An Analytical and Experimental Study ofNonlinearities in Hologram Recording

A. Kozma, G. W. Jull, and K. 0. Hill

A theoretical analysis of the effect of recording nonlinearities upon the image reconstructed from a holo-gram made of a diffuse object is presented. Extensive experimental evidence which supports this theoryis also given. In particular, it is shown that the magnitude of the nonlinearity noise can be calculatedknowing the shape of the amplitude transmittance-exposure (T - E) curve, the bias transmittance(Tb), and the ratio of the reference beam to the object beam intensity (K). It is further shown that fordiffuse objects the shape of the nonlinearity noise distribution can be calculated from the shape of theobject.

1. Introduction

Several studies of the effects of film nonlinearities inholography have been presented recently.'- 6 In thepresent paper, we approach this problem from a differ-ent point of view and derive a theory which predicts theeffect of the recording nonlinearities upon the recon-structed image of a diffuse object. Experimentalevidence is also presented.

The results given here are similar to those given byGoodman and Knight.4 However, in their work theFourier transform technique for analyzing nonlineari-ties is used, whereas here we shall use a power seriestechnique given previously.6

We derive the theory under the assumptions that theamplitude transmittance of the recording film is real, thefilm emulsion is thin, the object is small and planar, andthat the film is perfect in the sense that the modulationtransfer function is unity for all spatial frequencies.

II. Preliminary Considerations

According to the Fresnel-Kirchhoff formula, thecomplex amplitude, a(x) expj0(x), of the light scatteredto a hologram by a small planar object can be written as

A. Kozma was with the Institute of Science and Technologyof the University of Michigan when this work was done, and isnow with the Electro-Optics Center, Radiation, Inc., P. 0. Box1084, Ann Arbor, Michigan 48106; the other two authors arewith the Communications Research Centre, Department ofCommunications, Shirley Bay, Ottawa 2, Ontario.

Received 8 July 1969.

a(x)ejo(x) = fjf fO(u)ejR(xu)d,u, (1)

where R(x,u) = [(x - u)2 + (y - v)2]/2z, z is the dis-tance from the hologram to the object, x = x,y are thecoordinates of the hologram plane, 0(u) is the complexamplitude of the object in the u = u,v plane, and Ao isthe area of the object.

Let 0(u) be made up of a deterministic complexamplitude ao(u) and a random phase part expj*(u).For the case of a transparent object backed by a groundglass, ao(u) represents the transmittance of the object,and expjr(u) represents the phase shifts introduced bythe ground glass. For a reflective object, expj*(u)represents the phase shifts due to many small imperfec-tions on the surface of the object.

The types of objects described above are generallycalled diffuse objects. It is reasonable to assume thatfor this type of object, I(u) is a random variable which,when reduced modulo 27r, has a probability densityfunction which is uniformly distributed between 0 and27r. Further, it is reasonable to assume that F(ul) andF(U2) are independent random variables for any twopoints of the object. This assumption implies that therandom-like structure of the ground glass or the imper-fections of the surface of a reflective object are smallwhen compared to the spatial variations of the deter-ministic part of the transmittance or reflectance of theobject.

Thus, using the property of independence, we canwrite that

Efei[T(ul) -'P(U2) = 1; U1 = U2= 0; otherwise,

(2)

where E(x) denotes the average value of x.As a consequence of the first assumption, it is not

March 1970 / Vol. 9, No. 3 / APPLIED OPTICS 721

difficult to show that a(x)expj0(x) is a complex randomprocess which has real and imaginary componentswhich are each gaussian and independent of eachother.7 Further, we can determine the joint probabilitydensity function of a(x,) and a(x2) and from this densityfunction find that 9

E[an(x,)a-'(X2)] = (2,2)(n+m)/2 r (n + 2) r (m + 2)

X F (2' 2' ' 4o4) (3)

where o-2 = E[a2(x) ]/2, r(.) is the gamma function,F(a,b,c; z) is the hypergeometric function, and

O(x1 ,x2 ) = E{a(x 1 )a(x2)expj[0(xi) - (X2 )l} (4)

is the correlation function of the complex amplitudea(x) expj0(x).

Using Eq. (1), we can write 0 in terms of the complexamplitude of the object. Thus,

expj(k/2z)[(X1' - X2 2) + (yI2 - Y2 2)] P

(XZ)2 j. fu

X f ,fdu2a(u)ao*(u2)EJ expj[T(u) - (u,)] }Ao

X exp - j(k/z)(x~u, - X2U2 + yv - y2v 2)

X expj(k/2z)(uI2 - U22 + V1

2- V22). (5)

Applying Eq. (2) to Eq. (5), we have

J|(XIX2)J2 = (Xz)2 j Jdulao(u)i2

X exp - j(k/z)[(x - X2)U + (yi - 2)V]| . (6)

upon the Ta - E curve and the beam ratio K.6The function B(x) is the normalized deviation from

the average value of the intensity produced at the holo-gram by the object, or

B(x) = a2(x)/E[a2(x)] } - 1. (9)

Physically, the function B is produced by mutual inter-ference between pairs of object points and the effect ofmultiplying the linear transmittance by a polynomial inB is to produce intermodulation products in the holo-gram transmittance which, in turn, cause a distortion oran intermodulation product noise, in the reconstructedimage.

