Optimal pixel profiles for spatially discrete coherent imaging systems

15 June 2001

Optics Communications 193 (2001) 87±95

www.elsevier.com/locate/optcom

Optimal pixel pro®les for spatially discrete coherent imaging systems Mark A. Neifeld *, Binling Zhou ECE Department/Optical Sciences Center, University of Arizona, Tucson, AZ 85721, USA Received 25 September 2000; received in revised form 23 February 2001; accepted 16 March 2001

Abstract The term ``pixel pro®le'' refers to the spatial distribution of electric ®eld amplitude within the aperture of a single pixel of a spatial light modulator. We discuss the use of non-uniform pixel pro®les for increasing the energy throughput of spatially discrete coherent optical imaging systems. Prolate spheroidal pixel pro®les are shown to be optimal and can produce throughput gains of up to 32% for optical systems with rectangular symmetry and 27% for those with circular symmetry. These throughput gains are shown to impact the capacity of volume holographic data storage systems. A simple M=#-based analysis predicts a 15% capacity increase as compared with the use of uniform pixel pro®les. Ó 2001 Published by Elsevier Science B.V. Keywords: Optical imaging; Image ®delity; Optical data storage; Array illuminator; Spatial light modulator

1. Introduction Two-dimensional (2D) coherent imaging of spatially discrete arrays is a required function within many di€erent applications including optical interconnects, volume optical storage, optical computing, and optical signal processing. In all of these applications overall system performance can be enhanced via improvements in image ®delity. For example, improved image ®delity has been shown to result in increased storage capacity within a volume holographic storage environment [1,2]. Motivated by such system improvements, ®delity-sensitive imaging has been the subject of

*

Corresponding author. Fax: +1-520-621-8076. E-mail address: [email protected] (M.A. Neifeld).

several recent e€orts focusing on the relationships among noise, space±bandwidth product, bit-errorrate, and ®delity-based optical design [3±7]. Recent algorithmic studies have resulted in coding and signal processing techniques that facilitate high®delity imaging of binary-valued arrays [8±12]. In all of these studies the use of uniform rectangular modulator pixels is assumed. However, such a uniform pixel pro®le is suboptimal owing to two energy-loss factors: lowpass loss and detector loss. The ®rst factor arises from the lowpass nature of optical systems. The resulting loss of high spatialfrequency information carries with it an optical energy throughput loss. The second factor arises from the detection process. The blur that results from a loss of high spatial-frequency content can produce signal energy falling outside the area of the desired detector creating a further reduction in energy throughput. In this paper we present the

0030-4018/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 2 5 7 - 3

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M.A. Neifeld, B. Zhou / Optics Communications 193 (2001) 87±95

Fig. 1. 4F system used to model spatially discrete coherent imaging. System includes a laser source (LS), collimation optics (CO), array illuminator (AI), spatial-light-modulator (SLM), lenses (L1 and L2), aperture (W), and a detector array (DA).

2D coherent ``signalling'' problem. Treating the coherent imaging system as a linear (in ®eld), shift invariant, lowpass ®lter, we seek the spatial ®eld distribution within a single modulator element (i.e., pixel pro®le) that maximizes energy throughput and therefore o€ers the optimum pixelwise signal ®delity under the assumption of ®xed detector noise. We study systems with rectangular symmetry as well as systems with circular symmetry, o€ering theoretical and experimental results for both types of system. Spatially discrete coherent imaging systems comprize four basic components: (1) a coherent source together with illumination optics, (2) a pixellated object to be imaged, (3) some imaging optics, and (4) an image plane typically consisting of a 2D photodetector array. For the case of diffraction-limited imaging, all such systems can be conveniently modelled by the so-called 4f system shown in Fig. 1. This ®gure depicts the four components listed above: (1) a laser source and collimation optics together with a (optional) 2D array illuminator, (2) a 2D spatial-light-modulator (SLM), (3) two ideal lenses together with a ®nite aperture located in their common Fourier plane and (4) an array of square detector pixels each of which produces an output signal that is proportional to the spatial integral of the local intensity. There are several important reasons to include an array illuminator within the system of Fig. 1. Firstly we are motivated by the growing importance of so-called ``smart-pixel'' SLM technologies that co-integrate electronic (smart) processing with 2D modulation [13±15]. The circuit area required

