Computer generated holograms: an historical review
`
`G. Tricoles
`
`Since Gabor's first hologram, the meanings of that term have grown with increased use of the invention.
`Computers expanded applications further, and this paper surveys the methods and techniques of computer
`generated holograms.
`
`1.
`
`Introduction
`
`A. Holograms; Holography
`In the beginning, a hologram was a tangible record of
`an intensity pattern that was formed when a wave
`scattered by an object interfered with a coherent refer-
`ence wave. The first holograms were formed by Gabor
`in research on reducing aberrations of electron micro-
`scopes.' These holograms were utilized in optical ex-
`imaging
`a new two-step
`to demonstrate
`periments
`principle. In the first step, an illuminated object scat-
`tered a field which interfered with the reference wave;
`in Gabor's experiments the reference wave was the
`wave bypassing the object. The interference pattern
`exposed a photographic plate; the developed plate was
`called a hologram. In the second step, the hologram
`was illuminated to produce an image, which occurred
`on the optical axis.
`A hologram's appearance usually differs from the
`object's, but holographic images are remarkably realis-
`tic. Many holographic images are three-dimensional2;
`this property stimulated interest in holograms for dis-
`plays. Leith and Upatnieks used a reference field that
`was from a coherent beam that was distinct from the
`object wave; this procedure produced a pair of off-axis
`images.
`The term holography in analogy to photography has
`emerged to describe the expanding applications and
`techniques of holograms. In addition to imaging, ho-
`lograms are directly useful for diagnostics without im-
`age formation because they are interferograms. Holo-
`to correct
`grams are useful as optical elements
`
`The author is with General Dynamics, Electronics Division, P.O.
`Box 85227, San Diego, California 92138.
`Received 15 March 1987.
`0003-6935/87/204351-10$02.00/0.
`© 1987 Optical Society of America.
`
`aberrations; this kind of application motivated Ga-
`bor's original work. Holograms are also useful for data
`storage. Some of the applications exploit the separa-
`tion of the process into two steps; the separation of
`formation and reconstruction provides an interval for
`analysis and processing.
`Optical data processing for synthetic aperture radar
`stimulated interest in holography during the 1960s.
`Much of this work was done at the University of Michi-
`gan.3 Formation was done at centimeter wavelengths,
`which differ from visible reconstruction wavelengths.
`Gabor's experiments with visible light and their con-
`nection to DeBroglie waves further illustrate wave-
`length diversity between formation and reconstruc-
`tion.
`Since Gabor's 1948 publication, the meaning of the
`term hologram has grown to include many applications
`over a wide range of wavelengths in both electromag-
`netics and acoustics.49 The first half of the word
`hologram stems from the Greek word for all; the second
`half suggests a record. The idea is that a hologram
`records both phase and amplitude in the form of an
`intensity pattern. Polarization information was not
`considered in the early days. The potential of holog-
`raphy for applications has stimulated diverse forma-
`tion experiments, and as a result descriptions of holo-
`grams often include several adjectives; some are
`mentioned later.
`The meaning of the word hologram has grown also
`because holograms can be generated by computers.
`Computer generated holograms (CGH) have many
`useful properties. For example, an object need not
`exist; an ideal wavefront can be computed on the basis
`of diffraction theory and encoded into a tangible holo-
`gram. Such holograms can be optical elements.
`CGHs can be synthesized for optical filters. The syn-
`thesis of holograms is connected to inverse scattering,
`itself a broad discipline.
`This paper describes methods and techniques of
`CGH from an historical and broad view. Some details
`are mentioned but only partially; more complete de-
`scriptions are in the References and Bibliography.