If the hologram, after chemical processing, is il-luminated by the wave a, expjkfr, which is a replica ofthe original reference wave, the complex amplitude ofthe wave which will reconstruct the virtual image is,from Eq. (8),

Ud(X) = aD E C- B'(x). a() eio(x),n =0O 10 1

(10)

where

D = (K)1siC 1o/(K + 1). (11)

This can be divided into a signal wave (the wave whichwill produce an undistorted virtual image) and into anoise wave (the wave associated with the intermodula-tion products which will produce a distorted virtualimage). Thus,

(12)S(x) = a[D/(2)K] a(x)eJO(X),

N(x) = [ .CB(x) Dara(x) eio(x)Lne1 CIO (2)1c

(13)

Thus, we see that the correlation function of the com-plex amplitude, a(x) expjO(x), can be written in terms ofa Fourier transform of the intensity distribution of theobject. This is a result also obtained by Goodman.10

With the above preliminary considerations in mind,we can proceed to derive the formulae which describethe effect of the recording film nonlinearities on the re-constructed image.

111. Average Noise Due to Nonlinearities

The film nonlinearities change the ideal linear trans-mittance of the first-order portion of the hologram,given by the expression

T(x) = [2(K)I/(K + 1)]s,[l + B(x)]lI2 cos[kRr - (x)], (7)

to a distorted transmittance'

Td(x) = 2(K) SIC°0 CC1 B ](x)K +1 __ CIO

X [1 + B(x)]l1 2 cos[kRr - (x)]. (8)

Here, K is the beam ratio; the ratio of the exposure dueto the reference beam to the average exposure due to theobject. s is the slope of the linear portion of thenormalized amplitude transmittance vs exposure (Ta -

E) curve," Rr is the distance from the reference sourceto the hologram and C,,/Co are constants which depend

where S(x) and N(x) are the signal and noise waves, re-spectively. We can write the signal and noise waves interms of the complex amplitude of the object, by sub-stituting Eq. (1) in Eq. (10). Thus,

Ud(X) = D.i, j ffk ao(u)eiF(u)ejkR(x;u)du

+ [f C Bn(x) eikz J' f a(u)eIt(u)eikR(x;u)du}.

(14)

Thus, using the Fresnel-Kirchhoff formula, the complexamplitude at a reconstructed virtual image point i isfound to be

U(ui) = - ejk J f Ud(x)e -k2(x;ui)dx, (15)

where A, is the area of the hologram. Using Eq. (14),we have that

U(ui) = - Da, [S(ui) + N((u))],

where

S,(ui) = fA, f du fA f dxao(u)ei`(u)

(16)

722 APPLIED OPTICS / Vol. 9, No. 3 / March 1970

X exp[j(k/2z)(u2 - Ui + V2 -Vi2)]

X exp{ -j(k/z)[x(u - u) + y(v - vi)]} (17)

and

N.(ui) = X jdu X dxao(u)ei'(u) C B(x)L. At, n= CIO

X exp[j(k/2z)(u2 - Ui 2 + v2-vi2)

X exp{-j(k/z)[x(u-ui) + y(v-vi)I}. (18)The values S(u,) and N,(u,) are proportional to thecomplex amplitudes of the signal and noise, respec-tively, at an image point i.

The average intensity, (I)i is given by E[U(ui)U*(u,) ] or, using Eq. (16), we have

(I)i = D {E[Sv(ui)SV*(ui)] + E[S,(ui)Nv*(ui)](Xz)' 2 o'

+ E[S*(ui)NV(ui)] + E[N,(ui)N*(ui)]

or we can write that

(l)i = D22 [I(ui) + I(ui) + I"(ui)] (19)(Xz)'2a'

where Ir = ar2 is the intensity due to the reconstructionbeam and

I = E(SS,*), (20)

= E(SN,,*) + E(S,*Nv), (21)

In" = E(NvNv*). (22)

IV. The Average Intensity of the UndistortedSignal

The quantity I is proportional to the average intens-ity of the undistorted reconstructed virtual image.Using Eq. (17), we can write I as

1(u) = Lf Fduf fdU' f dxff dx'ao(u)ao*(u')JAo J Ao J At J AtS

X E{expj[l(u) - F(u')]} exp[j(k/2z)(u2 - u'2 + v -v12

X exp{ -j(k/z)[x(u - u) + y(v - vi) - x'(u' - u)

-y'(v'-vi)]}. (23)

Applying Eq. (2) to Eq. (23), we have that

I(ui) = Ao du dx J ('dx'lao(u)2JAoJ JAt . Ao J

X exp{-j(k/z)[(x - x')(u - ui) + (y - y')(v - v)]}. (24)

Performing the integration over x and x', Eq.becomes

(24)

1(ui) = Ao(LrLA)2 j J dulao(u)12 inc2 2z (u - u)

X sinc2 (v - vi), (25)2z

where L, and L, are the sides of the rectangular holo-gram aperture in the x and y direction, respectively, andsinex = sinx/x. We can also write Eq. (25) as

I(ui) = AO(L.L,)2JaO(Ui)J2*h(ui), (26)

where h(ui) = sinc2 (kL.u,/2z) sinc2 (kLvvi/2z). Thus,we see that the average intensity of the undistortedimage is proportional to a filtered version of the inten-sity of the object, as we would expect.

