M. 0. Hagler, R. J. Marks 11, E. L. Kral, J. F. Walkup, and T. F. Krile
In certain linear coherent processing techniques a temporal signal is spatially encoded on an amplitude
transmittance that serves as the processor input. In this paper a technique is presented whereby the tempo-
ral signal is alternately used to amplitude- and/or phase-modulate a raster scan of the processor's input
plane. Using the temporal integrating and summing properties of a hologram placed in the processor's out-
put plane, one can then regain the identical processor output that would have arisen from the spatial encod-
ing technique. Preliminary experimental results are presented along with the theory of the input scanning
technique.
1. Introduction
In certain coherent processing schemes a processorinput is received as a temporal electronic signal. Con-ventionally, this signal is spatially encoded as a 2-Damplitude transmittance that then serves as the pro-cessor input. It is, however, usually the correspondingprocessor output that is of interest.
In this paper we present a scheme whereby one canachieve an identical linear processor output by utilizingthe temporal signal to amplitude- and/or phase-mod-ulate the field amplitude of an input raster scan. Thetime-varying field amplitude at the system's output isthen temporally integrated and summed using holo-graphic techniques. Upon playback the hologram isshown to produce a diffracted term that is identical tothat which would be obtained by placing a corre-sponding input field amplitude transmittance mask atthe processor's input. This scheme therefore eliminatesthe necessity of spatially encoding the input. Use oferasable photographic media suggests possible imple-mentations near real time.
Input scanning has been used extensively in inco-herent processing to add the temporal degree of freedomto the already available spatial variables. Various in-coherent processors and corresponding references aregiven in the excellent review paper by Monahan et al.1Scanning techniques have also been extensively appliedin holography2 -7 as have the effects of time-varying fieldamplitudes. 8 -10
R. J. Marks II is with University of Washington, Department of
Electrical Engineering, Seattle, Washington 98195; E. L. Kral is with
Boeing Company, Albuquerque, New Mexico 87108; and the other
authors are with Texas Tech University, Department of Electrical
Engineering, Lubbock, Texas 79409.Received 21 December 1978.
We limit our scanning technique to those systemsthat are linear. Such systems can formally be expressedvia the superposition integral
g(xy) = slu(xy)I
= ff J Q,r/)h(. - #, y - n; (,-q)dtdrn, (1
where g is the system output corresponding to an inputu into a system S [-]. The point spread function is de-fined as
h(x - ,y - i,) = S[(x - (,y- )], (2)
where 6(.,.) denotes the Dirac delta. We are here usingthe Lohmann-Paris point spread function (impulseresponse) notational 2
Consider then the scanning geometry shown in Fig.1. For the fixed value of 7 = Em we scan the input planeover t at speed v.. Modeling the scanning point as anamplitude- and/or phase-modulated delta function, thefield amplitude to the right of the input plane at timet is
U(vt,n )6Q~ - vt,n - m (3)
From Eq. (2) the corresponding complex field ampli-tude incident on the output plane is
u(vt,7m )h(x - vt,y - 7im;Vt,7tm). (4)
Placed in the output plane is a photosensitive mediumon which is also incident a planar reference beamexp(jk ay), where a is a direction cosine.9 The corre-sponding intensity at time t is thus given by
Assuming the resulting hologram's amplitude trans-mittance is proportional to the exposing intensityfunction, we have for one scan an amplitude transmit-tance of8-10
where T is the exposure time for a single scan. For Mscans corresponding to various values of ?m, the holo-gram's amplitude transmittance is
i(x,y) = E tm(Xy) = t + t2 + t3 , (7)m=1
where
t = [fTu(vtm)h(x vty - im;vt,-m)dt] exp(-jkay)
t2 = tl,(8)
t3 = TM + fT U(vt,?lm)h(x -Vt,y - lm;Vt, 7m)l 2dt.
Here, * denotes complex conjugate. It is the t term inwhich we are interested. Making the variable substi-tution = t and assuming each scan covers the entireinput pupil at = m gives
ti =-y [5 u(Q,nm)h(x - y - 7m;t,7.)
X d] exp(-jkay). (9)
Playback is performed as shown in Fig. 2. Theplayback beam gives rise to three diffracted terms. Theterm t3 exp(jkay) is the zero order through beam, andt2 exp(jkay) is the twin image conjugate component.The term of primary interest is
This expression is recognized as a semidiscrete version
of the superposition integral in Eq. (1). That is, theintegral over X is approximated by a summation. Insome instances Eq. (10) will be an adequate approxi-mation for the true system output.