`
`15 October 1987 / Vol. 26, No. 20 / APPLIED OPTICS
`FNC 1032
`
`4351
`
`

`

`B. Computer Generated Optical Filters
`Computer generated holograms are similar to com-
`puter generated filters which were studied for optical
`signal processing of synthetic aperture radar data.10
`The detection problem involved filtering noisy 2-D
`signals or sets of noisy 1-D signals, which were record-
`ed on photographic film. Filtering was done with a
`coherent optical system consisting of a point source,
`collimating lens, Fourier transform lens, and imaging
`lens; a signal was inserted in the front focal plane of the
`transform lens and a filter in its back focal plane. For
`noise additive to the signal, the transform and filtering
`reduced the noise amplitude. The basis of the method
`was to modulate a spatial carrier, that is, form a de-
`formed grating. Modulated carriers were also de-
`scribed in Refs. 2 and 3. Kozma and Kelly, at the
`University of Michigan, constructed 1-D filters that
`operated on phase-only by computing the signal's Fou-
`rier transform, drawing a black and white 1-D grating,
`and photographically reducing the drawing to produce
`a filter transparency. These filters, which were devel-
`oped for the detection of signals in noise, were precur-
`sors of computer generated holograms.
`
`C. Overall Process of Computer Generated Holography
`An overall schema for computer generated hologra-
`phy is shown in Fig. 1. It includes the following enti-
`ties and processes:
`Object The object need not exist physically; it can
`be imaginary or idealized. The object can radiate or
`be illuminated by an external source.
`Wave Propagation Wave propagation is comput-
`ed with theories appropriate for Fraunhofer, Fresnel,
`or near-field diffraction; the theories may be vector or
`scalar.
`Hologram Surface The object-scattered field is
`evaluated on a hypothetical surface which is usually
`flat.
`Hologram Fabrication The field on the hologram
`surface is represented by a transparency produced by a
`computer driven plotter, laser beam, or electron beam
`on correspondingly diverse materials. Scale reduction
`is usually necessary for reconstruction with visible
`light or other radiation of similar wavelength.
`Hologram The computer generated hologram is a
`tangible mask with spatially variable transmittance.
`Reconstruction The hologram is illuminated, and
`diffracted energy propagates to a detector.
`Detector Photographic film is a common detector,
`but other sensors such as charge-coupled devices can
`be used. For microwaves small antennas and receivers
`have been utilized.
`
`D. Computer Generated Holograms
`Computer generated holograms were described by
`Brown and Lohmann in 1966.11 Motivation included
`optical spatial filtering, which they experimentally
`demonstrated for 2-D objects. Imaging also was dem-
`onstrated. In addition, Ref. 11 pointed outtwo impor-
`tant general aspects of computer generated holograms.
`
`4352
`
`APPLIED OPTICS / Vol. 26, No. 20 / 15 October 1987
`
`OBJECT
`
`WAVE PROPAGATION
`
`HOLOGRAM
`SURFACE
`
`I_
`
`_
`
`_ -O
`
`HOLOGRAM
`FABRICATION
`
`HOLOGRAM (
`( RECONSTRUCTION .
`Fig. 1. Schema of computer generated holography.
`
`*
`
`DETECTOR
`
`One aspect is that the object need not exist; there-
`fore, idealized wavefronts can be produced. This at-
`tribute underlies application to optical testing with
`holograms and the generation of optical elements.
`The second aspect is that hologram synthesis is the
`opposite from usual diffraction problems in which the
`diffracting object is given and the diffracted field is
`sought. Instead, the image is prescribed, and the dif-
`fracting object, the hologram, is sought; thus hologra-
`phy is connected to inverse scattering.
`The work of Brown and Lohman stimulated interest
`in applications and research on computer generated
`holograms.
`Research emphasized (1) methods for representing,
`or encoding, computed wavefront data into, tangible
`holograms and (2) the consequences of approxima-
`tions in formats and computer aided graphic methods;
`Sec. II briefly describes this work. Many applications
`have been made; Sec. III summarizes them and pre-
`sents a chronological distribution of publications as a
`means of showing trends.
