Multiple-reference-beam nonlinear holography with applications insuppression of intermodulation background noise

Hossein GhandeharianDepartment of Electrical Engineering, The University of British Columbia, Vancouver, B.C. V6T 1 W5, Canada

(Received 4 December 1979)

Double- and multiple-reference-beam nonlinear holography and their applications in the suppres-sion of halolike background noise of the faithful images of diffuse objects are analyzed. The insightsand conclusions are made through a study of the autocorrelation function of the hologram transmit-tance. An exact formulation is given for double-reference-beam holograms. For multiple-reference-beam holograms, however, only approximate closed forms are presented. It is shown that the addi-tion of extra reference beams during the recording step of the hologram may greatly amplify thefaithful images without increasing the background noise significantly. Furthermore, this amplifica-tion and the number of ways through which it could be affected increase with the number of addition-al references; however, more care should be taken in positioning the references so that there is not anoverlap of any of the newly produced intermodulation products with the desirable images. Experi-ments confirm the theoretical expectations.

INTRODUCTION

This paper is mainly concerned with nonlinear effects inmultiple-reference-beam holography (MRBH) versus sin-gle-reference-beam holography (SRBH), and it explores apossibility to improve reconstructed faithful images byemploying additional reference beams during the recordingstep of the hologram. This will be done through the evalua-tion of the autocorrelation functions of the transmittancefunctions of the holograms. The object to be imaged is as-sumed to be a diffuse object (or a diffuse object accompaniedwith strong nondiffuse reflectors).

Previous studies of the nonlinear effects in holography withdiffuse objects (e.g., Refs. 1 and 2) showed that the halolikenoise around the faithful images is due to those intermodu-lation terms that produce extra distorted images in the di-rections of the faithful images. In SRBH this backgroundnoise can be reduced, at the cost of the fringe visibility anddiffraction efficiency of the hologram, by keeping the objectirradiance sufficiently low so that the variation of exposureis confined to an almost linear region of the ta - E (amplitudetransmittance versus exposure) curve. Thus the importantquestion arises whether it is possible to reduce the inter-modulation distortion without much loss in efficiency. Theanswer to this question, as the following study of nonlineareffects in double-reference-beam holography (DRBH) andMRBH will show, is in the affirmative.

The derivation of the formulas for the problem under studyis done in two parts. First, an exact formulation is given forDRBH. Second, the analysis is extended to the general caseof MRBH, but in this case only approximate closed forms arepresented. In both cases, however, it is shown that a favorableredistribution of the intermodulation distortion is possible.The analytic procedure is the "transform" or "characteristicfunction" method of communication theory.3

1. PRELIMINARY CONSIDERATIONS

It has already been established that, from a mathematicalpoint of view, recording a hologram is analogous to the passageof electrical signals through a nonlinear electronic detector.For example, it was shown2' 4 that the transmittance of a ho-

logram of a diffuse object can be taken as equivalent to theresult of passing the sum of a real narrow-band normal processand a sinusoid (if the reference is a plane wave) or two realnarrow-band normal processes (if the reference is also a diffusesource) through a half-wave nonlinear detector.

More specifically, the problem in its original form is con-cerned with the exposure E [the irradiance of the coherentlyadded object wave and the reference wave(s) times the expo-sure time T] as the input to a nonlinearity whose transfercharacteristic is an approximate fit to the ta - E curve, e.g.,a polynomial in exposure,

ta(E) =:ao+a 1E+a 2 E2 +a 3 E3 +**. E>O. (1)

The output is the amplitude transmittance ta (E) of the ho-logram. The equivalent problem in communication theoryis concerned with the passage of an input x, which consists ofa real narrow-band noise plus sinusoidal signal(s), through amemoryless nonlinear device:3 followed by an ideal low-passfilter. The transfer characteristic of the device g( ) is ob-tained by replacing E, in ta(E), with x2, e.g.,

g(x) = ao + aix2 + a2 x 4 + a:3x 6+

N= L az, 2xv (veven).

v= O(2)

The quantity of main concern is the autocorrelation functionof the output. This problem in communication theory hadbeen evaluated and closed-form solutions for the outputcorrelation functions had been given when the input x consistsof one signal plus noise and for vth law devices,3 e.g., a half-wave vth law device, where

g(x) = axv x >Og0 x<0

(3)

