recent approaches to computer— generated hologramsja ...€¦ · use in image enhancement6,...

Recent Approaches To Computer - Generated Holograms*

D.C. CHU

Hewlett - Packard Co., 5301 Stevens Creek Blvd., Santa Clara, California 95050

and

J.R. FIENUP

Information Systems Laboratory, Stanford University, Stanford, California 94305

(Received February 4, 1974)

ABSTRACT

Computer holography is a subject of interest to a small but increasing number of researchers.This paper briefly describes the concept and history of computer holography, how it works, andits possible applications. Also presented are the recent approaches to computer holographydeveloped at Stanford University, and a procedure for synthesizing this new class of computerholograms. These new on -axis holograms are very efficient both in their use of reconstructionlight and in their use of display resolution elements during synthesis. They keep the advantagesof the kinoform, a previous approach by IBM, but do not have its limitations. One approachmakes use of color reversal film, such as Kodachrome II. Another approach only requires phase-

transmittance material, but preserves both amplitude and phase information. Experimentalresults are presented.

INTRODUCTION

In ordinary interferometric holography, the hologram ismade by recording the interference pattern of an object com-plex wave and a phase- coherent reference wave. Upon illumi-nation of the hologram, an image of the original object can bereconstructed.

In synthesizing holograms by a computer, no physicalobject exists. Instead, one starts with a mathematical descrip-tion of an object, usually in the form of an array of phase -coherent point sources. The complex wavefront from such anarray on a fixed plane some distance away can be computed.If a transparency can be synthesized whose complex transmit-tance is equal to the computed wavefront on that plane, thenupon illumination of such a transparency (or hologram) by aplane wave, light will be modulated by the hologram so thatit will emerge exactly as though it came from the object. Withthis technique, therefore, a visible image can be constructed,in three dimensions if desired, of an object which does notexist physically. The problem, of course, is the realization of

*This work was sponsored by National Science Foundation and theHewlett- Packard Company.

VOL. 13 -NO. 3

the transparency whose transmittance is proportional to acomputed complex function.

Black and white film can be used to control the amplitude,or modulus, of a light wave and a bleached film can be usedto control the phase of light, due to changes in refractiveindex and surface relief effects. A transparency of a givencomplex transmittance can be synthesized by making a"sandwich" of two transparencies, one controlling amplitudeand the other the phase. In practice, however, this is seldomdone because of the difficulty in obtaining proper registrationbetween the two transparencies. To overcome this problem,many ingenious approaches were devised to represent a com-plex function by using a recording medium which controlsonly amplitude or only phase. Some of these are describedlater on in this paper.

These holograms of non -physical objects are called "syn-thetic holograms ". Since a digital computer is often used intheir synthesis they are sometimes also called "computer -generated holograms ".

FOURIER TRANSFORM HOLOGRAMS

One particularly important class of synthetic holograms isthe Fourier transform hologram which essentially contains

OPTICAL ENGINEERING -MAY /JUNE 1974 189

___ ____ JA

Recent Approaches To Computer— Generated Holograms

D.C. CHU

Hewlett-Packard Co., 5301 Stevens Creek Blvd., Santa Clara, California 95050

and

J.R. FlENUP

Information Systems Laboratory, Stanford University, Stanford, California 94305

(Received February 4, 1974)

ABSTRACT

Computer holography is a subject of interest to a small but increasing number of researchers. This paper briefly describes the concept and history of computer holography, how it works, and its possible applications. Also presented are the recent approaches to computer holography developed at Stanford University, and a procedure for synthesizing this new class of computer holograms. These new on-axis holograms are very efficient both in their use of reconstruction light and in their use of display resolution elements during synthesis. They keep the advantages of the kinoform, a previous approach by IBM, but do not have its limitations. One approach makes use of color reversal film, such as Kodachrome II. Another approach only requires phase- transmittance material, but preserves both amplitude and phase information. Experimental results are presented.

INTRODUCTION

In ordinary interferometric holography, the hologram is made by recording the interference pattern of an object com plex wave and a phase-coherent reference wave. Upon illumi nation of the hologram, an image of the original object can be reconstructed.

In synthesizing holograms by a computer, no physical object exists. Instead, one starts with a mathematical descrip tion of an object, usually in the form of an array of phase- coherent point sources. The complex wavefront from such an array on a fixed plane some distance away can be computed. If a transparency can be synthesized whose complex transmit tance is equal to the computed wavefront on that plane, then upon illumination of such a transparency (or hologram) by a plane wave, light will be modulated by the hologram so that it will emerge exactly as though it came from the object. With this technique, therefore, a visible image can be constructed, in three dimensions if desired, of an object which does not exist physically. The problem, of course, is the realization of

This work was sponsored by National Science Foundation and the Hewlett-Packard Company.

the transparency whose transmittance is proportional to a computed complex function.

Black and white film can be used to control the amplitude, or modulus, of a light wave and a bleached film can be used to control the phase of light, due to changes in refractive index and surface relief effects. A transparency of a given complex transmittance can be synthesized by making a "sandwich" of two transparencies, one controlling amplitude and the other the phase. In practice, however, this is seldom done because of the difficulty in obtaining proper registration between the two transparencies. To overcome this problem, many ingenious approaches were devised to represent a com plex function by using a recording medium which controls only amplitude or only phase. Some of these are described later on in this paper.