to implement electronic functionality in these smart-pixel SLMs results in a signi®cant reduction in modulator ®ll-factor. This loss in ®ll-factor will produce an unacceptable loss in optical eciency unless an array illuminator is used to direct an appropriate set of addressing beams to/from the SLM [16±18]. Secondly, given the need to address modern smart-pixel SLMs with a spot array, we are motivated by the potential to customize the spatial pro®le of the resulting spots. In particular, if a Fourier-type array illuminator is used, then the pro®le of each spot in the array is given by the Fourier transform of the incident illumination. A simple input beam apodizer can thus be used to program the spatial pro®le of the spots produced by the array illuminator, providing us with the freedom to optimize this pro®le for the purposes of improving the detected signal ®delity. Determining the optimal ®eld pro®le with which to illuminate each pixel of the SLM is the goal of the present work. Note that modulator-based smart-pixel technologies are generally re¯ective; however, the system of Fig. 1 has been presented in a transmissive mode for convenience. Also note that diffraction-limited imaging systems that do not make explicit use of the 4f architecture can be represented by an equivalent 4f system via suitable choice of the Fourier aperture; however, systems that su€er from signi®cant aberrations cannot be considered shift invariant and do not ®t well within this formalism. An indication of how the results of this work might be applied to such an aberrated optical system will be provided in Section 5.

M.A. Neifeld, B. Zhou / Optics Communications 193 (2001) 87±95

2. Optimal pixel pro®les The 4f system shown in Fig. 1 is shift invariant and linear in electric ®eld. We can therefore limit our study to a single pixel on the SLM and its image in the output plane. We can model the input±output relationship for this system as a 2D convolution between the input pixel pro®le pin …r† and the system point-spread-function (PSF) hW …r†. The PSF is simply the Fourier transform of the Fourier-plane aperture AW …f† whose width is W. The input pixel pro®le is determined by both the illumination pro®le and the ®nite SLM pixel size X. Speci®cally, we include the e€ect of ®nite SLM pixel size by requiring that pin …r† ˆ 0 outside the ®nite 2D interval X. The output of the system in response to R such a single-pixel input is given by pout …r† ˆ X pin …r0 †hW …r r0 † dr0 . Note that any system magni®cation can be accommodated via use of these normalized quantities. We de®ne the output energy of this system to be proportional to the integral of the intensity over a single detector pixel. Taking the magni®cation of the system to be unity, and setting the detector pixel size equal to the SLM we compute the output energy R pixel size, 2 Eout ˆ X jpout …r†j dr. De®ning the input in R energy 2 a similar fashion we have Ein ˆ X jpin …r†j dr. Finding the pixel pro®le that optimizes energy throughput for this 4f system is equivalent to selecting pin …r† to maximize the criterion function lp ˆ Eout =Ein . A simpli®ed expression for this energy throughput criterion function is given by R R  pin …r0 †pin …r00 †KW …r0 ; r00 † dr0 dr00 lp ˆ X X ; …1† R jp …r†j2 dr X in where 0

00

KW …r ; r † ˆ

Z X

hW …r

r0 †hW …r

r00 † dr

…2†

is a kernal function that contains all information about the optical system PSF. We begin with the case of rectangular symmetry. In this case the array illuminator is assumed to produce a spot pro®le that is separable in rectangular coordinates so that pin …r† ˆ px …x†py …y†. We de®ne the ®nite interval X to be x, y 2 ‰ 0:5; 0:5Š and we assume the use of a Fourier aperture of the

89

form AW …f† ˆ rect…fx =W †rect…fy =W †. Under these assumptions, Eq. (1) for lp can be re-written in terms of a single spatial variable (say t) as " R 0:5 R 0:5 #2 0 00 0 00 0 00 p …t †p …t †K …t ; t † dt dt t t W 0:5 0:5 lp ˆ ; …3† R 0:5 2 jp …t†j dt 0:5 t where KW …t0 ; t00 † ˆ W 2