Under certain bandlimited assumptions on the inputand point spread function, we can obtain a better ap-proximation by performing a low pass filtering opera-tion in the y direction. This stems from space-variantsystem sampling theory.13"14 If
f u( ,) exp(-j27r-qv)d?7 0 for v > w, (11)
J h(x,y;,n) exp(-j27rf7v)dn1 0 for vJ > w, (12)
then the desired low pass filter is unity over the fre-quency band
-(w,, + wu) s v < (w + wu), (13)
As shown is Fig. 2, this filtering can be performed byconventional spatial filtering techniques.9
III. Experiment
To illustrate the temporal integration capabilities ofthe hologram, we consider the system in Fig. 3. A pointsource makes a single scan across the 1-D double-pulseinput aperture a (x). The linear processor in this ex-ample is the familiar Fourier transformer that consistsof the single lens L1. The scan is performed along theline = 0.
Following the previous model development, the fieldincident on the photosensitive medium is
Mirror where f is the focal length of lens L1. Under the pre-viously stated recording assumptions, the resultingholographic field amplitude is
Reference Beam i(x,y) = t + t2 + t, (15)
where ST/2\ \ Film t = JTba(vt) exp(-jkvtx/f)dt exp(-jk ax),
Moving A\ -T/2
r< Sourcet2 = t, (16)
pT/2
\ i\\ t3 = T ~~~~~~~~~+1J l a (vt )I2dt.Optical .= T. -T/2
Flat a(s) When the hologram is played back, the diffracted
term immediately to the right of the hologram corre-Experimental scanning configuration. sponding to t, is
1 a(t) exp(-jktx/f)d,, (17)
where we have made the variable substitution v = Vt andhave assumed the scan completely covered both inputpulses. Equation (17) is recognized as the 1-D Fouriertransform of a (x). Thus we should be able to regaina (x) by an additional Fourier transform. This is ac-complished by a single spherical lens. The result ofplayback is shown in Fig. 4 and, as can be seen, com-pares quite favorably with the theory. Similar resultsfor a single pulse (slit) input are given in Fig. 5. Thesomewhat discontinuous appearance of the slit may bedue to the fact that the rotation of the optical flat shownin Fig. 3, which produced the scanning point source, wasaccomplished by hand.
IV. Conclusions
We have demonstrated a technique whereby tem-poral signals can be linearly processed without firstbeing spatially encoded as an amplitude transmittance.The scheme makes use of the temporal integration and
Bxperimental output for a double slit input. summation properties of the hologram.This technique is potentially applicable to all linear
coherent processors. By using a scanning modulatedline source, it is also directly applicable to the recentlypresented glass of linear 1-D coherent processors.15"16
In addition, it has been shown that when coupled withthe availability of changeable Fourier plane masks, thetemporal holography approach may be used to imple-ment a 2-D space-variant processor.'7
The scanning technique described above is, of course,subject to the diffraction efficiency limitation of se-quentially recorded holograms.1-2 1 It should be noted,however, that these limitations need not be severe20 21
if appropriate exposure conditions are employed. Theissue of bias-induced limitations inherent in sequen-tially recorded holograms as compared to those obtainedin a single simultaneous recording is discussed in moredetail in the Appendix.
The authors want to acknowledge the assistance withthe experiments provided by Mike I. Jones, and thetyping of various versions of the manuscript by JudyClare, Jan Daniel, and Heidi Jackson.
This research was supported by the Air Force OfficeExnerimental output for a single slit input. of Scientific Research, Air Force Systems Command,
USAF, under AFOSR grants 75-2855 and 79-0076.Portions of this work were presented at the 1978 In-ternational Optical Computing Conference in Londonand at the 1978 Annual Meeting of the Optical Societyof America in San Francisco.
Appendix
When constructing sequentially recorded holograms,there is the potential for a bias build up problem andconsequent reduction in diffraction efficiency. Toexamine the extent of this problem and lead into pos-sible solutions proposed by other workers, we willcompare the signal-to-bias ratios for the simultaneousrecording case and the sequential recording case.
Assume an object composed of N points, and denoteby a (x,y) the amplitude of light on the hologram,produced by point k on the object. For simplicity the2-D spatial function ak (x,y) will be written as ak fromnow on.
In the case of simultaneous exposure of a hologramderived from all N object points, the total film exposureis:
Esim = TimlRsim + z2 ak 12= TsinI RsimI2 + Rim ak + Rima + 32:aiaJ, (Al)
where Tim is the exposure time, Rsim is the amplitudeof the reference beam, and the index in all sums runsfrom 1 to N. The signal-to-bias ratio is then givenby
where ae is the same number as before. Here I ah max isthe brightest spot in the recording plane when just thekth input point is active.
