THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING REDUNDANCY IN COHERENT IMAGING SYSTEMS by Robert Warren Lewis A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Electrical Engineering January, 1973 IP-850

Doctoral Committee: Professor Emmett N. Leith, Chairman Professor Frederick J. Beutler Associate Professor William B. Ribbens Professor Thomas B. A. Senior Doctor Carl C. Aleksoff Lecturer Juris Upatnieks

ABSTRACT REDUNDANCY IN COHERENT IMAGING SYSTEMS by Robert Warren Lewis Chairman: Emmett N. Leith Diffuse coherent illumination has a wide spatial bandwidth with no dominant frequency components. This type of illumination makes a holographic recording redundant since any small part of the hologram can reconstruct the entire object field. Because of this redundancy property, a diffuse holographic recording is rather insensitive to scratches and other system artifacts. Unfortunately, diffuse coherent illumination results in a multiplicative noise granularity called laser speckle. The spectral density of laser speckle has a significant low frequency contribution which limits both image quality and resolution. Mixed-integration processing can utilize excess system bandwidth to suppress speckle. The statistical speckle fluctuation is reduced by adding uncorrelated samples noncoherently. This paper establishes an equivalence between noncoherent techniques which continuously sample a signal's spectrum and a simple irradiance filter. Most optical noise suppression techniques for coherent spatial filtering systems involve transverse displacement of the optical signal relative to the optical axis together with noncoherent addition. An effective moving grating method is presented for introducing redundancy into coherent spatial filtering systems.

For holographic systems, special redundancy modulation is needed to obtain a high quality image of a continuous-tone transparency. Redundancy modulation must satisfy two constraints. First, the modulation must be free of low frequency speckle. Second, the spectral density of the modulation must be bandlimited and must satisfy certain uniformity conditions. Numerical results obtained compare the characteristics of several functions which appear to satisfy these constraints. Periodic functions are the most promising class of redundancy modulation for the high density storage of continuous-tone transparencies. The class of speckle-free periodic modulation proposed by Gabor has been studied and new solutions with uniform spectral characteristics have been determined using numerical methods. Several methods exist for synthesizing this speckle-free modulation. Two indirect synthesis methods are presented which require a spatial filtering operation. The noise suppression performance of periodic functions has been analyzed in terms of a family of redundancy parameters. Redundancy modulation with good diffraction efficiency characteristics can be obtained by using a coarse lens array in tandem with a periodic grating solution. Experimental results demonstrate that the addition of a lens array can improve the noise suppression characteristics of this class of functions.

ACKNOWLEDGMENTS I wish to thank the members of my dissertation committee for their service. A special note of thanks is given to Professor Leith for his interest and helpful discussions while serving as Chairman. Professor Senior provided important advice during the early stages of my dissertation work. Juris Upatnieks, Dr. Aleksoff and several other members of the Radar and Optics Laboratory provided helpful comments during the course of my research. Special appreciation is expressed to my wife Jill for her encouragement, and for editing the manuscript. Thanks are due to Carole Newton for typing the dissertation. A major portion of this dissertation work was supported by NSF Grant GK-31474. A portion of the work was supported by the Air Force Avionics Laboratory, Air Force Systems Command, Wright-Patterson Air Force Base, under Contract No. F3361570-C-1433.

TABLE OF CONTENTS Page ACKNOWLEDGMENTS........................... ii LIST OF FIGURES............................. v LIST OF TABLES.............................viii LIST OF SYMBOLS.............................ix CHAPTER 1. Partially Coherent Processing Techniques to Reduce Laser Speckle................ 1 1. 1. Introduction............................ 1 1.2. Characteristics of Laser Speckle............. 2 1. 3. Mixed-Integration Processing................ 3 1. 3.1. Introduction..................... 3 1. 3. 2. Description of Mixed-Integration Processing....................... 3 1. 3. 3. Effect of Mixed Integration on the Variance/(Mean)2 of Laser Speckle....... 7 1. 3. 4. Image Contrast.................... 12 1. 3. 5. Mixed Integration Using a Moving Diffuser. 16 1. 4. Optical Noise Suppression in Optical Filtering..... 20 1. 4. 1. Introduction......................20 1. 4. 2. Reduction of the Spatial Source Coherence.. 21 1. 4. 3. Noise Suppression Using Moving Gratings.. 24 1. 5. Summary............................ 28 CHAPTER 2. Redundancy Modulation for Coherent Imaging Systems............................ 30 2. 1. Introduction....................... 30 2. 2. The Self-Imaging Characteristics of Gratings.....30 2. 3. Optical Noise Suppression Using Gratings........34 2. 4. Constraints for Redundancy Modulation.........38 2. 5. Quadratic Phase Modulation..................... 42 2. 5. 1. Characteristics.................. 42 2. 5. 2. Experimental Results...............55 2. 5. 3. Optical Noise Dispersion.............60 2.6. Summary........................... 63 iii

Page CHAPTER 3. Redundancy Requirements for High Density Coherent Storage Systems.................. 64 3. 1. Introduction......................... 64 3. 2. System Constraints..................... 64 3. 3. Bandlimited Periodic Redundancy Modulation.... 66 3. 4. Noise Suppression Analysis................ 70 3. 5. Summary............................ 82 CHAPTER 4. Periodic Modulation with Optimum Redundancy Characteristics.................... 83 4. 1. Introduction........................ 83 4.2. Gabor's Solutions...................... 85 4. 3. Two-Dimensional Redundancy Modulation........ 87 4. 4. General Solutions to Equations (6)......... 88 4. 5. Even Function Solutions to Equations (6)...... 89 4. 5. 1. Solutions where arg {a a- +l} = It/2.. 89 4. 5. 2. General Even Functionlsorutions.. 100 4. 5. 3. Solution Summary............... 103 4. 6. Synthesis of Periodic Modulation with Optimum Redundancy Characteristics............ 106 4. 7. Diffraction Efficiency Considerations....... 113 4. 8. Increasing the Redundancy of Periodic Modulation by Reducing the Speckle Frequency Requirements. 116 CHAPTER 5. Conclusions......................117 REFERENCES........ 120 APPENDIX I. General Solutions to Equations (6)........ 122 APPENDIX II. General Even Function Solutions for N = 4.. 126 APPENDIX III. General Even Function Solutions for N = 5.. 130 APPENDIX IV. Pinhole Array Characteristics......... 133 iv

LIST OF FIGURES Figure Page 1-1 Block Diagram for a Processor with Type B or C Mixed Integration................ 4 1-2 Intensity Transmittance of a Positive Transparency as a Function of the Image Irradiance... 13 1-3 Image Showing Laser Speckle with and without Mixed Integration.................... 14 1-4 Construction of a Light Source for MixedIntegration Processing.................. 16 1-5 Coherent Spatial Filtering System........... 21 1-6 Original Transparency..................22 1-7 Filtered Image........................23 1-8 Construction of an Array of Spatially Noncoherent Sources.................. 24 1-9 Generation of Multicarrier Illumination....... 25 1-10 Filtered Image Using the Illumination of Fig. 1-9. 26 2-1 Coherent Imaging System.............. 35 2-2 Image Plane C without Redundancy..........36 2-3 Image with Ground Glass Illumination........37 2-4 Image with Phase Grating q(x, y)...37 2-5 Phase for qA (x).43 2-6 Phase for qB(x).............43 2-7 Spectral Density of qA(x)................. 44 2-8 Redundancy Characteristics for qA(x)......47 2-9 Redundancy Characteristics for qB(x)......48 2-10 (a) I(x) = J(x)/max{J(x)} for Entry V of Table 2-1....................... 54 (b) I(x) for Entry W of Table 2-1.......... 54 V

Figure Page 2-11 (a) I(x) = J(x)/max{J(x)} for Entry Q of Table 2-2...................... 55 (b) I(x) for Entry R of Table 2-2........... 55 2-12 Coherent Imaging System, where Lens B has a Focal Length of 100 mm and an F-Number of 2. 3. Two wires each 0. 23 mm in Diameter were Introduced at Plane A................... 56 2-13 (a) Image with Vertical Wires at Plane A of Fig. 2-12..................... 56 (b) Image with Two Horizontal Wires at Plane A whose Vertical Positions are Indicated with Arrows.......................56 2-14 Irradiance Near the Back Focal Plane of Lens B. 57 2-15 Coherent Imaging System.................57 2-16 (a) Image with Uniform Illumination.........58 (b) Image with Quadratic Phase Modulation.... 58 2-17 Image of Fig. 2-16(a) with Artifact at Near Fourier Transform Plane. (a) No Redundancy, (b) Redundancy qA..59 2-18 (a) Irradiance at Fourier Transform Plane of Fig. 2-15 with q (x, y) Present.......59 (b) Spectral Density of qA for Q = 9.23.....59 2-19 Coherent System Showing the Demagnification of Artifact Noise...................... 60 2-20 (a) Irradiance at the Image Plane without qA+. 62 (b) Image Plane when qA+ is Present........ 62 3-1 Image of USAF Resolution Chart with SingleSideband Imaging for the Horizontal Coordinate.. 67 3-2 Coherent Imaging System................ 71 3-3 Image of Diffraction Pattern for a Small Wire Artifact (w = 2wrf and f = 16 cycles per mm). 74 4-1 Redundancy and Modulation Characteristics for N= 2............................95 4-2 Redundancy and Modulation Characteristics for N = 3..................*.**....95 4-3 Redundancy and Modulation Characteristics for N = 4........................96 vi

Figure Page 4-4 Redundancy and Modulation Characteristics for N = 5............................ 97 4-5 Redundancy and Modulation Characteristics for N = 6.............................. 98 4-6 Redundancy and Modulation Characteristics forN= 3............................101 4-7 Redundancy and Modulation Characteristics for N= 4............................. * 102 4-8 Spectral Density Characteristics............ 105 4-9 Phase Profiles for Entries B through H of Table 4-1 and the Point BR = 1 of Fig. 4-4(c)... 109 4-10 Irradiance at System Fourier Transform Plane.. 114 4-11 Irradiance at the Image Plane with Two Wires Introduced as Artifacts................. 115 IV-1 Plane Wave Incident on Grating q(x).......... 134 IV-2 Amplitude Transmittance................ 138 IV-3 Sinc Function.......................... 39 vii

LIST OF TABLES Table Page 2-1 Characteristics of J(x) and RM for qA(x)..... 51 2-2 Characteristics of J(x) and RM for qB(x)..... 53 4-1 Solution Summary..................... 104 IV-1 Pinhole Array Data.................... 141 viii

LIST OF SYMBOLS a n-th complex Fourier series coefficient n arg{ ~ } denotes the argument of a complex number bR free real parameter represented as BR in computer R subroutines BR subroutine variable equal to bR BW bandwidth circ(r) defined on page 17 d(x) dust diffraction pattern E the percent of the light energy within the system bandpass E. } expected value operator Eq. (N, k) k-th equation of Eqs. (6) on page 84 F F-number of lens f focal length of lens f reciprocal of the period of a grating 0 FT{* } Fourier transform operator G general redundancy parameter (see Condition B, page 41) g(x) final average image irradiance gI(x) impulse response of low-pass irradiance filter H frequency transfer function h(x) impulse response hI(x) impulse response of irradiance filter h (t) impulse response for slit at position v h(fx) Fourier transform of h(x) Ik modulus of the k-th Fourier series coefficient of periodic q(x) 12

IIM output irradiance of mixed integrator I(x) output irradiance of imaging system J1 first order Bessel function J(x) output of low-pass irradiance filter (see Condition A, page -39) k equal to (27r)/X L constant often used as upper bound for V (page 41) M irradiance modulation (see page 49) m mean value A m(v) weighting function dependent on scanning position m /U defined for Condition (A) on page 39 N positive integer n(fx) Fourier transform of n(x) P(x) phase of q(x) p(x, z) impulse response for free space propagation (see page 71) Pcoh PDF for coherent image irradiance PMI PDF for image irradiance after mixed integration PDF probability density function Q constant usually representing the space bandwidth parameter of a quadratic phase signal (see page 42) qA periodic quadratic phase modulation (page 42) gqB periodic quadratic phase modulation with continuous dP(x)/dx (page 43) q periodic q(x) after one order a is attenuated o q(x) complex amplitude transmittance representing redundancy modulation q(n/X) n-th complex Fourier series coefficient of periodic q(x) RM redundancy parameter for periodic redundancy modulation (page 46) rect(x) defined on page 31 RDC 1/(duty cycle)

S power spectrum s DC term of s(x) 0 s(x) input signal amplitude sinc(x) sin (irx) (rx) SIN signal-to-noise ratio u diameter of a dust particle V width of an elementary element of q(x) (page 40) Var { } variance operator W constant often used for bandwidth w equals 2irf o o w(x) impulse response of coherent processor X period of grating Z output amplitude for white complex Gaussian input r(N) gamma function 6(x) Dirac delta function e small positive constant X light wavelength cr standard deviation xi

1 PARTIALLY COHERENT PROCESSING TECHNIQUES TO REDUCE LASER SPECKLE 1. 1. INTRODUCTION Coherent imaging differs from noncoherent imaging systems in several important respects. A coherent system is linear in complex amplitude, while a noncoherent system can be described as a linear mapping of intensity or irradiance. The lensless recording of the complex amplitude of light afforded by holography can be exploited for signal processing, data storage, and many other applications. Unfortunately coherent imaging has certain characteristics which limit its utility in many applications. For example, if non-diffuse objects such as transparencies are imaged with a uniform beam of coherent light, system artifacts such as a small scratch, bubble, or dust particle introduce dark diffraction patterns at the image plane. For diffuse objects with many random scatterers in a resolution cell or when imaging transparencies backlighted with diffuse illumination from fine ground glass, small system artifacts no longer limit image quality. However, now superimposed on the average reflectivity of the object is a multiplicative "noise" granularity referred to in the literature as laser speckle [ 1]. It will be seen that laser speckle limits both image quality and system resolution. It will be demonstrated that excess system bandwidth together with a controlled reduction in coherence can suppress optical noise and decrease speckle modulation. 1

2 1. 2. CHARACTERISTICS OF LASER SPECKLE For diffuse objects imaged with spatial bandlimited coherent optical systems, a multiplicative "noise" granularity is superimposed on the average reflectivity or transmittance density of the image. This granularity, known as laser speckle, limits the resolution and image fidelity of such systems. Speckle can be modeled as white, complex Gaussian clutter with a zero mean [2]. Such a model is a good approximation if large numbers of scattering centers exist in a resolution cell, in which case linear filtering (such as low-pass filtering) results in a zero-mean, stationary, complex Gaussian process. The spatial power spectrum of the image amplitude is usually bandlimited and uniform. The fineness of the image speckle is determined by its autocorrelation width, which is approximately equal to the reciprocal of the bandwidth of the image signal amplitude. The (modulus) of the image amplitude has an exponential probability density function (PDF), (1/c) exp (-u/v), with domain 0 < u<oo. This PDF has a variance a and a mean m = a and thus a (mean) /variance of unity. The ratio m 2/a can be considered a signal-to-noise ratio which measures image fidelity. It will be seen in the next section that mixed integration can be used to increase m /a by decreasing the achievable system resolution. The image irradiance at the output of a mixed integrator is the sum of N identically distributed, uncorrelated randomi variables, each with an exponential distribution. 2 2 For this sum m /a = N, and the PDF is the gamma distribution wt1 doi u -u/(u/N) (I/N) r (N) uINNwith domain 0 _< u < oo.

3 1.3. MIXED-INTEGRATION PROCESSING 1. 3.1. INTRODUCTION Mixed integration is a method of processing a coherent image which reduces speckle granularity but also decreases the achievable resolution [2, 3]. This processing method is equivalent to introducing a system diversity parameter. The method usually involves processing portions of the signal spectrum separately and then noncoherently summing the energy densities of a family of images. The noncoherent summation reduces the statistical fluctuation of the speckle. An equivalence will be established between sampling portions of the signal's spatial spectrum and sampling the coherently processed image to accomplish the noncoherent addition. 1. 3.2. DESCRIPTION OF MIXED-INTEGRATION PROCESSING There exist several discrete and continuous types of mixed integration. Three types will now be considered. Case A: Discrete Sampling in the Frequency Plane Mixed integration can be introduced into an optical processor by processing N equal nonoverlapping portions of a signal's spectrum separately and then by summing N output image irradiances noncoherently. Let s(x) be the input signal and represent bandlimited white complex Gaussian speckle. Since the N sections of the spectrum do not overlap, mixedintegration processing performed in this way results in the noncoherent addition of N independent samples. The output resolution-cell width is N times the width that would be obtained with coherent processing.