These holograms of non-physical objects are called "syn thetic holograms". Since a digital computer is often used in their synthesis they are sometimes also called "computer- generated holograms".

FOURIER TRANSFORM HOLOGRAMS

One particularly important class of synthetic holograms is the Fourier transform hologram which essentially contains

VOL. 13-NO. 3 OPTICAL ENGINEERING-MAY/JUNE 1974 189

Downloaded From: http://spiedigitallibrary.org/ on 11/25/2015 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx

the Fourier transformt of the desired image. Upon illumina-tion of the hologram in front of a lens, the desired image isfound one focal length behind the lens. Fourier transformholograms have many superior properties; for example,lateral translation of the hologram produces no change in theintensity distribution of the image, which remains stationary.This property is a direct result of a property of the Fouriertransform where translation in one domain is accompaniedonly by a linear phase shift in the other, without affectingthe modulus. Another desirable property of the Fouriertransform is its ability to distribute localized disturbances inone domain more or less uniformly in the other. As a result,localized defects of the hologram due to dust particles andscratches do not result in localized blemishes on the image,but rather result in a slight general loss of fidelity of theentire image. By simply repeating the Fourier transform anumber of times side by side on the hologram, the image isnot repeated, but approaches a sampled version and appearsin a dot structure. This last property is particularly useful ifthe desired image is that of an array of binary data. Thisdiscrete dot structure facilitates detection by discrete elec-tronic detection devices such as photo- diodes, and reducesinterference between adjacent data elements. The repeating ofthe transform increases the redundancy, suppresses thespeckle, and increases the signal to noise ratio. Notice alsothat the increased redundancy is accomplished without anaccompanying increase in the number of detection devices inthe image domain.

APPLICATIONS OF COMPUTER HOLOGRAMS

The word "hologram" immediately brings to mind three-dimensional images; but that is just one of many possible usesfor computer- generated holograms. A general wavefrontmodulator such as a hologram can also be a coherent opticalspatial filter, an optical memory, a component in the testingof optical elements, or an optical element itself.

A computer hologram can be made of any one, two orthree- dimensional object which can be described mathemati-cally.1 The description can be a set of equations or a sequenceor array of numbers obtained from such things as a scannedphotograph or an acoustic microscope. Color images may beobtained by making three color- separation holograms for thered, green and blue information and illuminating them withthe respective colors of coherent light.2 The principal draw-back is that the large number of resolvable elements in a highquality image, typically 105 or 106 for a two -dimensionalpicture and many times that for a three -dimensional one,combined with the extra redundancy desirable in a hologram,requires time -consuming and expensive computation anddisplay.

A promising use of computer holograms is for archivalstorage of data from computer outputs. The data to be storedmay exist in core, and in this case the digital Fourier transfor-mation step would represent the only extra step in compari-son with writing the data directly on film. The benefits oftranslation invariance, burst noise resistance and redundancymentioned earlier make this extra step worthwhile, particu-larly if the memory is to be machine -read later. The outstand-ing features of holographic memory are high storage density,quick accessibility due to parallel processing and low cost.

t Strictly speaking it should be the inverse transform. Thedistinction, however, depends only on the definition ofspatial coordinates and is conceptually unimportant.

190

The construction of a memory based on computer hologra-phy has recently been completed at Radiation, Inc., inMichigan.

Another area in which computer holography has been putto practical use is in the testing of optical elements.3 Manu-facturing errors in an optical element can be detected byobserving the difference between a wavefront from theoptical element and a reference wave in an interferometer.Computer- generated holograms can be used to generate thereference wavefront standard. This is particularly useful in thetesting of aspheric surfaces, where the complexity of the wavemakes it difficult to synthesize by any other means.

In addition to their use in testing optical elements, com-puter generated holograms can be the optical elementsthemselves. Examples of possible uses include computer -generated Fresnel lenses, diffraction gratings, zone plates,and diffusers. With the ability to control amplitude as well asphase transmittance, a whole new class of optical elements ispossible, including an imaging device which resolves beyondthe diffraction limit.4

Still another application of computer -generated hologramsis their use as optical spatial filters. Although many types ofcoherent optical spatial filters can be optically generated5,computer generation is often the most convenient - andsometimes the only way to make them. Spatial filters finduse in image enhancement6, restoration of degraded images7,matched filtering8, and code translation.9 With computer -generated holograms, matched filtering with incoherentillumination is also possib1e.1 O

PREVIOUS APPROACHES

Most display instruments used for synthesizing holograms,such as digital plotters or computer controlled CRT's, havean upper limit to the number of addressable locations, orresolution elements, restricting the ultimate complexity ofthe hologram as expressed by the space- bandwidth product. Avery important parameter in the synthesis of holograms istherefore the number of such resolution elements required forencoding each Fourier coefficient in a Fourier transformhologram.

The early attempts at computer generation of hologramswere merely simulations of the process of the optically gen-erated Leith and Upatnieks hologram.11,12,13 The complex-ity of the computation involving a reference beam, and thelarge number of display resolution elements needed to encodethe resulting high frequency interference fringes made theseapproaches unattractive.