V. Interference Between the Virtual Image andthe Intermodulation Products

I,' is proportional to the average intensity of theinterference between the undistorted virtual image andthe distorted image produced by the intermodulationproducts.

Using Eqs. (17) and (18), that quantity can be writ-ten as

I,'(ju) = fA f du f du' fA f dx f f dx'

ao~~~~u E C. -Ix'X aO(u)ao*(u/) Elexpj[T(u) -(u')]} E n = n B(xC )

X exp[j(k/2z)(u2 - u2 + v2 -v2)] exp{ -j(k/z)[x(u - ui)

+ y(v - vi) - x'(u' - u) - y'(v' - vi)} + c.c., (27)

where c.c. means complex conjugate of the previousterm.

Applying Eq. (2) to the above, we obtain

In'(ui) = Ao f . dx fr f dx'f L f dulao(u)12

X E[ C' Bn(x')] exp{-j(k/z)[(x - x')(u - ui)n= CO

+ (y - y')(v - v)]} + c.c. (28)

Using Eq. (9) and (3) in Eq. (28) and intergrating over xand x', we have that

In'(ui) = 2Ao(L:L,)2Ca| f dufao(u)|2h(u - ui, v - vi) (29)

or

I.'(ui) = 2Ao(LxLv)2alao(ui)12*h(ui), (30)

where

n! C-n= E 57 -Jnp

n = p = (n - p)! C,,(31)

From this expression, we see that the average inten-sity of the noise due to the interference between the un-distorted image and the intermodulation products isdirectly proportional to a filtered version of the objectintensity. Thus, this term can be called an object in-tensity-dependent noise.

VI. Self-interference of the IntermodulationProducts

Consider the term I,,". This term is proportional tothe average intensity of the self-interference of theimage produced by the intermodulation products.Using Eq. (18), I,," can be written as

March 1970 / Vol. 9, No. 3 / APPLIED OPTICS 723

Int(ui) = ( I du X f du' F f dx Fr dx'ao(u)

X a(u') E(exp{j[hT(u) - 'I(u')] 1) E E (C )2n=l m=i -c+_ )2

X Bn(x)Bm(x')] explj(k/2z)(u2 - 2 ± 2-V2)

X exp{ -j(k/z)[x(u - ui) + y(v - v) - x'(u' - ui)- y'(v' -Vi -

Applying Eq. (2) to this expression yields

(32)

In"(ui) = A f f dulao(u)12 f f dx f fi dx'

X E (C O)2 (X'(X)

X expj(k/z)[(x - x')(ui - u) + (y - y')(Vi - v)]}. (33)

In Eq. (33), consider the expression giving the ex-pected value of the product of the two polynomials.Using Eq. (9), we can write this expression as

E, E C 2 1Bn(x)Bm(x')In=l m=l (Clo)2

- E E i ( (x _ 1((Cio)2 20.2 2a i1)j] (34)Thus, using the relation given in Eq. (3), we have that

E[ZIn= 1

E C BnCm(x)B-()rn-i1 (CIO)2 I

+ x 1 x'(,) + b(x, I )2 +4o4 4a-4

where the coefficients of this expression are

n c~~~(L...i (Z EC (1)n+ n!(a1 E CIO(l~~ (n! _)

n= p=0C1o (n - p)!

n = ( E Cn (n! (-p)(-p +=1 P CIO )~ (n -p)! 2 1)

(35)

(36)

(37)

(38)

By comparing Eqs. (31) and (36) we can note that fi =a 2

If we substitute Eq. (35) in Eq. (33), we obtain

I.'(ui) = Ao f f dula,(u))2 f j' dx f f dx'

X , + ±Y tO(X,X')12 ± 8[It(x,x)I212±X ( 4+e 4 L 4y4]

X exp{-j(k/z)[(x-x')(u-ui) + (y-y')(v-vi)]}. (39)

An expression for the magnitude squared of the corre-lation function, J0(x,x')J2 , is given by Eq. (6) in termsof the object intensity function, ao(u) 2. Thus, usingEq. (6) in Eq. (39) and integrating over x and x', we ob-

tain the expression

Inl(ui) = Ao(L.Lo)2 [i3 LO f dujao(u)l'h(u - i, v - v)

± C J f dulao(u) ao(u-ui, v-

*Iao(u - Ui, v -vi)2*h(u -ui, v -vi)

C2 O f dula.(u)12ao(u -Ui, V - i)12*lao(U - Ui, V - i)12

*Iao(u - ui, v - i)12*Iao(u - ui, v - vi)12

*h(u-u,v-vi) ... ] (40)

where

C = [f f dulao(u)2]2.In the above equation, the * (pentagon) denotes a cor-relation while the *, as usual, denotes convolution.

For K about four or larger, the constants Cln/C1obecome such that we can neglect and , and thus wecan simplify Eq. (40) as follows:

A)= °(LCv)27 fj dulao(u)2ao(u - ui, v -Vi)2

*Iao(u - ui, v - vi)12*h(u - ui, v - vi). (41)

The term laoI2*lao 2*h is a filtered version of the auto-correlation function of the intensity distribution of theobject. This quantity will closely resembled ao 2* ao 2and will occupy about the same area, namely, about twicethe area of the object. From Eq. (41), we see that thenoise due to the self-interference of the distorted imageis computed by positioning the center of the function('ao 2*ao 2) at the object point in question, us, and thentaking the product of the object intensity and the auto-correlation function of the object intensity and integrat-ing over the overlapped area of these two functions.Thus, we see that the noise term Int, is an integral overthe area of the overlap of the two functions and can benonzero even if the particular object point u, has zerointensity.