`
`II. Types of Computer Generated Holograms
`
`A. Detour Phase Holograms
`The first computer generated holograms were made
`by Brown and Lohmann.11 The holograms, intended
`for Fraunhofer diffraction, were formed by computing
`the image's Fourier transform and representing the
`transform values in a mask that had transparent aper-
`tures in an otherwise opaque screen. Because the
`mask's transmittance had values of zero or unity, the
`holograms were called binary. The transform plane
`was subdivided into regions of equal size, or cells.
`Three ways of representing the data were presented.
`In two, each cell contained an aperture, whose height
`or width depended on the transform magnitude at the
`center of the cell; in the other, each cell contained two
`apertures whose total width depended on the trans-
`form's magnitude. In all three representations, the
`lateral positions of an aperture was proportional to the
`transform's phase at the center of its cell. This lateral
`
`

`

`m8v
`
`vY
`
`<'V
`
`FREG
`1MG
`HOLO
`SOU
`Fig. 3. Optical setup for reconstruction with a point source (SOU)
`the hologram,
`at x0. HOLO, IMG, and FREQ are, respectively,
`and frequency planes.
`
`VX oximage,
`
`(Av/v) 2 > (Ax/6x) 2 = (AxAv)2.
`(4)
`In fact, 6v was taken to be (Ax)-1 on the basis of the
`sampling theorem.
`To develop expressions for aperture dimensions,
`consider Eq. (1) and apply scalar diffraction theory to
`the left-hand side. The hologram's transmittance is
`
`E rectl[vx -
`H(vx,vy) = E
`n m
`
`(n + Pn00 )bv]/cb5vl
`
`X rect[(vy - m5v)/Wnmbv]I
`The rectangle functions describe the aperture dimen-
`sions in a cell as in Fig. 3. Illumination by a plane wave
`produces complex amplitude
`h = f fH(vP,vy)E[(x + xo)v, + yvyldvxdvy
`
`(6)
`
`(5)
`
`= c(av)2 sinc[cbv(x + Xo)] By Wnm
`
`X sinc(yWnmbv)Et6v[(x + x)(n + Pnm) + ym].
`For the right-hand side of Eq. (1) the described image
`u(x,y) in its Fourier representation with the tilde signi-
`fying the transform is
`-_ ~~~~u(xy) = f f (v.,vY)E(xvx + yvy)dpxdvy.
`(1) Consider the Fourier sampling theorem
`a = uja(n/Ax,m/Ax)
`ted
`n m
`
`(7)
`
`(8)
`
`nav
`Fig. 2. Typical cell in a binary hologram. Position indices are
`and m. Width is CMv, height Wnmbp, and lateral displacement frc
`the cell center is PmbIv.
`
`shift led to the name detour phase, in analogy to d
`fraction gratings with unequally spaced rulings.
`Brown and Lohmann's 1966 paper was significa
`It demonstrated imaging and matched filtering; it a]
`stimulated much research and many applicatioi
`However, according to Lohmann and co-workers, t
`paper was intuitive so in 1967 they presented a mc
`analytical connection between reconstructions and I
`logram configurations and a discussion of approxin
`tions.
`To describe the approximations we collect and into
`pret some formulas given by Lohmann and Paris
`The reason for the approximations is to describe t
`amplitude reconstructed from the hologram in t
`form of a Fourier series, which is equated to a Fouri
`series representation of the image. In the procei
`formulas result for dimensions of cells as shown in F
`2.
`
`The aim is a hologram with transmittance H(v.,v
`which produces an image amplitude u(x,y) when ill
`minated by a plane wave exp(i27rvxxo) = E(vxxo) in t
`arrangement of Fig. 3, where XH = Xfvx and Yh = Xf
`The requirement is that h(xy) the amplitude diffra
`ed by the hologram be proportional to that of the ima
`in a finite region; that is,
`h(x,y) = const. u(x,y).
`With the image width Ax and height Ay, the diffract
`amplitude is
`h(x,y) = rect(x/Ax) rect(y/Ay)
`
`(2)
`SSH(vx,vy)E[(x + xo)vx + yvy]dv~dvy,
`where rectz has a unit value for Izi < 1/2 and zero
`otherwise.