Perhaps the best known computer- generated hologram,one which departs from a mere simulation of the interfero-metric hologram, is the detour -phase binary hologram ofBrown and Lohmann.14 In this technique exposure controlis not critical since only binary amplitude transmittance isrequired. The method divides the hologram into cells, eachone representing a complex Fourier coefficient. Within eachcell is an aperture, with its size related to the amplitude andits position within the cell to the phase of the coefficient.When illuminated, the desired image is obtained off -axis inthe first diffraction order. In this method, a large number ofdisplay resolution elements are used for each Fourier coeffi-cient. Any attempt to reduce the number results in increasedamplitude and phase quantization error. There are othervariations of the detour -phase binary holograms by Haskell15as well as gray -level detour -phase holograms by Lee16 andBurkhardt.17 However, they are all off -axis techniques and

OPTICAL ENGINEERING- MAY /JUNE 1974 VOL. 13-NO. 3

the Fourier transformt of the desired image. Upon illumina tion of the hologram in front of a lens, the desired image is found one focal length behind the lens. Fourier transform holograms have many superior properties; for example, lateral translation of the hologram produces no change in the intensity distribution of the image, which remains stationary. This property is a direct result of a property of the Fourier transform where translation in one domain is accompanied only by a linear phase shift in the other, without affecting the modulus. Another desirable property of the Fourier transform is its ability to distribute localized disturbances in one domain more or less uniformly in the other. As a result, localized defects of the hologram due to dust particles and scratches do not result in localized blemishes on the image, but rather result in a slight general loss of fidelity of the entire image. By simply repeating the Fourier transform a number of times side by side on the hologram, the image is not repeated, but approaches a sampled version and appears in a dot structure. This last property is particularly useful if the desired image is that of an array of binary data. This discrete dot structure facilitates detection by discrete elec tronic detection devices such as photo-diodes, and reduces interference between adjacent data elements. The repeating of the transform increases the redundancy, suppresses the speckle, and increases the signal to noise ratio. Notice also that the increased redundancy is accomplished without an accompanying increase in the number of detection devices in the image domain.

APPLICATIONS OF COMPUTER HOLOGRAMS

The word "hologram" immediately brings to mind three- dimensional images; but that is just one of many possible uses for computer-generated holograms. A general wavefront modulator such as a hologram can also be a coherent optical spatial filter, an optical memory, a component in the testing of optical elements, or an optical element itself.

A computer hologram can be made of any one, two or three-dimensional object which can be described mathemati cally. 1 The description can be a set of equations or a sequence or array of numbers obtained from such things as a scanned photograph or an acoustic microscope. Color images may be obtained by making three color-separation holograms for the red, green and blue information and illuminating them with the respective colors of coherent light. The principal draw back is that the large number of resolvable elements in a high quality image, typically 105 or 106 for a two-dimensional picture and many times that for a three-dimensional one, combined with the extra redundancy desirable in a hologram, requires time-consuming and expensive computation and display.

A promising use of computer holograms is for archival storage of data from computer outputs. The data to be stored may exist in core, and in this case the digital Fourier transfor mation step would represent the only extra step in compari son with writing the data directly on film. The benefits of translation invariance, burst noise resistance and redundancy mentioned earlier make this extra step worthwhile, particu larly if the memory is to be machine-read later. The outstand ing features of holographic memory are high storage density, quick accessibility due to parallel processing and low cost.

t Strictly speaking it should be the inverse transform. The distinction, however, depends only on the definition of spatial coordinates and is conceptually unimportant.

The construction of a memory based on computer hologra phy has recently been completed at Radiation, Inc., in Michigan.

Another area in which computer holography has been put to practical use is in the testing of optical elements. 3 Manu facturing errors in an optical element can be detected by observing the difference between a wavefront from the optical element and a reference wave in an interferometer. Computer-generated holograms can be used to generate the reference wavefront standard. This is particularly useful in the testing of aspheric surfaces, where the complexity of the wave makes it difficult to synthesize by any other means.

In addition to their use in testing optical elements, com puter generated holograms can be the optical elements themselves. Examples of possible uses include computer- generated Fresnel lenses, diffraction gratings, zone plates, and diffusers. With the ability to control amplitude as well as phase transmittance, a whole new class of optical elements is possible, including an imaging device which resolves beyond the diffraction limit. 4

Still another application of computer-generated holograms is their use as optical spatial filters. Although many types of coherent optical spatial filters can be optically generated 5 , computer generation is often the most convenient and sometimes the only way to make them. Spatial filters find use in image enhancement6 , restoration of degraded images7 , matched filtering8 , and code translation. 9 With computer- generated holograms, matched filtering with incoherent illumination is also possible. 10

PREVIOUS APPROACHES

Most display instruments used for synthesizing holograms, such as digital plotters or computer controlled CRT's, have an upper limit to the number of addressable locations, or resolution elements, restricting the ultimate complexity of the hologram as expressed by the space-bandwidth product. A very important parameter in the synthesis of holograms is therefore the number of such resolution elements required for encoding each Fourier coefficient in a Fourier transform hologram.

The early attempts at computer generation of holograms were merely simulations of the process of the optically gen erated Leith and Upatnieks hologram. 1 1 »12,13 j^g complex ity of the computation involving a reference beam, and the large number of display resolution elements needed to encode the resulting high frequency interference fringes made these approaches unattractive.