VII. Average Intensity of the ReconstructedImage

The average intensity of the reconstructed image,(1)i, can be found by substituting Eqs. (26), (30), and(41) in Eq. (19). Thus, the average intensity of thereconstructed image at point i, is

(I) = { ao(u)J ± 2 alao(ui)l2 + yd(ui)} *h(ui).2a2 (XZ)4

(42)

The first term of Eq. (42) is the signal; i.e., the undis-torted reconstructed image intensity at point i. Thesecond term is an object intensity-dependent noise dueto interference between the undistorted image and thedistorted image. The third term, which is due to theself-interference of the distorted image, is an object

724 APPLIED OPTICS / Vol. 9, No. 3 / March 1970

LU

Lv

V

4

* - Lu - I

(a)

LL ao4

-hV

u= 2Lu/3 12LV .

(B)(A)

(b)

Fig. 1. Bar pattern object. (a) Shape of pattern. (b) In-tensity autocorrelation function along a central horizontal sec-

tion, and along two vertical sections through (A) and (B).

form of the object intensity), and on the film transfercharacteristics and beam ratio through the values Cin/Cio contained in the constants a and y, and is inde-pendent of other considerations such as object andhologram size.

Vil. Computation of S/N

A. Example 1

Suppose ao(u) 2 is given by the distribution shown inFig. 1(a) i.e., ao(u)2 = ao2 in the shaded region and iszero otherwise. From this intensity distribution we cancompute the autocorrelation of the intensity. Thisfunction is shown in Fig. (b).

For Kodak 649F plates, developed for 5 min in D-19at 20'C, and forK = 4, Tb = 0.5; a = 0.036 and y 0.04(Figs. 6 and 7). Using these values, we can compute thenoise and thus determine the average intensity of the re-constructed image. This is shown in Fig. 2 for a sectionof the reconstructed object along the u axis (v = 0).We see from this figure that the intensity-dependentnoise and the shape-dependent noise both add to thesignal in the region of the reconstruction where the ob-ject intensity is nonzero; but in regions where the ob-ject intensity is zero, only the shape-dependent noisecontributes to the average intensity.

The shape-dependent noise extends out to 3 L0 /2,three times the width of the original object. The reasonfor this can be explained by examining Eq. (43), whichgives the distribution of the shape-dependent noise.This equation is a convolution of the object intensitywith the autocorrelation of the object intensity. Thus,since the autocorrelation is twice the size of the object,the convolution of this function with the object intensitywill produce a function three times the size of the object.

shape-dependent noise; i.e., a noise not directly de-pendent on the intensity of the original object at pointi, as is the intensity-dependent noise, but dependent ona weighted average over the object-intensity, object-intensity-autocorrelation product. The weighted aver-age is given by

d(w) = c [f f dujao(u)|2 lao(u- U, v-

*Iao(u - ui, v - )I2. (43)

We can find the average signal-to-noise ratio by dividingthe undistorted reconstructed object intensity at point iby the sum of the intensity-dependent and the shape-dependent noise intensities, or

SIN = Iao(ui)2*h(ui)[2.Jao(ui)J2 + yd(ui)]*h(ui) (44)

Thus, we see that the average signal-to-noise ratiodepends on the object intensity at the point in question,on the form of the intensity distribution of the object(since the shape-dependent noise is a function of the

SIGNAL-DEPENDENTNOISE

SIGNAL

18 r64

-I.O+ 2a+627Y64

1.0 + 2a

-3Lu -Lu -Lu 0 Lu Lu 3Lu2 2 2 2L . .. i1

-I

Fig. 2. Calculated image intensity levels for bar pattern object.Object shape-dependent noise (1); object intensity-dependent

noise (2); and signal (3).

March 1970 / Vol. 9, No. 3 / APPLIED OPTICS 725

Hi -a 1 - ( | \ Ad -4 > u

r - ;>ouI

L.r 3 1

B. Example 2

As a second example, we take the intensity of theobject, ao(u)2, to be the distribution shown in Fig. 3.The object is a 36-point regular array of points with theninth and twenty-first points missing. Point 21 islocated at uf = -L,/10, v = L,/10. Suppose we wantto find the noise at a position corresponding to point 21in the reconstructed image. Since ao(u) 2 = 0 at thisposition there is no intensity-dependent noise. Thenoise at this point is obtained by integrating over theoverlap of the object-intensity and the filtered nor-malized object-intensity-autocorrelation function whichis a 121 point regular array. Thus, for y - 0.04, thepeak noise is, I,," = 0.022a2 (-16.5 dB down from thesignal).

The distribution of the noise around the twenty-firstposition will be

Inl(ui) = 0.022a2 sinC2 [i (Ui + 1)] sinc2[ (Vi --

for a rectangular hologram.Suppose now we want to find the shape-dependent

noise at ui = L./10, v = 7 L,/10, a point just outsidethe reconstructed image. Again, we integrate over theoverlap. Thus, I,, = 0.0122a2 (-19dB down from thesignal).

