1978 International Optical Com~utina Conference

Transcription

1 M.O. Hagler, E.L. Kral, J.F. Walkup and R.J. Marks II, "Linear coherent processing using an input scanning technique", Proceedings of the 1978 International Coputing Conference, London, England, 1978, pp Proceedings of the 1978 International Optical Co~utina Conference I ;I Septeber 5-7,1978 London, England DIGEST OF PAPERS Sponsored by the Coputer Society of the Institute of Electrical and Electronic Engineers in cooperation with the Office of Naval Research and the Society of Photo-Optical instruentation Engineers

2 n. 0. Hagler, E. L. Ma1 and J. P. Walkup Departent of Electrical Engineering Texas Tech university Lubbock, Texas and R. J. ELarks 11 ~epartent of Electrical Engineering University of Washington Seattle, Washington In certain linear coherent processing techniques, a teporal signal is spatially ended as an aplitude transittance which serves as the processor input. In this paper, a technique is presented whereby the taporal signal is altertely used to aplitude and/or phase laodulate a raster scan of the processor's input plane. Using the teporal integrating and suing properties of a hologr placed in the procesaor's output plane, on0 can then reqain the identical processor output which would have arisen fro the spatial ending technique. Preliinary experiental results are presented along with the theory of the input scanninq technique. Theory we linit our scanning technique to those systes which are linear. Such syste can forally be expressed via the superposition integralr g(x,yl - siu(x.yil - u(e.q)h(x-e.y-?it.?) dedn 4 (11 where g it the syste output corresponding to an input u into a syste The point spread function is defined as Introduction In certain coherent processin9 cheas, a processor int is received as a t&ral electronic signal. Conventionally, this signal is spatially encoded as a two-diensional aplitude transittance which then serves as the processor input. It is, however, usually the corresponding processor output which is of interest. In this paper, we present a schee whereby one can achieve an identical linear procesaor output by utilizing the teporal signal to aplitude and/ or phase odulate the field aplitude of an input raster scan. The tie-varying field aplitude at the syet's output is then teprally integrated and sued using holographic techniques. upon playback, the hologra is shown to produce a diffracted tern which is identical to that which would be obtained by placing a corresponding input field aplitude tzsittance ask at the processor's input. This schee, then, eliinates the necessity of spatially encoding the input. Use of erasable photographic edia suggests possible Mleentations near real-tie. where 6(-..I denotes the Dirac delta. We are here using the Irohnann-Paris point spread function (ipulse response) notdtion Consider, then, the scanning geonetry shown in Fig. 1., we scan the R input plane over E at speed v. Modeling the scanning point as an loplitude and/or phase odulated delta function, the field aplitude to the right of the input plane at tie t is For the fixed value of n - rl Pro Eq. (21, the corresponding ceaplex field aplitude incident on the output plane is Placed in the output plane is a photosensitive ediu on which is also incident a vlanar reference be- exp(jkay). where a is a direction cosine. 9 The corresponding intensity at tie t is thus given Input scanninq has been usad axtensively in incoherent processing to add the twral degree of freedo to the already available spatial vari- (5) ablea. var.ious incoherent processors and corres- ponding references are given in the excellent re- 1 view paper by nahan, Brley Md Bocker. Scanning techniques have also been extensively applied by I(x,yitl - lu(vt,~ih(x-vt,y-~~vt,?~) + axpokay1 I 2 Assing the resulting hologra's aplitude transittance is proportional to the exposing intensity function, we have. for one scan, an aplitude trans- ittance of 8-10 in holography2-' as %!pothe effects of th-vary- - T ing field aplitudes. tx1-1 I(x.yit1 dt (61 t-0