Z

0:5 0:5

sinc‰W …t

t0 †Šsinc‰W …t

t00 †Š dt: …4†

Finding the function pt …t† that maximizes lp would be simple if we could ®rst ®nd a complete set of orthonormal basis functions on the ®nite interval ‰ 0:5; 0:5Š, that also happen to be eigenfuncR 0:5 tions of the linear operator L‰pt …t†Š ˆ 0:5 pt0 …t†  KW …t0 ; t00 † dt0 . Such a set of basis functions exist and are given by the angular prolate spheroidal functions of the ®rst kind [19]. These are functions that are self-similar under Fourier transform on a ®nite interval and ®nd important uses in many areas of science and engineering. Writing ptP …t† in terms of 1 these basis functions yields pt …t† ˆ iˆ1 ai Si …W ; t†, where fSi …W ; t†g is the set of eigenfunctions associated with the kernal whose width parameter is W and fai g are the projections of the pixel pro®le pt …t† onto each of these basis functions. Substituting this expression for pt …t† into Eq. (3) yields P1 2 P1 2 2 lp ˆ … iˆ1 kWi jai j = iˆ1 jai j † where fkWi g are the eigenvalues associated with the eigenfunctions fSi …W ; t†g: The maximum value of lp occurs for a1 ˆ 1 and ai ˆ 0 8 i 6ˆ 1, in which case the opti2 mum energy throughput becomes lp ˆ …kW1 † . Fig. 2 depicts the optimum pixel pro®les for several values of W. Notice that as W is made larger, higher spatial frequency content is present in the optimum pro®le. Also shown in Fig. 2 are the corresponding output functions that result from passing these optimum pro®les through the 4f system. Because these pro®les are eigenfunctions on the ®nite interval, the output functions are proportional to the input pro®les in the range t 2 ‰ 0:5; 0:5Š. Note that these output functions cannot be identically zero outside this ®nite interval due to the ®nite aperture in the Fourier plane; however, when squared and integrated over this

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M.A. Neifeld, B. Zhou / Optics Communications 193 (2001) 87±95

Fig. 2. (a) Examples of optimal pixel pro®les in systems with rectangular symmetry for values of W ˆ 1:7 (Ð), W ˆ 1:3 (± ± ±) and W ˆ 0:8 (


). (b) Corresponding output signals.

domain, they yield the largest possible values of energy throughput lp . The above results indicate that the optimum 2D pixel pro®le for the 4f system shown in Fig. 1 in the case of rectangular symmetry is pin …r† ˆ W popt …x; y† ˆ S1 …W ; x†S1 …W ; y†. In order to quantify the bene®t of using such a pixel pro®le, we have

computed the energy throughput as a function of Fourier aperture width for both the optimum pixel W …x; y† and the more traditional uniform pro®le popt pixel pro®le pin …r† ˆ rect…x†rect…y†. We de®ne the energy throughout in the traditional case to be lr , and the ratio of these two criteria c ˆ lp =lr , represents the gain in eciency that is obtained via

Fig. 3. Energy throughput gain achieved by use of optimal signalling, as compared with uniform rectangular pixels. Overall eciency gain c (Ð), and component gains cin (- - -), and cout (


) are also shown along with experimental data (symbols).

M.A. Neifeld, B. Zhou / Optics Communications 193 (2001) 87±95

optimum signalling. Fig. 3 presents a graph of c versus W. From this ®gure we see that for small values of W there is little bene®t to the use of optimum signalling; however, for intermediate values of W, there can be signi®cant throughput gains. We see a peak gain of 32% at a Fourier aperture of 1:25 times the Nyquist aperture WNyquist ˆ 2kf . The eciency gains depicted by the solid curve in Fig. 3 are composed of two components: gain arising from a reduction in lowpass loss and gain arising from a reduction in detector loss. We label these two components of c as cin and cout respectively, so that c ˆ cin cout . The dashed curves in Fig. 3 represent these two components of c as a function of W. So that our results might be applicable to systems that use circular lenses we next consider the case of circular symmetry. For this case we assume that the array illuminator produces a spot pro®le that is separable in polar coordinates so that pin …r† ˆ pr …r†ph …h†. We also de®ne the ®nite interval X as r 2 ‰0; 0:5Š and h 2 ‰0; 2pŠ. Note that the use of this interval carries with it an implicit assumption of circular SLM and detector pixels. Although this assumption is somewhat unrealistic, it is necessary to avoid mixing coordinate systems and giving rise to an intractable solution. Let the