Perhaps the best known computer-generated hologram, one which departs from a mere simulation of the interferometric hologram, is the detour-phase binary hologram of Brown and Lohmann. 14 In this technique exposure control is not critical since only binary amplitude transmittance is required. The method divides the hologram into cells, each one representing a complex Fourier coefficient. Within each cell is an aperture, with its size related to the amplitude and its position within the cell to the phase of the coefficient. When illuminated, the desired image is obtained off-axis in the first diffraction order. In this method, a large number of display resolution elements are used for each Fourier coeffi cient. Any attempt to reduce the number results in increased amplitude and phase quantization error. There are other variations of the detour-phase binary holograms by Haskell 15 , as well as gray-level detour-phase holograms by Lee 16 and Burkhardt. 17 However, they are all off-axis techniques and

190 OPTICAL ENGINEERING-MAY/JUNE 1974 VOL. 13-NO. 3


the light diffracted to the desired image is typically about 1%of that incident on the hologram.

Another well -known wave -front reconstruction device isthe kinoform18 which is not a hologram in the sense that thetotality (holo) of an amplitude and phase information is notpreserved. For the kinoform, the Fourier (or Fresnel) trans-form amplitude information is discarded under the assump-tion that for a "well- diffused" object, the transform ampli-tude can be approximated by a constant. Accordingly, thekinoform is a transparency whose phase transmittance is thetransform phase and whose amplitude transmittance is unity.Since there is no general method to diffuse any given objectperfectly, significant loss of fidelity often results from theamplitude quantization.

There are other computer -generated holograms describedin the literature. The interested reader is referred to twosurvey papers on the subject19,20 as well as a chapter of thetex t Optical Holography. 21

RECENT APPROACHES

Despite its fidelity problem, the kinoform has manyadvantages. It is efficient in its use of display resolutionelements, each one of which can generate a Fourier coeffi-cient (without amplitude). Also, all incident light on thekinoform is diverted to one useful image, and the efficiencyis ideally 100 %. Our work at Stanford University has beendirected at making holograms which retain most of theadvantages of the kinoform, but remove its limitations.

ROACH

One of our approaches is to seek a hologram materialwhich not only controls the phase, as with bleached film, butadds amplitude control without having to align two trans-parencies. The referenceless on -axis complextt hologram22(ROACH) uses multi -emulsion film, such as Kodachrome II,in which different layers can be exposed independently bylight of different colors. Upon illuminating with light of agiven color, one layer will absorb, while the other two layerswill be predominantly transparent, but will cause phase shiftsdue to variations in film thickness and refractive index(Figure 1). Since the complex transmittance, both amplitude

Source

Hologram

Amplitude Phasecontrol control

Imageplane

On -axis

image

Figure 1. A schematic diagram showing the reconstruction ofa ROACH. The amplitude and phase are independently con-trolled by different emulsion layers.

fi t The word "complex" here refers to the transmittance offilm which simultaneously controls amplitude by absorp-tion and phase through film thickness variation.

and phase, can be controlled at each point, only one resolv-able element is needed to encode one complex Fouriercoefficient. In the reconstruction, there are no conjugateimages or central -order spot, so the efficiency is high.

To make a ROACH for reconstruction with, for example,the red 632.8 nm light from a helium -neon laser, we simplyexpose the amplitude pattern with red light since only thered -sensitive layer will absorb red after processing, and exposethe phase pattern with blue -green light to generate the reliefpattern. The red exposure is calculated to give a final ampli-tude transmittance to red proportional to the amplitude ofthe transform. The blue -green exposure is computed to give afinal density to blue -green light (which we assume to belinearly related to the phase of red light) proportional to thetransform phase, subtracting an appropriate amount tocompensate for the undesirable relief due to the amplitudepattern. Detailed experimental procedure is given later inthis paper.

The hologram constructed with this method is simply akinoform with amplitude control. The improvement due toamplitude control can be seen by comparing the kinoformand ROACH reconstruction in Figure 2. The non -uniformityof the dot intensities in the kinoform reconstruction shownin Figure 2a is due primarily to the loss of the hologramamplitude information.

SPECTRUM SHAPING METHODS

Another approach for improving the kinoform is byencoding or modifying the computed Fourier transformmathematically in such a way that each element has the sameamplitude, with various phases. Thus, a hologram witheffective amplitude and phase control can be synthesizedwith a phase -only transparency. Unlike the kinoform, thehologram phase is now no longer equal to the transformphase, but modulated in accordance with the transformamplitude. The reconstruction is the optical Fourier (inverse)transform of this modified array. This procedure is illustratedin the block diagram of Figure 3. Here f represents the dataarray. After a fast -Fourier transform (FFT) digital processingstep, the complex transform array F is obtained which is thenencoded to give the modified array H. H has uniform modulusand the hologram I is synthesized whose phase transmittanceis made equal to the phase of H. The array h represents thefield reconstructed from the hologram although only themagnitude image is observable.

Immediately, the question is raised that since H is differentfrom F, then h is different from f; in other words, thereconstructed image is other than the one desired. The key tosuccess, however, lies in the fact that h contains all theelements of f, uncorrupted, together with some extraneous"parity" elements introduced by the modification of theFourier spectrum. Their locations are clearly separated fromthose of the data and can be ignored. Also, these parity ele-ments act as a light ballast, drawing light into or away fromthe data in such a manner that the intensity of each data bitis constant, regardless of the number of "on" bits in thearray. Alternatively, if constant intensity is not important,one can design for maximum efficiency.