4 Case B: Discrete Sampling at the Image Plane Instead of sampling the signal's spectrum to achieve N independent additions, a mixed-integration operation can be introduced as shown in Fig. 1-1. The signal s(x) is first coherently processed by means of a filter with impulse response w(x) and is then square-law envelope detected to yield an output irradiance If(x) 12. The output irradiance! f(x) 12 is filtered with a linear filter with an impulse hI(x) to yield a (sin ffx) final image irradiance g(x). Let w(x) be the irx impulse response of a coherent processor. Then if N-1 hI(x) = 6(x- n) n=O the resolution-cell width will increase by a factor N. Since sin (irx) the autocorrelation function for the speckle is (X) and ox the speckle has Gaussian statistics, the output irradiance for each point x of g(x) is the sum of the irradiances of N inde0 pendent coherent image points. SWft().12 I gxl..... Coherent Image Final Image Intensity Intensity Figure 1-1. Block Diagram for a Processor with Type B or C Mixed Integration

5 Case C: Continuous Sampling of the Signal Spectrum with a Scanning Slit Case C, which utilizes continuous sampling of the signal spectrum, can be represented by the same block diagram as that shown in Fig. 1-1. This type of mixed integration is implemented by scanning the signal spectrum with a slit, while integrating the output irradiance over one scan time. One scan time is the time it takes for the slit to pass completely across the spectral window. To simplify the derivation of the mixedintegration transfer function, let f(x) = s(x) * w(x) represent the coherently processed signal amplitude used as the input to a mixed integrator, where the symbol * denotes the convolution operator. Let h(x) be proportional to the Fourier transform of the scanning slit, and let mn(v) be a weighting function dependent on the scanning position v. Then the mixed integrator has the following form: f(x) is the input signal, and g(x) is the average output irradiance averaged over one scan time of the scanning aperture. Thus g(x).S a(x; v) I2m(v)dv -o0 where a(x; v) = f(x - t)h (t)dt -oO and where h (t) = the impulse response for the system when the scanning slit is centered at position v in the frequency plane. Hence, g(x) can be written as

00 g(x) = 55 f(x - t)f(x - T) j 2n & h (t)h*(T)Am(v) dvl dt dT -00 -00oo Let Q(t, T) = 2 r hv(t)h*(T)m(v) dv -00 For the mixed integrator being discussed, -ivt hv(t) = h(t)e and m(v) = constant Then, to within a constant of proportionality Q(t, T) = h(t)h(T)*6(7 - t) and g(x)= ~ If(x - t) 2h(t)2 dt 00 Thus, the output of a processor using Type C mixed integration is g(x) = s(x) *c w(x)12 9 h(x) 2

7 where s(x) is the amplitude of the input signal to the processor. For the discrete sampling method discussed in Case A, m(v) takes the form co jj 6(v- nD) n=-oo where D represents the width of the scanning slit. In summary, f(x) = s(x) * w(x) is the processor's output amplitude that has been coherently processed over the full input signal bandwidth. The mixed integrator that continuously samples the Fourier transform of the signal performs a simple low-pass video filtering operation on the output signal intensity of the form g(x) = Jf(x) 12 * h(x) where hi(x) ='h(x) |2 and h(x) is simply the Fourier transform of the scanning aperture, if the usual scale factors introduced by lenses are set equal to unity. 1.3. 3. EFFECT OF MIXE] INTEGRATION ON THE VARIANCE/(MEAN) OF LASER SPECKLE Mixed integration compresses the dynamic range of laser speckle. For complex Gaussian speckle, mixed integrators A and B reduce the variance/(mean) of the processed image irradiance from 1 to 1/N, if N independent samples are noncoherently added. Calculations in this section will show how mixed integrator C affects the variance/(mean) characteristics of complex Gaussian speckle.

8 Before determining the effect of mixed integrator C on the variance/(mean) of speckle, the power spectrum of complex Gaussian speckle must be determined. Let Z(x, y) be the output amplitude of an imaging system with a linear transfer function H(fx, fy) and an input which is a stationary white complex Gaussian random process with zero mean. Since the statistics of Z are also Gaussian, the following theorem [4] will be used to calculate the correlation function for IZ 2. If Z1' Z2' Z3, and Z4 are complex jointly Gaussian random variables with zero means, then E(Z 1Z2Z3Z4) = E(Z1Z2)E(Z3Z4) + E(Z1Z3)E(Z2Z4) + E(ZlZ4)E(Z2Z3) where E is the expected value operator. Let the irradiance of the imaging system be I(x, y)= Z(x, y) 12 when this theorem is used E[I(x, y)I(x + t1, y + t2)] = {E[I(x, y)]} 2 + E[ Z*(x + t1, y + t2)Z1(x, y)]E[Z*(x, Y)Zl(x + t1, Y+t2)] where t is a constant. Let the power spectrum of Z(x, y) be S (f, f ). Then the power spectrum S, for I(x, y) is Zx y Si(fx, f ): {IE[I(x, y)]'} 6(f, f )+ S(fx f ) * S*(-f, -f)

where 6 is the Dirac 6 function. Assume S(fx fy)= S(-f, -fy) = 4H(fx f )2 Then SI(f f) = E 2(I)6(fx f) + c(JH 2 I H 12 x'y x y 2 Let aI = variance of I, and pI = E(I) = mean of I. Then 0I = C H(fx f) * df dfy -VIf - _'2 = C[FT-(1H12)] at x = y= = C s (f f2dfxdf 2 2 Since PI = aI from Section 1.2 2 C= I LSS IH2 df dfjI -oo00 2 rIHI2 * IHI2 SI =I (fx f)+ (s H2 fdfY)d L10

Let the output intensity IM result from a mixed-integration operation with inipulse response hI(x, y). IM I(x, y) *hI(, y) The power spectrum for IM is 2 IM(fx' fy | I SI where HI FT(h ) Thus, the (mean) /variance ratio for IM is M HI(O., 0) 120 H(f' fy)2 df df 2 (1) I~c fx y2IH(fx fy)2 *Hl(fx' fY) ]df df -o00 A sample calculation for a Gaussian aperture derived from Eq. (1) is as follows: 2 2 -rf H(fl, f2) = e i=l then 2 2 _rf. HI2* 2IHI21 1 e 1 i=l -1 h=FT (H)

11 2 2 h(xl1 x2)= e e 1 i=l 2 2 Ihl%2 e 1 i=1 Let 2 hI(x1, x2) = I I e i=l Note that hI is scaled N. wider than I h 2 along the xi. I 1 coordinate. H 21 j2~e 7[ - e-T(Nifi) i=l when these expressions are substituted in Eq. (1) 2 o_ 2rf2 e df. (P i)1 1 a 2 002 2 - 7r(Nifi) - 7rfi e i=(1/2)e df T[ 0

12 Using the fact that Ae-1/2(X/C) dx = C h -o00 Then (P2 2= 22 N+1 N +1 asN N1,- (, N1N2(a2/2) 1. 1. 3.4. IMAGE CONTRAST The nonlinearity introduced by recording film or by a human observer affects the final image contrast, when an estimate of the average image irradiance is sought for a small image patch containing many resolution cells. Consider an image patch which has an average irradiance a. If there are many scatterers in a resolution cell, the PDF for the irradiance, w, at a single image point is 1 -w/u p (w) e Pcoh(w) a for a coherent image, and N-i PMI(W) (c/ N) r(N) (7) for an image processed by mixed integration, which noncoherently adds N independent samples. p h(w) and pMI(w)

13 are both zero for w < 0, and the mean of each distribution is 2 a. The variance-to-(mean) ratio is 1 for pcoh and 1/N for coh PNMf These probability density functions can be found in PMI" Ref. [5]. Let nT(w), shown in Fig. 1-2, represent the transfer function relating w to the intensity transmittance of a positive transparency used to record a processed image. Let C be a positive constant. Then the space-averaged irradiance W of the light transmitted by the image transparency is 00 coh _C 0pcoh(w)nT(w) dw for the coherent image, and 00 WMi = CS PMI(W)nT(w) dw for the image processed by means of mixed integration. nT(w) orb a Figure 1-2. Intensity Transmittance of a Positive Transparency as a Function of the Image Irradiance

Consider points w = oa and w = ab in Fig. 1-2. If the ava b erage irradiance of the patch is a, the mean a for PDF's Pcoh and PMI is set equal to'a and it is found that WMI > Wch. If Yb is the average irradiance, WMI < Wch An example of this characteristic is shown in Fig. 1-3, in which a resolution chart was coherently imaged with and without mixed-integration processing. Complex Gaussian speckle was introduced by placing a ground glass plate against the resolution chart. The pictures show that mixed-integration processing can improve contrast. (a) Coherent Image Without Mixed Integration Figure 1-3. Image Showing Laser Speckle With and Without Mixed Integration

1 5: vi: -If~~~~~~~~~~~~~ — (b) Image With Mixed Integration Figure 1-3. Image Showing Laser Speckle With and Without Mixed Integration (continued)

16 1. 3.5. MIXED INTEGRATION USING A MOVING DIFFUSER It was seen in Section 1. 3. 2 that mixed integration is equivalent to a noncoherent spatial filtering operation. A spatially noncoherent source can introduce this operation in holographic image reconstruction since a lateral shift in the reference beam source point causes a corresponding shift in the position of the image [ 2]. It will be seen in Section 1.4. 2 that reducing the source coherence helps suppress artifact noise. An efficient means of achieving a spatially noncoherent source using laser light will now be given. Consider the light distribution shown at Plane B of Fig. 1. 4. Position of Mask m(x, y) Coherent Collimated Beam Radius R Plane A Plane B Lens L H-dy -:,,, 1 Focal Length. -e 1-4. Construction of a Light Source for Mixed-Integration Processing

17 Let us examine the cross correlation of the light amplitude at two points in Plane B if the mask is moving. Let the coordinates of Plane A be the Cartesian coordinates x and y. If a mask, mi(x, y), is centered at point (ac,,3) of Plane A, then the light distribution at Plane B is -i2ir(f x+f y) u(f f ) = Ce Sicirc( m(x- yH- )e dxdy -00 where 2 x kd J2 y Xd 2 +y2 r= x + y FT[ m(x)] = m(fx) 1, rl circ r = c, otherwise Then 212 2 -iwt _____ _____ ___ u(f, f ) CR e-i x y f2 + f2 x y Lmn[f. f )e j

18 Defining i 2 7rR x f 2 1x..... C(ff ) CR2 J R f +f x y Let w(f f a, 3) C-i (ft f * m(fx, fy)e x(fx' fy L'x p When the mask m is moved to various positions during reconstruction or optical processing with the source formed at Plane B, it is necessary to determine the cross correlation R(fxl, fyl X2 f2) = S (fxl f y1 ~lfx2x 2 y l (a, 13)EAXB w* (fx 2' fy2',' ) da d3 where the integration is over the Cartesian product A X B, and A = B = (-oo, oo). Or A = N (spatial mask period along x coordinate) B = N (spatial mask period along y coordinate) if the mask is periodic in each coordinate, and N is a positive integer or N >> 1. It is necessary to determine how far apart source points must be for the cross correlation to be zero. By interchanging integrals and integrating first over A X B it is found that

19 00 R(fxl fyl; f2' f) = C( xl yl x2' y2 xl' yl -00 CT x + (fx2 fx l)' yl (fy2 -fyl B (fof ixl X' fyl - Ty )12 dto 1 dt Sincer, C(T X1 etyl) T for Tis + Tyl > 1/R, and the correlation coefficient R(fc f f fd xl' yl x2' y2' R(fx xl' yl )R(fx2' fy2 fx2' y2) is approximately zero if ( x2 -x fX + (fy2 - fyl) > This result shows that the intensities of the images produced by source points spaced more than 2kd/R apart on Plane B of Fig. 1.4 add together noncoherently to form the total output image irradiance. To achieve this noncoherent addition, the image irradiance must be time averaged over some time interval of length T. T is dependent on the time required for the integration over A X B. In practice a spinning lens array is a suitable choice for m, since lens arrays with rather narrow circ function spectral densities exist, and good light efficiency can be achieved with such a diffuser. Amplitude weighting at

20 Plane B can help control the source shape. Section 1. 4. 2 will show the optical noise suppression characteristics of this type of spatially noncoherent source. 1.4. OPTICAL NOISE SUPPRESSION IN OPTICAL FILTERING 1.4. 1. INTRODUCTION Unless special techniques are employed, optical noise limits the performance of coherent optical spatial filtering systems. Because the output of a spatial filtering system is usually square-law envelope detected, this type of system is amenable to noncoherent noise suppression techniques. Various methods have been used to reduce optical noise. One method which has proved effective in processing synthetic aperture radar data is to move the input and output films in synchronism [6]. Optical elements with rotational symmetry can be rotated during processing to reduce optical noise effects. In one technique the spatial carrier frequency of a filtering system is time varying and a spatial filter must track the transform plane motion [7]. If the spatial source coherence is reduced, it will be seen that optical noise can be dispersed and noncoherently added at the output plane. Unfortunately, decreasing the spatial coherence causes a loss of resolution at the Fourier transform plane of a filtering system. In Section 1. 4. 3 a moving grating technique for suppressing optical noise is presented. This technique does not reduce the resolution at either the image or the transform plane, and in general is one of the simplest and least expensive techniques to implement. This latter technique gives insight into the noise suppression techniques required for fully coherent systems which are considered in Chapter 2.

21 1.4.2. REDUCTION OF TEHE SPATIAI. SOURCE COHE RENCE An experiment was conducted which illustrates the fact that reducing the spatial coherence of a source helps suppress optical noise in spatial filtering systems. Figure 1-5 shows a Lens fy Lens Y2 Illumination B ns pnFi t f f,, Image Transparency Fourier Transform Plane Figure 1-5. Coherent Spatial Filtering System typical coherent spatial filtering system. In this example, a high-pass filter was created by placing a 1)C stop at the Fourier transform plane. The input transparency to be filtered in this example is shown in Fig. 1-6. Figure 1-7(a) shows the filtered output image when illumination B is a coherent collimated beam. It is seen that optical noise limits the output image quality. Instead of using a collimated coherent beam, consider illumination B shown in Fig. 1-8. Here a crossed 16 cycle per mm sinusoidal phase grating gl was used to generate 9 equal carriers. A spatially noncoherent source was created at Plane B using the technique described in Section 1. 3. 5. Since the DC

22 Figure 1-6o Original Transparency component of the transparency now forms illumination cfi' (fx,) 2 (f - 1B) cit e n=-i m=-l consisting of an array of circular dots each with diameter 1 mm at the Fourier transform plane, an array of DC stops is now required in order to high-pass filter the input transparency. The filtered output using this illumination is shown in Fig. 1-7(b), and clearly illustrates the fact that the system with reduced spatial coherence gives superior performance.

23 NNI (a) Single Coherent Collimated Beam (b) An Array of Spatially Noncoheren't Sources Figure 1-70 Filtered Image

24 x Laser I Illumination Beam l Plane!. Spinning I Lens Array g Figure 1-8. Construction of an Array of Spatially Noncoherent Sources 1. 4.3. NOISE SUPPRESSION USING MOVING GRATINGS' A moving grating technique for suppressing optical noise will now be presented. This method does not reduce the resolution at the frequency plane of a spatial filtering system. Consider Fig. 1-9 where a linear grating in2rf x gl(x) = a e 0 n= -oo is illuminated with a collimated beam of coherent light and is set in motion with constant velocity v in the x coordinate direction, If the temporal frequency of the beam is ft, the n-th order plane wave generated by the grating has temporal frequency ft + nvf. Redundancy can be introduced in an optical system such as Fig. 1-5 by using the above property to generate multiple carrier illumination B. Multicarrier illumination requires that an

25 i o k SX~t + nVfo Illumination Collimated - B Coherent Light ft I 1 t- nvf Figure 1-9. Generation of Multicarrier Illumination array of spatial filters be placed at the Fourier transform plane of Fig. 1-5. Each carrier generated by the moving grating has a distinct temporal frequency. With the proper choice of recording parameters, it is possible to noncoherently add the output signal energies from each spatial carrier. Consider the temporal frequency difference vfo for adjacent orders and let T equal the observation or exposure time. Then if Tvf >> 1 or Tvf is an integer, the output irradiance variation due to the presence of the grating is negligible since Tvf equals the number of cycles of translation of the grating. If more than one grating is used in tandem, care must be used to assure that each carrier has a distinct temporal frequency. A cross grating should be translated in a direction. that is not parallel to the lines of the grating; a direction forming a 30 degree angle with a grating line would suffice. Experimental results using moving grating illumination are given in Fig. 1-10. Here the filtering experiment of Section 1.4. 2

26 was repeated using the moving grating technique. In this experiment, the spatially noncoherent source at source Plane B of Fig. 1-8 was replaced with a point source. Figure 1-10(a) shows the output of the system with a crossed phase grating stationary in the system. Figure 1-10(b) shows the same output when the gratings were translated during the exposure time. It is seen that translating the gratings removes the fringes shown in Fig. 1-10(a). By comparing Figs. 1-7(a) and 1-10(b), it is seen that although only nine carriers were employed, the moving grating method was quite effective in suppressing optical noise. (a) Grating Stationary Figure 1-10. Filtered Image Using the Illumination of Figo 1 9

27 (b) Moving Gra-ing Figure 10. Filtered Imnage Using the Illumination of Fig. 1-9 (continued) Although it is permissible to use real amplitude gratings, phase gratings are more desirable to efficiently utilize the available light. The modulation provided by the moving grating technique can also be achieved by the generation of sound waves in a fluid gate to produce traveling wav:Tes which propagate across the optical channel. The moving grating technique is a simple practical means of achieving optical noise suppression. Since the gratings are translated during processing, the gratings' design requirements are reduced and the need for precise positioning of the gratings along the optical axis is usually eliminated. The moving grating method was presented at the 1972 Fall OSA meeting [8] Closely related work by Brandt was also presented at that meeting [9, 1010

28 1. 5. SUMMARY Diffuse coherent illumination has a wide spatial bandwidth with no dominant frequency components. This type of illumination makes a holographic recording redundant since any small part of the hologram can reconstruct the entire object field. This redundancy property makes a diffuse holographic recording rather insensitive to scratches and other system artifacts. Unfortunately, diffuse coherent illumination results in a multiplicative noise granularity called laser speckle. The spectral density of laser speckle has a significant low frequency contribution which limits both image quality and resolution. Mixedintegration processing can utilize excess system bandwidth to suppress speckle. The statistical speckle fluctuation is reduced by adding uncorrelated samples noncoherently. Synthetic aperture radars can use both angle and temporal diversity to obtain independent samples [3]. Temporal diversity can also be employed at optical wavelengths [ 1 1]. Optical artifact noise can be suppressed in coherent optical filtering systems by reducing the source coherence. It is thus advantageous that the source coherence not exceed the system requirements. Noncoherent techniques are permissible in coherent optical filtering systems since the output is usually square-law envelope detected. Most noise suppression techniques for such systems involve transverse displacement of the propagating signal relative to the noise scatterers together with noncoherent addition. Many of these techniques are costly and difficult to implement. Others place limits on processing speed and frequency plane resolution. A moving grating technique was presented which does not have the above limitations. This method can achieve noncoherent addition by translating a grating one grating period.

