. A/Vol. 1, No. 2/February 1984
scalar product. Thus Eq. (A3) and consequently Eq. (Al) areproved.
APPENDIX B
The correlation coefficient rJe (Ax, Ay) of the irradiance dis-tribution le (x, y) in the normal speckle pattern and the cor-relation coefficient rEg (Ax, Ay) of the resultant exposure ofn successively recorded normal speckle patterns are equal ifthese patterns are not correlated, possess the same mean (Ie)and autocorrelation RIe(Ax, Ay), and are exposed for the sameduration. To show this it will be proved first, by using themathematical induction principle, that under our assumptionsthe autocorrelation of n normal speckle patterns summed onan intensity basis is given by
thus, for n = 1, Eq. (Bi) is fulfilled. Then, taking into accountEqs. (Bi) and (B2) and the fact that, for the sum of uncorre-lated random fields X(x, y) and Y(x, y), the autocorrelationRX+y(Ax, Ay) is given by 31
Thus it is shown that, from the validity of Eq. (BI) for a cho-sen n, one obtains its validity for n + 1. The above statementand Eq. (B2) show that Eq. (BI) is proved for arbitrary n.
The autocorrelation in Eq. (Bi) corresponds to the corre-lation coefficient re(Ax, Ay). Since the resultant exposureand the summed intensity are proportional, with the pro-portionality coefficient being the exposition time, we concludefinally that their correlation coefficients are equal to eachother and equal to rne (Ax, A)).
ACKNOWLEDGMENT
This research was carried on under research project MR1/5.
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