91

lowpass Fourier aperture be written in the form AW …f† ˆ circ…fr =W †: Because this aperture can be characterized by a single spatial coordinate, we can perform the h-integral to arrive at a simpli®ed expression for the energy throughput " R 0:5 R 0:5 lp ˆ 2p

0

0

# pr …r0 †pr …r00 †KW …r0 ; r00 †r0 dr0 r00 dr00 ; R 0:5 2 jpr …r†j r dr 0 …5†

where KW …r0 ; r00 † ˆ 2pW 2

Z 0

0:5

J1 …2pW ‰r r0 Š†J1 …2pW ‰r …r r0 †…r r00 †

r00 Š†

r dr: …6†

Finding the function pr …r† that maximizes lp is once again reduced to ®nding a complete set of orthonormal basis functions on the ®nite interval ‰0; 0:5Š, that also happen to Rbe eigenfunctions of 0:5 the linear operator L‰pr …r†Š ˆ 0 pr …r†KW …r; r0 †r dr. These basis functions also exist and are given by the generalized prolate spheroidal functions fSiG …W ; r†g [20,21]. The maximum value of Eq. (5)

Fig. 4. (a) Examples of optimal pixel pro®les in systems with circular symmetry for values of W ˆ 1:8 (Ð), W ˆ 1:5 (± ± ±) and W ˆ 1:1 (


). (b) Corresponding output signals.

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M.A. Neifeld, B. Zhou / Optics Communications 193 (2001) 87±95

Fig. 5. Energy throughput gain achieved by use of optimal signalling, as compared with uniform circular pixels. Overall eciency gain c (Ð), and component gains cin (- - -), and cout (


) are also shown along with experimental data (symbols).

is once again achieved by choosing pin …r† ˆ W popt …r† ˆ S1G …W ; r†, resulting in a maximum energy 2 throughput lp ˆ …kW1 † . Figs. 4 and 5 are analogous to Figs. 2 and 3. Fig. 4 depicts the optimum circularly symmetric pixel pro®les for several values of W together with their corresponding output functions. Fig. 5 presents the eciency gains obtained through use of the optimum pro®les relative to a uniform aperture pin …r† ˆ circ…r†. Although somewhat smaller than the eciency gains seen for the case of rectangular symmetry these gains are signi®cant, reaching a peak of 27% at an aperture whose width is 1:4 times the Nyquist aperture. 3. Application example: volume holographic data storage We expect that the gains in energy throughput derived in the previous section will be bene®cial for many applications. In this section we quantify the improvements in storage capacity that accompany these throughput gains for volume holographic data storage (VHDS) systems. A VHDS system typically comprises two optical subsystems: a ref-

erence-beam subsystem and an object-beam subsystem. The reference arm is responsible for de®ning a holographic storage address during recording and readout; while, the object arm is responsible for imaging a data-bearing SLM through some holographic material and onto a 2D detector array. In Fourier-domain storage, the object arm often resembles the 4f system shown in Fig. 1, with the Fourier-plane aperture modi®ed to include a volume storage material (e.g., a photorefractive crystal). Throughout this discussion we will assume a rectangular aperture placed in front of the storage material for the purposes of optimizing storage density in an angularly multiplexed VHDS system [22]. There are two mechanisms through which VHDS system performance may be enhanced via use of optimal pixel pro®les: the ®rst mechanism is associated with the holographic recording process; while, the second arises during data retrieval. During hologram recording a reference beam with amplitude R0 interferes with an object beam with amplitude B0 giving rise to an interference pattern in the storage material. This recording process may be characterized by the so-called M=# of the system [23]. The M=# is an important system