29 Most noncoherent noise suppression techniques are not useful for diffraction-limited holographic storage systems. The remaining chapters consider redundancy modulation for holographic systems. Mixed integration can be used to remove high frequency residual fringes which are introduced by periodic redundancy modulation. This is accomplished by using a spatially noncoherent source for image reconstruction.

2 REDUNDANCY MODULATION FOR COHERENT IMAGING SYSTEMS 2. 1. INTRODUCTION For the coherent optical processing systems considered in Chapter 1, transverse displacement of the propagating signal relative to the processing optics together with noncoherent addition was effective in suppressing optical noise. For fully coherent systems where noncoherent addition is not permitted, optical noise reduction is more difficult and special types of speckle-free redundancy modulation must be considered. In this chapter, redundancy modulation for imaging continuoustone transparency signals will be investigated. This class of signals is especially susceptible to optical noise because a continuous-tone signal has a dominant DC frequency component. Leith and Upatnieks showed that the self-imaging characteristics of phase gratings are useful for generating diffuse speckle-free illumination, and for coherently imaging a volume [ 12]. This characteristic is analyzed in the next section. 2.2. THE SELF-IMAGING CHARACTERISTIC OF GRATINGS It is well known that gratings have a self-imaging characteristic. This property is sometimes referred to as the Talbot effect [13]. Self-imaging will now be analyzed for a onedimensional grating. Consider the periodic grating function

31 rN ii2 rnx q(x) = a e n=-N with complex Fourier series coefficients X/2 i27rnx n X an X q(x) e dx -X/2 If coherent collimated light is incident normal to the grating surface, let q(x, z) be the complex light distribution a distance z from the grating. Since q(x) is 00 i27rf x q(x, 0) = q(f )e df -00 where f denotes spatial frequency, the free space transfer function H(f ) relating q(x, z) to q(x, 0) is [14] x i2f1rz 2 x X 2(l-(Xf H(fx) = recti e (2) where rect(x) - 1 if x < 1/2 0 otherwise. Since 1- (Xf)2 = 1 1 2 1 ) x 2' 8 xoe(3) H(fx) becomes

32 i27rz 2 12r i27rzXf X2 H(f ) = e e ect (4) if f is sufficiently small. Let f = 1/X be the fundamental freX 2 quency of the function q(x) and let 2zXf = J. Then 0 N /! \ N 2 i27rnf x J iJ7rn o q 2f-x, a e e 2Xf n=-N i27rz where complex constants such as e are ignored. If J = 1, q(x, z) becomes ( 1\ N i i27rnf(x 2f+1) 2Xf 2 ) -N an e = (x + X o) Thus q(x, 0) has a self-imaging distance L= 1 2f 2f 0o where the same diffraction pattern forms, but is shifted transversely a distance X/2 which is one-half the period of q. From Eq. (3) the self-imaging property requires that thp approximate inequality 2 rz X(Nfo 14 q 8 20 or 3 4 5X Nf ) 0)

33 be satisfied. If the bandwidth BW equals 2Nf, the above 0 inequality becomes 16 z < 3 4 5X (BW) The approximate number of self-imaging planes is thus 0. 4 2f2N ). For a grating with a 100 cycle per millimeter bandwidth and a 0. 1 millimeter period, the number of selfimaging planes is approximately 16 if X, equals 0. 6328 microns. In practice, the flatness and uniformity of a grating often limit the number of image planes. Equation (4) can be used to determine the approximate focus sensitivity of self-imaging planes. A self-imaging plane is in focus if 27r I|zIXNf < 0.05 75 or L! 0. 05 N 2 where the self-imaging distance L equals (2kf2 ). In terms of the bandwidth BW = 2Nf this inequality becomes ]IzI 1 1 10 X(BW)2

34 This is comparable with the approximate focal tolerance, Lz ~ where BW 1 2X(BW)2 X(F-number) given for a point target in Ref. [ 15]. A computer program was written which uses Eq. (2) to calculate q(x, z) 12 given N i2 rnf x q(x, 0) = ane n= - n n=-N and z. The program can be used to determine the characteristics of a particular function q without using the above approximations. 2.3. OPTICAL NOISE SUPPRESSION USING GRATINGS Leith and Upatnieks considered the problem of coherently imaging a three-dimensional object s(x, y, z) such as a transparent volume with various scattering centers; where the goal was to image a desired plane of the object without incurring the noise due to the scattering points of other planes or system artifacts [12]. They formulated criteria for image improvement and demonstrated how the self-imaging characteristic of gratings can be used to suppress the effects of system noise occurring throughout the system. Consider the optical system in Fig. 2-1. If redundancy modulation were introduced by placing a mask with complex amplitude transmittance q(x) against s(x) at Plane B, noise introduced by system artifacts between Planes B and C could be suppressed. However, this would have no effect on noise effects from artifacts to the left of Plane B. A periodic mask with some

35 desirable complex amplitude transmittance q(x) can be placed at Plane A, and Plane B can be a self-imaging plane of the grating. Now noise effects due to grating defects and system artifacts occurring both to the left and right of Plane B can be attenuated. Leith and Upatnieks demonstrated that quasidiffusers such as phase gratings offered an improvement over imagery formed when the illumination is produced by an ordinary diffuser such as ground glass [12, 16]. Collimated Coherent Imaging Light I System Plane A Plane B Image Plane C Figure 2-1. Coherent Imaging System The results of an experiment which utilizes the selfimaging characteristics of phase gratings for optical noise suppression are now presented. The bandlimited coherent imaging system shown in Fig. 2-1 was used to image a continuous-tone transparency. Sinusoidal crossed phase gratings generated the redundancy modulation function inw x)( inw o w 1 where f = o and {a} = {i, 1, i}. The bandwidth of ~o 27 n'n - n = - 1I the system was 4f - e for each of the Cartesian coordinates x 0

36 y at the image Plane CO The positive constant c was just large enough to block the second order terms of the phase grating. Figure 2-2 shows the image with no redundancy modulation present. Note the dark artifact diffraction pattern beneath the spool in the upper right hand corner. Figure 2-3 shows the speckle pattern introduced at the image plane if ground glass is placed at Plane Ao Figure 2=4 shows that only high frequency speckle results if the crossed phase grating is placed at Plane A and signal Plane B is a self-imaging plane. The diffuser noise resulting from the redundancy modulation is seen to limit the image resolution. Figures 2-1, 2-2, and 2-3 show that by carefully choosing the redundancy modulation, optical noise can be suppressed without introducing low frequency laser speckle. Figure 2-2. Image Plane C without Redundancy

3 7 F e2 mage'with round -lass Illumination 7 sa.444)'P.~'~~~~~' -'1.'"i "...~-~,~~~ 44 ~ ~ W Figure 2-3. Image with roPhaseun Grating q(x, y)llumination i~~~~~~~~~~~'ar v,~ Ii r~~~~~~~~~~~~wa:~~ AP9 -jg~~~~~~~~~~~~~~~~~~~~~~~~~~it~~~~~~~~~~~~~~~~~~~~~~~~i

38 Although the self-imaging characteristic of gratings provides a convenient means of generating redundancy illumination, it is not absolutely required. If redundancy modulation with the proper characteristics is generated by a complex mask at the plane of the mask, the far field of a point source created with a lens pinhole assembly can provide a clean beam for illuminating the mask. In order to achieve a more compact system, a hologram of the above illumination can be made using a point source reference beam. If the hologram to mask distance is not too small, the quality of the illumination reconstructed by the hologram will not be sensitive to scratches and other artifacts on the hologram surface. Thus many holograms for generating redundancy modulation can be made from one high quality, extremely clean phase grating. 2.4. CONSTRAINTS FOR REDUNDANCY MODULATION Constraints for redundancy modulation will now be specified and certain types of complex modulation will be considered which satisfy these constraints. Let q(x)s(x) be the complex input signal amplitude for a bandlimited coherent imaging system where s(x) is the amplitude transmittance of a continuous-tone transparency and q(x) represents complex modulation used to increase the spatial bandwidth of the transparency. Suppose h(x) = sine (Wx) is the impulse response of the system, where sinc(x) denotes sin Then the frequency transfer function 7rx h(f ) = FT{h(x)} of the system is rect (f /W), where FT{ - } denotes the Fourier transform operation. In order to achieve high resolution and adequate image quality, it is necessary that the output image irradiance | [q(x)s(x)] * h(x)12

39 be free of low frequency speckle. Since s(x) has a dominant DC frequency component, it is only necessary to place constraints on | q(x) * h(x) 2 to satisfy this condition. Suppose gI(x) is the impulse response of a low-pass irradiance filter and consider J(x) = I q(x) * h(x)j 2 * gI(x) with mean m and variance r. Then if the image is to be free of low frequency speckle, Condition (A) below must be satisfied. 2 m2' C1 (A) 2-1 where C1 represents some minimum signal-to-noise ratio. For best resolution, gI(x) must equal the Dirac delta function 6(x). When q(x) represents diffuse illumination from fine ground glass and gI(x) equals rect [x/(Q/W)1 m /r Q for Q >> 1 [ 2]. For a two-dimensional system, Condition (A) requires that the linear resolution be W. Although plane wave illumination iwx q(x) = e satisfies Condition (A), large dust diffraction patterns are superimposed on the reconstructed image when q(x)s(x) has a dominant frequency component. Therefore, a second condition must be established to assure that high redundancy and thus adequate noise suppression is obtained. There are two mechanisms which improve image quality when optical noise occurs. The law of conservation of energy predicts

40 that'dispersion of the noise diffraction patterns uniformly over the image space will improve the local signal to noise ratio S/N. Here, S/N is defined as the space average image irradiance to the change in the average signal irradiance for a local region of the image. A second mechanism is to demagnify or increase the spatial bandwidth of the noise diffraction pattern irradiance distribution. This decreases the area of the image affected by optical noise and helps preserve low frequency image detail. High frequency detail has appreciable bandwidth and is less susceptible to optical defects, but can be obscurred if the average irradiance in a local region is too low. It will become evident when quadratic phase modulation is considered that although dust diffraction patterns can be demagnified, their size cannot approach the optical system resolution. For this reason the principal mechanism for optical noise suppression is dispersion. In order for optical noise to be dispersed evenly over as large an image area as possible, it is necessary to establish certain uniformity conditions for the spectral density of q. Let S(fx, V, x) = FT rect ( V q(x - x~'i where V represents the width of an elementary element of q. Condition (B) requires that Envelope of S(fx, V, x0) rect (fx/W) (B) or that the parameter

41 W/2 S S(f, V, x ) df W/2 a 2VV max 2 over f S(f -fc, V, x0) df be large for all x e L. The constant WL 2 be large for all x where e < 1 and V _< L. The constant WL is a system design parameter which is determined by certain system constraints. In general, L is dependent on the size and position of artifacts and the function q(x). If q(x) is periodic with period X and L/X is greater than three or four, then Condition (B) is satisfied if Envelope of IFT {q(x)} 2 = rect (fx/W) If X exceeds a certain upper bound, optical noise will not be dispersed uniformly at the image plane. If q(x) is periodic and a system artifact is several times larger than the period X, the noise diffraction pattern will be multiply imaged with unity magnification at the image plane. If the main lobes of the multiple noise diffraction patterns do not overlap, dispersion and thus attenuation of the noise is assured. In Chapter 3 it will be shown that if a dust particle is in the far field relative to the image plane, overlap will not occur if X is smaller than the diameter of the dust particle. When many coherent overlapping patterns are formed, one cannot always be assured that even dispersion of the optical noise will occur. Careful examination of Fig. 2-4 shows that optical noise introduces fringes iat the image plae with the same periodicity as q(x). This property also makes it desirable to limit the size of X.

42 2.5. QUADRATIC PHASE MODULATION 2. 5. 1. CHARACTERISTICS A class of phase modulation will now be considered which under certain conditions has a rather uniform, bandlimited power spectrum, and thus appears to satisfy Conditions (A) and (B). A logical function q(x) to consider is a train of quadratic phase signals, 2 I 7rx 0o q (x) = L e rect (x/X) * 6(x - nX) (5) n= -oo where 6 denotes the Dirac delta function. The phase P(x) for qA (x) = eiP(x) is shown in Fig. 2-5. It is well known that such a signal has a bandlimited uniform spectral density if the space bandwidth parameter Q = X/(XF) >> 10 [17, 18]. If the maximum phase variation (irQ)/4 is not too large, the modulation qA(x) can be realized using a fly' s eye lens array consisting of a train of positive or negative lenslets. It will be seen that it is advantageous to dP(x) remove the discontinuity of dx where the lenslets join by defining a new function w8 cn L 00.50 1.00 1.50 2.00 X = NORMRLIZED DISTRNCE Figure 2-5. Phase for qA (x)

43 r i7rQ rx2 1' 1-ix —-- q (x) = e e rect (x/X) 7Q. 7(X - X) oc + e e XF )rect n= -oo with alternating positive and negative lenslets as shown in Fig. 2-6. IC g -.00.50 1.00 1.50 2.00 X = NORMALIZED DISTRNCE Figure 2-6. Phase for qB(x) Let qA (n/X) be the n-th complex Fourier series coefficient for qA (x) if qA is periodic with period X. Figure 2-7 shows the dependence of IqA(n/X)I2 and IFT{rect (x/X)qA(x)}l2 on Q.

44 o ~ o 0.oo 6.00 12.00 0 -30.00 -S.O0.00 15.00 30.00 Z FREQUENCY (1/PERIQO) z FREQUENCY (I/PERIO0) (a) Q = 10 (b) Q = 21 LJO a: CC w w (c) Q= 4i (d) Q= 60 Figure 2-7. Spectral Density of q, (x)

45 The function qA_(n/X) can be expressed in terms of Fresnel integrals. Let L = -Q/2 + n 42/Q where Q = X/(XF) and define xL 2 + 2 - n XL jQ and x V Q+ 2n + 2n2 XU 2 + 2 Using the Fresnel integrals C(x) = cos 2 u du 0 and S(x) = & sin ( u du 0 qA_(n/X) becomes 2 ^ Q-1/2 iT qA_(n/X) = e f(xU. XL) where f(xU, XL) = [C(XU) + C(xL)] - i[S(xu) + S(xL)] if L < 0 and f(xu, XL) = [C(xu) - C(xL)] - i[S(xu) - S(xL)] if L > 0 By taking the Fourier transform of the first convolution term in Eq. (5) and noting that the form is similar to Fraunhofer diffraction of a sinc function if Q >> 10, the envelope rect (kFf ) is obtained for qA(fx). This implies that qA has Q dominant orders when Q is very large. For negative lenslets qA+(n/X) = qAsince

46 qA is even. If qB(n/2X) represents the n-th complex Fourier series coefficient of qB(x), it can easily be shown using superposition that (i TQ 4 qB(n/2X) 2 Re qe A_ (n/2X) if n is even and i Q-, =2 Im le qA (n/2X) if n is odd. Re and Im denote the real and imaginary parts respectively of a complex number. Although Fig. 2-7 shows that AqA(n/2X)12 is almost constant for in < Q when Q >> 10, it is not clear that qB(n/2X)| has this same property. It is clear that /[qB(n/2X)12 c 2 A 41 qA(n/2X) I and this indicates that qB may have a sharper cutoff than qA. As discussed earlier, the bandwidth of qA and qB is approximately Q/X if Q is large. In order to satisfy Condition (A), it is necessary that the bandwidth of the optical system exceed Q/X. Since qA and qB have periods of X and 2X respectively, the maximum number of dominant orders for Q >> 10 is approximately Q and 2Q respectively. In order to consider the spectral density uniformity of qA and qB, it is helpful to define the redundancy parameter N n= -N R - n JiZ an+j in2irf x for periodic bandlimited functions q(x) = a e with n- -N

47 bandwidt 2Nf. The parameter RM is directly related to G O M of Condition (B) by the equation RM = G_1. Since the optical system must pass almost all the light diffracted by qA or qB if Condition (A) is to be satisfied, it is convenient to set N equal to infinity in order to remove the dependence of RM on the system M bandwidth. The parameter RM thus becomes Kax ( )ja j2\ oo since i a 2 equals unity. Figure 2-8 shows RM versus Q n= -mo for qA' Note that RM must be bounded above by Q/M for Q >> 10, since qA has Q dominant Fourier series coefficients. 0 0 A'= LL Q = SPACE BANNIDWTH PROOUCT Q = SPACE BRNDNIDTH PRODUCT Figure 2-8. Redundancy Characteristics for qA(x) The Symbols 0, A, +, y, Z denote R., R.,... R= Respectively z z 00 IhO.00 80.00 120.00 160.00 200.00.oo 8.00 16.00 24.00 32.00 40.00 O = SPACE BANDWIDTH PRODUCT Q = SPACE BANDWIDTH PRODUCT (a) (b) Figure 2-8. Redundancy Characteristics for q Ax) The Symbols 0, A, +, Y, Z denote R1, R,... R5 Respectively