`The hologram is represented by sampled values.
`The number of samples depends on image size and
`image resolution. If Ax is the resolution element size in
`the image, the number of resolvable points in the im-
`age is, with Ax = (Av)-1,
`2 = (AX4A)(Av)2 .
`N 2 = (A)/(bX)
`(3)
`N is called the number of degrees of freedom. The
`hologram is assumed to have at least N 2 points to
`preserve information; thus
`
`X sinc(vxAx - n) sinc(vyAx - m).
`
`(9)
`
`With Eq. (9), Eq. (8) gives
`u(x,y) = rect(x/Ax)
`
`X rect(y/Ax) B
`n m
`
`u(n6v,m5)E[6v(xn + ym)].
`
`(10)
`
`To satisfy the condition in Eq. (1), Eq. (7) is approxi-
`mated to equal Eq. (10) term by term. The result is
`(51v)2 WnmE [xobv(n + Pnm)]
`(11)
`const. a(n~v,mbv),
`where const. 11(n6v,mbv) is defined as c(5v) 2AnmE('kmm/
`27r), and Anm and cnm are the magnitude and phase of
`the transform C1. Thus
`
`15 October 1987 / Vol. 26, No. 20 / APPLIED OPTICS
`
`4353
`
`

`

`Wnm Anm;
`
`Pnm + n
`nm/27rx6v.
`(13)
`In words, Wnm controls magnitude; Pnfl controls phase.
`If x0 is chosen so that x05v is an integer M,
`Pn- ' n./2,W.
`(14)
`Equation (14) shows that the lateral displacement is
`proportional to the phase of the transform, an impor-
`tant relation, which has intuitive meaning as the phase
`difference of waves originating from two separated
`Huygens sources.
`In equating terms on both sides of Eq. (1), in the
`forms in Eqs. (7) and (10), some approximations are
`necessary in Eq. (7). These are as follows:
`
`sinc[cbv(x + x0 )J
`
`constant in 4x1 < x/2;
`
`sinc(yWnmnv)
`
`1 in y < (Ax/2);
`
`(15)
`
`(16)
`
`E(xPnm.v)
`
`1 in x < Ax/2.
`
`(17)
`The approximation of Eq. (17) has been called detour
`phase error; it has been extensively studied.12"14 It can
`give inhomogeneous image intensity, which can be
`more noticeable near image edges. The approxima-
`tion of Eq. (16) reduces image brightness but does not
`affect quality. The approximation of Eq. (15) pro-
`duces inhomogeneous intensity; it has been called ap-
`erture error.
`Despite the approximations, detour phase holo-
`grams are quite useful in imaging and filtering. In
`practice, distortions can be reduced by reducing cell
`size. However, as applications progress and require-
`ments become more stringent, the approximations
`should be recalled. Reference 14 discusses a relation-
`ship between image error and the SNR.
`Detour phase holograms contain additional approxi-
`mations that can degrade images. Some of these er-
`rors also occur in other types of computer generated
`holograms, but for convenience the errors are de-
`scribed in this section.
`Computer generated holograms represent sampled
`data. Transform values are computed at discrete
`points, and plotters often have a finite number of
`positions. This sampling can introduce aliasing error,
`which causes intensity in higher-order images to ap-
`pear in lower orders. 1 51 6
`An error, called a quantization error, occurs when a
`finite number of phase or intensity values is plotted in
`the hologram.17"18 The consequence
`is false images.
`An error known as gap and overlap occurs when phase
`exceeds jar. This ambiguity can be eliminated by
`restricting the spacing between adjacent apertures.' 9
`Finally, a truncation error occurs when the hologram is
`smaller than the spatial extent of the transform. 4
`B. More Detour Phase Holograms
`Lee developed a method that is based on decompos-
`ing the Fourier transform of the object into four quad-
`rature components, which were represented by the real
`non-negative functions.20 The phase of each compo-
`
`4354
`
`APPLIED OPTICS / Vol. 26, No. 20 / 15 October 1987
`
`nent was coded into sampled functions, and the sum of
`the four functions represents the sampled function.