M.A. Neifeld, B. Zhou / Optics Communications 193 (2001) 87±95

metric for VHDS owing to the well-known scaling law that governs ®nal di€raction eciency g as a function of the number of stored data pages 2 M:g ˆ ‰…M=#†=MŠ . For most cases of interest we have the reference-beam intensity much larger than the object-beam intensity (for any single pixel) so that M=# becomes proportional to object-beam amplitude M=# / B0 [24]. Because the use of an optimal pixel pro®le allows cin more p light to pass the Fourier-plane aperture, a cin increase in object-beam amplitude accompanies p W the use of popt …x; y†. This cin increase in M=# corresponds to a cin increase in ®nal system diffraction eciency. During the retrieval process a reference beam reconstructs the stored hologram to produce a data signal at the detector array. Not all of this di€racted light is detected by the desired detector element; some of it is detected by neighboring elements giving rise to crosstalk. The impact of crosstalk is twofold: the energy reaching the desired detector is reduced and the signal reaching neighboring detectors is corrupted by W additional ``noise''. The use of popt …x; y† mitigates both of these corrupting in¯uences. Focusing on the former, we see that the optimal pixel pro®le will increase the intensity measured on the desired detector by a factor cout as compared with the use of uniform pixel pro®les. Taking these two e€ects together we can write the VHDS system M=# in the case of optimum signalling as …M=#†opt ˆ p c…M=#†, where c ˆ cin cout as de®ned in the previous section. A working VHDS system must satisfy some end-user bit-error-rate (BER) constraint (e.g., BER < 10 12 ). The details of how an acceptable BER is achieved for a speci®c storage channel will depend upon numerous factors related to coding and signal processing; however, once these factors have been decided the required BER can be cast in terms of a minimum acceptable signalto-noise ratio …SNRmin †. It is instructive to consider the case of a VHDS system dominated by thermal detector noise. In this case we can write SNRmin / gmin so that data ®delity constrains the maximum number of holographic pages according p to Mmax ˆ …M=#†= gmin . From this expression we can see that an increase in M=# carries with it a proportional increase in storage capacity. From

93

the curves in Fig. 3 we can predict capacity gains p as large as 1:32 ) 15% through the use of optimal signalling. We should note that not all VHDS systems will bene®t from both the cin and cout factors described above. In some cases for example, it may be undesirable to modify the recording beam ratio. For these cases, the capacity gain will be limited to p cout . During recording the increase in object beam power achieved via the use of optimal pixel pro®les would be o€set by redistributing laser power between the object and reference arms. The net recording power in this case would therefore increase resulting in reduced holographic recording time. 4. Experimental results In this section we provide details of the experimental procedures that were used to verify the predicted improvements in energy throughput. Our experimental apparatus was designed to evaluate the energy throughput for a single pixel. Uniform and prolate spheroidal pixel pro®les were tested in the presence of three di€erent Fourier apertures. Each pixel was spatially quantized to a 100  100 array and amplitude quantized to 64 grey levels. Our simulation study suggested that these quantization levels would be sucient to approximate continuous pixel pro®les by use of randomized area modulation in an absorptive transparency. In any practical implementation of this technique however, a di€ractive apodizer would be required in order to eliminate absorption losses associated with creating the optimal pro®les. Our apodizer masks were fabricated by PhotoSciences Corporation using an inexpensive (resolution of 2±5 microns) lithographic process on glass substrates. Our pixel pro®les were 6:4  6:4 mm2 and the 4f system setup utilized a He±Ne laser (k ˆ 633 nm) and two f ˆ 200 mm doublets with clear apertures of / ˆ 75 mm. Rectangular Fourier apertures were used with widths of 32, 52, and 68 lm corresponding to normalized widths of W ˆ 0:81, 1.3, and 1.7 respectively. Results from our experiments with these rectangular apertures are shown by the isolated symbols in Fig. 3. Circular Fourier apertures were used with diameters