It is clear, of course, that there is no noise except inthe close vicinity of the mesh points of the object-in-tensity, object-intensity-autocorrelation function, sincethese are the only places where these functions over-lap.

IX. Experimental Studies

A. Procedure

Experimental studies were conducted to determinewhether the foregoing theory is adequate to describe

31

25

Iv32 33 34

I S26 27 S

28

Lu

* S35 36

* *4 a2 8(U-U30,V-V30)29 30 (typical)

Lv * I * * S19 20 21 22 23 24

- _______ - - _________ ~- - - - - - -- 0

* * * * * S - -13 14 15 16 7

Lv

* * * .S7 8 9 10 11 12

I 2 3 4 5 6

Fig. 3. Regular 16 X 16 array object, with points 9 and 21missing.

M

REFERENCEBEAM

HOLOGRAM

M w LASER

M

,OBJECT

CAMERA ]

Fig. 4. Hologram recording system for nonlinearity noisestudy. The objects (2.5 cm X 2.5 cm) were located 53 cmin front of hologram recording plane (M-mirror, BS-beamsplit-

ter).

observed-nonlinearity recording noise. The objectsused were all simple geometrical shapes on photographictransparencies which were backed by randomly-scat-tering glass plates. Measurements confirmed that theobjects were diffuse, as defined in Sec. II. The shapesof the objects were (a) a square, (b) a step wedge, and(c) a regular array of point sources. The calculationof the intensity-autocorrelation function for these ob-jects is relatively straightforward.

Holograms were made with the recording system il-lustrated in Fig. 4. The objects subtended smallangles at the hologram recording plane, and the refer-ence beam illuminated the hologram plane at a rela-tively small angle (11°). Thus, the holograms could beconsidered to be recorded on a thin emulsion whichresponded equally to all spatial frequencies. Record-ings were made for reference beam-object beam ratios,K, from 1 to 30, and bias transmittances, Tb, from 0.3 to0.7. The holograms were recorded on Kodak 649Fspectroscopic plates (backed), with 0.6 32 8-,g radiationfrom an He-Ne laser. The plates were developed for 5min in D19B developer at 20'C.

The reconstructed images were scanned using a OO-upinhole and a photomultiplier. Calculations showedthat intensity measurements were accurate to within4 1 dB.

B. Calculated Noise Coefficients

A normalized amplitude transmittance-exposure(Ta - E) curve, appropriate for our data, was con-structed and used to calculate nonlinearity noise co-efficients. This curve is shown as (a) of Fig. 5 and issimilar to curves reported by others for 649F plates,e.g., Goodman and Knight. 4 The full curve represents

726 APPLIED OPTICS / Vol. 9, No. 3 / March 1970

the least mean square fit of a ninth order polynomial tothe experimental data, and is valid over a normalizedexposure range -1 < E < 7. A second Ta - E curve(measured earlier and reported elsewhere6 ) is shown as(b) in Fig. 5. The polynomial representing this lattercurve is valid over the range -1 < E < 3. Differencesin calculated noise coefficients for these two Ta - Ecurves are discussed in Sec. X.

Figures 6-8 present calculated values of the noise co-efficients (a3), (y), and (), respectively. These co-efficients are defined by Eqs. (31) and (36)-(38).Comparison of these coefficients provides an immediateassessment of the relative importance of various types ofnonlinearity noise which the foregoing theory predicts.We can see that for K > 5, first-order shape-dependentnoise (determined by the value of y) will be more im-portant than other predicted types of noise. Forexample, for holograms made with K = 5, Tb = 0.5, theintensity-dependent noise (determined by the value ofa) is predicted to be about 0.047. This amount of noiseresults in a change in reconstructed image intensity ofonly about 0.4 dB (which is less than our experimentalaccuracy). Also for these holograms the second-ordershape-dependent noise coefficient () is 6.5 X 10-4, or2.4% of y.

C. Bias Transmittance

The bias transmittance (Tbm) of each hologram usedin the study was measured and the correct bias trans-mittance, Tb, was calculated using the equation shownbelow 6' 9 :

Tb = Tb - [2K/(K + 1)2]s2To,

, 1,-3-

Qt

I -

I. 4

Z

0 -

X o:F-

-6

Tb=0.3

-Tb= 0.5

'Tb = 0.6

205 10 15.K

T =0.7

25

-x.10.-

aI

I.

2

7

30

Fig. 6. Calculated values of the noise coefficients a and ,, wherea = 13a (a determines the magnitude of the object intensity

dependent noise).

(45)

where

To = [1 + 3K S4 1OK286+ 35K3 s8

(K + 1)2 s2 (K + 1)4 s 2 (K + 1)6 S2

(46)

t F(E)

+.5I-,1.4

T3t.2

, t

b-.3t4I-

1.0-

0.1-

I

IL

8

uzZ

0.0I-

-I -0.5 0 0.5 .0 1.5 2.0 2.5 3.0 3.5 4.0

NORMALIZED EXPOSURE E -I

Fig. 5. Normalized T - E curves: (a) measured and used inpresent study; (b) reported earlier.6 (649F plate, developedin D-19 for 5 min.) E is the normalized exposure, T(E) =Tb + F(E), where Tb is the bias transmittance (in this case 0.5).Crosses on curve (a) indicate points of experimental fit to the

9

polynomial: F(E) = E sEl, where s = -0.65742, 2 = 0.24179,v=i

s3 = 0.15012, 84 = -0.17341, 5 = 0.069183, 8 = -0.014483,87 = 0.0016806, 8 = -0.000-10149, and s = 0.0000024587.