48 Figure 2-9 shows RM versus Q for qB(x). Since the period of qB is 2X, RM is now bounded above by the quantity (2Q)/M. To compare the redundancy characteristics of qA and qB, RM of Fig. 2-8 should be compared with R2M of Fig. 2-9. Figures 2-8 and 2-9 indicate that qA and qB both have rather uniform spectral densities for Q >> 10. Although qA appears to have better spectral characteristics, it will be seen that qB results in a much higher signal to noise ratio mn /u (Condition A). 8 8 =. o I.00 40.00 80.00 120.00 160.00 200.00 - 00 6.00 16.00 2.00.00 10.00 Q = SPRCE BANOWIOTH PRODUCT 0 a SPACE BANDWIDTH PROnJCT (a) (b) Figure 2-9. Redundancy Characteristics for q(x) The Symbols, + Y Z denote R R Respectively

49 Condition (A) requires that the speckle noise from q be low. f X Let h(f ) = rect BQ be the frequency transfer function of the coherent system where B > 1. In order to achieve the best redundancy in the available system bandwidth BQ X B should be almost unity. However, B must be greater than unity if m /(x of Condition (A) is to be sufficiently high. A digital computer was used to examine J(x) = q(x) * h(x) 2 gI(x) of Condition (A) as a function of Q, W and the width of g (x). Let g1(x) = rect [ A X This filtering operation is easily attained if a spatially noncoherent(X/Q) reference source is used to reconstruct a hologram. If A is greater than two, the filter gI approximately increases the coherent resolution cell width by the factor A. Tables 2-1 and 2-2 give the characteristics of J(x) when q(x) equals qA and qB respectively. The variables given in these Tables are defined below. 2 2 m /a = the signal to noise ratio defined by Condition (A) M 100 max{J(x)} - min{J(x)} and represents irradiance max {J(x). modulation due to speckle noise

50 N l|an12 n=-N R where 2N+1 represents the total RM rm ax n ] an+j j=l number of orders within the bandpass of the system N E = 100 lan 2 and represents the percent of the light energy n= -N within the system bandpass W = (BQ)/X is the system bandwidth and equals (2N)/X for qA and N/X for qB'

Table 2-1. Characteristics of J(x) and RM for qA(x) 2 2 Entry Q B A rnm M E R R R R 2N+1 A 6.78 1.77 0 6.85 30.27 99.604 4.29 2.60 13 B 6.78 1.77 1 12.76 15.72 C 10. 1. 0 2.43 67.33 97. 052 6.53 3.57 11 D 10. 1.2 0 3.33 54.53 98. 645 6.63 3.63 13 E 10. 1.4 0 4.37 44.27 99. 288 6.68 3.65 15 F 10. 1.4 1 8.14 23.16 G 10. 1.4 2 32.71 18.43 H 10. 2. 0 7.88 26.92 99. 825 6.72 3.67 2.12 21 I 20. 1. 0 1.79 78.1 97. 321 13.25 7.09 4.20 21 J 20. 1.2 0 2.71 62.41 99.05 13.48 7.22 4.28 25 K 20. 1.4 0 3.76 49.65 99..566 13.55 7.26 4.30 29 L 20. 1.4 1 6.72 26.46 M 20. 1.4 2 22.09 19.26 N 20. 2. 0 7.31 27.74 99. 914 13.60 7.28 4.32 2.95 41 0 20. 3. 0 14.61 16.04 99. 994 13.61 7.29 4.32 2.95 61 P 20. 3. 1 24.85 8.96 Q 40. 1. 0 1.43 85.06 97. 738 27.73 13.96 7.71 5.83 41

Table 2-1. Characteristics of J(x) and RM for qA(x) (continued) Entry Q B A 2 2 R1 R R R 2N+1 R 40. 1.2 0 2.40 66.94 99. 405 28.21 14.19 7.84 5.93 49 S 40. 1.4 0 3.51 52.62 99. 757 28.31 14.24 7.87 5.95 57 T 40. 2. 0 7.13 28.60 99. 956 28.36 14.27 7.88 5.97 81 U 50. 1. 0 1.34 86. 71 99. 883 34.22 17. 74 9.57 7.08 51 V 50. 1.2 0 2.33 67.89 99. 499 34. 79 18.03 9. 73 7.20 59 W 50. 1.2 1 4.10 34.48 X 50. 2. 0 7.10 28.82 99. 964 34.95 18.12 9.77 7.23 101 Y 50. 2. 2 55.18 12.09 AA 10. 3. 0 15.07 15.52 31 AB 10. 4. 0 23.73 9.87 41 AC 10. 6. 0 42.88 5.08 61 AD 10. 8. 0 63.93 3.18 81 AE 9.23 1.082 0 2.81 60.49 97. 849 5.84 3.53 11 AF 9.23 1. 082 1 5.21 34. 75 11 AG 9.23 1. 082 2 15.77 25.24 11 AH 9.23 1.30 0 3.81 48-. 03 98. 973 5.91 3.57 13 AI 9.23 1.30 1 7.07 25.77 13 AJ 9.23 1.30 2 26.09 20.14 13

Table 2-2. Characteristics of J(x) and RM for qB(x) Entry Q B A rn/ M E R2 R R R12 s 2 4 8 12 A 10. 1. 0 3.08 51.21 98. 379 6.42 4.12 10.5 B 10. 1.2 0 7.56 23.46 99.749 6.51 4.18 12.5 C 10. 1.4 0 17.92 10.59 99.954 6.53 4.19 14.5 D 19. 1.4 1 25.16 9.66 E 10. 1.4 2 56.50 6.66 F 10. 2. 0 116.89 2.62 99.998 6.53 4. 19 2.31 20.5 G 20. 1. 0 2.42 53.30 98.661 12.05 6.99 4.29 20.5 H 20. 1.2 0 8.57 18.82 99.901 12.20 7.08 4.34 3.05 24.5 I 20. 1.4 0 26.24 7.22 99.988 12.21 7.09 4.34 3.05 28.5 J 20. 1.4 1 35.16 6.59 K 20. 1.4 2 71.26 4. 80 L 20. 2. 0 224.92 1.40 99.9991 12.22d 7.09 4.34 3.05 40.5 M 40. 1. 0 2.10 53.61 98.948 21.66 12.80 8.21 5.68 40.5 N 40. 1.2 0 12.57 12.91 99.971 21.88 12.93 8.30 5.74 48.5 0 40. 1.4 0 52.14 4.30 99.998 21.89 12.93 8.30 5. 74 56.5 P 50 1. 0 2. 10 54.69 99.04 25.96 15.80 9.73 7.13 50.5 Q 50. 1.2 0 15.28 11.25 99. 990 26.20 15.95 9.82 7.19 60.5 R 50. 1.2 1 19.34 10.65

54 The functions qA and qB can be compared in Tables 2-1 and 2-2 for identical values of Q. Since the period of qB is 2X and the period of qA is X, RM should be compared with R2M in Tables 2-1 and 2-2, respectively. Note that RM is bounded above by (2N+1)/M. The function qB has a much higher value of m /a than qA if entries with the same Q, B, and A values are compared. Figures 2-10(a) and 2-11(a) show J(x)/max [J(x)] for entries V and Q in Tables 2-1 and 2-2, respectively. By comparing Figs. 2-10 and 2-11 with Figs. 2-5 and 2-6, it is seen that the principal place where diffuser noise occurs is the point at which lenslets join. Since dP(x) for qB is continuous, Fig. 2-11 shows less speckle noise than Fig. 2-10. C1 o o o 0 CCUS - CC... I I II.00.50 1.00.00.50 1.00 x x (a) (b) Figure 2-10. (a) I(x) = J(x)/max{J(x)} for Entry V of Table 2-1, (b) I(x) for Entry W of Table 2-1

55 o 0 o o c: cc ( (b FIge-11. (xI l-2 cI: cI: (b) I(x) for Entry R of Table 2-2 X2 2 better spectral density uniformity and thus higher redundancy. 2 2 2.5 2. EXPERIMENTAL RESULTS Experiments were performed which show the noise reduction characteristics of qB in a coherent imaging system. Figure 2-12 shows a coherent imaging system where the function qB(Y1) was used to introduce redundancy along the Yl coordinate; X, Q, and X for qB(Yl) have values 0. 5 mm, 142 and 0. 6328 p respectively. Since there exists no redundancy along the xl signal coordinate, two wires placed in Plane A parallel to the Y2 axis introduce dark diffraction patterns at the image plane as shown in Fig. 2-13(a). The maximum number of cycles per mm of the w signal bar pattern is 7. However, when the two wires are placed parallel to the x2

56 Input Signal qB (Y1) (x1'SY1) Lens Plane A Lens B Lens Y2 Image Pinholel Plan Laser 3 7 x 10 15.2 20.4 20.4H cm cm cm cm Figure 2-12. Coherent Imaging System, where Lens B has a Focal Length of 100 mm and a F-Number of 2, 3. Two Wires each 0. 23 mm in Diameter were Introduced at Plane A axis, the wire diffraction patterns are highly attenuated as shown in Fig. 2-13(b). Figure 2-14 shows the irradiance at the back focal plane of Lens B. The function qB(yl) had a spatial bandwidth of 284 cycles per mm. (a) (b) Figure 2-13. (a) Image with Vertical Wires at Plane A of Fig. 2-12, (b) Image with Two Horizontal Wires at Plane A, whose Vertical Positions are Indicated with Arrows

57 Figure 2-14. Irradiance Near the Back Focal Plane of Lens B qA"(x'Y s( x-y) I f-~-~ i | Image Lens Pinhole iot 10 FT Plane Noise Plane Figure 2-15. Coherent Imaging System

58 The optical system shown in Fig. 2-15 was used to test how qA (x, y) = qA (x)qA (y) would perform in a near Fourier transform holographic storage system. Function qA- corresponds to entry AE of Table 2-1 and consisted of positive lenslets with Q = 9. 23 and X = 0. 58 mm. A high quality lens array with a Q of 9. 23 was obtained by glueing two lenticular gratings with their lines crossed rough side to rough side with a high index Kodak assembly cement. The bandwidth of qA- was 15. 9 cycles per mm and the system bandwidth was 18. 2 cycles per mm. Figure 2-16(a) shows an image of a standard USAF resolution chart without any redundancy modulation. Element number 1 of group 1 represents two lines per mm. Figure 2-16(b) shows the same image if lens array qA is placed against the signal transparency. Note that the dust particle diffraction pattern which occurs at the top central region of the picture is multiply imaged and dispersed. This phenomenon will be discussed in the next section. Figure 2-17(a) shows how the image is affected by a system artifact in the near Fourier transform plane when the signal is illuminated with a uniform beam. The artifact consisted of a black letter "O'' 0. 8 mm high on a transparent glass plate. Figure 2-17(b) shows the image quality improvement when redundancy modulation qA is introduced. (a) (b) Figure 2-16. (a) Image with Uniform Illumination, (b) Image with Quadratic Phase Modulation

59 (a) (b) Figure 2-17. Image of Fig. 2-16 with Artifact at Near Fourier Transform Plane. (a) No Redundancy, (b) Redundancy qAFigure 2.18(a) shows IFT{s(x, y)qA_(x, y)} 2 at the Fourier transform plane of the system. Figure 2-18(b) shows both FT'[qA (x)}| and IFT{rect (x/X)qA (x2)}2 for Q 9. 23. (a) (b) Figure 2-180 (a) Irradiance at FT Plane of Fige 2-15 with qA (x, y) Present, (b) Spectral Density of qA- for Q - 9o 23

60 Although a lens array qA can improve the image quality, it can be seen from entry AH of Table 2-1 that objectionable diffuser noise occurs where the lenslets join if B is near unity. It is also seen that R1 is only one-half the maximum limit. A new class of redundancy modulation will be investigated in Chapter 4 which does not have these limitations. 2. 5. 3. OPTICAL NOISE DISPERSION It was seen in Fig. 2-16 that a lens array can demagnify dust diffraction patterns. Geometrical optics or the direct correspondence between the signal space and the frequency space of qA predicts this effect. Fig. 2-19 shows how a negative lenslet can demagnify a dust particle diffraction pattern. Geometrical optics Plane A Pla e B L_ -:1:1 Telescope F e Artifact L___ _ -J Image Plane With Diameter u Figure 2-19. Coherent System Showing the Demagnification of Artifact Noise

61 predicts that the shadow of the particle at the image plane is u[f/(f + z)]. The number of images of the particle diffraction pattern at the image plane is O(z + f)/X or approximately (z + f)/f if 0 is small. When diffraction theory predicts that the particle diffraction pattern will be significantly larger than u[f/(f + z)], the local signal-to-noise ratio is improved. If 2 2u f < x z(z + f) it can be shown using stationary phase approximations developed in [19] that the particle diffraction amplitude becomes d(x) exp i-ik 2 2 f \2 i u i.65u X z(z-+f) r ~~~~2 12-1 If 2 2u f X z(z + f) the width of jd(x) 2 reduces to approximately (Xz)/u. This becomes obvious if it is noted that the Fourier transform of the dust particle is formed at the back focal plane of the lenslet and has a width (z + f)/u ~ (z/u) if f << z; and the above inequality is rewritten as z (+ f -(z << 1

When this condition holds and (Xz)/u a X, good dispersion of the noise will occur~ If (Xz)/u - X and f << z, the above inequalities require that Q - X/(XF) >> 2 and X << (uz)/(XF) If Q is large, the bandwidth of qA is i/(AXF) and Condition (B) is satisfied. For many practical display systems, the spatial bandwidth must be low, and the above inequalities cannot be met. Figure 2-20(a) shows the irradiance at the image plane of Fig. 2-19 when randomly placed black dots on a glass slide were introduced at Plane B. Figure 2-20(b) is the same picture when a lens array qA+ with 0. 377 X 0. 377 mm lenslets is placed at Plane A. The values for u, z, and f were 0. 323 mram, 12. 5 cm, and 3. 35 cm, respectively. The demagnification factor f/(f + z) was thus 0. 211 2u f and ~ equaled 0. 558. z(z +f) (a) (b) Figure 2-20. (a) Irradiance at the Image Plane -without qA, (b) Image when qA+ is Present

63 2. 6. SUMMARY In this chapter the redundancy requirements for suppressing optical noise in coherent systems was considered. The selfimaging property of gratings provides a convenient but not essential means for generating redundancy illumination. Constraints were established for the redundancy modulation needed to coherently image a continuous-tone transparency. It was seen that quadratic phase modulation can satisfy these constraints when the space bandwidth parameter Q is large. For some high density holographic storage systems, certain imaging constraints require that Q be small. In this case the spectral density of a lens array is no longer uniform, and a new class of functions must be considered.

3 REDUNDANCY REQUIREMENTS FOR HIGH DENSITY COHERENT STORAGE SYSTEMS 3. 1. INTRODUCTION Redundancy modulation for coherent imaging systems with very narrow spatial bandwidth will now be considered. One such system is Fourier transform holography, where a lens or converging wavefront is used to form the Fourier transform of an input signal near the hologram plane. Such a system has desirable properties for the high density storage of continuous-tone transparencies. Individual points on the hologram can record information concerning the entire image space. The storage of continuous-tone signals presents special difficulties since the dominant DC component of this class of signals makes the signal especially sensitive to artifact noise. For example, a small emulsion defect on the hologram may remove the DC portion of the signal. Special redundancy modulation will be investigated which can improve system performance and increase hologram diffraction efficiency by providing more uniform illumination at the hologram plane. 3.2. SYSTEM CONSTRAINTS The resolution for a Fourier transform holographic system is inversely related to the size of the hologram. If a diffraction limited hologram with dimension D is recorded, the system resolution RH is Xf H D 64

65 where f and X correspond to the focal length and wavelength respectively. To meet the system resolution requirements, the hologram width must exceed some lower limit DR. Redundancy modulation should not require the hologram width to exceed DR by more than a small factor such as two if a holographic system is to compete with present storage systems such as microfiche. The resolution of the eye and constraints on the use of an output enlarging lens place a lower bound on RH. An output enlarging lens could increase this bound. However, the magnification of an enlarging lens is limited for a practical system since coherent noise suppression techniques are not effective for artifacts on an optical surface close to an image plane. Noncoherent noise suppression methods such as rotating the enlarging lens are too cumbersome to be helpful. Since Condition (B) of Chapter 2 limits the period X for quadratic phase modulation, this modulation cannot provide adequate redundancy for high density storage applications where the space bandwidth product XW and thus the parameter Q must be low. Consider Condition (B) for nonperiodic redundancy modulation q(x) when the space bandwidth product VW must be small. Since S ( > = L e sinc (Vf~fI, S(fx v Xo) sinc (Vfx) cq(fx) should be approximately equal to rect (fx/W), the importance of the sinc term when VW is small makes it difficult to determine a function which satisfies Conditions (A) and (B). For periodic modulation with period X where L/X is greater than three or four, Condition (B) is satisfied if Envelope of iq(nf ) rect (f /W) where n is an integer. Since only discrete points must be considered,