It has been shown,5 however, that the mathematical procedurefor an arbitrary type of nonlinearity is basically the same asthat of the Pth law devices. Hence, in this paper, too, for thebrevity of the mathematical formulas, but without any lossof generality, the nonlinear characteristic of the recordingmedium is taken to be of (half-wave) vth law type, with v beingan even integer. Furthermore, the reference beams (and theglints on the object, if any) are presented as if they produced

835 J. Opt. Soc. Am., Vol. 70, No. 7, July 180 0030-3941/80/070835-08$00.50 9 1980 Optical Society of America 835

uniform plane waves on the hologram plane. However, thisassumption also does not bring about any limitations to thegenerality of the study and the extension of the analysis to thecases where some, or all, of the references are spherical israther straightforward.

With reference to the aforementioned background, thesubject under study is the equivalent of the following classicalproblem of signal detection theory: Find the autocorrelationfunction of the output of a half-wave vth law detector in re-sponse to the input

Px(r) = sp (r) + n (r), (4)

p=i

where

sp (r) = Sp cos(w r + apr) (p = 1,2, . . . P) (5)

and n (r) is a sample function of a real narrow-band zero-meannormal process

n(r) = N(r) cos[wcr + 0(r)], (6)

where co, is an arbitrary carrier frequency 2 so chosen that thevariations of N(r), (r), and cosapr are slow compared withthose of coscowr. s 1,S2, . .. sp are fictitious representationsof the P reference beams, with constant amplitudes S1,S2 ,.... ,Sp, incident on the hologram plane at angles

-q, = sin'I(Xap/27r) (p = 1,2,.. P). (7)

Similarly, n (r) is a fictitious representation of the object's fieldat the hologram plane with its amplitude N(r), Rayleighdistributed and its phase, 0(r), uniformly distributed 6 be-tween 0 and 27r. With these notations the real presentationof the scalar monochromatic field diffracted by a diffuse objectin the Fresnel region can be written as

ReIN(r) exp(jQt)l= ReIN(r) expUj(r)] exp(jQt)l

= N(r) cos[Qt + 0(r)], (8)

where Ret I denotes "the real part," N(r) is the magnitude,and *h(r) is the phase of the complex field N(r); Q is the tem-poral radian frequency of the coherent wave, t is the timevariable, and r is the space variable over the hologram plane.It can be shown 6 that the autocorrelation function of N(r),R&(rl;r2 ), is proportional to a Fourier transform of the radi-ance distribution of the object Ih~j(p);

is continuous and that the complex field diffracted by theobject over the hologram plane is a zero-mean circular complexnormal process. 6 However, often this might not be the case,that is, the process might not be normal and/or zero-meanand/or the nonlinear characteristic might be discontinuousat some points. These cases can be covered by a direct ex-tension of the results reported in Ref. 5 in conjunction withRef. 7. Furthermore, this study considers the interactingwave amplitudes over the hologram plane as scalar quantities;and, any change of polarization in the reflected light from theobject is ignored. In an earlier paper,8 however, the afore-mentioned analogies were extended to study the nonlineareffects in holography when a cross-polarized component ispresent in the reflected light from the object.

II. TRANSMITTANCE AUTOCORRELATION OF ADOUBLE-REFERENCE-BEAM HOLOGRAM

With x (r) of (4) as the input to the detector and P = 2, theautocorrelation of the hologram transmittance, R0 , may bedivided into six terms:

; Ro = T 2+ R,1X12 + Rnxn + Rnxs + Rnx' 2 + RnxslX, 2, (12)

where Ta is the mean transmittance and where each index ofthe remaining terms is indicative of those signals that are es-sentially (but not always solely) responsible for the corre-sponding term. For example, RsIXS2 represents that part ofthe transmittance autocorrelation that is essentially due tothe interaction of the two reference waves.