Combining Eqs. (A5), (A6), and (A7), we find that
hsim N|ak|max
tbseq | 2ak I max(A)
We see from Eq. (A8) that the relative sizes of thesignal-to-bias ratios in the simultaneous and sequentialrecording cases clearly depend on the nature of the totalsignal at the recording plane and how it is partitionedfor sequential exposures. For example, at one extreme,if the total signal at the recording plane is uniform andis partitioned into disjoint spatial regions for sequentialrecording, then
I lak max = I ak max, (A9)
so that
=im N.
aseq(A10)
At the other extreme, if the signal at the recordingplane is uniform and is partitioned for sequential re-cording such that the ak} are uniform and equal,then
sim = Rim lak = a k ,I I R aim12 I R i. |I;ak I max = NIak max,
(A2)
where we assume 22aia << IRsiml J2For the sequential recording case the total film ex-
posure is
Eseq = TseqZ IRseq + aI 2
= TseqJNIReq12 + ReqZak + ReqZa + 2; Zaia, (A3)
where Tseq is the total film exposure time over all ex-posures. Note that Rseq is not necessarily the same asRsim, but we do assume Rseq is the same for each indi-vidual exposure. In this case the signal-to-bias ratio
(All)
so that
= 1.bseq
(A12)
As a more realistic intermediate case, we can considerthe ak I to be phasors with equal amplitudes and uni-formly distributed random phases, in which case wewould expect that on average
where ae is a number on the order of unity. Since akrepresents the signal in the recording plane from theentire object, z2ak I max is the brightest spot in the re-cording plane when all object points are active.
For the sequential recording case it is only necessaryto require that
IRscql = ala5 I max, (A7)
bsim-=/iseq
(A14)
In practical cases, therefore, we can expect the sig-nal-to-bias ratio for the sequential case to degrade as the-IN. It is important to note, however, that the deg-radation in seq can be reduced significantly by opti-mizing Rseq for each exposure. Such conditions arediscussed in Refs. 18-21 and lead to situations where5sim can be made equal to 6seq,
To add credence to these observations, we have in ourown laboratory successfully recorded 100 sequentiallymultiplexed holograms of the Fourier transforms ofpoint sources for the purpose of holographically repre-senting space-variant optical systems.22 The systemplays back well so we are confident that we are achievingadequate diffraction efficiencies. At this point thenumber of holograms successfully recorded in sequencehas been limited mainly by our patience.
Uranium Enrichment. Edited by S. VILLANI. Springer-Ver-lag, Heidelberg, 1979. 322 pp. $49.00.
This book is a collection of review articles on the state of uraniumisotope enrichment technology. The greatest amount of space (130pp.) is devoted to a thorough survey of all aspects of the presentworkhorse process, gaseous diffusion (this process accounts for 95%of the uranium separative work presently produced). Next in im-portance is a discussion (60 pp.) of the theory of the gas centrifugetechnique, with emphasis on the hydrodynamics and computermodeling of the flow fields and separation effects. Physical andtechnical aspects of the separation nozzle process, now entering ademonstration phase, are discussed (23 pp.). Advanced separationconcepts that are presently the subjects of intense investigation, in-cluding laser isotope separation (20 pp.) and plasma separation pro-cesses (24 pp.), are each reviewed from the point of view of physicalaspects of the various techniques under investigation. Detaileddiscussions of these processes are necessarily limited by the prelimi-nary nature of their development and by proprietary and securityclassification restrictions.
In Chapter 1, S. Villani presents a concise review of the status ofenrichment technology, beginning with a brief assessment of projectednuclear growth, and forecasts through 1990 for uranium enrichmentdemand vs capacity. The conclusion is that in the Western Worldexisting and planned capacity should meet demand through 1990.Work on improving existing techniques and investigating advancedtechniques is therefore justified not on the basis of urgency but on thebasis of an opportunity to develop alternative methods that may beless capital intensive and require less operating power than the gas-eous diffusion process. The general natures and the capacities ofvarious industrial and demonstration plants based on gaseous dif-fusion, centrifugation, and aerodynamic (separation nozzle and vortextube) techniques are reviewed. New processes including laser,plasma, and chemical exchange are noted. The brief discussion inthis chapter of the French chemical exchange process is the only onethat appears in the book. A somewhat more detailed expositionwould be desirable (if possible), for comparison with the other ad-vanced techniques.