There are many ways of encoding the Fourier transformto achieve data and parity separation. We shall describe twomethods used in our experiments. The first is simply calledthe parity sequence method.23 If jfn q} is an NXQ dataarray whose complex Fourier transform'is {Fm }, also NXQin size, then we encode ¡Ft to generate a 2Nx2Q hologram

VOL. 13 -NO. 3 OPTICAL ENGINEERING -MAY /JUNE 1974 191

the light diffracted to the desired image is typically about 1% of that incident on the hologram.

Another well-known wave-front reconstruction device is the kinoform *8 which is not a hologram in the sense that the totality (holo) of an amplitude and phase information is not preserved. For the kinoform, the Fourier (or Fresnel) trans form amplitude information is discarded under the assump tion that for a "well-diffused" object, the transform ampli tude can be approximated by a constant. Accordingly, the kinoform is a transparency whose phase transmittance is the transform phase and whose amplitude transmittance is unity. Since there is no general method to diffuse any given object perfectly, significant loss of fidelity often results from the amplitude quantization.

There are other computer-generated holograms described in the literature. The interested reader is referred to two survey papers on the subject 19,20 as wejj as a chapter of the text Optical Holography. 2 *

RECENT APPROACHES

Despite its fidelity problem, the kinoform has many advantages. It is efficient in its use of display resolution elements, each one of which can generate a Fourier coeffi cient (without amplitude). Also, all incident light on the kinoform is diverted to one useful image, and the efficiency is ideally 100%. Our work at Stanford University has been directed at making holograms which retain most of the advantages of the kinoform, but remove its limitations.

ROACH

One of our approaches is to seek a hologram material which not only controls the phase, as with bleached film, but adds amplitude control without having to align two trans parencies. The referenceless on-axis complextt hologram22 (ROACH) uses multi-emulsion film, such as Kodachrome II, in which different layers can be exposed independently by light of different colors. Upon illuminating with light of a given color, one layer will absorb, while the other two layers will be predominantly transparent, but will cause phase shifts due to variations in film thickness and refractive index (Figure 1). Since the complex transmittance, both amplitude

Hologram

Amplitude control

Phase control

Figure 1. A schematic diagram showing the reconstruction of a ROACH. The amplitude and phase are independently con trolled by different emulsion layers.

ttThe word "complex" here refers to the transmittance of film which simultaneously controls amplitude by absorp tion and phase through film thickness variation.

and phase, can be controlled at each point, only one resolv able element is needed to encode one complex Fourier coefficient. In the reconstruction, there are no conjugate images or central-order spot, so the efficiency is high.

To make a ROACH for reconstruction with, for example, the red 632.8 nm light from a helium-neon laser, we simply expose the amplitude pattern with red light since only the red-sensitive layer will absorb red after processing, and expose the phase pattern with blue-green light to generate the relief pattern. The red exposure is calculated to give a final ampli tude transmittance to red proportional to the amplitude of the transform. The blue-green exposure is computed to give a final density to blue-green light (which we assume to be linearly related to the phase of red light) proportional to the transform phase, subtracting an appropriate amount to compensate for the undesirable relief due to the amplitude pattern. Detailed experimental procedure is given later in this paper.

The hologram constructed with this method is simply a kinoform with amplitude control. The improvement due to amplitude control can be seen by comparing the kinoform and ROACH reconstruction in Figure 2. The non-uniformity of the dot intensities in the kinoform reconstruction shown in Figure 2a is due primarily to the loss of the hologram amplitude information.

SPECTRUM SHAPING METHODS

Another approach for improving the kinoform is by encoding or modifying the computed Fourier transform mathematically in such a way that each element has the same amplitude, with various phases. Thus, a hologram with effective amplitude and phase control can be synthesized with a phase-only transparency. Unlike the kinoform, the hologram phase is now no longer equal to the transform phase, but modulated in accordance with the transform amplitude. The reconstruction is the optical Fourier (inverse) transform of this modified array. This procedure is illustrated in the block diagram of Figure 3. Here f represents the data array. After a fast-Fourier transform (FFT) digital processing step, the complex transform array F is obtained which is then encoded to give the modified array H. H has uniform modulus and the hologram H is synthesized whose phase transmittance is made equal to the phase of H. The array h represents the field reconstructed from the hologram although only the magnitude image |h| is observable.

Immediately, the question is raised that since H is different from F, then h is different from f; in other words, the reconstructed image is other than the one desired. The key to success, however, lies in the fact that h contains all the elements of f, uncorrupted, together with some extraneous "parity" elements introduced by the modification of the Fourier spectrum. Their locations are clearly separated from those of the data and can be ignored. Also, these parity ele ments act as a light ballast, drawing light into or away from the data in such a manner that the intensity of each data bit is constant, regardless of the number of "on" bits in the array. Alternatively, if constant intensity is not important, one can design for maximum efficiency.

There are many ways of encoding the Fourier transform to achieve data and parity separation. We shall describe two methods used in our experiments. The first is simply called the parity sequence method. 23 If -jfn q j- is an NXQ data array whose complex Fourier transform'is jFm p >, also NXQ in size, then we encode |Fx to generate a 2NX2Q hologram



..+a . ....a. .. + . .

. ; +.'.a{ ..* ....... ..} ,. ..

.a. ¡.j. 4/ ra.. OP.wr

a ./P .*, +s 4,0 . a..rr .íi r ...r. r . . . r a a

. .

aa. ..