Without going into details of the algebraic manipulation,but referring the interested reader to the rich literature onclassical signal detection theory,:3' 9-12 we evaluate the finalresults as follows:

Ta2 =H o2o

v/2Rslxs2 = E H 2 Icos[l(al - a2 )Arl,

1=1

x, = H=ooRk (MAr),k=2

(k even)

v-I kRnx., = ;' Z_ H 1oR (Ar) cos[l(aiAr - 0)],

I(k+/)evenl 1=1

(13)

(14)

(15)

(16)

Rj,(rl;r2 ) = exp (1d (r - r2 )) J(Tr), (9)

v-m

Rnxs2 = ;=iI(k+m)evenj

Z H20omRN(Ar) cos[m(a 2 Arm=1

where J( ) denotes the Fourier transform of Iobj(p) and Tr= 27rAr/Xd, with Ar =_ ri - r2, d the distance of the hologramplane from the object, and X the wavelength of the illumina-tion. The variance of N(r),a2, is equal to the mean irradianceof the field

o2 = IJ(0)I = const. (10)

The real presentation of the reference waves may also be givenas

Re[Sp (r) exp(jQt)] = Sp cos( Qt + apr),

(p= 1,2,...,P). (11)

In summary, it is assumed that the detector's nonlinearity

(17)

v-I-mn

RnXSIXS2 = Ike-[(I1/m1|+kh)even1

v-k-m v-k-IZ f kI 1nHk1/RN(Ar)

1=1 m=1

X cos [(1-m) (a, - ma2 Ar - )]

[(k+1+rn)evevn

k-m k-lZ Z H21mRnR(Ar)

1=1 m=1

X cos ±( + m) | -Mar + Oilw+ (er

where

836 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980

- 0)],

for v > 4, (18)

Hossein Ghandeharian 836

RN(Ar) = IRz(ri;r2 )I,

e - O(ri;r2) = tan-1 Im[Rf(r'r2)]Re[fl&(r1;r2 )]

-Sim = [(k + I + m)/2]![(k -I -m)/2]

[(k + I - m)/2]![(k - + m)/2]!

and where

H 2m = h1im/c2(V,0)[(k + I + m)/2j!

x [(k - I -m)/2]!22k-,

with, from Ref. 3, p. 287,

(19) reconstructing beam by a thin hologram transparency couldbe assumed as the action of a linear system. The impulse

(20) response of this system for Fresnel diffraction is proportional(2) to

(21)

(22)

c(p,o) = P(p + 1)/2P+1P2(1 + #12) (23)

and

ar(v + 1) r -k'hlIi(oSi)Im(CcS2)2h 2] c+

X exp(42co 2/2)dwo, (24)

for which a closed-form solution may be obtained as

-a(v + 1)(S2/a72)m/2 (S2/r2)1/2 v-k-I-m

= !(a2)(k -P)/22(v+1-k) z

h(r;d) = exp(w7rr 2/Xd), (29)

that is, a linear shift-invariant filter. By using the input-output relationships of the autocorrelation functions in linearshift-invariant systems14 one could show that, for example,the contribution of such a term as RN (Ar) expiU (apAr - 0)]to the autocorrelation of the field diffracted by a hologram atplane z = d is given by

Rri(pl;p2 ) = lobj(-Pl - 7p) 6 (P1 - P2), (30)

where -yp = apXd/27r and 5( ) is the Dirac delta function.Noting that the autocorrelation function of the original diffuseobject may be shown by (e.g., Ref. 15)

R0 jb(P1;P2) = Iobj(PI)P1 - P2), (31)

(30) implies a replica of the object at the plane z = d, invertedand shifted sidewise by yp. A similar operation performedover RN (Ar)exp[-j (ap Ar - 0)], however, with negative d inthe impulse response of (29), yields

Ji(pi;p2) = Iobj(P1 - Yp)(P1 - P2), (32)

(S2/af2)ihF1 (k + I + m + 2i - v)/2; 1 + 1;-S2/a2)

2 i!(m + i)!r[ 1-(k + I + m + 2i-)/2] .(25)

Here I, ( ) denotes a modified Bessel function of the first kindand order q and ri ) is the gamma function and is replaceableby the factorial of its argument minus one when the argumentis a positive integer, e.g., r(v + 1) = v!; when the argument isa negative integer the value of the gamma function is -. a isthe scaling constant of the vth law nonlinearity and a2

-24, 1F1(;; ) is the confluent hypergeometric functiondefined by the series

F v;z)= (Zr = 1 + - Z + A(i + 1) Z2r=O (v)rr! v 1! v(v + 1) 2!