`Lee called this method delayed sampling. It does not
`require phase quantization.
`The four functions are represented in a hologram by
`apertures at four laterally displaced, or shifted, posi-
`tions within each cell of the hologram plane. The
`method does not require phase quantization because
`the transmittance of each cell varies. In computing
`the object's transform, the sampling rate along the
`horizontal vx axis in Fig. 2 is 4 times that in the orthogo-
`nal v direction. Plotting was quantized because the
`hologram plotter was quantized. Lee demonstrated
`images for a binary object and a continuous tone ob-
`ject.
`Hsueh and Sawchuck2l developed binary holo-
`grams, called double-phase holograms, by decompos-
`ing the hologram transmittance into two-phase quan-
`tities. Each cell was divided into two subcells, which
`were laterally separated. Each subcell contained two
`transparent slits in an opaque background; slit widths
`were half of the cell width for diffraction efficiency.
`The vertical position of each slit was determined by the
`decomposed phase of the Fourier transform according
`to the detour phase principle. Consequently, even
`and odd diffraction orders were vertically displaced.
`Noise caused by subcell displacement and phase cod-
`ing were analyzed; noise reduction methods were dis-
`cussed. Visible images from holograms were present-
`ed.
`Burckardt developed an approach that utilized
`three components.22
`
`C. Nondetour Phase Holograms
`Lee developed binary computer generated holo-
`grams that did not utilize the detour phase concept.23
`The holograms were based on considering the comput-
`ed hologram as an interferogram. Positions and
`widths of the fringes were determined as the set of
`points satisfying inequalities involving the phase of
`the reconstructed wave. Experiments demonstrated
`reconstruction of spherical, conical, and helical wave-
`fronts.
`Lee later applied the view of computer generated
`holograms as interferograms to two problems of detour
`phase holograms.24 One problem was sampling at dis-
`crete points in the hologram plane. The other prob-
`lem was to phase variations exceeding 27r. Methods
`for encoding both amplitude and phase were de-
`scribed.
`The spatial bandwidths of Lee-type nondetour
`phase holograms have been analyzed with frequency
`modulation theory; a quantization error model was
`presented. 2 5
`Burch developed holograms that encoded the Fouri-
`er sine and cosine transforms of real functions that
`describe objects.26 Holograms are optically recorded
`by adding a bias term to the sum of the transforms.
`The reconstructions give pairs of off-axis holograms,
`but the on-axis light, corresponding to the object's
`
`

`

`is lower than that for holograms
`autocorrelation,
`formed with an inclined reference beam.
`A wavefront reconstruction device that is similar to a
`lens has been developed.2 7 2 8 This device, called a
`kinoform, resembles a blazed dielectric transmission
`grating. Kinoforms are useful as optical filters with
`noncoherent light. They have high diffraction effi-
`ciency compared with many kinds of holograms. They
`diffract on-axis rather than into off-axis orders as do
`sampled holograms or holograms formed with an in-
`clined reference beam.
`
`Ill. Literature Survey
`
`A. Temporal Distribution of Publications
`An analysis of the temporal distribution of pub-
`lished papers suggests some trends. (Publications are
`only one possible measure of activity; significance is
`another matter.) This analysis included approxi-
`mately 200 papers published in the 21 years from 1966
`through 1986. The bibliography is somewhat arbi-
`trary. It includes papers from the following journals:
`Applied Optics;
`Chinese Physics;
`Electronics and Communications in Japan;
`Electronics Letters;
`IEEE Transactions on Computers;
`Journal of the Optical Society of America, Includ-
`ing A;
`Laser Optoelektronika;
`Nouvelle Revue d'Optique
`Onde Electronique;
`Optica Acta;
`Optical Engineering;
`Optics Communications;
`Optica Pura Appl;
`Optik
`Proceedings of the IEEE;
`Review of the Electrical Communication Laborato-
`ries (Tokyo).