J r.e ge ra.

Figure 2a. Image reconstructed from a kinoform (only spec-trum phase is controlled). The nonuniformity of the dotintensities is due to the loss of the spectrum modulusinformation.

..,.

11. ...11.

111 ,

Y 1 4 .' 1 1 . 1

.41.14144

. v 1 11.v 1.

.

U.U. ... : 1

411

Figure 2b. The same image reconstructed from a ROACH(both spectrum modulus and phase are controlled).

array H whose phase transmittances are as follows:

Hm,p = Hm +N,p +Q =e xp i [Oin,p + ( -1)m+p am p11 ' (1)

Hm +N,p = Hm,p +Q = exp i [em,p - (-1)m +P

am,p J ' (2)

where

Fm,p0

lFm pI[exp i em,p] (3)

192

am,p = cos 1 iFm pI , if A IFm,pIA

= 0, ifA<IFmpI

DATAARRAY

f DFT

FOURIERSPECTRUM

F

(4)

HOLOGRAMCONFIGURATION

H

RECONSTRUC-O RECONS- TION.(DFT-1

TRUCTEDARRAY

h

ENCODING

l --4SYNTHESIS 14--HOLQGRAM

H

Figure 3. Block diagram representation of the synthesis andreconstruction of a computer - generated hologram involvingcoding of the Fourier spectrum.

The constant A is a parameter which can be chosen totradeoff fidelity and efficiency. For A = IFm,plmax, one canideally obtain 100% fidelity of data, but a significant propor-tion of light energy may go toward the parity sequences ascompared with the energy in the data. As A is lowered, oneobtains more light to the data at the expense of parity ele-ments, but increased loss of data fidelity can be observed asA is lowered too much. For constant intensity per bit, how-ever, A should be a constant for all arrays. As A approacheszero, the resulting hologram is simply the kinoform, which isa limiting case of maximum efficiency for minimum fidelityin the trade -off.

In the reconstruction, the image is represented by a 2NX2Qarray hn The original data .{f }can be found in locationsgiven by

h2n 2q = fn,q

The parity elements are located at

h2n +1,2q +1 = Pn,q

(5)

(6)

and can also be easily computed in advance, if desired. Inaddition, it can be shown using the inverse Fourier transformthat

h2n+1,p = hp,2 q+l = 0 (7)

Figures 4a and 4b show the pattern for a 3X 3 data arrayand the reconstructed (6X6) array. The alternating sign ofam ,p in Equations 2 and 3 effectively insures that the parityelements immediately next to the data elements are weak.

Figure 5 shows a reconstruction from such a hologram.Note that the data pattern and the parity elements are offsetfrom each other by one space and can clearly be distinguished.

Another coding method is the synthetic coefficientmethod24, where we code as follows:

H2m,2p = H2m+1,2p+1 = exp i em,p + am,p

H2m+l 2p = H2in,2p+lrr

=expiLem,p m.p]

(8)

(9)

OPTICAL ENGINEERING- MAY /JUNE 1974 VOL. 13 -NO. 3

^cos4 |Fm 'pl , if A > |F,

Figure 2a. Image reconstructed from a kinoform (only spec trum phase is controlled). The nonuniformity of the dot intensities is due to the loss of the spectrum modulusinformation.

% « • . f<l. f

*%%%

» % % % % * * % % «%

i %-4i * «

rn,p'

0, if A<|F.rn.p 11 (4)

0 —————

0 —————RECONS TRUCTED ARRAY

DATA ARRAY

*1

RECON TION .

FOURIER SPECTRUM

i iti-i- • ^ PI

|J J. 1VUV»» •m ^*

(DFT~ ) HOLQGRJH

HOLOGRAM CONFIGURATION

^CODINGJ ———— - ————— |

\M l ———————

Figure 3. Block diagram representation of the synthesis and reconstruction of a computer-generated hologram involving coding of the Fourier spectrum.

The constant A is a parameter which can be chosen to tradeoff fidelity and efficiency. For A = |Fm> p! max , one can ideally obtain 100% fidelity of data, but a significant propor tion of light energy may go toward the parity sequences as compared with the energy in the data. As A is lowered, one obtains more light to the data at the expense of parity ele ments, but increased loss of data fidelity can be observed as A is lowered too much. For constant intensity per bit, how ever, A should be a constant for all arrays. As A approaches zero, the resulting hologram is simply the kinoform, which is a limiting case of maximum efficiency for minimum fidelity in the trade-off.

In the reconstruction, the image is represented by a 2NX2Q array hn q . The original data |fn q lean be found in locations given by

h2n,2q = fn,q (5)

The parity elements are located at

h 2n+l,2q+l ~ Pn,q (6)

and can also be easily computed in advance, if desired. In addition, it can be shown using the inverse Fourier transform that

Figure 2b. The same image reconstructed from a ROACH (both spectrum modulus and phase are controlled).

array H whose phase transmittances are as follows:

m,p

A AHm+N,p = H m,p+Q = ex P

where

(1)

(2)

h2n+l,p ~ hp,2q+l (7)

i/9m,p]

192

Figures 4a and 4b show the pattern for a 3X3 data array and the reconstructed (6X6) array. The alternating sign of O^p in Equations 2 and 3 effectively insures that the parity elements immediately next to the data elements are weak.