(26)

Note that when , is a negative integer =-q, 1F1 will terminateafter q + 1st term. C+ in integral (24) is a contour of inte-gration that lies to the right of the imaginary axis in the wplane (cf. Ref. 4, Chap. 13).

In solving (24), the power-series expansion Of I (S 2W) hasbeen employed13 ; that is, the right-hand side of

= 2) m (oS 2/ 2 )2i (27)i-O i! (m +i!

is substituted for I, (WS2) in (24) to give

aF(v + 1)(S2/2)m (S2/2) 2i

im - 27rj ii!(m + i)!

sc wck+m+2i-,-lIi(wS1) exp(2CO2/2)dco. (28)

A detailed and precise procedure to solve such integral as in(28) is given in Chapter 13 of Davenport and Root.3 Followingthat procedure gives the result of (25).

Equations (13)-(18) now may be used to predict some im-portant properties of the hologram images including theirirradiance distributions, since the scalar diffraction of the

that is, a replica of the object shifted sidewise by yp (see Fig.1). When we note that the diffracted field is traveling in thepositive z direction, the implication of having an image at theplane z = -d is that the image is virtual. Thus it is seen thata term such as

RN(Ar) cos(apAr - 0) = RN(Ar)texp[-j(apAr - 0)]+ expUj(apAr - 0)31/2 (33)

signifies two images, a virtual image at z = -d and, the other,a real image at z = d.

III. PHYSICAL INTERPRETATION OF TERMSINVOLVED IN Ro

The first term in (12), T2, the mean squared transmittanceof the hologram, is responsible for the undiffracted portionof the transmitted light (e.g., the bright spot on the optical axisin the reconstruction process).

The second term, R81xs2, is the autocorrelation function forthe sum of v/2 sinusoidal gratings with the frequencies l(a,1- a 2 ); 1 = 12, . . .v/2. This term is responsible for the arrayof the pointlike images (in the Fraunhofer region) that areformed in directions dictated by l(yi - 72). (Note that, ingeneral, ap and -yp must be treated as vectors.)

The third term, Rnxn produces an image on the optical axis.The irradiance distribution of this image, which sometimesis called the ambiguity function, is proportional to ZYk=2[(k - 1)-fold convolution of the object irradiance] (k even).

The fourth and fifth terms, Rgxsl and R tXS2, represent thecontributions that are essentially due to the active involve-ment of the object wave with the first reference wave (nxsI),and with the second reference wave (nxs 2). For I = 1, k = 1,and m = 1, k = 1 each is responsible for a pair of faithful im-ages (the usual real and virtual images), while for k > 1 theygenerate distorted images. Some of these distorted images

837 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980 Hossein Ghandeharian 837

(a)

(h)

FIG. 1. Each term in Ro(r 1 ;r2 ) signifies an image, and it indicates the rel-ative strength, position, orientation, and width of that image. Roughsketches of some of these images along with the terms signifying them aregiven for two recording geometries: (a) DRBH, the geometry is basicallythe normal off-axis holography and the additional background wave origi-nates from a hole in the plane of the object; and (b) SRBH.

are located in the same direction as the faithful ones (thosefor which k > 1, but I = 1 and m = 1). This mechanism wasshown" 2 to be responsible for the halolike noise around thefaithful images reconstructed from nonlinearly recordedsingle-reference-beam holograms [see Figs. 2(b)-2(d)]. It hasalso been established that an amplification of the faithfulimages in SRBH is usually associated with stronger back-ground noise. However, it will be shown that in DRBH (andMRBH) it is possible to amplify the faithful images withoutsignificantly changing the level of the background noise.

