R.J. Marks II, J.F. Walkup and M.O. Hagler "Volue hologra representation of spacevariant systes", in Applications of Holography and Optical Data Processing edited by E. Maro, A.A. Friese and E. Wiener-Aunear, Oxford: Pergaon Press, pp.105-113 (1977).
VOLUME HOLOGRAM REPRESENTATION OF SPACE-VARIANT SYSTEMS Robert J. Marks 11, John F. Walkup and Marion 0. Hagler Departent of Electrical Engineenizg, Texas Tech University, Lubbock, Texas 79409 ABSTRACT Methods of characterizing linear space-variant systes by their responses to various sets of inputs are discussed. Recording these responses within a volue hologra results in a filter which is approxiately equivalent, in an input-output sense, to the space-variant syste. The approaches considered are (1) storing the transfer functions of the syste for point sapling of the input plane on playback: (2) a piecewise isoplanatic approxiation approach based on division of the input plane into isoplanatic regions, with storage of the transfer function for each isoplanatic patch; and (3) storage of the syste's responses to eleents of an orthonornal basis set. The potential advantages and liitations of each of these approaches, as well as experiental results, are discussed. INTRODUCTION It is well known that a hologtaphically recorded filter can be placed in the Fourier plane of a coherent optical processor and used to represent the transfer function of a linear, space-invariant syste. This single filter displays the input-output characteristics of the corresponding syste (1). Unfortunately, any optical systes which one ight desire to represent holographically are space-variant. For exaple, even an ideal iaging syste with non-unity agnification is rigorously space-variant. Space-variant systes ay, however, be characterized by cataloging the syste's responses to a nuber of separate inputs, as contrasted with the use of the single point spread function required to characterize the space-invariant syste. If a thick ediu is used to store the syste responses, the resulting volue hologra can subsequently act as a space variant filter which exhibits the input-output characteristics of the original syste. Such a representation can, in principle, significantly reduce the weight and size of a coherent processor, and should also iprove its orientation stability. For atheatical and notational siplicity attention will be restricted to one diensional linear, space-variant systes. For additional details, the reader is referred to soe recent papers by the authors and their colleagues (2-6). S2ace variant systes,
106 R. J. Marks 11, J. F. Walkup, M. 0. Hagler characterized by the linear operator S[-1, ay be described by the superposition integral g(x) = SIf(x)l = f(s)h(x-s;s)ds (1) where f (x) is the input, g (x) is the output, and h (x-<;<) represents the space-variant line spread function. That is h(x-5;5) = S[G (x-s) 1 (2) where 6(x) is the Dirac delta function. This for of the line spread function has soe advantages in describing space-variant systes (6-7). In the event that S is space invariant, we find that h(x-s;s)+h(x-s) (3) which says that h(x-5;s)i.s independent of its second arguent. We now consider three approaches to representing the effects of the space-variant syste S described by Eq. (1). Each of these approaches ay, at least in principle, be ipleented holographically. Soe details on the holographic ipleentation of the sapling theore approach are given in a later section. THE SAMPLING THEOREM APPROACH A odification of the Whittaker-Shannon sapling theore (1) perits us to design a holographic representation for space-variant systes based on line source sapling.of the input plane of S, and subsequent angle-ultiplexed holographic storage of the transfer function of S for each input plane saple. The sapling theore for space-variant systes (5) is based on the concept of the syste's variation spectru, which is defined by A Hg (x;v) = F5 [h(x; 5) I (4) where F [a] denotes Fourier transforation with respect to the in- 5 put variable 5, and where v is the frequency variable associated with 5. The variation spectru is a easure of how the line spread function changes for with respect to the input variable 5. In brief, the theore states that if the input, f(c), is bandliited to bandwidth 2Wf, and if h(x;c) has a variation spectru of width 2Wv (i.e. HS(x; v) = 0 for Ivl>wV for all x), then their product f(~)h(x;sj will have bandwidth 2W = 2Wf + 2Wv. As a result, the sapling theore states that by sapling the input line to the syste S at a rate of 2W saples per unit length, the output g(x) ay be written as the infinite su where En = n/2w. Equivalently, in the frequency doain,
Volue hologra representation 107 where, sinc x = - sin sx, and sx * 2, IxI< S rect x = 0, 1x12 4 -. Equation (6) states that G(fy) ay be obtained as a weighted su of individual transfer functions Hx(fx;(n), where the nth of which is weighted by the nth input saple f (5,). The presence of rect (fx/2w) in Eq. (6) indicates the low pass interpolation filter (bandwidth 2W) function necessary to reconstruct the continuous output g(x). One obvious proble with Eqs. (5) and (6) is that they require us to store a countably infinite nuber of hologras for an exact reconstruction of g(x). In practice we would expect to approxiate g(x) by storing a finite nuber deterined by space-bandwidth product considerations. The advantage cf the sapling theore approach is that it specifies a technique for exact reconstruction of the continuous output g (x) (i. e. low pass filtering). A disadvantage is that it. requires that we saple the input to S at a iniu rate deterined by the su of the variation bandwidth of S and the input band-, width. One technique for cutting down on the density of input plane saples required is to eploy the piecewise isoplanatic approxiation (PIA) approach described next. THE PIECEWISE ISOPLANATIC APPROXIMATION (PIA) APPROACH The piecewise isoplanatic approxiation, or PIA approach (4) akes the assuption that the space-variant syste S is piecewise spaceinvariant (see Ref. 1). It effectively divides the input line into segents (or the plane into "patches"), and each segent is characterized by its own line spread function. Matheatically we ay rewrite the input f (6) as where rect ((;C,u) represents a rectangle function of unit height, extending fro 5 = 1, to 5 = urn, and l,( 5, 2 u. Since g(x) = S[f (511 we obtain or equivalently g(x) = 1 SZf(E-S)1 (9