66 the problem of synthesizing a function which satisfies Conditions (A) and (B) is more tractable. The parameter XW for periodic modulation represents the number of Fourier series components of q(x) recorded by the hologram. Since periodic phase modulation 00 q(x) jj inwx n n= -oo requires that the number of non-zero orders be infinite and that lan - 0 as n -- oo, this class of functions cannot provide maximum redundancy for systems with very limited spacial bandwidth. 3.3. BANDLIMITED PERIODIC REDUNDANCY MODULATION To achieve high redundancy when VW << 100, consider periodic bandlimited redundancy modulation -N in27rf x q(x): j a e i where 2N+1 < XW n=-N Since this means the signal information will be placed on a multiplicity of carriers, the effect of the spatial frequency coordinates of each carrier on the system resolution will now be examined. The Fourier transform of a real signal s(x) has the following property: If 0o i2irf x S(fx) = s(x) e dx -oO then X [ x= ]

67 This property means that the signal information can be recovered if a hologram records the DC term and only one sideband of the signal. Maximum resolution for a bandlimited system is achieved if only one sideband of the signal is recorded. Consider a bandlimited system such as Fig. 1-5, with frequency transfer function h(fx, f ) rect (fx/W) rect (fy/W)o Coordinates f and f are the Cartesian frequency plane coordinates. Figure 3-1 shows the image of a USAF resolution chart when plane -wave illumination eir(/ ) is incident on the signal. Since there was only singles sideband plus DC transmission along the horizontal x coordinate, higher frequency information was passed and the vertical bars are better resolved than the horizontal barso Figure 3-1. Image of USAF Resolution Chart with Single-Sideband Imaging for the Horizontal Coordinate

68 To account for the low contrast of the vertical bars, let -the real signal 00 i27rf x s(x) =j s(f )e dfx -00 be written as s(x) + s 1( + 1 (x) where s is positive real constant and 0 oo i2rf x x s (x)= s(fx)e df 0 For double-sideband imaging, the image irradiance is s2 + 4 ReIs1 + 4sR Re{s } and for single-sideband imaging the image irradiance becomes 2 + Re 2s + Im sl} + 2s Re{s1 The major effects of single-sideband imaging with a bandlimited optical system are to improve the resolution and to decrease the image contrast. For multicarrier imaging these effects must be averaged on an energy basis for each carrier. For example, an average channel bandwidth <W> is defined below where W represents the maximum single - sideband bandwidth for each carrier and

69 in27rf x q(x)= X an e n=-N is the multicarrier illumination. N I (I1a 2W) n=-N - N n= -N Additional complexities occur in using multicarrier illumination. Let s(x)q(x) be the input amplitude to a coherent imaging system with frequency transfer function h(f) = rect (fx/W) where s(x) is a real input signal and q(x) represents redundancy modulation. If the spectrum of s(x)q(x) falls completely within the system bandpass, the square-law detected output is simply js(x) 2 jq(x) 2 The example discussed below illustrates the case where the bandwidth of s(x)q(x) exceeds W. Consider the real signal s(x) s + (x) + (x)* and redundancy modulation

70 iwlx -iwlx e + e q(x) e + consisting of two carriers at each edge of the system bandpass. The output irradiance [q(x)s(x)] * h(x) j2 becomes 1 2 s 2)e 2 (s+ s1)e + (s+s1 )e 1s+S1(x) I -i2w x i "2-i2w x + Cs R(2w x)+ 2s Re s (x)e + Re s1 (x)e 1 The first term simply represents the output of a single-sideband system. The second term is the fringe pattern due to q(x) weighted by the DC component of the signal. Terms three and four represent interaction of the carriers with the signal. The last term is potentially dangerous; since if s1(x) has a dominant term cei(W+~)x a low frequency moire fringe pattern occurs at the image plane if e is small. This illustrates that caution is required if carriers are placed near the edge of the system bandpass. It was seen in Chapter 2 that the frequency distribution of the image speckle resulting from coherent addition of the carriers limits the system resolution. Fortunately, classes of periodic bandlimited redundancy modulation which have no low frequency speckle exist and are presented in Chapter 4. 3.4. NOISE SUPPRESSION ANALYSIS The noise suppression characteristics of periodic, bandlimited modulation will now be formulated in terms of a family of redundancy parameters. Dispersion of artifact noise uniformily over the image is the principle mechanism for improving the local signal-to-noise

71 ratio. If the density of artifacts is too high, coherent dispersion can no longer improve image quality. Consider the noise reduction for a single isolated artifact located at Planes 1 or 2 of Fig. 3-2. 1: 1 Telescope Plane 2 Collimated Beam q(x) s(x) Plane 1 FT Plane Image Figure 3-2. Coherent Imaging System First to be considered is the effect of artifacts at Plane 1. Let h(x) be the impulse response of the system and 1 - n(x) represent the amplitude transmittance at Plane 1 where n(x) is zero if no artifacts are present. The propagation phenomenon may be regarded as a linear dispersive spatial filter [ 14]. For Fresnel diffraction the impulse response p(x) for this filter is k 2 ikz. x e ~ 27r p(x) = eiz e where k- 2 If z is constant, the z dependence will usually be ignored. The output image amplitude v(x) can be written v(x) = v1(x) + V2(X)

72 where Vi(X) = [q(x)s(x)] * h(x) is the signal term and v2(x) = - { [q(x)s(x)] *_ p n(x) * p' * h(x) is the noise term. If q has a rather uniform spectral density, the dispersive action of p will distribute the noise effects due to n(x) uniformly over a wider area of the image plane. The amount of dispersion is directly related to the distance z and the spectral width of q. The output irradiance is lv(x) 12 = lj2 + 2 Re IV1v2} + 1v21 If v2(x) << v1(x) for all x, Iv(x) 2 lv1l + 2 Re {vlv*} A1though the redundancy modulation q(x) does not change the signalto-noise ratio if the whole signal space is considered, it can improve the local S/N over a local region Aof the image. One measure of the local signal-to-noise ratio (S/N)A is SA] vl(x) 2 dx (S/N)A 2 Re {V 12}I 21 dx This measure will prove convenient since it can be expressed in terms of the redundancy parameter RM. If periodic redundancy modulation

73 N inw x q(x) = ae n=-N is considered and only the dominant DC component s of a continuous-tone signal is retained, v2(x) can be simplified. The expres sion N i27r ff x+ -(Xnf ) ] Soq(x) *p(x, z)= s 1 a0n n=-N becomes -N i2ir(nf x) s a a'e o i, n n= -N where i7Xz(nf )2 0 a' =a e n n for Fresnel diffraction. The artifact function n(x) typically takes the form rect (x/u) for a dust particle with dimension u, hence v2 becomes N k 2 N 0 i27rnf x -i 2zX n= -N If u is several times larger than the period 1/f, this expression predicts that multiple artifact diffraction patterns spaced Xzf apart appear at the image plane. Noise suppression is dependent on the sequence aI2,. Since the irradiance of the n-th

74 artifact diffraction pattern is proportional to the coefficient an of the plane wave that prod-uced the image, noise suppression is maximum if all sequence terms are equal. Consider the case where the main lobe of the m-ultiple diffraction patterns do not overlap. Figure 3-3 below shows this condition for a small wire artifact when iw x -iw x 0 0 q(x) - ie - 1 - ie Figure 3-3. Image of Diffraction Pattern for a Small Wire Artifact (w = 2wfo and fo 16 cycles per mrnm) If (S /No) is the local signal-to-noise ratio when q(x) = 1, the local signal-to-noise for the n-th artifact diffraction pattern is N (S/ )n nN (S/Ny jaj|

75 It is necessary to maximize the minimum local signal-to-noise ratio. This is equivalent to maximizing the parameter N I vI aI n=-N max jan[j n J since min i (S/N)A R(S o/ N)A Noting that ai max I anl it is seen that R1 equals the maximum limit 2N+1 if all terms I a 2 are equal. If the grating N ~N inw x q(x) = a e 0 n=-N which maximizes R1 can be selected from the class of functions satisfying Condition (A) of Chapter 2, the optimization problem is over. However, what if P multiple noise diffraction patterns overlap and a grating with R1 = 2N+1 cannot be found? In this case it it necessary to consider the redundancy parameter

76 N I ljan1 n= -N R =. RP P max an+j n j=l The sum in the denominator is maximized over the index N for -N < n < N - P + 1. The minimum local signal-to-noise ratio is now Rp(S /N )A' and thus optimum redundancy is achieved if Rp is maximized. A simple example will help illustrate the condition for no dust diffraction overlap. If the object plane is in the far field of a dust particle at Plane 1 of Fig. 3-2 and n(x) = rect (x/u), the diffraction pattern at the object plane is sine R(ux)/(Xz) with main lobe width (Xz)/u. Let 0 be the propagation angle of plane wave i27rf x 0 e relative to the optical axis for a grating with period X = /f. Then the condition for no overlap is O0 ~8z. Az u since f 0/X if 0 << 1, this implies 0 X < u The above inequality requires that the period of the grating be smaller than the artifact width. Thus if X is 1/5 mm, fo must be greater than or equal to 5 cycles per mm if no overlap is to occur. If the periodicity of the multiple wire diffraction patterns is examined in Fig. 3-3, it can be seen that the increased diffuser

77 noise that occurs is due to attenuation of one of the three coefficients a 1' a, or al. The increase in the low frequency speckle noise due to a noise particle will be examined quantitatively for artifacts at Plane 2 of Fig. 3-2. This work will also be useful for estimating the low frequency speckle introduced by artifacts at Plane 1. Consider artifacts at Plane 2 of Fig. 3-2. For Fourier transform holography, Plane 2 represents the hologram plane. Let h(fx) be the Fourier transform of the impulse response h(x) of the system. The output amplitude v(x) for amplitude transmittance [1 - n(fx)] at Plane 2 is v(x) = [q(x)s(x)]:" h(x) - [q(x)s(x)] * h(x) * n(x) If h(f) =rect (f /W) and n(fx) rect aW where a << W and If < W/2, v(x) becomes i27rf x - v(x) = W[ q(x)s(x)] * sine (Wx) - aW[ q(x)s(x)] * ei sine (aWx)j Artifact noise at the Fourier transform plane affects the entire image space. If no redundancy modulation is employed, a small artifact could remove the DC component of the signal. Although periodic redundancy modulation q(x) with no low frequency speckle is determined in Chapter 4, the blockage of a spectral order of q(x) reduces m /U of Condition (A), since low frequency speckle is introduced at the image plane. This effect will now be formulated in terms of the redundancy parameter RM. Since continuous-tone signals s(x) have a dominate DC component, only q(x) need be

78 considered in the following analysis. A small scratch or artifact attenuates one or more Fourier series components of q(x). Let (1 - B) be the attenuation factor due to such a noise term. If in27rf x 0 a =ae n n then N q(x)|2= 1j an a n, m=-N Let qo represent q(x) after one order an is attenuated by the factor (1 - B). Then q (x) = q(x) * [6 (x) - n(x)] -!Fa Ba (n=-N and N 2 qo(X)2 2 = 21 a + B a a n=_-N N -2 a~, |an B cos [41(X) - 2(x)] where!1(X) = arg {q(x)}

79 and 2(x)= arg {n B In Chapter 4 a class of even functions q(x) with no low frequency speckle are examined [20]. These functions have the characteristic N 2 iq an n=-N = 1 + D cos (2ir2Nf x) 0 As R increases, D becomes small relative to 1. For bandlimited phase modulation such as qB of Chapter 2, D is essentially zero. If D << 1, N aCn a 1 + (D/2) cos (27r2Nf x) n= -N If gI(x) [(2N)/Xlrect [(2Nx)/X] is the impulse response of a lowpass irradiance filter which removes the high frequency speckle term cos (2N27rf x), Iqo(X) I gI(x) = 1 + 1Ba2 N 1 o j 1B221 ~ 2 n=-N n —N Xi1 + D cos (2r2Nf x)] cos ( -) (x) o ~~~~~~1(x)

80 For small D the expression above is bounded above by +LR + -2-I R and from below by 1B12 2!BI 1+ thus q o(X)| * gI(X) = 1+ + +(x) where (X) I 2I B 4R1 Speckle modulation is thus very low if 2 -B- <<1 If an artifact is not at the Fourier transform plane, the effective B is often much less than one. Also, if R1 is the redundancy for q(x), this same parameter has the value R1 for two-dimensional modulation q(x)q(y). A limit will now be established for the magnitude of Fourier

81 series coefficients of jqo!2 which are introduced by this type of artiface noise. If lq = 1 + D cos (2ir2Nf x) attenuation of the complex coefficient an by the factor (1 - B) o will introduce new irradiance frequency terms of the form - I B a * a + a a J 1* * -2Ik cos 27rkf x + arg B* a a + a ) where k I|n nn +k n n 0 0 0 0 Since 2eter XB small. I <n k N these new frequency coefficients are bounded by the inequality 2Ik' 4/R1 As stated in Chapter 2, the frequency of the speckle noise is never lower than f = 1/X for periodic functions, hence it is advantageous to keep the number of orders 2N + 1 or the space bandwidth parame-ter XW small.

82 3. 5. SUMMARY This chapter has considered the redundancy requirements of high density coherent storage systems for continuous-tone transparencies. Periodic functions appear to have the best characteristics for maximizing the redundancy in this type of system. The noise suppression characteristics of bandlimited periodic redundancy modulation were formulated in terms of a family of redundancy parameters. These parameters will prove useful in Chapter 4 in determining periodic modulation with optimum redundancy characteristics. Chapter 4 will show that uniform hologram illumination required for high diffraction efficiency in Fourier transform holography can be achieved by using periodic modulation in tandem with a coarse lens array. Experimental results presented indicate that this technique can improve noise suppression.

4 PERIODIC MODULATION WITH OPTIMUM REDUNDANCY CHARACTERISTICS 4. 1. INTRODUCTION In Chapter 3 the redundancy requirements of high density storage systems for continuous-tone transparencies were considered. There it was shown that periodic functions are the most promising class of redundancy modulation for systems where the spacebandwidth product VW << 100. Periodic modulation with optimum redundancy characteristics will now be investigated. In order to satisfy Condition A of Chapter 2, redundancy modulation q(x) should have no low frequency speckle noise. It is well known that a grating which generates three orders can produce a diffraction pattern which is twice as fine as the grating period [ 21]. Gabor [20] generalized this property by showing that it is possible for a grating with 2N + 1 orders to produce a fringe pattern that is 2N times as fine as the grating period. Consider the class of bandlimited functions N i i2wrnf x q(x) = e 0 n=-N having 2N + 1 orders. The irradiance I(x) = Iq(x)12 thus becomes 83

84 2N I(x) = I0+ 2 I cos (27rnf0x n) n=l If N-k c -k a a* k / in n+k n=-N then Ik- Icki and = arg {Ck} where arg {( } denotes the angle of a complex number. Low frequency speckle modulation is eliminated if the system of nonlinear equations given below is satisfied. N-k aa*+k= 0 for k = 1, 2,..., 2N-2, 2N-1 (6) n=-N N Since only relative values of the coefficients {a } are of n=-N interest, let aN =i. Assuming that only even functions are of interest, a = a, and the system of Eqs.(6) above reduces to -n n 2N-1 equations and 2N unknowns. This leaves one free parameter which can be varied to maximize the redundancy parameter N Rn=-N R - I maxHj Pan l2}

85 in order to obtain a grating with the best spectral uniformity. The irradiance I( x) can be written as N-1 1 I(x) = 2 + a + 2 cos (27r2Nf x) n=-(N- 1) Thus if q(x) has a rather uniform spectral density, it is clear that as N -* oo that Var {I(x)} -0O and thus q(x) - eia(x) E { I(x)} For fixed N, the free parameter value could be selected to make Var I(X)} small rather than simply maximizing RM. In this E { I(x) case redundancy would be sacrificed in order to achieve diffraction limited resolution performance. 4.2. GABOR'S SOLUTIONS Gabor obtained even function solutions to equation system (6). Noting the solution pattern that developed, he attempted to derive a recurrence relation which would yield a higher order solution N+1 N sequence {an} from the solution sequence {an} n=-(N+l) n=-N The recurrence relation was formed by considering the new terms introduced for equations I1 = 0 and I2 = 0 when the number of grating orders increases from 2N+1 to 2(N+l)+l. It appeared that to obtain a solution

86 from the, lower order solution sequence {a_N, a_(N,1)... aN}' it was only necessary to calculate a new DC term a from terms a0 and.al and let a1 = a and anl = a for n > 1. The recurrence a1 -ndlet aa0 a n+l n relation obtained can be written as laol - jail 2 a = ia (7) 2Im amal *} where Im{' } denotes the imaginary part of a complex number. Solutions given by Gabor are listed below [20]. Note that aN was set equal to 1 instead of i. 0 N N la{ n} N' KNotea =a for{a } -n n n 1 1 ik 2 1 ik k 3 1 ik -.2 k2 -1 ik(k2 -4) 1 2 1 2 1 4 2 4 1 ik(k -4) - (k - 24k + 16) k is a real free parameter. Solution sequences for N = 3 and 4 were obtained using recurrence relation (7). If the above solution sequences are checked in Eqs. (6), it is found that although {a } given above satisfies Eqs. (6), 4 n=-3 {an} does not. For N = 4, the frequency coefficient I4 of n=-4 1q12 is

87 1 k4 2 I4 3 —2 (k + 8k + 16) / 0 for real k The class of solutions obtained by Gabor possess the 90 degree phase property, arg {a a*+l} = ~7r/2. Although recurrence relation (7) does not yield a solution sequence for N > 3, solutions for N > 3 do exist and will be determined in Sections 4. 4 and 4. 5. Before obtaining even function solutions to Eqs. (6), let us regress a step and consider two-dimensional redundancy modulation and one-dimensional solutions where a: a -n n 4.3. TWO-DIMENSIONAL REDUNDANCY MODULATION Let x and y be Cartesian coordinates. Two-dimensional redundancy modulation q2(x, y) can easily be attained by forming the separable function q2(x, y) = q(x)q(y). The redundancy RM for q2 is then simply equal to the square of the redundancy for the one-dimensional case. If q2(x, y) is not separable, the problem of selecting q2 such that [q21 has no low frequency speckle is much more difficult. However, it is natural to ask if nonseparable functions can be found with better redundancy characteristics than separable functions. Consider the function N i(nwx+mwy) q2(x, y) nm n, m=-N Suppose a function q2(x, y) exists with the following characteristic: All of the Fourier series coefficients of I q21 are zero except the DC term and the terms due to the interference of the two orders a-NN and aN, N and the two orders a N N and aN N. This characteristic makes the spatial frequency of the diffuser noise a

88 factor dJ greater than that attained with separable solutions.'Imposing this condition requires more equations than unknowns. For example, such a condition results in 20 equations and 16 unknowns for N = 1, and 60 equations and 48 unknowns for N = 2 if no assumptions about symmetry are made. It is therefore doubtful that a nontrivial solution exists. Trivial solutions such as { 1, 1' a_ 1,1, al 1' al 1} = {1, i, 1, i} and all other a = 0 can, of course, satisfy the above conditions. n, m If the frequency condition attained using separable functions is imposed for N = 1, this results in a nonlinear system of 16 equations and 16 unknowns. If these conditions are relaxed, the number of free parameters increases rapidly as N increases. Because of the above difficulties, only separable functions will be considered in the next section. Fortunately, there are also practical reasons for considering separable functions since they are easily synthesized by using two one-dimensional gratings in tandem with their lines crossed. 4.4. GENERAL SOLUTIONS TO EQUATIONS (6) Appendix I investigates general solutions to Eq. (6). It is shown that considerable effort is required to obtain solutions for even small values of N. For N equal to two, a two-dimensional solution space results and a nonlinear system of equations must be solved using numerical algorithms which require an initial guess. Fortunately, one-dimensional even functions are easier to synthesize; and it will be seen that with a modest computer effort an even solution sequence with R1 near the maximum limit 2N+1 can be found for N = 4.