Another interesting feature of the nonlinear double-refer-ence-beam recording is due to the simultaneous interactionof the object wave with both of the reference waves. Thisinteraction is represented by the last term in (12), that is,R11xsXS2, and is divided into two parts [see (18)]. Note thatthis interaction is present only when nonlinearity is present;that is, for a linear hologram where v = 2, Rnxsixs2 = 0, as isexpected. An important role played by Rnxsxs2 is due to thefirst part of it which shows the production of faithful images.Depending on the value of v, there are several terms thatproduce these faithful images, that is, those terms for whichI I - m I = 1 and k = 1. For example, for v = 4 there are twofirst-order terms, that is, I = 1 and m = 2, and 1 = 2 and m

= 1 with k = 1. In this example, (v = 4), there are not anyterms with noise contribution to these faithful images due toRnxsi.xs2 (i.e., no terms with k > 1 when I1-m I = 1). There-fore, if the nonlinear characteristic could be approximatedwith a second-order polynomial in exposure (which wouldmean v = 2 and 4), noiseless images of the object may be re-constructed in regions determined by factors (2-yi - y2) and(#Yl -

2 Y2)-

The second part of Rnx.,XS2 does not produce linear imagesbut only distorted higher-order images, in the regions deter-mined by the factors (1y1 + mY2 ).

To have a better grasp of the physical interpretations of thedifferent terms involved in R0, an illustrative example is givenin Fig. 1(a). In this example v is 6. One of the reference wavesis coming through a small opening beside the object. Thisopening and the object that is a diffuser in the shape ofnumber 2 are on a plane parallel to the plane of the recordingmedium (lensless Fourier transform recording geometry).The spatial radian frequency of the interference of this ref-erence wave and some point of the object on the same hori-zontal line is referred to as a2. The interference of the otherreference wave with the same point on the object is chosen toproduce fringes with the spatial frequency a,, but this one inthe vertical direction Thus the images of this point due to theindividual reference waves will be located at coordinates (72,0)

and (O,yl). Figure 1(a) shows several images that have beensketched approximately to scale. The images would be pro-duced by such terms as RN(Ar) cos(a 2Ar - 0) of Rn-s 2;RN(Ar) exp[-j(a1Ar - 0)] of Rnx,1 ; RN(Ar) expl-j[(al -2a 2)/Ar + O)11, RN(Ar) expl-j[(2a, - a 2)/Ar - O]1, andRN(Ar) expl-j[(2a, - 3a 2) Ar - 01} of the first part ofRnxs.XS2; R 2 (Ar) exp(-i1[(a, + a 2)Ar - 20]1 of the second part

(a)

(b) (d)

FIG. 2. Reconstructed images of nonlinear SRBH: (a) 0th harmonic; (b)1st and 2nd harmonics; (c) 2nd harmonic enlarged and more in focus; and(d) 1st and 2nd harmonics of a hologram recorded with severe nonlin-earity.

838 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980 Hossein Ghandeharian 838

of RnxslXS2; exp[-jl(ai - a2 )Ar] of R2 x,2 for 1 = 1 and 2 (notindicated on the figure); and the mean transmittance T2. Forrough sketches of such terms as R' (Ar) of Rnxn and R' (Ar)exp[-j(2aAr - 20)] of Rnxsl see Fig. 1(b).

It is perhaps of interest to refer to the fact that the auto-correlation function for the transmittance of a SRB hologrammay be obtained from (12) by considering it a special case ofDRBH in which the magnitude of one of the reference beams,say s2, is approached zero. In that case, hklm is zero for m> 1, since

mmŽ11 m=0

Therefore, the only nonzero terms in (12) are T2, Rnx,, andRnxsl, that is,

Ro= T2 + Rnxn + Rnxsl,

with

av!(Sp /f 2)'1 2 Fi[(k + I - v)/2; 1 + 1;1-hkl = 1!( 0y2)(k-,)/22v+1-kF[1 - (k + I- - /2]

(34)

(35)

For the purpose of a partial comparison of the imagingprocess in the two cases of SRBH versus DRBH, the illus-tration of Fig. 1(b) is given.

IV. SUPPRESSION OF INTERMODULATIONDISTORTION OF FIRST-ORDER IMAGES

The main purpose of this section is to compare the firstharmonic images in the two cases of SRBH versus DRBH.This comparison will show that the addition of a second ref-erence beam in the recording of a hologram is bound to im-prove the contrast of the faithful images compared with thosethat would have been obtained using one reference beam only.To simplify the matter and to present the central ideas andresults in a more tractable fashion, the nonlinearity is assumedto have a half-wave vth law characteristic with v = 4. Notethat nonlinear effects are present only for v > 4, and that inpractice when the t0 - E curve is approximated by a polyno-mial in exposure, the most significant effects are due to theterms up to the quadratic term (i.e., v is to be taken 2 and4).