R. J. Marks 11, J. F. Walkup, M. 0. Hagler When copared with the sapling theore approach, the PIA approach appears to offer the advantage of being independent of the bandwidth of the input function f(x). Rather it is the anner in which the line spread function h(x-6;s)changes which deterines the nuber of hologras which ust be stored to represent the syste. Depending on the relative sizes of the syste's variation bandwidth and the input bandwidth, this advantage ay or ay not be significant. It would appear to be difficult, in general, to copare the perforance of a PIA-based ipleentation with one based on the sapling theore in situations where systes are not truly piecewise isoplanatic. In the next section we discuss the orthonoral response approach, where the nature of the input function ust be considered, but where the nature of the space-variant syste is in general, not a deterining factor. THE ORTHONOR-L RESPONSE APPROACH A third approach to characterizing the space-variant syste is to expand the input function f(x) as a weighted su of the eleents of an orthonoral basis set. Thus we write where the eleents of the set C$n(~): n = 1,2,...I are assued orthonoral, i.e. with 6 being the failiar Kronecker delta function. Substituting Eq. (11) into the relationship g (x) = S [f (x)] we obtain g (x) = 1 an S Mn (XI (13) n where the expansion coefficients are found by Note now that if we let On(x) be the response of S to tha input $,(XI, @,(XI = SIOn(x)l then Eq. (13) ay be rewritten
Volue hologra representation 109 It is worth noting here that the eleents of the syste response set {Qn(x) are not necessarily orthonoral, as are the eleents of the input set {$,(XI 1. To illustrate this approach, if we assue that f(x) is bandliited to bandwidth 2Wf, then by the sapling theore f (x) = 1 f (x n=- n sinc 2Wf (x-xn) where xn = n/2wf. We can view Eq. (17) as an orthonoral expansion with @n(~) = sinc 2Wf (x-xn) and with a =- I f (xn) nq The syste's sinc response is then given by S [sinc 2Wf (x-xn) I = - q On(x) and the syste output, g(x), ay be written as g (x) = 1 f (xn) S [sinc 2Wf (x-xn) 1 n Note here that the advantage of the orthonoral response characterization is that only the input bandwidth deterines the iniu required input plane sapling rate, = the variation bandwidth of S. In addition, since the sinc function is just the Fourier transfor of the rectangle function (Ref. 11, physical generation of a coherent syste's-sinc response is easily ipleented. A possible approach to ipleenting the sinc response approach is discussed at the end of the next section. We have presented three approaches to characterizing the perforance of space-variant systes. In principle, each of these approaches can be ipleented experientally. In our experiental work to date we have assued point sapling of the input plane of a syste, with the idea of ipleenting the sapling theore approach. In the next section we briefly discuss soe of the practical liitations present when one attepts to angle ultiplex the sapled transfer functions of a space-variant syste within a thick recording ediu. IMPLEMENTATIONS Our experiental work to date has concentrated on the holographic ipleentation of the sapling theore approach (2-3). Figure 1 illustrates (in one diension) the basic approach. Point sources are used to saple the input plane of the space-variant syste S,
110 R. J. Marks 11, J. F. Walkup, M. 0. Hagler resulting in the syste's point spread function appearing at the output of S. The nth reference point source, shown of set by a, also lies in the output plane of S. After the lens L perfors a Fourier transf~r~, the interference of the reference plane wave and the transfer function of S for the nth point source input is holographically recorded in the thick ediu. When we perfor this operation sequentially, with a different reference point source for each object point source, we are angle ultiplexing a nuber of transfer function hologras into the ediu. By using the extinction angle concept we can guarantee essentially noninterfering hologras. I / Reference Fig. 1. Recording volue hologra: sapling theore approach One places the volue hologra in the Fourier plane of a coherent optical processor for playback, as indicated scheatically in Fig. 2. On playback one spatially saples the input plane using a duplicate of the reference array. Each input point accesses the hologra which represents the transfer function of S for that input point location. Neglecting crosstalk between the stored hologras, coherent addition of the outputs then gives the desired response. The experients have ainly been perfored using a DuPont holographic photopolyer (8) as the recording aterial, 0 and an Argon laser operating at 5145 A (2-3). Experiental ipleentations of siple one- and.two-lens agnifiers for objects consisting of siple arrays of point sources have been produced and found to yield the correct agnifications though the iages contained soe aberrations. Additional experiental work is in progress.