89 4. 5. EVEN FUNCTION SOLUTIONS TO EQUATIONS (6) With the even function assumption, a = a and the problem of maximizing the redundancy parameter RM becomes tractable. 4. 5. 1. SOLUTIONS WHERE arg{a a* } = ~7r/2 n n+l First to be considered are the class of even function solutions, with the 90 degree phase property arg{a a* +} = ~7r/2. The 90 degree phase property makes each of Eqs. (6) represent the addition of collinear vectors. This together with the symmetry condition a n = a greatly simplifies these equations. Since the solutions given by Gabor are correct for N = 1, 2 and 3, these solutions will simply be listed below. These solutions differ by a multiplicative factor i from those listed earlier, since a N is set arbitrarily equal to i for the new solutions to be developed in this chapter. N l Na (where a = an and arg{a na*} = + r/2) 1 {i, bR_.} 2 {i, b,. -i 0.5bR2'} 3 {i, bR, -i 0. 5bR -0. 125b (b2 4), } R bR R R bR is a real number and represents the free parameter discussed earlier Solutions will now be obtained for N = 4, 5 and 6. As a shorthand notation, let Eq. (N, k) denote equation N-k aa a, = 0 n=-N of the 2N - 1 Eqs. (6). Relation (C) is defined below in order to

90 reduce the work required for higher order solutions. In general, Eqs. (N, 2N-1), (N, 2N-2),... (N, N+1) are equivalent to Eqs. (1, 1), (2, 2),... (N-l, N-l) (C) Equations (6) will now be solved for N = 4. Using Relation (C) 4 { a} becomes n= -4 (i, bR, -i 0. 5bR 2 d i e All variables such as bR, dR, and eI are understood to be real in the ensuing discussion. 4 Eq. (4, 4) eI = Rd - 0. 125 bR Eq. (4, 3) is satisfied because of symmetry conditions. 2 2 2 Eq. (4, 2)= dR + 2bRdR bR eI - bR O Solving Eqs. (4, 4) and (4, 2), dR and eR become dR(b) = (-1 21/2)b + 0. 5(-1 ~ 2 1/2 )b 3 R R R 1/2 2 1/2 4 eI(bR) = (1 F 2 )bR + 0. 5(0. 75 T 2 1/2)b The solution sequence for N = 4 is thus {i, bR' -i 0. 5 bR dR(b eI(bR) ~~~~R d'R)'''~R,..

91 Next to be considered, is the solution sequence 5 (a} = {a, b, c, d, e, f, e, d, c, b, a} n=-5 Let the subscripts R and I denote the real part and the imaginary part respectively of a complex sequence term. Using Relation (C), 5 {a } becomes n=-5 i, bR -i 0.R5 bR R R R. {i bR, i 0. 5 R v d + 125R )' fR'..... } Equations (5, 5), (5, 3), and (5, 1) are satisfied because of symmetry conditions. Eq (5, 4) -2(bRdR + 0. 125 b ) + 2bRfR + bR (bRdR + 0.125 b )+d = R R R R R After simplifying the above two equations, dR and fR can be obtained as a function of bR by solving the equations below. d3 _ 2b 2 _ b (2 + 1.25 b + 0 125 b 4)dR R R R R (2+ + 125bR - bR (-1 + 0.125 bR + bR /64) = 0 f -0 5 b1d 2 + (1 - 0. 5 b 2)dR + (bR /16)(2 - bR2) R -.bR R It is seen that the amount of work required to obtain a solution increases rapidly as N increases. Finally, the solution

92 N sequence {an} will be considered for N = 6. After applying n= -N Relation (C), this sequence becomes {i, b. -i 0.5 bR dR, -i(bRdR + 0 125 bR ) fR i gI4''' } iR,9 -i RdR.bR R0. ) bfR., g... After considerable work, the equations below which express the sequence terms as functions of bR are obtained. Let 3 9 12 3 10 27 8 5 6 co b b +-b + b +-b c = bR 4096 R 128 R 128 R bR 7 4 2 16 bR 4bR b 3 8 9 6 21 4 2 C1 R b R 3 b + 9. 5 b 8 R *R C = bR 2 bbR+ b6 + 8. 25bbR + 29 b + 24 2 bR 16 R R R c3 = 2b + b + 4b + 16 R R R c4 =bR 2 + bR2) c 5=4 Then solutions for dR can be obtained by solving the polynomial 5 i c d n 0 n=0

93 Then if 2dR + b R(2 + bR ) 2, f =2 4 fRR 2 ) R 2dR + bR(2 + bR ) 2.42 gI = -bRfR - 0. 5 bR (bRd + 0.125 b) 0. 5 dR No solution exists if 2dR + bR(2 + bR ) = 0. It was shown in Chapter 3 that the redundancy parameter N Z lan2 n=-N R - n=-N l where M= 1, 2,... M -M M n l I |an+j12{ is a useful measure of the spectral density uniformity of periodic functions. The parameter M = 100 I (I - I. ) where max max min I(x) = q(x) 2 is a measure of the high frequency fringe modulation -1 term. The percent modulation A = 100 (I + I ) (I -1 max min max I min ) equals M(2 - M) for sinusoidal modulation. The redundancy parameters R1, RR,.3o.., R[N/2]+1 and the modulation parameter M were plotted in Figs. 4-1 through 4-5 as a function of BR = bR for each of the solutions above using a digital computer. The redundancy and corresponding modulation characteristics are adjacent to one another. The parameter bR was varied from -4 to 4 in 0. 01 increment steps. The computer program which plotted

94 these characteristics also determined the solution sequence N (an} which maximized R1. These results are tabulated in n= -N Section 4. 5. 3. It is not necessary to vary bR over a very large interval in order to determine which values maximize RM. For example, since Ia_N = 1 and a-N+1 = bR' R1 can never exceed 2N - 0. 875 if IbR > 4. It is sometimes necessary to examine the functional dependence of the sequence coefficients in order to minimize this interval, however. The K roots from solutions involving K-th order polynomials were stored in K different storage arrays. The discontinuities and lack of symmetry for some of these figures is due to a cross over of roots during array assignment. Arrays with very low redundancy were not plotted. Since the functional relationship of the sequence coefficients reveal that RM and M are even functions of bR, negative values of bR could have been omitted. The next section shows that it can be advantageous not to retain the functional relationship of the coefficients. Since Figs. 4-1 through 4-5 show that R is well below the maximum limit 2N + 1 for N = 2, 3,... 6, the condition arg{a a*+l} = +7r/2 will be removed in order to see if R1 can be increased. It will be seen later, however, that functions with the 90 degree phase property are somewhat easier to synthesize.

95 8. ooi8~~~~ 8 8 _-4.00 -2.00.00 2.00 4. 00'-.O0 -2.00.00 2.00.LOO BR BR Figure 4-1. Redundancy and Modulation Characteristics for N = 2. The Symbols 0 and L Denote R1 and R2 Respectively. -1 M = 100(I - I )I max min max 0 0 %g~~~ 0 =~r..x ~L0 "-.00 -2.00.00 2.00 4.00' -'.00 -2.00.00 2.00 4.00 BR BR Figure 4-2. Redundancy and Modulation Characteristics for N 3. The Symbols 0 and A Denote R1 and R2 Respectively. 1 2 M = 100(I - I.)I max min max

96 8s~~ d~~o 8 a -4.00 -2.00.00 2.00 4.00'I..00 -2.00.00 2.00 4.00 BR BR (a) o Co o 3 o, I4 o00 -2.00.00 2.00 OO ~ -e.00 -2.00.00 2.00 4Q.00 BR BR (b) Figure 4-3. Redundancy and Modulation Characteristics for N = 4. The Symbols 0, A, and + Denote R1, R2, and R3 Respectively. M = 100(I - I.)I max mrn max

97 Uo co o0 -4.00 -2.00.00 2.00 t4.00'-4.00 -2.00.00 2.00.00O BR BR (a) Co 8 o cc 4.00 -2.00.00 2.00 4.00 -4-.0 0 -2.00.00 2.00 4.00 BR BR (b) Figure 4-4. Redundancy and Modulation Characteristics for N = 5. The Symbols 0, A, and + Denote R1, R2, and R3 liespectively. -1 M = 100(I - I. )I max rnml r rn ax

98 M t.00 -2.00.00 2.00 00 00 -2.00.00 2.0 -2.0. BR BR (c) Figure 4-4. Redundancy and Modulation Characteristics for N = 5. The Symbols 0, A6, and + Denote R1, R2, and R3 Respectively. M= 100(I - I. )I (continued) max mmn max o 8 0 -.00 -2.00.00 2.00 4.00 -4.00 -2.00.00 2.00 4.OO BR BR (a) Figure 4-5. Redundancy and Modulation Characteristics for N = 6. The Symbols 0, A, +, and Y Denote R1, R2, R3, and R4 Respectively. M =100(I I. )I max mrin max

99 ~~8 o ~8 8 8 (b) 0 ~80~~ ~O~ -4. 0 -0.0. 00 2.00 toO'-.0oo -2.00.O 0 2.00. O0 BR BR (c) Figure 4-5. Redundancy and Modulation Characteristics for N 6. The Symbols 0, A, +, and Y Denote R1, R2, R3, and Ra4 Respectively. M = 1 OO(I - I. )I (continued) max min max mnax rni max

100 4.5.2. GENERAL EVEN FUNCTION SOLUTIONS Although added complexity results if the 90 degree phase condition is not imposed, significantly higher redundancy can be attained. Let subscripts R and I again denote the real and imaginary parts respectively of a complex sequence term. Again consider N the solution sequence {a } to Eqs. (6), where a = a and n=-N a-N = i. For N =1, if {a} = {i, b, i}, Eq. (1, 1)bI = 0 nn n=-l For N = 2, after applying Relation (C) the solution sequence be2 comes {i, bR, c, bR, i}. Eq. (2, 2)=cI = -0. 5 bR and Eq. (2, 1) c —R = 0 if bR f 0. This is the same solution as before, since arg{a a*n+l = +(r/2). For N = 3 the solution sequence simplifies to {i, bR, cR - i 0. 5 b d,... } if Relation (C) is employed. Eq. (3, 3)dI = -bRCR. This result together with Eq. (3, 1) yield the solution sequence 3 1/2 an} = i, bR, 3/2b R(1 + 0.25 bR ) n=-3 2 2 - i 0. 5 bR - b(1 + 0. 5 bR ) 2 1/2herea i 31/2bR 2(1 + 0. 25 bR ) where a = a R R ) I. a -n n For N = 4, Relation (C) reduces the solution sequence to 4 2 a n} = {i, b R - i 0. 5 b R bc e, } n=-4R' R dR- ibRCR e,.... n= -4

101 Appendix II explains the manner in which the remaining equations were solved. A function labeling technique incorporated into the computer program eliminated some very tedious algebra. A computer search which maximized R1 resulted in a solution sequence with an R1 of 8. 49. Since R1 is bounded above by 9 when N = 4, this is a very encouraging result. A grating with 9 orders should be quite suitable for a high density storage system since Chapter 3 showed that the number of orders must be limited for this type of system. The redundancy characteristics for the principle roots to Eqs. (6) for N = 3 and 4 are given in Figs. 4-6 and 4-7. Removing the 90 degree phase condition increased the maximum value of R1 from 4.64 to 5.3 for N = 3, and from 4.99 to 8.49 for N = 4. The max {R1} upper bound for R is 2N + 1. Note that N + 1 increased from 0. 75 to 0. 94 when N increased from 3 to 4. It is logical to ask if max {R1} 2N + 1 will approach unity as N is further increased. Appendix III shows that maximizing R1 for N > 4 is a very difficult task. 83~~ 0~~o 8U;E, -_, \. "-.oo -2.00 2.0 2.00 q.'-1.oo00 -2.00.00 2.00 4.00 BR BR Figure 4-6. Redundancy and Modulation Characteristics for N = 3. The Symbols O and Z Denote R1 and R2 Respectively. M = 100(I - I )I max rin max

102 0 8 8:q4.000'-. -2.00 0 2.0 0 o - 00.00 2.00 q.00 BR BR (a) o M= 0 o(I - I..I 4.o00 -2.00.00 2.00 4.00'-4.00 -2.00.00 2.00 4.00 BR BR (b) Figure 4-7. Redundancy and Modulation Characteristics for N = 4. The Symbols 0, z6, and + Denote R1, R2, and R3 Respectively. M=100(I - I )I max min max

103 4. 5. 3. SOLUTION SUMMARY N Table 4-1 lists even function solutions {a } to Eqs. (6) n= -N which maximize the redundancy parameter R 1' Let FT{ } denote the Fourier transform operation. Figs. 4-8(a) to 4-8(g) show the coefficients Iq(n/X)j2 = an 2 and the envelope I FT{rect(x/X)q(x)} I2 for entries B through H of Table 4-1. Figure 4-8(h) shows the spectral density corresponding to point BR = 1 of Fig. 4-4(c). In this example R1 is not maximum, but the modulation parameter M is quite small.