It has been shown that the first harmonics can be split upinto two main parts, one responsible for the faithful images[where in (16) and (17) k = 1 and I or/and m = 1)], and theother the background halos (where k > 1 but I or/ and m = 1).In the following the former is often referred to as "signal" andthe latter as "noise." As usual, it is desired to increase thesignal-to-noise ratio (to improve the contrast) while keepingthe depth of modulation (fringe visibility) high.

In SRBH, high fringe visibility is attainable when the objectand reference irradiances at the film have roughly equalstrengths. But increasing the irradiance of the object isusually the cause of more pronounced nonlinear effects in-cluding an increase in the background noise which could be-come troublesome [see Figs. 2(b)-2(d)]. In DRBH there arethree beams interfering, and a rough definition for fringevisibility may be given, for example, by assuming the combi-nation of the object wave and one of the reference waves onthe recording medium as one wave front and the other refer-

ence wave as the interfering second wave front. Therefore,a high fringe visibility will be obtained if the total strength ofthe object wave and one of the reference waves on the holo-gram plane is roughly equal to the strength of the other ref-erence wave. In any case, to keep the background noise lowthe object wave ought to be kept low. In SRBH this meansa low fringe visibility, while it does not have to be so in DRBH.For example, if one of the reference waves is K times strongerthan the object wave, the second reference wave should beroughly K + 1 times stronger than the object wave to satisfythe high fringe visibility condition.

To be more specific, for SRBH, the part representing thesignal is given by [see (16), (22), and (35), with v = 4]

25a 2 S2(a 2 )2 lF2(-1;2;-S2/o2 )RN(Ar) cos(aAr - 0) (36)

and the part representing the background noise is given by

24 a 2S2R3(Ar) cos(aAr - 0) (37)

(which also shows that if the t0 - E curve is approximated bya second-degree polynomial in exposure, the background noisehas a width three times that of the faithful images).

In DRBH with v = 4 there will be four sets of faithful images(two contributed by Rnx0 l and R0 X12, and the other two byRnxslXS2). These images are, respectively, presented by [see(16)-(26)]

25 a 2 S1(U 2 )2 [iFi(-1;2;-S2/a 2 ) + S/IU2]2

X RN(Ar) cos(a 1Ar-0), (contributed by Rnxs), (38)

25

a 2S2(U

2)

2[iFi(-1;1; -S1/f

2) + S2/(21

2)]

2

X RN(Ar) cos(a 2Ar-0), (contributed by RnXJ2) (39)

and

23

a2(S2)(S1) 2

RN(Ar) cos[(2a - a2)Ar - 0] (40)

and

23a 2(S2)2(S2)RN(Ar) cos[(al - 2a 2)Ar + 0](contributed by Rn,1xsX). (41)

The background halos for the first two sets of faithful imagesare, respectively, given by

and

24a 2 Sl2R3 (Ar) cos(ai1 Ar - 0)

24 a2 S2R3(Ar) cos(a 2Ar - 0).

(42)

(43)

Since v is assumed to be 4, there is no background noise as-sociated with the last two sets of images.

As can be easily seen from a comparison of (38) and (42)with (36) and (37), if the same ratio S1/C 2 is used in bothSRBH and DRBH, the background halos of the correspondingfaithful images will be of the same strength in both cases, butthe corresponding faithful image in DRBH is amplified up toan additive factor of

2(S2/a 2) 1F 1(-1;2; -S2/U2 ) + (S2/U2)2

= (S2/U2)[2 + S2/U2 + S2/U2]. (44)

Furthermore, note that this amplification factor is propor-tional to both (S2/U2) and (S2/U 2), while the correspondinghalolike noise is only proportional to (S2/U2), but independent

839 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980 Hossein Ghandeharian 839

(a)

(b)

(d)(c)

FIG. 3. Reconstructed images of nonlinear DRBH; S» >> S2 and a2

. Im-ages are mainly due to: (a) Rnxs, (centered around the optical axis); (b) Rnsand R,,.,,"2; (c) lower part of (b) enlarged and more in focus; it consists ofa faithful image due to Rn and the 2nd harmonic of Rnx5 1; and (d) upperpart of (b) enlarged and more in focus.