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M.A. Neifeld, B. Zhou / Optics Communications 193 (2001) 87±95

of 44, 60, and 72 lm corresponding to normalized widths of W ˆ 1:1, 1.5, and 1.8 respectively. Results from our experiments with these circular apertures are shown by the isolated symbols in Fig. 5. Alignment diculties with some of the smallest apertures produce some experimental errors for small values of W; however, the overall agreement between theory and experiment is quite good. 5. Conclusions We have presented the use of non-uniform pixel pro®les for increasing the energy throughput of spatially discrete coherent optical imaging systems. Prolate spheroidal pixel pro®les were shown to be optimal and can produce throughput gains of up to 32% for optical systems with rectangular symmetry and 27% for those with circular symmetry. These throughput gains were shown to impact the capacity of volume holographic data storage systems, producing capacity gains up to 15% as compared with the use of uniform pixel pro®les. We should note that the 15% capacity gains predicted in Section 3, were based on an analysis of energy loss alone. In addition to a loss in energy throughput, the use of uniform pixels will also increase crosstalk noise (intersymbol interference), resulting in reduced image ®delity and lower capacities than those predicted by the simple M=# scaling rule invoked herein. Optimal pixel pro®les with their reduced crosstalk levels may therefore o€er somewhat larger gains than those predicted in Section 3. Other applications might also bene®t from increased energy throughput; however, the degree to which optimal pixel pro®les might impact these other optical imaging applications will depend upon the extent to which these applications su€er from the lowpass losses described in Section 1. The work presented here is applicable to diffraction-limited coherent imaging systems. The prolate spheroidal functions represent optimal solutions in such cases owing to their property of self-similarity under Fourier transformation on a ®nite domain. Optical systems that are shift-variant may still admit the analysis presented here for

those pixels at which the local PSF remains a sinc… † function. Each pixel pro®le is simply customized to that particular prolate spheroidal function optimized for some local value of W. Such a scheme is theoretically possible but implementationally cumbersome. For pixels at which the spatial frequency response departs from an ideal lowpass characteristic however, the prolate spheroidal functions will no longer maximize energy throughput and a new kernal function must be inserted into Eq. (1). Most such insertions will result in pixel pro®les for which no closed form representation exists. In such cases we must resort to numerical techniques to ®nd the optimal pixel pro®les. References [1] G.W. Burr, W.-C. Chou, M.A. Neifeld, H. Coufal, J.A. Ho€nagle, C.M. Je€erson, Experimental evaluation of user capacity in holographic data-storage systems, Appl. Opt. 37 (1998) 5431±5443. [2] M.A. Neifeld, W.-C. Chou, Information theoretic limits to the capacity of volume holographic optical memory, Appl. Opt. 36 (1997) 514±517. [3] J.A. O'Sullivan, R.E. Blahut, D.L. Snyder, Informationtheoretic image formation, IEEE Trans. Information Theory 44 (1998) 2094±2123. [4] E. Shusterman, M.I. Miller, B. Rimoldi, Rate-distortion theory applied to automatic object recognition, IEEE Trans. Information Theory IT-46 (2000) 1921±1926. [5] F.O. Huck, C.L. Fales, R. Alter-Gartenberg, S.K. Park, Z. Rahman, Information theoretic assessment of sampled imaging systems, Opt. Eng. 38 (1999) 742±762. [6] W.-C. Chou, M.A. Neifeld, R. Xuan, Information-base optical design for binary-valued imagery, Appl. Opt. 39 (2000) 1731±1742. [7] R. Alter-Gartenberg, Information metric as a design tool for optoelectronic imaging systems, Appl. Opt. 39 (2000) 1743±1760. [8] G.A. Betzos, J.F. Hutton, M. Porter, P.A. Mitkas, Evaluation of array codes for page-oriented optical memories, Opt. Comput. Tech. Digest 8 (1997) 204±206. [9] J.F. Heanue, K. Gurkan, L. Hesselink, Signal detection for page-access optical memories with intersymbol interference, Appl. Opt. 35 (1996) 2431±2438. [10] G.W. Burr, J. Ashley, H. Coufal, R.K. Grygier, J.A. Ho€nagle, C.M. Jeferson, B. Marcus, Modulation coding for pixel-matched holographic data storage, Opt. Lett. 22 (1997) 639±641. [11] V. Vadde, B.V.K. Kumar, Channel estimation and intrapage equalization for digital volume holographic data storage, Proc. SPIE 3109 (1997) 245±250.

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