Table 4-1. Solution Summary Assumption N Entry arg {a an+ 1} = +/2 N R1 jq(x) 2 wherew= X {a } where a =a ~n n~~l X n n=~-n - n n A 1 3 1. + 0.66 cos 2wx {i,l,i} B 2 4.25 1. + 0.47055 {i, 1.00007, -iO.50007,..} cos (4wx) C * 3 4. 64 1. + 0. 43094 {i, 1. 00007, -iO. 500007, cos (6wx) 0.375009,... } D 3 5.3 1. + 0.37734 {i, 0.550074,0.988133 cos (6wx) -iO. 151290, -0. 633294 -iO.543546,... } o E * 4 4.99 1. + 0.10022 {i, -1. 999953, -il. 999907, cos (8wx) 0. 343029, -il. 312095,....} F 4 8.49 1. + 0. 23231 {i, -0.999931, 0.792831 cos (8wx) -i0.499931, 0.621247 +iO. 792776, 0. 889049 +iO.181948,.... } G * 5 7.40 1. + 0.13162 {i, 1.100068, -iO.605075, cos (10 wx) -1. 432878, ii. 393206, -1.433349,....} H * 6 8.47 1. + 0.19489 {i,-1.099933,-iO.604926, cos (12 wx) 0. 799482, iO. 696409, 0.908429, il.100901,......}

105 LLI8 LwJ.:- 1.-4 o o tJI cr a: C3 g cc cc cc0 cc0 ~~~~~~~~~~~~~~C3' zc F R QECY(/EI}zFRUN CY(/RI MC Ig.. I I =. 4 a cc -2.00.00 2.00.00 o-6-. 00 -3. 00.00 300 80'-..o -3.o 3o3D O o'8.qo o 8. oo'100 FREQUENCY (1/PERIOD) z FREQUENCY (I/PERIOD) (a) Entry B of Table 4-1 (b) Entry C of Table 4-1 LLJ8~~~~~~~~~~~~~'' w~~~~~~~~~~~Z"' a: ~~~~a: 0. C3 cm3 a: cc cc cc0 cc0 W 0 CC ~' rQ n~~~~~~~~~~~~~j ~ ~ ~ ~ ~ ~ Ccc cc. 2:40 ~ ~~~ ~ ~~~~~~ I'I o I o - -- I. oo.:;.~ ~~~~s~~" — t.00 o) -6.00 -3.00.00 3.00 8.00 0 -4.0.00'1.00 8. 00 Z FREQUENCY (i/PERIOD) Z FREQUENCY (1/PERIOD) (c) Entry D of Table 4-1 (d) Entry E of Table 4-1 0 r a: a: 0 0 a:a cc cc -i a: a:0 o uo,o uoo so 8 -4.. i.00. 8.00 o -10.00 -5.00.00 5.00 10.00 Z FREQUENCY (1/PERIOD) Z FREQUENCY (1/PERI00) (e) Entry F of Table 4-1 (f) Entry G of Table 4-1 w8~~Fgr 4-.SetalDniyCarceitc

106 Lu8 %t. o oo.oo 6.00 z FREQUENCY. I1/PEBIOD) z FREUENC~:1/PERI OO cr [20]. Here, the complex modulation q(x) is synthesized i2rnf x q(x) = X a e n=-N Z FREQUENCY (1/PERIOD) Z FREQUENCY (1/PERIOD) Figure 4-8. Spectral Density Characteristics (continued) 4.6. SYNTHESIS OF PERIODIC MODULATION WITH OPTIMUM REDUNDANCY CHARACTERISTICS The first method of synthesis to be considered is that suggested by Gabor [20]. Here, the complex modulation q(x) is synthesized by constructing a special filter with an amplitude transmittance proportional to i2rnf x q(X) Z ene n=-N Such a filter is constructed using two distinct filters with amplitude transmittances of t1(x) and t2(x) respectively. These functions are defined below:

107 i2irnf x t1 (x) = ae n=-N N-I 1/2 2 + 1 janJ + 2 cos (27r2Nf x) n= -(N-l) and t2(x) = eiP(X) where 0 N i27rnf P(x) = arg ane n=-N Two such filters in tandem produce the complex amplitude transmittance q = t1t2. The first filter has a real amplitude transmittance and can be produced photographically. The second filter is a pure phase function with a period 2N times as coarse as that of t1, and can be produced using a ruling engine. New technologies being developed, such as electron beam thermoplastic recorders, may provide other means of synthesizing t2. The self-imaging characteristics of periodic modulation make it unnecessary to place the two filters in contact with one another. This property should also prove useful when two such filters are crossed at right angles to provide two-dimensional redundancy. The phase profiles for t2 representing entries B through H of Table 4-1 are given in Figs. 4-9(a) to 4-9(g). Computer program logic produced a smooth curve for P(x) instead of P(x) modulo 27r. The

108 Fast Fourier Transform algorithm [22] was used to perform the i27rnf x complex summation a e efficiently for a large number n=-N of points. Figure 4-9(h) represents the phase profile for the solution sequence whose spectral density is shown in Fig. 4-8(h). For this particular example, R1 is not maximum but the fringe modulation M is only 3 percent. The triangular profile shown in Fig. 4-9(d) should be one of the simplest to fabricate. The second synthesis technique to be considered requires optical spatial filtering, and is most applicable for solution sequences with the property arg{ana al} = ~ir/2. Some special properties of this n n+l class of solutions will now be given. Consider functions of the form i2wrnf x q(x) = a e with period X = 1/fo, and let Iq(x)[ = m(x) n= -N N and arg{q(x)} = P(x). If q(x) is even and {a } satisfies Eqs. (6), n=-N then m(x) and P(x) are both even and m(x) has a period of length X/(2N). Let an = fR(n) + ifI(n) and suppose P(x + X/2) = -P(x). Since m(x + X/2) cos [P(x + X/2)] = m(x) cos [P(x)] and m(x + X/2) sin [P(x + X/2)] = -m(x) sin [P(x)] fR(n) = O for n odd and fT(n) = O for n even or zero

109.~.! M n w O.00 l.. 1.00 1.50 2.00.00.50 1.0.so 2.00 X = NORMALIZED DISTANCE X = NORMALIZED DISTANCE (a) Entry B, P = 0. 7048 X (b) Entry C', P = 1. 1268 X rnax rax -J..00.50 1.00 1.50 2.0o 00.so t. 0 t.so 2.00 X = NORMALIZED DISTANCE X = NORMALIZED DISTANCE LWJs LPJS m Thex Parameters P and Represent rn1 ~ax the Maximum Phase Variation and 27r Radians Respectively

110 isa wa. La,,AL 0 0,,g. LU,., XQ- 0. so t1.00 1.50 2.0 00. s0 1.00 1t. 2. 00 X NORMALIZED DISTRNCE X = NORMRLIZED DISTRNCE (g) Entry H, P = 1. 4655 X (h) Point BR = 1 of Fig. 4-4(c) max P = 0. 7425 X max Figure 4-9. Phase Profiles for Entries B through H of Table 4-1 and the Point BR = 1 of Fig. 4-4(c). The Parameters P and X Represent the max Maximum Phase Variation and 27r Radians Respectively (continued) ie It is thus clear that q(x) or e q(x) has the property arg{anan+} = +7r/2 if 0 is a constant. Note that the phase plots in Fig. 4-9 satisfy the equation a(x + X/2) = -a(x) where X/2 Ca(x) = P(x)- (1/X) / P(x) dx x' 2 The sinusoidal phase function eA cos(wx) also has this 90 degree phase propertyand canbe fabricated rather easilyusing photographic bleaching techniques. Thus, some of the solutions to Eqs. (6) can be formed by using a simple discrete optical filter to properly weight the Fourier series coefficients of a sinusoidal phase grating. As a

111 simple example, consider the function eiA cos (wx) where the constant A is chosen so that the lower order Fourier series terms are al(x) = -0. 097 + i 0. 942 cos (wx) - 0. 92 cos (2wx) - i 0. 47 cos (3wx) The DC term of al(x) can be removed by spatial filtering to yield a2(x) = al(x) +,0. 097 and hence, la2(x) 2= 0. 977 + 0. 001 cos (2wx) - 0. 02 cos (4wx) + 0. 11 cos (6wx) This particular example was achieved experimentally. N A third method of synthesis for functions q(x) = 2 a e n=-N with the property argt ana*+ 1 = ~r/2 becomes evident if the Fresnel free space transfer function H(fx, f ) is examined. ikz -irXz(f 2+f 2 H(f f e e x y where f and f are spatial frequencies for the x and y coordinates respectively of a Cartesian coordinate system. For separable functions only a one-dimensional transfer function need be considered. For

112 periodic functions with period X = 1/f, f = nf. and H becomes 2 2 -ikr Xzf n ikz o H(nf ) = e e 0 Now if 7rXz f = 2rJ - 7r/2 where J is an integer, O O - ikz e 0H(nf ) -i if n is odd - 1 if n is even or zero The Fresnel diffraction pattern at a distance z from q(x) becomes N -N -in27rf x iO =e Of(x) i0 where f(x) is a real function and e o is a complex constant. Since f(x) is real, a filter with amplitude transmittance t(x)= f(x) - min {f(x)} t(x) max {f(x)} - min {f(x)T can be fabricated photographically. Then if an optical filter is used to remove the bias term -min {f(x)} [max {f(x)} - min {f(x)}]-, the complex modulation q(x) is achieved by imaging a plane a distance z from the filter. 0 Optical filtering noise will limit the quality of the illumination obtained by the second and third synthesis methods.

113 4.7. DIFFRACTION EFFICIENCY CONSIDERATIONS In order to record holograms with high diffraction efficiency in Fourier transform holography, it is necessary that IFT{q(x)} 2 be rather uniform. This condition can be met if the solutions given in Table 4-1 are used in tandem with a coarse lens array to obtain a new modulation function q(x) = q1(x)q2(x); where q1(x) is a solution to Eqs. (6) such as Entry F of Table 4-1, and q2(x) is a lens array with a period X2 large enough to satisfy the equations given below. N FT{q1(x)} I a6 ( X1 ) n= -N f is a spatial frequency variable. If x FT{q2(x)} rect (Xlfx) then 2 l fx FT{q1(x)q2(x)} 12 ret 2N+ 1 An experiment illustrating this concept was performed using the parameters listed below 2 7rnx 1 i,.1 q1(x) = an e, where {a} = {i, 1, i} and n=-1 n=-l X1 = 1/(16) mm q2(x) = qA(X) of Chapter 2 where X2= 0.58mm and Q = 9.28

114 Figures 4-10(a) and 4-10(b) show FT{ql }2 and FT ql q2} respectively at the Fourier transform plane of a bandlimited coherent imaging system. The imaging system was the same as that shown in Fig. 3-2 and had bandwidth (4 - c)/X1 where E << 1. Figures 4-11(a) through 4-11(d) show the image plane of the system when q(x) equals 1, q2(x), ql(x), and ql(x)q2(x) respectively. In each of these figures, two wires were placed between the imaging system and image plane. These two wires introduce two dark horizontal lines when q(x) = 1. The upper and lower wires were 0. 235 mm and 0. 112 mm respectively in diameter, and were placed at the respective distances 5. 5 cm and 10. 0 cm from the image plane. (a) Spectral Density of ql (b) Spectral Density of qlq2 Figure 4-10. Irradiance at System Fourier Transform Plane

115 ( a) q(x) = 1 (b) q(x)- q2(x) (c) q(x) - ql(X) (d) q(x) - ql(x)q2(X) Figure 4-11o Irradiance at the Image Plane with Two Wires Introduced as Artifacts

116 Figures 4-11(c) and 4-ll(d) Show that the addition of a coarse lens array can increase noise dispersion uniformity and thus aid the noise suppression process if the focal length of each lenslet is small relative to the scatterer to image distance. If the period of the lens array is chosen sufficiently large, its self-imaging distance willbe large relative to the optical system dimensions. Geometrical optics predicts that a choice of negative lenslets for q2 is sometimes necessary to assure that q2 does not reduce noise suppression at some points in the system. 4.8. INCREASING THE REDUNDANCY OF PERIODIC MODULATION BY REDUCING THE SPECKLE FREQUENCY REQUIREMENTS In the previous sections the low frequency speckle noise was minimized by imposing Eqs. (6) which require that each of the low frequency coefficients It, I2,...I2N 1 be zero for a grating with 2N + 1 orders. Solutions were then found which optimized the spectral density.uniformity. It was seen that solutions with good spectral uniformity exist. For cases where solutions with good redundancy characteristics do not exist, it is logical to increase the dimension of the solution space to J by requiring that only the first 2N - J of the coefficients 2N-1 {In} be zero, where J > 1. Of course, if J is not small. relative n=l to 2N, the storage density of an optical system will be significantly reduced. An alternative approach is to impose spectral uniformity by requiring that R1 equal 2N + 1. Equations (6) can then be studied to see if J can be made small. A group at RCA reported work which is related to this latter approach [23]. They examined the Fresnel diffraction pattern generated by a pinhole array. The characteristics of a one-dimensional pinhole array are derived in Appendix IV. Computer calculations reported in this appendix show that the redundancy characteristics of a pinhole array are not optimum for high density storage systems.

5 CONCLUSIONS Chapter 1 investigated the limitations that laser speckle places on the image quality and resolution of coherent imaging systems. Mixed-integration processing which reduces the speckle fluctuation can improve image contrast. Mixed integration can be achieved efficiently using a spinning lens array. An important result given in Section 1. 3. 2 was the equivalence between noncoherent techniques which continuously sample a signal's spectrum and a simple irradiance filter. This result helps clarify what performance can be expected from various mixed integration methods. An analysis of speckle reduction techniques led to the moving grating method which proved to be an effective means for introducing redundancy into coherent spatial filtering systems. Since optical artifact noise can be suppressed in coherent optical filtering systems by reducing tie source coherence, it is advantageous that the source coherence not exceed the system requirements. Most noncoherent noise suppression techniques are not useful for diffraction limited holographic storage systems. In order to coherently image continuous-tone transparencies, special types of bandlimited redundancy modulation must be considered. Redundancy modulation must satisfy two constraints. First, the modulation must be free of low frequency speckle. Second, the spectral density of the modulation must be rather uniform. There exist two mechanisms for optical noise suppression. One mechanism is to increase the spatial bandwidth of the noise irradiance. The principle mechanism for noise suppression is to disperse optical noise 117

118 more uniformily at the image plane. Quadratic phase modulation has good redundancy characteristics if the space bandwidth parameter Q is much greater than ten. Computer results and experimental evidence were presented which show the redundancy characteristics of quadratic phase modulation as a function of Q. Periodic functions are the most promising form of redundancy modulation for the high density storage of continuous-tonle transparencies. The noise suppression performance of this class of functions was analyzed in terms of a family of redundancy parameters. When speckle-free periodic redundancy illumination is perturbed by a system artifact, some low frequency speckle is introduced. Limits on the low frequency speckle modulation were established which show how speckle modulation is related to the spectral uniformity of the redundancy illumination. Recently a system was described in the literature which used a pinhole array filter to generate redundancy illumination [23]. Appendix IV discusses the redundancy characteristics of a pinhole array illumination and shows that this type of modulation is not optimum for the high density storage of continuous-tone transparencies. A computer program developed to investigate selfimaging characteristics on an arbitrary periodic function from its complex Fourier series coefficients was used in this analysis. A system of nonlinear equations must be solved in order to determine periodic modulation which is free of low frequency speckle. For the class of even function solutions obtained by Gabor, a 90 degree phase difference existed between adjacent Fourier series coefficients. The recurrence relationship given by Gabor to generate higher order solutions is not valid for a function with more than 7 nonzero Fourier coefficients. Solutions do exist, however, and have been calculated for functions with up to 13 nonzero orders. Numerical techniques were required to solve these equations. The spectral characteristics of the solutions considered can be predicted from

119 the redundancy parameter plots. Although even function solutions without the 90 degree phase condition are more difficult to obtain, more general solutions have better spectral uniformity and thus higher redundancy. In particular, a 9 order grating defined by entry F of Table 4-1 appears to meet quite well the redundancy requirements of holographic microfiche systems which store continuous-tone transparencies. One method of synthesizing these solutions is to directly fabricate a filter with the proper complex amplitude transmittance. A computer program was developed to plot the continuous phase function for a particular solution from its Fourier series coefficients. For solutions with a 90 degree phase difference between adjacent complex Fourier coefficients, two indirect synthesis methods requiring a spatial filtering operation were presented. Redundancy modulation with good diffraction efficiency characteristics can be obtained by using a coarse lens array in tandem with a periodic grating solution. Experimental results demonstrated that the addition of a lens array can improve the noise suppression characteristics of this class of functions.

REFERENCES 1. L. I. Goldfisher, J. Opt. Soc. Am., 55, 247 (1965). 2. R. W. Lewis, "Mixed-Integration Processing of Synthetic Aperture Radar Data, " Technical Report AFAL-TR72-215, Air Force Avionics Laboratory, Air Force Systems Command, Wright-Patterson Air Force Base, Ohio, August 1972. 3. L. J. Porcello, N. G. Massey, R. B. Innes, and J. M. Marks, "Diversity and Mixed-Integration Processing in Synthetic Aperture Radar (U), " Fourteenth Annual Tri-Service Radar Symposium Record (U), Vol. II, Willow Run Laboratories of the Institute of Science and Technology, The University of Michigan, Ann Arbor, July 1968, pp. 114-142 (SECRET). 4. L. H. Enloe, Bell Sys. Tech. J., 46, 1479 (1967). 5. J. I. Marcum and P. Swerling, Special Monograph Issue, IRE Trans. Inform. Theory, IT-6 (1961). 6. A. Kozma, E. N. Leith, and N. G. Massey, Appl. Opt., 11, 1766 (1972). 7. C. E. Thomas, Appl. Opt., 7, 517 (1968). 8. J. Upatnieks and R. Lewis, Opt. Soc. Am. Meeting Program, Oct., M13 (1972). 9. G. B. Brandt, Opt. Soc. Am. Meeting Program, Oct., B16 (1972). 10. G. B. Brandt, U. S. Patent 3620598. 11. M. Elbaum, M. Greenebaum, and M. King, Opt. Commun., 5, 171 (1972). 12. E. N. Leith and J. Upatnieks, Appl. Opt., 10, 2085 (1968). 13. F. Talbot, Phil. Mag., 9, 401 (1836). 14. J. W. Goodman, Introduction to Fourier Optics, McGrawHill Book Co., Inc., New York, 1968. 15. M. Born and E. Wolf, Principles of Optics, Pergamon Press, London, 1965. 16. J. Upatnieks, Appl. Opt., 6, 1905 (1967). 17. J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, Bell Sys. Tech. J., 39, 745 (1960). 18. E. N. Fowle, IEEE Trans., IT-10, 61 (1964). 120

121 19. S. S. Hsiao, "Analysis on Information Reduction and Beamwidth Reduction in Holography, " Ph. D. Dissertation, University of Michigan, Ann Arbor, Michigan, p. 198, 1972. 20. D. Gabor, IBM J. Res. Develop., 14, 509 (1970). 21. F. Zernike, J. Opt. Soc. Am., 40, 326 (1950). 22. J. W. Cooley and J. W. Tukey, Math. Comput., 19, 297, (1965). 23. A. H. Firester, E. C. Fox, J. Gayeski, W. J. Hannan, and M. Lurie, RCA Review, 33, 131 (1972). 24. H. J. Gerritsen, W. J. Hannan, and E. G. Ramberg, Appl. Opt., 7, 2301 (1968).