Volue hologra representation 111 The extinction angle, be, is the ajor factor affecting syste resolution (2-3,9). One can show that even in the worst case where the syste point spread functions associated with sapling the input plane of S overlap each other copletely, it is possible, by properly spacing the reference array point sources, to obtain noninterfering hologras in the thick recording ediu. If we assue (see Fig. 1) that the transforing lens has focal length ft>>a, the reference array offset, we find that based on the assuption that the extinction angle is essentially invariant as one oves the reference and object beas over their respective arrays (a reasonable approxiation in any cases), the iniu reference array eleent-to-eleent spacing is given by Axin it A0 (22) Fig. 2. Playback schee: sapling theore approach This spacing will assure iniu hologra crosstalk on playback. To illustrate, we found that for a 100 icron thick layer of the DuPont holographic photopolyer, with ft = 10 c. and with an on axis object array, plus a 30 reference array offset, the extinction angle was A8E2". Based on Eq. (22) this predicts that Axin = 3.5. Since on playback one illuinates the input transparency through a duplicate of the reference array, the axiu spatial frequency present in the object for which we could achieve the Nyquist sapling rate would be 1/2Axin = 0.14 cycles per illieter = 3.6 lines per inch. While this is not a very high resolution, it should also be noted that thick recording edia such as photochroic glasses exist with extinction angles one and two orders of agnitude saller than the exaple just cited (101, so that the state of the art predicts uch higher syste capabilities. It is clear, however, that storing large nubers of well require, poses nuerous probles. These include signal-to noise ratio probles, crosstalk probles., and dynaic range probles, to ention just a few. An additional proble involves the difficulty in obtaining resolutions, for two diensional representations, which are equal to the one diensional syste resolutions predicted on the basis of the extinction angle concept (11).
112 R. J. Marks 11, J. F. Walkup, M. 0. Hagler A coent should be ade concerning ipleentation of the sinc response approach discussed in Eqs. (17)-(21). Based on Eq. (21) and Figs. 1-2, we see that by replacing the Dirac delta function by a sinc function at the input to S, one can effectively use the sae recording geoetry for ipleenting the sinc response approach as was used in ipleenting the sapling theore approach. The three approaches presented for synthesizing and using volue hologras to represent space-variant optical systes appear proising, despite soe obvious practical liitations. The potential savings offered by such holographic optical systes representations should, however, be sufficient to otivate further research into overcoing soe of the probles identified to date. Additional work is underway at present to explore the potential and liitations of these and alternative techniques for optically representing space-variant optical processors. ACKNOWLEDGEMENTS The authors wish to express their appreciation to Dr. Thoas F. Krile of the Rose-Hulan Institute of Technology, Terre Haute, Indiana, for his suggestions during the course of the research. This work has been supported by the Air Force Office of Scientific Research, USAF, under Grant AFOSR-75-2855A. REFERENCES (1) J. W. Goodan, Introduction to Fourier Optics, XcGraw-Hill, New York, 1968. (2) L. M. Deen, J. F. Walkup and H. 0. Hagler, Representations of space-variant optical systes using volue hologras, Appl. w., 14, 2438 (1975). (3) J. F. Walkup and M. 0. Hagler, Volue hologra representations of space-variant optical systes, Proc. of Tech. Prgr., Electro-Optical Systes Design Con.-1975, Anahei, Calif., Nov. 11-13, 1975, pp. 31-37. (4) R. J. Marks I1 and T. F. Krile, Systes theory for holographic representation of space-variant systes, Appl. Opt. (to appear). (5) R. J. Marks 11, J. F. Walkup and M. 0. Hagler, A sapling theore for space-variant systes, J. Opt. Soc. A., 66 (1976- to appear). (6) R. J. Marks 11, J. F. Walkup and M. 0. Hagler, On line spread function notation, Appl. Opt., 15 (1976-to appear). (7) A. W. Lohrnann and D. P. Paris, Space-variant iage foration,
J. Opt. Soc. A., 55, 1007 (1965). Volue hologra representation 113 (8) B. L. Booth, Photopolyer aterial for holography, Appl. Opt., 14, 593 (1975). (9) H. M. Sith, Principles of Holography, 2nd edition, Wiley- Interscience, New York, 1975. (10) A. A. Friese and 3. L. Walker, Thick absorption recording edia in holography, Appl. Opt., 9, 201 (1970). (11) R. J. Collier, C. B. Burckhardt and L. H. Lin, Optical Holography, Acadeic Press, New York, 1971.