Chapter 2 is devoted to a discussion by B. Brigoli of the theory ofcascading elementary separation units. Such considerations are offundamental importance in overall plant performance when the ele-mentary enrichment fact of a single unit is only slightly larger thanone. As its author indicates, most of the material of this chapter isavailable in other places in the literature, most recently in the ANSMonograph "Isotope Separation" by S. Villani. The original con-tribution here by Brigoli is a treatment of ideal nonsymmetric cas-cades; this type of stage connection is required for optimum perfor-mance when the separation factor and the cut (head fraction) arestrongly interdependent. This is the case, for example, with theseparation nozzle and vortex methods.
Chapter 3 is a review by D. Massignon of all aspects of the gaseousdiffusion process. The discussion ranges from molecular flow con-siderations of the elementary separation effect in various types ofdiffusion barrier, to economic and cost considerations and summariesof various operating plants. A large section (47 pp.) is devoted todiscussions of diffusive flow through porous barriers and the effec-tiveness of theories of the separation effect. Details of staging for theUF 6 diffusion process are considered. The author describes unclas-sified details of barrier design and testing and plant hardware suchas compressors and heat exchangers. Economics and plant designare covered in the final sections, along with a description of existingand projected plants in the U. S. and France. This chapter is the mostcomplete, lucid, and up-to-date summary of uranium gaseous diffu-sion technology available today.
The discussion of centrifugation by Soubbaramayer in Chap. 4is limited to the theoretical analysis and computer modeling of thecountercurrent gas centrifuge. An elementary discussion of theseparation effect in a simple gas centrifuge is expanded to includeconsiderations of countercurrent flow and other means of enhancingthe separation effect. The largest part of the discussion is devotedto the hydrodynamics of the flow, the boundary layer conditions, andthe analysis of the flow. Numerical methods for optimizing theseparative power for specific configurations are discussed. Propri-etary and security classification considerations severely restrict dis-cussions of specific rotor design, materials, experimental results, andcost analyses.
The preeminent aerodynamic technique, the separation nozzleprocess, is reviewed concisely by E. W. Becker in Chap. 5. The dis-cussion includes a review of various nozzle configurations, theory ofthe nozzle separation effect, and a rationale for optimization of thesystem operating conditions. Cascade staging details of the non-symmetric type are discussed. A section on nozzle fabrication andseparation stage construction is included. The design and operationof a ten-stage pilot plant is described. Economics and future devel-opment are also discussed briefly.
Chapter 6 is a review of laser isotope separation (LIS) methods byC. P. Robinson and R. J. Jensen. The high spectral irradiance oflaser outputs provides a means for selectively existing isotopic speciesin various mixtures, and a variety of schemes have been used to per-form isotope separation experiments with lasers since the first ex-periments almost 10 years ago. The authors provide a survey of theexperiments on many molecular isotopic species and the laser excitedseparation schemes on which they are based. Laser characteristicsare reviewed with respect to application in LIS. The two contendinguranium LIS schemes, namely, the atomic vapor and molecularschemes, are discussed. Both schemes have demonstrated feasibilityand are rapidly being developed to the point where commitments fordemonstration facilities can be made. As in the case of the centrifuge,proprietary and security classification considerations limit the amountof detailed information that can be published on either process.Nevertheless, the authors give a concise description of each processincluding advantages and problem areas. Economics are brieflydiscussed; no thorough cost analysis is presented.
In the final chapter, F. Boeschoten and N. Nathrath reviewplasma separation effects, including the plasma centrifuge and ioncyclotron resonance processes. Work reported in the literature sofar has emphasized the plasma centrifuge process, and the reviewconcentrates on that process. Theoretical considerations are followedby descriptions of experimental results with rare gas arcs, in which82 Kr/ 6 Kr separation factors as high as 1.1 have been obtained, and
with uranium metal arcs, where local separation factors (23 5U/ 238U)of about the same magnitude have been measured. The importantconsiderations here are the ratio of rotational to thermal energy andminimizing the energy required for vaporization. Various arcgeometries are considered, including ring mode and hollow cathodeconfigurations. The authors give the basis for the ion cyclotron reso-nance process and estimate the requirements on magnetic field uni-formity, magnetic field intensity, and plasma densities. Tenfoldenrichments of potassium isotopes have been demonstrated; infor-mation on the results of experiments with uranium plasmas is subjectto the same proprietary and classification restrictions as i the ura-nium LIS cases.
I believe this book will be of great value to those desiring to reviewthe state of the art in uranium isotope separation. Its major drawbackis a somewhat inconsistent breadth of coverage from chapter tochapter, although some of this is unavoidable as noted. Particularlyvaluable are the detailed reviews of the gaseous diffusion technologyand the separation nozzle process, and the overviews of the advancedprocesses.