of (Sl/a 2) (under the assumption of v = 4). That is, the powerof the signal can be increased, by increasing (S2/a 2), inde-pendently of the level of the background noise. [Note that,strictly speaking, this is true only for v = 4; when v > 4 thebackground halo contributed by Rn,81 will be dependent on(S1/a 2 ), too, but the effects in most practical cases are not verysignificant due to, among other things, the further amplifi-cation of the faithful images caused by higher-order non-linearities.] Therefore, the addition of a second referencebeam in the process of recording the hologram not only am-plifies the faithful images without affecting the halo level(image contrast improvement), but also provides us with acontrol factor over this amplification. For example, one maykeep the background noise below some acceptable level, byadjusting the ratio (S2/a 2 ), while having control over theamount of the inherent amplification of the signal due to thepresence of the second reference wave by adjusting (S2/a2 ).The same arguments apply to the other set of images, with theroles of S2 and S2 interchanged. When S2 = S2 both sets ofimages due to Rn,81 and RnxS2 are of the same irradiance. Inthis latter case, the strengths of the sets of faithful images dueto Rnxsx11 2 are also the same.

V. EXPERIMENTAL RESULTS

A partial insight into the above-mentioned theoretical ex-pectations may be gained by considering the following ob-

servations. Figure 3 shows the images reconstructed from ahologram for which S» >> S' and a2. The object was a diffuserin the shape of number 2, and the extra reference wave wasprovided through an opening beside it. Since S2 >> S2, theimages due to Rnxs, are much stronger than those due to RnxS2[cf. Figs. 3(b)-3(d) with 3(a)]. For the same reason, one wouldexpect, from a comparison of (40) with (41), that out of thefaithful images due to Rnxslxs2 the one centered at coordinates(--y2,2-yl) to be the strongest. This was undoubtedly the case,as is manifested in Figs. 3(b) and 3(c).

Next, we consider a case in which S2 S S2 + a2 and S2 wasslightly larger than a2. Images were sharp and easily ob-servable (an indication of high fringe visibility). Nonlineareffects were also more pronounced, although a2 was not yetstrong enough to produce strong background noise. Since S2is now comparable with S2, the images due to Rnxsl and Rnxs2are closer in strengths. The image formed in the center of Fig.4(b) is due to the first part of Rnxsixs2 with I = m = 1 and k= 2,4,. . .,v-2. That, is

v-2hk1/2V0

(k 2 [(k/2)!]222 k I R'(Ar) cos(ai - a 2)Ar, (45)

which is a shifted version of the ambiguity term Rnxn of (15).This shifted ambiguity function is producing a stronger image[at coordinates (-Y2,-Y1)] than the original one produced byRnxn (centered at the optical axis), since for this hologram hk1 1> hkoo [see (15), (18), and (25)]. A comparison of the centralregions of Figs. 4(a) and 4(b) clearly confirms this expectation.

(a)

FIG. 4. Reconstructed images ofnonlinear DRBH; S S + a

2,

and S2 is slightly larger than a-2.

Images are mainly due to: (a)

Rnxs2 (centered around the optical

axis); (b) Rn,, and Rnxsxs 2; and (c)

Rnxs,,s2 and the 2nd harmonic ofRnxsl.

(b)

(C)

840 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980 Hossein Ghandeharian 840

The inverted faithful image in the left-hand side of Fig. 4(c)signifies the effects of the nonlinear terms for which v > 6 (e.g.,a polynomial approximations of ta - E curve must at least bea third-order polynomial in exposure). The right-hand sideof Fig. 4(c) shows a faithful image due to Rnx8XS1 2 and thesecond harmonic of Rlxsl. Further experimental results aregiven in Sec. VI.