Figure 5 shows a reconstruction from such a hologram. Note that the data pattern and the parity elements are offset from each other by one space and can clearly be distinguished.

Another coding method is the synthetic coefficient where we code as follows:

(8)

(9)

VOL. 13-NO. 3

H2m,2p = H2m+l,2p+l = exP <

H = H(3) ll2m + l,2p ==H2m,2 P 4-l = ex P •

OPTICAL ENGINEERING-MAY/JUNE 1974


where 0m and amp are the same as in Equation 1.It can be shown mathematically that the reconstructed

array is given by

hn 9= exp CiTr(Ñ +Ql [fn cos 7M cos 2Q

7f n it q+ pn sin 2N sin 2Q

f 00 f01 f02

f 10 fll fl2

f20 f21 f22

ORIGINAL DATA ARRAY

f00 f01 f02

P00 POl P02

f10 f11 f12

P10 Pll P12

f20 f21 f22

P20 P21 P22

RECON STRUCTED

MAGE FIELD

(10)

Figure 4. The original data array shown together with theexpected reconstruction array from a parity sequence holo-gram. The original data is preserved, interlaced diagonallywith extraneous parity elements.

. .... . . .

: 55:::5

.. .5.... .. :.5 :... ..

. . ._. . ... . '.:.:' .... . .:: . '. . « 5

.. .. ,. .555 5... .e....

Figure 5. Image reconstruction from a parity sequence holo-gram. Only phase transmitting material is needed. Note thatthe parity noise and the data are offset in both the x and they directions.

VOL. 13 -NO. 3

where f and p are the data and parity elements, respectively.Because of the sine type coefficient of the parity elements{p }, they are suppressed near the origin, while the data ismultiplied by a cosine type function and is strongest in thisregion. Hence, the data is reconstructed with good fidelitynear the origin and surrounded by parity elements in theouter field region. This is confirmed in Figure 6, which showsan actual reconstruction of a hologram synthesized by thismethod.

. r.....

e.

' .M.

..

.. :: ....... .....5i .. ' s...' .. .....

55

11,

.5.5......M. esewN.NOW...,,

:: ::55. ..':,:::,.

Figure 6. Image reconstruction from a synthetic coefficienthologram. Again, only phase- transmitting material is needed.

For all three holograms discussed, it is advantageous tointroduce arbitrary phases to the object array to improveefficiency by lowering the spectrum dynamic range. Thisis analogous to diffusion with ground -glass in ordinaryinterferometric holography. The phases may be selectedrandomly, iteratively through numerical methods25, ordeterministically. 26

SYNTHESIS PROCEDURE

To synthesize phase -only holograms as well as ROACHs,we expose Kodachrome II directly from a computer -controlled CRT and have the film processed by EastmanKodak. The combination of one -step synthesis and depend-able commercial processing is a convenient way to makecomputer holograms. Since we use red 632.8 nm helium -neonlaser illumination to reconstruct the image, the phase infor-mation is displayed on the CRT and photographed through ablue -green filter. In a second exposure, the amplitude infor-mation is displayed and photographed through a red filter.For phase -only holograms, in which the amplitude is con-stant, this re- exposure step is more economically accom-plished by photographing a uniformly illuminated surfacethrough a red filter. Determination of the correct exposureto achieve a desired complex transmittance requires a calibra-tion procedure described here in detail.

INITIAL PREPARATION

We control exposure by the length of time that the CRTbeam is turned on at each point. To minimize exposure times,the highest intensity that will not damage the phosphor is


where O m p and am p are the same as in Equation 1.It can be shown mathematically that the reconstructed

array is given by

n q r. 7m TTq [fn ,q cos-cos JL

, TTn . TTq 1 sm^sm—— J, (10)

f oo£ 10f 20

foif llf 21

f02

f !2

f 22

ORIGINAL DATA ARRAY

foo

fio

f20

poo

PIO

P20

f oi

f ll

f 21

P01

PII

P2 1

f02

f !2

f22

P02

Pl2

?22

RECONSTRUCTED IMAGE FIELD

Figure 4. The original data array shown together with the expected reconstruction array from a parity sequence holo gram. The original data is preserved, interlaced diagonally with extraneous parity elements.

Figure 5. Image reconstruction from a parity sequence holo gram. Only phase transmitting material is needed. Note that the parity noise and the data are offset in both the x and the y directions.

where f and p are the data and parity elements, respectively. Because of the sine type coefficient of the parity elements|pK they are suppressed near the origin, while the data is multiplied by a cosine type function and is strongest in this region. Hence, the data is reconstructed with good fidelity near the origin and surrounded by parity elements in the outer field region. This is confirmed in Figure 6, which shows an actual reconstruction 'of a hologram, synthesized by this method.

Figure 6. Image reconstruction from a synthetic coefficient hologram. Again, only phase-transmit ting material is needed.

For all three holograms discussed, it is advantageous to introduce arbitrary phases to the object array to improve efficiency by lowering the spectrum dynamic range. This is analogous to diffusion with ground-glass in ordinary interferometric holography. The phases may be selected randomly, iteratively through numerical methods25 , or deterministically. 26

SYNTHESIS PROCEDURE

To synthesize phase-only holograms as well as ROACHs,we expose Kodachrome II directly from a computer- controlled CRT and have the film processed by Eastman Kodak. The combination of one-step synthesis and depend able commercial processing is a convenient way to make computer holograms. Since we use red 632.8 nm"helium-neon laser illumination to reconstruct the image, the phase infor mation is displayed on the CRT and photographed through a blue-green filter. In a second exposure, the amplitude infor mation is displayed and photographed through a red filter. For1 phase-only holograms, in which the amplitude is con stant, this re-exposure step is more economically accom plished by photographing a uniformly illuminated surface through a red filter. Determination of the correct exposure to achieve a desired complex transmittance requires a. calibra tion procedure described here in detail.