`It also includes papers in SPIE Proceedings and
`Proceedings of the Optical Computing Conference.
`However, it excludes abstracts of the annual meeting
`of the Optical Society of America and summaries of
`papers presented at the International Commission of
`Optics meetings.
`No claim is made that the set of approximately 200
`papers includes all on computer generated holography.
`Some recent books contain useful reviews.29-31
`As a basis for discussion consider Fig. 4, which shows
`a plot of the number of papers published in each year
`from 1966 through 1986. In 1966 one paper by Brown
`and Lohmann was published. The number per year
`has an increasing trend until it reaches a maximum of
`fifteen papers in both 1974 and 1975. A minimum
`occurs in 1977 and 1978 with an increase to a second
`maximum of twenty-five papers in 1983. The maxi-
`mum is followed by a decrease to thirteen papers in
`1986; this maximum exceeds the minimum of seven
`papers in 1977 and 1978. The number of papers may
`not reliably show interest in and vigor of a field, but it
`
`n LOAN
`
`. .~ -
`66
`
`-
`
`-
`68
`
`70
`
`.
`72
`
`|----t-s---
`.
`.
`.
`.
`74 76
`YEAR
`Number of CGH papers published per year.
`
`30
`
`25
`
`20.
`
`PAPERS PER YEAR is 4.
`
`10 4.
`
`5.
`
`I IliiiiilI
`
`.
`
`.
`.
`.l
`78 80
`
`82
`
`II
`
`86
`
`84
`
`I
`
`Fig. 4.
`
`is affected in a com-
`does show activity. Publication
`plicated way by the maturity of a field because applica-
`tions stimulate research and vice versa. Research
`funding may also influence activity.
`As a matter of interest we note that the 1985 Annual
`Meeting of the Optical Society of America had three
`papers on CGH; the 1986 meeting had five.
`Although Sec. III.B describes applications, we antic-
`ipate it and state that the maxima in 1975 and 1983
`In 1975, eight of the fifteen
`have an interpretation.
`papers were on techniques (methods and theory) of
`computer generated holography, and four were on op-
`tical elements with one or two on other applications.
`In 1983, ten papers of twenty-five were on optical
`processing, eight on holographic techniques, and seven
`on optical elements. This analysis is somewhat ap-
`proximate because some papers fall into two classifica-
`tions. Nevertheless, the nature of papers in the two
`years suggests that in 1975 emphasis was on improving
`CGH and that in 1983 the application of optical pro-
`cessing emerged.
`
`B. Applications
`Computer generated holograms now have several
`diverse uses which are grouped into broad categories as
`follows. Specific topics illustrate the nature of the
`categories.
`Diagnostic and Testing acoustic mapping of the
`earth, determining particle sizes and scattering prop-
`erties, analyzing fiber optical modes, analyzing vibra-
`tions, visualizing aberrations.
`Digital and Optical Interconnects sequential op-
`tical logic operations, digital optical architecture and
`computing.
`High-Energy Physics character detection, proces-
`sor alignment.
`Imaging and Display map displays, map transfor-
`mations, hologram scaling, colored displays, 3-D dis-
`plays, reduced quantization error, electron microsco-
`py, image processing and deblurring, stereoscopic
`displays.