VI. HOLOGRAMS MADE WITH MORE THAN TWOREFERENCE BEAMS

When there are more than two reference waves present inthe recording of the hologram, equations similar to those of(12)-(18) could be written, with new coefficients hk, 1 2 ..

given as

hka1 2. -p (v + 1) C wk 1 lI1 (wSi)hh112- 2iwj Jc+

(46)

The intensity in each term will be proportional to h2 . Theterms that are of concern here are those for which one of thesubscripts l, is 1 but the other are 0, while k is 1 for a faithfulimage and k > 3 for its background halolike noise; e.g., one ofthe faithful images will be proportional to ho10 . o while theterms contributing to its background noise each is propor-tional to h21 . 0; k = 3,5,. - 1. The approximate solu-tions for these coefficients may be obtained employing amethod introduced by Gyi16 and later used by Shaft12 to an-alyze the passage of signal plus noise through a vth law non-linearity.

Without loss of generality, the images due to Rnxs, will betaken as the ones of interest. The approximate solutions forhkloo .0 (k = 1,3,. . . v-1) are obtained using the approxi-

(b)

(a)

(d)

FIG. 6. Images of a nonlinear triple-reference-beam hologram. Exceptfor the addition of a third reference wave, that was provided through a holebeside the object, the arrangement was the same as for Fig. 5. Figure 6(c)shows that nonlinear intermodulation of two (or more) reference beams andthe object wave can produce noiseless images [cf. Fig. 3(c)].

mate equality

io(sco) - exp 22-)

in (46). That is,

hk 100.. .0

(v + 1) c cok- lII(cSl) exp at c) dco,J+2

where

2t= 2 + E -S2Zp=2 2

(47)

(48)

(49)

that is, all the reference waves except the first are lumpedtogether with the object wave.

The integral (48) is the same as the integral in (28) with I-= 1. Following Davenport and Root in Ref. 3 p. 303,

ar(v + 1)(S'/2 U2)1/2

hkl ... o0 Pr[1 - (1 + k - v)/2](o.2/2)(kv)/2

X F1 (1 + k - P; 2; 2 s)(c)(a)

FIG. 5. Images of a nonlinear double-reference-beam hologram. Carelesspositioning of the reference beams have caused the overlap of some of thenewly produced intermodulation products with the main faithful ones.

Hence, taking v = 4 as before, the faithful images will be rep-resented by

841 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980

(b)

(c)

(50)

2 2

X I12(COS2) ... It, (wS,) exp 0-'Co �d co.2w

Hossein Ghandeharian 841

FIG. 7. Another view of the reconstructed images of a triple-reference-beam hologram.

252S(2 r2),F,2(-1;2 1S/2 f2

X RN(Ar) cos(aAr - 0) (51)

and the part representing the background noise is given by

24

a2S'Rk(Ar) cos(aAr - 0), (52)

that is, equal to that of SRBH or DRBH [cf. (37) and (42)], and

independent of all the other reference waves, in agreement

with our earlier result obtained for DRBH. It could also be

shown that when P = 2, that is, when only two reference beams

are present, (51) which has been obtained through an ap-

proximate method, is equal to (38), that was obtained earlier

by an exact evaluation of the integrals involved.

Looking back at (49), (51), and (52) one sees that by in-

creasing the number of reference waves the amount of the

consequent amplification of the faithful images against their

almost constant background noise and the degree of freedom

over this amplification will be increased. Of course, the

number of reference waves that could be accommodated

without introducing some complications is not unlimited.

Among these complications is one due to the increase of the

intermodulation products with the increase in the number of

the reference waves. These newly produced harmonics could

be disturbing if some of them formed images in the same re-

gions as the desirable faithful images. Examples of this type

of problem are given in Figs. 5-7 where the relative position

of two reference beams were not properly arranged.

Vll. SUMMARY

By a direct application of "characteristic function" method

of communication theory, expressions were evaluated for thetransmittance autocorrelation functions of multiple-refer-ence-beam holograms of diffuse objects. These expressionscan be directly used to acquire useful information about theirradiance distribution over the image plane. Proper use ofadditional reference beam(s), during the recording step of the

hologram, in conjunction with the nonlinear nature of therecording media, can improve the contrast of the recon-structed faithful images.

ACKNOWLEDGMENTS

This research was supported by the National ResearchCouncil of Canada. The encouragement and assistance fromthe author's Ph.D. supervisor, Professor Michael P. Beddoes,is also gratefully acknowledged.

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842 J. Opt. Soc. Am., Vol. 70, No. 7, July 1980 Hossein Ghandeharian 842