Tb = 0.3

Tb 0.5

Tb= 0.6

Tb 0.7

I1111 1,,, 1,,, 1'1111 I I IrI I 5 10 1 20 25 30

K

Fig. 7. Calculated values of the noise coefficient -y (-y determinesthe magnitude of the first-order object shape-dependent noise).

March 1970 / Vol. 9, No. 3 / APPLIED OPTICS 727

................ l ll

xl

u.uul1,........................._._ _. I

-1.10-2

I

. . .

T,,1

lxi

a

0

4

Tb' 0.3

Tb~ 0.5

Tb' 0.7Tb 06

RI 5

1 1 1 1 I I I I j I I I I I I I I-0 5 10 15 20 25 30

K

Fig. 8. Calculated values of the noise coefficient a ( determinesthe magnitude of the second-order object shape-dependent

noise).

In the above equations, 82, s4,... are the coefficients ofthe terms E2, E4,. . . in the power series representationof the normalized experimental T - E curve of therecording film.

The term [2K/(K + 1)2]s2 To is a shift in the bias re-sulting from the even portion of the Ta - E curve aboutthe bias point, and must be taken into account in themeasurement of the nonlinearity noise.

D. SquareTransparency ImagesThe reconstructed image of the square transparency

from a hologram with K = 1 is illustrated in Fig. 9.Here, the shape-dependent noise was distributed overthe image in the form of circular glow, with a bell-shaped intensity cross section. (The object had anopaque square central section, to allow observation ofthe noise within the image. The photographs of the re-construction were taken on high-contrast film and aseries of increasing exposure times provided a series ofdecreasing intensity thresholds, as shown.)

Figure 10, presents photometric scans along a centralcross section of reconstructed images from hologramsmade with (a) K = 1.8, Tb = 0.35; and (b) K = 9.7,Tb = 0.64. Also shown are calculated distributions ofshape-dependent noise, ya(u) [Eqs. (42) and (43) ].The value of y was determined from Fig. (7) and a(u) wascalculated from the intensity-autocorrelation functionof this object. The calculated and observed noise levelsdiffered by 4 dB for the K = 1.8 hologram image.

Comparison of (a) and (b) of Fig. 10 shows the differ-ence in observed noise distributions between K = 1.8and K = 9.7 hologram images. For the K = 1.8 holo-gram image, the shape-dependent noise with its bell-shaped intensity cross section was the dominant noisearising from the hologram recording process. For theK = 9.7 hologram image, the observed noise had amuch flatter distribution. For this case the calculatedshape-dependent noise, IN, was added to the observedbackground noise, IB, to give a resultant noise, IN + IB-Figure 10 shows that (Iv + IB) is only about 1 dB belowthe observed noise.

E. StepWedgeTransparency Images

The step wedge transparency used in the study con-sisted of a group of adjacent rectangles of decreasingtransmittance. A photometric scan across the imagefrom a hologram made with K = 33 and Tb = 0.49 isshown in Fig. 11. I.... ] indicate the intensities forvarious steps of the object. The nonlinearity noiselevels are relatively low for this value of K, and the uni-form background noise level (IB) was 27 dB below thepeak signal level 1.

Photometric scans across images from hologramsmade with the same value of K (K = 3.8) and differentvalues of Tb (Tb = 0.47 and 0.72) are shown in Fig. 12.

-9 dB -12 dB

THRESHOLD BELOW SIGNAL MAXIMUM (dB)

-1 dB

Fig. 9. Reconstructed images of the square transparency, il-lustrating object shape-dependent noise (K = 1 hologram).The thresholds are determined with reference to the exposurefor which the image was just visible, and are expressed as decibels(e.g., doubling the exposure time drops the threshold of visibility

by 3 dB for the high contrast film used).

I I 1 I i I ' I ' I I -5"2 -L .1'2 "I I' * 'l 3L"'2 -L -LQ O Lap L ,L

Fig. 10. Photometric scans across the square transparencyimages. Holograms made with (a) K = 1.8, Tb = 0.35; and(b) K = 9.7, Tb = 0.64. IN, calculated nonlinearity noise levels(- -); IB, observed background noise level (... ); observedintensity levels (-); (IN + B) (0-0-0); and calculated

(Is + IN + IB) (- ).728 APPLIED OPTICS / Vol. 9, No. 3 / March 1970

.T-1�I

"j

�JJ

�11 -II

I 111I.-,I I

H|- 5

- 10.

>. IS-

zIu

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-0 I I ' I ' I ' I ' I ' I I I I T -O.8L -0.6L -0.4L -0.2L 0 0.2L 0.4L 0.6L 0.8L

DISTANCE

Fig. 11. Photometric scan across the step wedge image. Holo-gram made with K = 33, Tb = 0.49. The five steps of the ob-ject intensity are indicated I1.. .I5, each with width of 0.2 L.IN, calculated nonlinearity noise level (- -- ); IB observedbackground noise level ( ... ); observed intensity levels (-);

calculated signal and noise intensity levels (- -* ).