3 whue T is the exposure tbe for a single scan. bilities of the holoqr, ve oonsider the syste in For U scans correspondir to various values of Il-. Piq. 3. A int source kes a sirle scan across 1 th; one-d&sional double- slit in&t aperture the holoqra's aplitude traosaittance is a(%). The linear proccesor in this exaple is the. H ^ failiar Fourier transforer which consists of the thy) - 1 t,(x.~l - tl + t2 + t3 (7) 1-1 single lens I,. The scan is perforwd along the line Il - 0. where T Following the previous odel developo~nt, tl 1 [~u(vt,\lh(x-vt,~-il~ivt,\ldt1urpl-jkay) the field incident on the photosensitive wdiu is o (8) t2 elf a(vt) ew(-jmx/f) + urp(jkax) (141 T 2 where f is the focal lenqth of lens L1. t3 - TM + ~(u(vt,\)h(x-vt,y-~ivt,\)i dt Under the o previously stdtad rerdinq aasu~tions, the re- Here. "*' denotes coplex conjugate. It is the tl sultinq holographic field srnplitude is " ter in which we are interested. Uaking the vari- t1x.y) - tl + t2 + t 3 (15) able substitution 5 - vt and assuing each scan covers the entire input pupil at t) - qives where T/2 tl - - v ~u(~,\)h(x-t,y-n~it.~)ds)cxp(-jlay) / a(*) ew(-jkvtx/fl dt e(-jkaxl -TI2 (9) t2 - tlf (16) Playback is perfored as shown in Pig. 2. The T/ 2 playback bea gives rise to three diffracted ters. t, - T + / la(vt)12 dt The tern t,ejkdy is the zero order through beaa and t2ejkay is the twin inage conjugate coponent. The When the hologra is played back, the difter of priary interest is Eracted ter Mistely to the right of the holo gra corresponding to tl is tlexp(jkayl - $ - 11 u(c,\)h(x-e,y-\r<,a) dt (101 I a(t) w(-jkcx/f) d6 (17) " This expression is recognized as a sei-discrete version of the superposition integral in Eq. (1). where we have rade the variable substitution That is, the integral over Il is approxiated by a f, - vt and have assued the scan c@apletely covered sursation. In sot instances, Eq. (101 will be an hcth input pulses. Wuation (171 is recoqnizad as adequate approxiation for the true aystea, output. the one-diensional Fourier transfor of a(x). Thus we should be able Co regain a(%) by an additional Under certain bandliited assuaptions on the Fourier transfor. This is accoplished by a single input and point spread function. we can obtain a cylindrical lens. The result of playback is sha better approxiation by perforing a low pass Iil- in Fig. 4 and, as can be seen, canpares quite favortering operation in the y direction. This stes ably with the theory. Siilar results for a single fro space-variant syste sapling theory pulse (slit) input are given in Fig. 5. * conclusions j u(t,q)exp(-j2nnv)dq = 0 tor lvl > v u (11) a and / h(x,yr~,il)exp(-jzwv)dr) * 0 for Ivl > wv (12) a then the desired low pass filter is unity aver the frequency band -T/2 We have deonstrated a technique whereby t-a1 signals can be linearly processed without tirst being spatially ended as an aplitude transittance. The schee akes use of the teporal integration and su~tion proprties of the hologra. This technique is potentially applicable to all linear coherent proceseors. By using a scan- -(Wv+ W ) < v z (wv+ WuI U - (13) ninq cdulated line source, it is also directly applicable to the recently presented class yf As sha in 2, this filring can be linear one-dinsionel coherent processors. 'I6 by conventional spatial filtering techniques. 9 Acknwledgents experient illustrate the teporal integration cap- The authors want to acknowledge the assistance with the experients provided by Ur. Hike I. Jones. and the typing of the anuscript by Us. Judy Clare. 10 9

4 This research was sugported by the Air Force Office of Scientific Research, Air Force Systes Corand. USAP, under Grant AfOSR References 1. M. A. nonahan, K. Branley and R. P. Bocker, "Incoherent Optical Correlators," Proceedings of the IEEE, 65, p J. C. Palaia, "Scanned Bea Holography," Applied *=, 2, p. 709, J. C. Palais and I. C. Vella, "Scne Aspects of Scanned Reference Bean Holography." I\ppllcd Optics, 14, p. 481, D. Kerisch, "Partially Coherent Insge Processing by Laser Scanning." - Journal of the Optical Society of Aerica, e, p. 887, D. Kerisch. "Principle of Equivalence Between scanning and Conventional Optical Iaging Systes." Journal of the %tical Society of Aerica, 67, p. 1357, K. Yap, "Effects of Object and Detector Sizes on the Spatial Frequencies of a Ona- Diensionally-Scanned Optical Syste," %tics, 14, p F. 0. Huck and 5. K. Park. "Optical-Mechanical Line-Scan Irasging Process: Its Inforation Capacity and Efficiency," Applied Optics, 14, p. 2508, J. W. Goodan. Introduction to Fourier Optics. New York: HcGrav-Hill, 1968, p. 247, Eq R. J. Collier, C. 0. Burckhardt and Lin. Optical Holography. Ntw Yorkr Acadeic Press, 1971, p. 440, Eqs through A. W. Lohnann and D. P. Paris, "Space-Variant Ilasge Foration." Journal of the Optical Society of &rica. 55, p. 1007, R. J. Harks 11. J. F. Walkup and M. 0. Hagler, "Line Spread Function Notation.. Applied Optics, 15, p. 2289, R. J. Harks 11, J. P. Walkup and M. 0. Hagler, "A Saplinq Theore for Space-Variant Systes," Journal of the Optical Society of Aerica, 66, p. 918, R. J. Elarks 11, J. F. Walkup and M. 0. Hagler, "Sapling Theores for Linear Shift-Variant Systes," IEEE Transactions on Circuits and %ss, CAS-25. p J. w. Goodnan, P. Kellan and E. W. Hansen, "Linear Space-Variant Processing of I-D Siqnals." &&cd Optics. 6, p. 733, R. J. Harks 11. J. F. Walkup, M. 0. Hagler and T. P. Wile, "Swce-Variant Processing of One- Diensional Signals:' fipplied Optics. 16. p. 739, J. w. Goodan, "Tepera1 Filtering Properties of Hologras,. %plied Optics, 6. p. 857, Reference Bea Laser Bea - Maddole! Sconnar, ' \ In C 1 lnpul Plane output Plane Figure 1. General input scanning geoetry. 150

5 Playback Bea exp [Bay) Low Pass Filter, Figure 2. Playbeck using a low pias filter. Fiqure 3. Experiental scanning configuration.

6 Flqure 4. Fxperiental output for a double slit Figure 5. Experiental output for a einqle slit input. input.