APPENDIX I GENERAL SOLUTIONS TO EQUATIONS (6) N Consider a solution sequence {an} to Eqs. (6) of Chapter n=-N 4 when a._ a and N = 1. If -n n {a} = {i, a, b} n=-l Eqs. (6) imply that b = a a a* Thus if ie a=aje a the solution sequence 1 I i i20 J. I I a. a {a} = i a e ie n=-l is obtained. When the two free parameters 0a and I a J are set equal to zero and one respectively, the sequence {an} becomes {i, 1, i} n=l This particular sequence is an even function solution and can be generated efficiently using the first three orders of a sinusoidal phase grating [ 24]. 122

123 Next, consider a grating for N = 3, which has four nonzero 3 coefficients. If {a} n = {b1, 0, b2, 0, b3, 0, b4}, the maxin=-3 mum limit for R1 is 4. Since b2 can be set arbitrarily equal to 1, equations system (6) consists of 4 equations and 6 unknowns. The solution sequence ~b~~~~~ b - 1 lb312, 0, 1, 0, b3, 0,' 2{ for these equations has two free parameters: lb31 and arg {b3}. It is clear that the maximum limit for R cannot be attained since b3 I = 1 implies that Ibl and Ib4l are equal to infinity. Finally, consider the solution sequence {bl, b2, b3, b4, bs} for N = 2. By setting b1 equal to 1 and by reducing equation system (6) to 4 unknowns and 2 equations, this sequence becomes b*b -b 2 3 3 1, b b2, 23' b* - b* b* - b*2 3 2 3 2 where b2 and b3 must satisfy the complex equation lb31 2b2 Ib312b* b + b b* + - (8) (b3 - b2) b3 - b2 (8) Suppose b2 = 1 + i. After reducing Eq. (8) to a cubic equation, the following roots are obtained: b = i, i(2 + 2,12), i(2 - 242) One possible solution sequence is

124 2 {a} = {1, 1 + i, i, 1 - i, -1} where a - bn+3 n=-2 The value of R1 for this sequence is 3. 5. It will now be established that no solution exists with R1 equal to the maximum limit 5. Suppose R1 = 5. Then 1b21 = Ib31 = 1, and the solution sequence becomes i 2 3o i(03-02) io3 2 e3 e e J I e 31 3 2 3~3 2 1 e, e 1 e e Since 1b41 and j b51 also must be unity, this requires that (03 - 202)= It can be shown using Eq. (8) that this implies 2 cos (02 t 7r/3) = cos 02 and 2 sin (02 7 ar/3) = sin 02 A contradiction is now apparent since these equations imply 1 = 4. The above examples have shown that obtaining general solutions to Eqs. (6) is difficult even if N is small. A computer search to maximize RM must be over a two-dimensional parameter space. The system of nonlinear equations differs for each point in the

125 parameter space; and algorithms to solve these equations require an initial guess. Fortunately, it will be seen that if a = a, a -n n more modest computer effort will result in a solution sequence with R near the maximum limit 2N + 1 for N = 4. 1

APPENDIX II GENERAL EVEN FUNCTION SOLUTIONS FOR N = 4 The computation required for obtaining general even function solutions to Eqs. (6) when N = 4 is described here. Relation (C) 4 of Chapter 4 reduces the solution sequence {a } to n= -4 {i, b cR i 0. 5 bR dR - ib e, R' R R' R RR As a shorthand notation, let Eq. (n, k) denote equation N-k anan = 0 n n+k n= -N of the 2N - 1 Eqs. (6) of Chapter 4. Eq. (4, 4)=zeI = -(bRdR+ 0.5 cR + 0.125 bR ) 3 Eq. (4, 3) -.. bRCR+2b bRC R + 2bRe R + 0.5R )= and 2beR+ 2cR(d+ 0.5 bR - bR) = 0 Using eI from Eq. (4, 4) Eq. (4, 2)~bR (-1 + 1.5 cR + bRd + 0.125 b ) + 2bRd + dR + 2cReR = O 126

127' Eq. (4, 1),2bRcR(1+ bd + 0.125 bRdR + 5 cR) + 2cR(dR + 0.5 bR ) + 2eRdR = If d = 0, it can be shown that R1 < 6, hence this case will be R 1 ignored in the programming described below. Assuming dR differs from zero, the equations above reduce to the equation 4 2n (Zn lcR )= 0 n=0 The coefficients Z(n + 1) are defined in terms of the parameter BR = bR by the algebraic steps in the subroutine COEFPN(BR, Z) given below. SUBROUTINE COEFPN(BR, Z) DIMENSION Z(11) BR2=BR*' BR BR4=BR2*BR2 BA1=BR2*(2. +BR2+0. 1875*BR4) BA2=0. 5*'BR2 AL=0. 25''BR*(4. +BR2) AL2=AL*AL AL3=AL*AL2 C2=BR*(2. +BR2) AC1 =BR2*(- 1. +. 125'-BR4) BC1=. 5''BR2-2. ACO=-2. BR(1. +. 5+0.5*BR2+0. 125*BR4) El = 3. -,AL2+C2`'2. *AL+AC1+BAl E2=BA2+BC1 E12=E2*E1 E22=E2*E2 E 11 =El 1'E1 P0=BA1'*E 1 P2 =BA2 *E 1 1 +BA1 9*2. *E1 2 P4 = BA1 *E 2 2+2. *BA2 *E 12 P6 = BA2*'E22 F31 = AL3+3. *AL*BA1

128 G31=3. *AL'BA2 HO= F3 1+C2'*(AL2+BA1 )+A LsAC 1 H2 = G 3 1+C2 *BA2+A L*B C 1+ACO Q0O=H HO H0 Q2= HO*H2 *2. Q4= -B R*H0*2. +H2*H2 Q6=-2. *B BR*H2 Q8=BR2 Z(1)=Q0-PO Z(2)=Q2- P2 Z(3)=Q4- P4 Z(4)=Q6- P6 Z(5)=Q8 RETURN END If the variable ZREAL denotes a positive real root of the polynomial 4 Z wn= 0 n+1 n=0 then subroutine ZORDGN given below calculates the sequence 4 { an where n= -4 a = ZAR(n + 5) + iZAI(n + 5) Choosing - ZREAL instead of 4ZREAL for cR in this subroutine would simply change the sign of a few of the sequence terms ZAR(n) and ZAI(n).

129 SUBROUTINE ZORDGN(BR, ZREAL, ZAR, ZAI, ZMOD) DIMENSION ZAR(13), ZAI(13), ZMOD(13) CR= SQRT( ZREAL) BR2=BR*BR BR4=BR2*BR2 AL=O. 25*BR*(4. +BR2) BA1 =BR2*(2. +BR2+0. 1875*BR4) BA2=0. 5*BR2 DR= AL+SQRT(BA 1+BA2*ZREAL) EI=-(BR*DR+O. 5*ZREAL+BR4*0. 125) ER=-BR*(CR/DR)*(1. +BR*DR+O. 5*BR2 1+0. 125*'BR4+0. 5*ZREAL)-CR ZAR(1)=0. ZAR(2)=BR ZAR(3)= CR ZAR(4)=DR ZAR(5)=ER ZAI(1)=1. ZAI(2)=0. ZAI(3)=-BR2*0. 5 ZAI(4)= -BR*CR ZAI(5)=EI DO 20 I=1, 4 ZAR( 5+I)= ZAR( 5-I) 20 ZAI(5+I)= ZAI(5-I) DO 30 I=1, 9 30 ZMOD(I)=ZAR(I)*ZAR(I) + ZAI(I)*ZAI(I) RETURN END The main program used these subroutines to plot the redundancy characteristics of solution sequences when BR was varied from -4 to 4 in 0. 01 increment steps. Appropriate program logic, in the form of a subroutine, monitored all the conditions imposed in the algebraic steps above. The solution sequence which maximized R1 was part of the output data from the program. Finally, to assure that no errors were made, an additional subroutine determined directly that each sequence generated satisfied Eqs. (6) of Chapter 4.

APPENDIX III GENERAL EVEN FUNCTION SOLUTIONS FOR N= 5 The difficulty in obtaining general even function solutions to Eqs. (6) for N greater than 4 is presented here. It will be shown that iterative methods requiring an initial guess are necessary to solve the system of nonlinear equations when N is 5. After imposing Relation (C) of Chapter 4, {a } becomes n= -5 {i, b cR - i 0.5 bR de - ib e Rd + 0. 5 c2 R' R R' R RCR; R RR R + 0. 125 bR ), f + if Eqs. (5, 5), (5, 4),... (5, 1) eventually reduce to a system of two nonlinear equations and two unknowns. The two unknowns are cR and dR, and the free parameter is bR. The variables a, z, eR1 and eR2 are functions of cR and dR, and are defined below in order to reduce the burden of listing the two equations. Let 2 4 a(bR, CR dR) - bRdR + 0. 5 cR + 0.125 bR 22 2 2 z(bR, cR, dR, a) bR (a -1 + bR cR 2d 2 R dR 2 +a -— R (-c + 0. 5 dR ) 130

131 eRl tbR. 0R'. dR>~c (1~+bR2 ibR) bR, CR, dR~ R1 R R R dR'Z) R lbR b \ ~~R 2 eb (b R c, d2 z) 1+b 2 R R'\D CR-PR R' R\ R b/ \../~R Then the system of two equations that must be solved provided bR t 0 is /2 2 2'" 2' 2c 2/ 2CR V 3 R2d R eR2 -b b + -b2 i ~R b R R R1 R R b 72 2 Rb\2 + acR - +2bRb1 4 2 cR dR 2]2 +bcR0. 5 b4 +bRdR -2 ) RR R bRCR /bR R R - — /R and 4e C \+5d 2>2 2 2d R1 R 2 2 2 2 R2LR R RbR R R bR R~~~~R r' 3~~~~~ 2cR bR3 + dR+ 2e d 2e / 22 2 2 2 -a+ 0.5 d + 0. 5 dRb c + c b R 2RR R R1 R__ 2 + ab e + 2a bRC + C d + 0 5 b 3R b R RR2 RR1 R R R' R R

132 Once bR is chosen and these equations are solved, the remaining orders can be determined from equations: R R1 R2 f fR + 0 5d + 0. 5 b C + CRe bR' R' R R RR1 CReR2 + 0.5 bR a fI -(bReR1 +bReR2 + CdR + 0.5 bR CR Iterative methods such as Gauss-Seidel or Newton-Raphson must be used to solve the remaining equations. Since such techniques require an initial guess, the computer time and programming effort to maximize R1 are very great. The work required to maximize RM can be reduced somewhat if it is decided that the only solutions of interest are those where RM is near the maximum limit. For example, R1 equals 2N + 1 if and only if bR J I c i, and I d I equal unity. Imposing this condition narrows the choice for (CR, dR) to either (~0. 53, ~0. 5) or (~0. 543, T 0. 5). If it is found that I f I is near 1, the values for (cR, dR) suggested can be used as initial guesses for the iterative techniques required to solve the two equation system. If interesting solutions develop, bR can be varied slightly in order to maximize R1. This approach should help reduce the programming effort, if general even function solutions which maximize R1 are required for N > 4.

APPENDIX IV PINHOLE ARRAY CHARACTERISTICS This appendix evaluates work performed by a group at RCA [23]. This group reported that a pinhole array with period X could generate redundancy illumination with period X/J where J is an integer. It will be shown that this redundancy modulation is not optimum for high density storage applications since J cannot exceed certain limits. N Consider a grating q(x) = a e with complex Fourier N n=-N coefficients {a }.. It can be shown that n=-N 2N |q(x) 2 = IO + 2In cos (2rnf0x- xn) n=l where w = 2rf0 N-k Ik = a"a k n n+k n=-N and N-k l = argt ( a a+ ( n=-N 1 133

134 iC~z Plane Wave I I q(x,O) q(x,z) Figure IV-1. Plane Wave Incident on Grating q(x) The Fresnel diffraction pattern q(x, z) 2 will now be determined when a plane wave is incident on the grating q(x) as shown in Fig. IV-1. N 2 -iirXz(nf0) inwx q(x, z)= e a e n n= -N -irXz(nf )2 Since the new Fourier coefficients are now a e z(nf n=-N N-k i rXzf02[n2-(n+k)2] I = a*a e n=-N ~ N-k aa*eilrf z(2nk) anan ke i n= -N 6

135 N N The investigators at RCA imposed the condition {a} = {1} n=-N n=-N This condition can be easily achieved experimentally using a low duty cycle pinhole array. This means the redundancy parameter Z' aj 2 R1 - is equal to 2N + 1 (the number of plane waves). max{ an12} The coefficient Ik becomes k N-k 2 i i27rXf znk Ik=Z e n —N Let 2 rXzfO = 7rc Then c Xf If c = 1, z represents the self-imaging distance and N-k i2wkn Ik= e i n=-N Note that if k << N and N becomes very large

136 I w k i e ( j 6(c k _fn o n= - oo n= -oo thus if c = 1/J where J is a positive integer, 00 Ik I j 6(k - nJ) n= -oo and Ik O if k / nJ Let N-k 2kn S= - el~r J n=-N If k f 0, ( 27k j i27rk (N-k+l) -i27k N S e.-1 =e -e and sin[ 27rk N+- (1 - k) I -S sin (.) If J = 2, it is clear that Ik = 0 for k = 1, 3, 5,.... Let N = L(J/2), where L is a positive integer. Then if

137 /~2 2N I X, Jxf2 I0~ 2In cos (27rnf0x - n) Jf 0 n=1 I = 2N + 1 and sin ~.-1 — {1 - k) Ik[ if k > 0 ~k sin r Thus, in general, the condition Ik 0 if k f nJ does not hold unless J = 2. It can easily be shown that J = 2 implies arg {a a*+1} = +~/2. For example, if N = 3 and J = 2, 3 {a} = {-i, 1,-i, 1, -i, 1,-i} n=-3 Since a = a and there are an odd number of sequence terms, -n n the condition Ik = 0 if k / 2n is rather obvious. Consider the case N = 3 where 3 3 {a} = {1} n=-3 n=-3 If z = 0, 2 6 2N+ {Ik} = {1.71, 1.43, 1.14, 0.86, 0.57, 0.29} If J = 2 or z = 1/2, the self-imaging distance

138 2N +i = { 0., 1.43, 0., 0.86, 0., 0.29} 2N + 1 kk'' If J = 3 or z = 0. 33, the self-imaging distance of the grating 2 + {Ik} = {0., 0. 29, 1. 14, 0., 0.29, 0. 29} 2N+ i k k.l For J = 3, note that 2N +I 0. 29 is rather large, since 1 2N + 1 Io = 1. Instead of demanding that R1 equal the maximum limit 2N + 1, 00 consider a grating q(x) = rect (x/A) * 6(x - n) where the =-o00 constant A < 1. The amplitude transmittance of q(x) is shown below. co Fiur2:x Figure IV- 2. Amplitude Transmittance Let RDC = 1/A be the reciprocal of the duty cycle of the function q(x). If

139 oc q(x) = a e where w - 2-f 11 0 n= -c then a oc sine -RDC where sine x =. -. X n RDC 7rx This function is shown in Fig. IV-3. _ -RDC n Figure IV-3. Sinc Function If RDC is a positive integer, the number of nonzero orders within the main lobe of the sinc function is 2 * RDC - 1. Suppose that a low-pass optical filter is used to filter out all but the lowest 2N + 1 orders generated by this grating. Now N-k i27rkn Ik L n n=k n=-N

140 if 2N q(x' 2 o + 2I cos (27rnfx - ) ~0 In=1 If 2N + 1 does not exceed 2' RDC - 1, then N N {arg (an)} = { 0 n=.-N n=-N and j an j decreases monitonically as Inl increases. Table IV-1 given below shows the dependence of {2Ik} on RDC, J, and the number of plane waves 2N + 1. It is seen that the condition Ik ~ if k f nJ is more accurate if the Fourier series coefficients are heavily weighted. In Table IV- 1 N 2 Z/a n=-N max{la I2} n and q(x, z) has been normalized so that I 1. In Table IV-1 it is shown that if RDC is adjusted so that R1 is one-half the maximum limit 2N + 1, the frequency of the diffuser noise is about one-half the maximum limit 2Nf0. Note that if R= (2N + 1), (2 * RDC - 1) 2N + 1. This 12 implies that 2 - 0 as | n - N and thus the power spectrum of q is not uniform. Since RCA's actual system did not require

141 a fringe pattern finer than the basic period of the pinhole array, 10 the coefficients {a } used for their system were not heavily n=-10 weighted. A computer program which was written to evaluate the selfimaging characteristics of gratings was used to obtain the numerical data presented in this appendix. Table IV-1. Pinhole Array Data RDC (2 RDC -1) (2N+ 1) R1 J {2Ik} for I 1 k=1 5 9 9 4.51 2 {0., 1.63} 5 9 9 4.51 3 {0.0086, 0., 1.27} 5 9 9 4. 51 4 {0.02, 0.02, 0.0096, 0. 879} 5 9 9 4.51 5 {0.106, 0.023, 0.026, 0.216, 0.532} 5 9 9 4.51 6 {0.33, 0., 0., 0.059, 0.2933, 0.269} 5 9 9 4.51 8 {0.78, 0.03, 0.006, 0.04, 0.05, 0.11, 0.097, 0. 024} 50 99 9 8. 92 4 {0.001, 0.22, 0. 31, 1.11} 11 21 21 9.93 9 {0.013, 0.004, 0.0029, 0. 0043, 0. 001, 0. 0053, 0.013, 0.042, 0.86} 20 39 21 15.87 9 {0.06, 0.05, 0.004, 0.06, 0.020, 0.0034, 0.030, 0.21, 1.15} 40 79 21 19.50 9 {0.15, 0.082, 0.001, 0.087, 0.030, 0.001, 0.05, 0.261, 1.15}

UNIVERSITY OF MICHIGAN 111111111 0346 46111111111111175111111 3 9015 03466 4675