The importance of Tb in determining the level of shape-dependent noise can be seen by comparing parts (a) and(b) of Fig. 12. With an increase in Tb from 0.47 to0.72: (a) The level of shape-dependent noise droppedby 2 dB to 3 dB. (b) The contrast of the low intensitystep wedge sections, 14 and Is, increased, due to the de-crease in noise.

The measured levels of shape-dependent noise agreewith calculated values within 1 dB (Tb = 0.47 image)and 3 dB (Tb = 0.72 image). If allowance is made forthe background noise, the measured levels agree withthe calculated total noise to within 1 dB.

F. ArrayTransparency Images

The array transparency was 16 X 16 regular array ofpoint sources with one-half of the points missing. Theposition of the point sources was chosen with the aid ofa random number table. The noise levels in the recon-struction of this half-filled array was calculated to beabout 3 dB below the noise levels for the filled arraydescribed in Sec. VIII. Photographs of the recon-structed images of the array (from K = 1 and K = 29holograms), and the array intensity-autocorrelationfunction are shown in Fig. 13. The reconstructedimage from the K = 1 hologram illustrates the distribu-tion of noise for array images.

Figure 14 presents photometric measurements of sig-nal and noise levels through a central cross section ofthe array images. (The particular cross section scannedis indicated in Fig. 13.) Shape-dependent noise wasobserved only at array points, and the shape of the noiseintensity envelope was similar to that observed for thesquare transparency. This arises from the similarity ofthe envelopes of their respective intensity-autocorrela-tion functions. (The array object transparency also

// L I..INI . . .. .-DEL ,4L -DXL - DXL O.RL DEL DEL

70- i\I

-DEL -04L -X2L 0 X2L DL DEL. DEL

Fig. 12. Photometric scans across the step wedge images.Holograms made with (a) K = 3.8, Tb = 0.47; and (b) K =

3.8, Tb = 0.72. (Captions on Figs. 10 and 11 apply.)

(a) (b) (c)

Fig. 13. Characteristics of a regular half-filled, 16 X 16 array. (a) Image of the array (hologram made with K = 29), showing cen-

tral cross section scanned. (b) Image of the array (hologram made with K = 1.0, Tb = 0.35), illustrating object shape-dependentnoise. (c) Intensity autocorrelation function.

March 1970 / Vol. 9, No. 3 / APPLIED OPTICS 729

-

I I A _ II'l II,-I I I "

30-

-L. -05L, 0 05L L.

ISTANCE

Fig. 14. Photometric scans across central cross sections of arrayimages. Holograms made with (a) K = 1, Tb = 0.6; and (b)K = 4, Tb = 0.7. Filled array points (-); missing array points(-). IN, and IN2, calculated nonlinearity noise levels at centerand edge of array. In, background noise between array points.

transmitted light between array points. This accountsfor the observation of background intensity, IB, be-tween the array points, which was about 25 dB belowthe signal level, Is.)

The calculated shape-dependent noise levels areshown for the center (Ni,) and edge (2) of the arrayimages. The calculated noise levels agreed with ob-served noise levels within 3 dB and 1 dB for the K = 1and K = 4 hologram images, respectively.

G. Comparison of Observed with Predicted NoiseCoefficients

The foregoing examples, and others obtained in thisstudy, provide a body of data which was used to com-pare values of the predicted noise coefficient, y, withvalues determined from measurements. The com-parison is presented in Fig. 15, for data in four ranges ofTb. The agreement between observed and predictedvalues of y varies with K:

(a) For 1 < K < 4 hologram images, the observednoise coefficients were consistently lower than predicted.(The observed values in the ranges 0.4 < Tb < 0.55 and0.55 < TB < 0.65 were the order of 3-4 dB below pre-dicted values.)

(b) For 4 < K < 10 hologram images, the observedvalues were in close agreement with predicted values.Differences of 3 dB were observed when the levels of thenonlinearity noise was the order of that for the back-ground noise. These discrepancies were reduced to theorder of 1 dB when the background noise levels wereallowed for (as in Figs. 10 and 12).

(c) For K > 10 hologram images of the square andstep wedge objects, the levels of nonlinearity noise weremuch less than those for the background noise. There-fore, no comparison of observations and predictions was

possible. The characteristics of the background noiseleads us to suggest that film grain noise was moreimportant than nonlinearity noise for our K > 10 data.Calculated values of film grain noise levels, based onmeasurements by Burckhardt," 3 are in general agree-ment with our observed background noise levels.

X. Discussion and Conclusions

Two major predictions of the theory have been testedby the experimental study. The first is that an accu-rate measure of the magnitude of the noise levels can becalculated, knowing the shape of the Ta - E curve, Kand Tb. The second is that for diffuse objects the shapeof the noise distribution can be predicted from a knowl-edge of the intensity distribution of the object. Thesepredictions are also implicit in other studies of non-linearity noise4 and has been confirmed in the presentexperimental work.