`Improved Holographic Techniques computa-
`tional efficiency, photographic film and alternate re-
`cording materials, detour phase and quantization er-
`
`15 October 1987 / Vol. 26, No. 20 / APPLIED OPTICS
`
`4355
`
`

`

`High-
`Diagnostic
`Inter
`energy
`Imaging Holographic
`Year and testing connects physics display
`techniques
`1966
`1967
`1968
`1969
`1970
`1971
`1972
`1973
`1974
`1975
`1976
`1977
`1978
`1979
`1980
`1981
`1982
`1983
`1984
`1985
`1986
`
`Table 1. Number of Papers on Each Application Area per Year
`
`Optical
`Inverse
`data
`Optical
`scattering Medicine Memories processing elements Scanners
`1
`2
`
`1
`1
`
`1
`1
`
`2
`1
`3
`
`1
`1
`1
`5
`10
`8
`5
`3
`
`1
`
`4
`
`4
`1
`
`1
`2
`2
`3
`3
`7
`4
`7
`2
`
`2
`
`1
`3
`
`1
`
`1
`2
`4
`1
`
`3
`4
`8
`3
`4
`7
`5
`4
`1
`2
`8
`2
`4
`2
`
`1
`
`1
`1
`
`2
`
`1
`1
`
`8
`2
`4
`2
`1
`1
`2
`1
`4
`2
`
`2
`
`1
`1
`
`1
`1
`3
`1
`2
`
`rors, signal-to-noise ratio; nondetour phase holograms,
`hologram recording methods, diffraction efficiency,
`large object formats, encoding representations, phase
`encoding, iterative methods for incomplete data, for-
`mation with electron or laser beams, sampling meth-
`ods, plotters, multiple-exposure holograms, Fresnel
`diffraction, polar geometries, multiplexing.
`Inverse Scattering
`acoustic imaging, wave aberra-
`tion retrieval, eigenfunctions expansion of scattered
`field and synthesis.
`Medicine diagnostics, therapy.
`Memories synthesis, error rates.
`Optical Data Processing filtering, correlation,
`pattern recognition, map transformation, imaging on
`curved surfaces, coordinate transformations, integral
`transforms, image deblurring, digital optical computa-
`tion, hybrid computation, rotation-invariant process-
`ing, shift-invariant processing, spectrum analysis,
`nonlinear processing, code translation.
`Optical Elements
`synthesis of optical elements,
`application to testing optical elements, interferome-
`try, real-time interferometry, wavefront synthesis,
`aspheric surface testing, aberrations, laser beam uni-
`formity, wavelength diversity, partitioned elements,
`Fresnel region CGH, volume holograms, wave polar-
`ization, diffraction pattern sampling, tandem ele-
`ments.
`Scanners
`field flatness, curvature correction,
`graphics scanners, beam steering.
`A perspective on the growth of computer generated
`holography can be obtained from Table I which plots
`the number of papers in the bibliography for each
`category of application by year.
`Work on improving holographic techniques has per-
`sisted for twenty years. An obvious reason is the need
`for better performance for new applications; another
`reason may be intellectual motivation to understand
`
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`
`APPLIED OPTICS / Vol. 26, No. 20 / 15 October 1987
`
`approximations.
`Interest in optical data processing
`has continued from the start of computer generated
`holograms; activity has increased in the past several
`years. Work continues on imaging and display but at a
`decreasing pace. Optical elements have been studied
`consistently; the number of papers has recently in-
`creased. A new application,
`interconnects
`for com-
`puters, has emerged.
`Although forecasting the future is risky, it seems
`plausible that interest will continue in computer gen-
`erated holograms. Research is being done on increas-
`ing the efficiency of methods for producing holograms
`and controlling diffraction for interconnects and pro-
`cessing. Finally, the educational potential of comput-
`er generated holograms merits consideration.
`It is a pleasure to thank Reiner Eschbach of the
`University of California, San Diego, and Olaf Bryng-
`dahl of the University of Essen for helpful comments
`and discussions. It is also a pleasure to thank my
`thesis advisor and mentor Adolf Lohmann of the Uni-
`versity of Erlangen-Nuremberg. Professor Lohmann
`introduced me to computer generated holograms al-
`most twenty years ago during a course on optical infor-
`mation processing at the University of California, San
`Diego.
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`5. G. W. Stroke, An Introduction to Coherent Optics and Hologra-
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`6. J. B. DeVelis and G. 0. Reynolds, Theory and Application of'
`Holography
`(Addison-Wesley, Reading, MA, 1967).
`
`

`

`of Holography
`
`to Acous-
`
`7. H. J. Caufield and S. Lu, The Applications
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`IBM
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