The question now arises as to the application of thequantitative calculations to other similar, but notidentical, T - E curves. A related question is theaccuracy of noise prediction if y' = (Cl/C 0 )2, the firstterm in the expansion of -y, had been used to approxi-mate y, rather than the first eight terms, Yc, used in thisstudy. A comparison of predicted value of y are pre-sented in Fig. 16 for Tb = 0.5. It is seen that yc and y'differ by less than 2 dB for K > 4. The use of y' tocalculate the noise levels fails to predict the rapid rise ofnoise for K < 4, but is otherwise a good, although em-pirical approximation. (Indeed, the experimental datado not exhibit the very high increase in noise levels forK < 4 which is predicted by ye.) We conclude that(Cli/Clo)l provides a satisfactory first-order estimate

00I-

(a) WbI

TbO07

s 10 15 20 25 I I I I I 2 2 I -5 10 15 .o 25

Fig. 15. Comparison of observed with predicted noise coeffi-cient, y. (a) Predictions for Tb = 0.3 and 0.6 compared withdata taken with 0.2 < Tb < 0.4 (X) and 0.55 < Tb < 0.65 (0).Observed values of -y for which the background noise B, wasallowed for (). (b) Predictions for Tb = 0.5 and 0.7 comparedwith data taken with 0.45 < Tb < 0.55 () and 0.65 < Tb <

0.75 (X).

730 APPLIED OPTICS / Vol. 9, No. 3 / March 1970

-L, -05L, D 05 L, L.DISTANCE

.0_: .0-

01-

.01-

01.

IL-9III

I

The authors wish to acknowledge the helpful supportand encouragement of Dennis Gabor. A part of thiswork was carried out in his laboratories at ImperialCollege, London in 1967, while A. Kozma and G. W. Jullwere on sabbatical leave. The continued interest ofJ. Brown of Imperial College is also acknowledged.R. T. Lowry contributed to this study by assis-tance in the experimental program. A. Kozma wassupported by an IST Fellowship from the Institute ofScience and Technology of the University of Michigan,while at Imperial College, and G. W. Jull was an Aca-demic Visitor to Imperial College from the DefenceResearch Board of Canada.

References1. A. Kozma, J. Opt. Soc. Amer. 56,428 (1966).

2. A. A. Friesem and J. S. Zelenka, Appl. Opt. 6, 1755 (1967).

- '

I I I I l I I I I I I I I0 5 10 15

K

I . . I . . . . l20 25 30 3. J. M. J. Tokarski, Appl. Opt. 7, 989 (1968).

Fig. 16. Comparison of predicted values of noise coefficient,y. c represents the first eight terms in the expansion of -y.

- = (Clo/Co0)2 is the first term in the expansion of y (valid forboth -1 < E < 7 and -1 < E < 3). -y = (Cn' /Co1')2 is thefirst term in the expansion for appropriate for the T - E

curve of Fig. 5(b).6

of -y for the range of K values (K > 4) normally used forholographic recording.

There still remains the question (which will occur tomany) of the validity of using the coefficients derivedfrom the Ta - E curve of Fig. 5(a), for predicting noiselevels in other experiments. Briefly, what is the sensi-tivity of the noise coefficients to change in Ta - E char-acteristics? We have attempted only one comparison,using the Ta - E curve presented earlier6 with the T, -E curve used in this study. These two Ta - E curves,shown in Fig. 5, differ in amplitude transmittance by upto 40.05 over the range -1 < E < 3. However, thecalculated values of Y,, wy', and 71, agree within 1.5 dBas we can see from Fig. 16.

We are therefore led to conclude that the calculatedvalues presented here can predict the noise withinseveral decibels for similar Ta - E curves. The self-consistency of our experimental data confirms this.The holograms used in the present study were made on649F plates possibly from different batches, and theTa - E curves for various holograms therefore cannot be considered to be identical. In spite of this, thevalues of the noise coefficient, y, determined from theexperiments were in agreement within several decibels.

4. J. W. Goodman and G. R. Knight, J. Opt. Soc. Amer. 58,1276 (1968).

5. 0. Bryngdal and A. Lohmann, J. Opt. Soc. Amer. 58, 1325(1968).

6. A. Kozma, Opt. Acta 15, 527 (1968).

7. D. Middleton, Introduction to Statistical CommunicationTheory (McGraw-Hill Book Company, New York, 1960),p. 356.

8. Ref. 7, p. 402.

9. A. Kozma, Ph.D. Thesis, University of London, 1968.

10. J. W. Goodman, Proc. IEEE 53,1688 (1965).

11. The normalized Ta - E curve is obtained from the ordinaryTa - E curve by shifting and scaling the exposure coordinateso that a normalized exposure E = 0 produces a desired biastransmittance Tb and a normalized exposure E = -1produces unity transmittance.

12. The area of the points was large enough to leave unaffectedthe distribution of the intensities in the spectrumw of therandomly-scattering plate which backed the transparency.

13. C. B. Burckhardt, Appl. Opt. 6, 1359 (1967).

March 1970 / Vol. 9, No. 3 / APPLIED OPTICS 731

1.0-

0.1-

0

11

LI

y

tII

0.01-

OPTICISTS

Henry S. Rowen, president of The Rand Corporationand newly elected director of Itek Corporation.

Kennett W. Patrick is general manager of the In-strument Division of the Perkin-Elmer Corporation.

Leonard Mandel University of Rochesterphotographed in 1968 by D. L. MacAdam.

Rudolf E. Thun of the Raytheon Company.

B. J. Tucker, director for research andengineering at EMR-Photoelectric.

732 APPLIED OPTICS / Vol. 9, No. 3 / March 1970