INITIAL PREPARATION

We. control exposure by the length of time that the CRT beam is turned on at each point. To minimize exposure times, the highest intensity that will not damage the phosphor is

VOL. 13-IMO. 3 OPTICAL ENGINEERING-MAY/JUNE 1974 193


determined. To get consistent results, we measure the spotbrightness directly with a light meter after the CRT is fullywarmed up.

The ideal display is a grid of adjoining squares with uni-form illumination over each of the squares and no gaps be-tween the squares; thus, the X and Y scales on the CRT areadjusted so that adjoining lines just overlap. One of two addi-tional methods may be employed to approximate the ideal:the electron beam may be slightly defocused so the spotattains a square shape; or four adjoining positions (in a 2X 2square) may be used to represent one coefficient. The lattermethod, however, reduces the number of available displaypositions by a factor of four.

The distance from the camera to the CRT is determinedso that the finest detail is recorded at spatial frequencieslower than 50 lines /mm, in order not to exceed the resolutionof the film. We have used only 20 lines /mm. To minimizeexposure time, the widest lens opening should be chosen sothat the finest detail can still be resolved. The CRT is shieldedfrom room light with a cardboard tube.

Since both the blue -sensitive and the green- sensitive layerscontribute after processing to the phase information, it isdesirable to choose a color "phase" filter that exposes bothlayers about equally. Thus, both layers can be simultaneouslyexposed in the linear parts of their density vs. log of exposure(H &D) curves. In addition, ROACHs require the red -sensitivelayer to be relatively unexposed by the phase -controllingexposure in order to avoid exposure cross talk. For type P31phosphor, Wratten filter No. 38A satisfies these requirements.Similarly, a red filter for the amplitude -controlling exposureis required that leaves the blue and green- sensitive layersunexposed. Wratten filters No. 26 and 92 are found to besatisfactory. If the available CRT phosphor were to emitpredominantly blue or green, then just the one layer can beused to control the phase. A "white" phosphor is mostdesirable. The relative exposures of the three layers can bedetermined by numerical integration over the product of theCRT phosphor's spectral curve, filter spectral transmittancecurve, and film spectral sensitivity curve. A visual idea of theeffect of a given filter can also be obtained simply by lookingat the light from the CRT through it. Naturally, if the holo-gram is to be illuminated by a color other than red, then adifferent set of filters is required.

For phase -only holograms, the processed red -absorbinglayer should be as transparent as possible, implying thatthe red -sensitive layer must be heavily exposed selectively. Weaccomplish this by photographing a uniformly illuminatedwhite piece of paper through a deep red (Wratten No. 92)filter, using eight times the exposure indicated by an unfilter-ed meter reading from the white paper.

ESTABLISHING THE PHASE -CONTROLLING EXPOSURE

A gray scale is displayed and photographed using theparameters and phase filter determined previously. Then thefilm is rewound in the camera and re- exposed to red light.The resulting densities to blue and green light are measured,and H &D curves are plotted to determine Emin, the minimumexposure at which both the blue and green H &D curves startto become linear.

Assuming that phase excursions are linear with density27,we expose over the linear part of the H &D curve. The moreheavily exposed parts of Kodachrome II are thinner, givingpositive phase. If an exposure range Emin to Emax gives 27rphase, then the exposure E for a phase i is given by

or

log E = log Emin + log (Emax /Emin) ' 0/27r (11)

E = Emin (Emax/Emin 0/

2 7r (12)

Determination of the value of Emax corresponding to 27r

phase, the "phase matching" condition, can be accomplishedby displaying and photographing a series of patterns throughthe phase filter, with exposure ranges Em in to various Emax;the film then is re- exposed to red light. The pattern displayedshould be one with a known diffraction pattern, such as asawtooth or sine wave pattern or a phase -only hologram.Emax is determined by observing which hologram's image isnearest to the desired result. For a sawtooth pattern, whichis nothing more than a prism modulo 27r, phase- matchingoccurs when the maximum intensity is diffracted into thefirst order. For phase- matching, log(Emax /Em is is found tobe approximately 0.7 if both the blue and the green- absorbinglayers are used. After this step, phase holograms can be madeusing the filters and the values of Em in and Emax determinedin the calibration procedure.

ADDING AMPLITUDE CONTROL FOR ROACHs

The desired amplitude transmittance tr is obtained bydividing the calculated amplitude A by a normalizing constantAmax so that tr is less than 1.0 everywhere. By displayingand photographing gray scales through the red filter, theH &D curve for red light can be obtained. The exposure Erneeded to get a desired amplitude transmittance tr is deter-mined from the H &D curve using the relation

Dr-Drmin-- log tÌ=-2 log tr, (13)

where Drmin is the density to red for the maximum exposureErmax. If, for simplicity, only the linear part of the H &Dcurve is used, then

or

log Er = log Ermax + B log tr (14)

BEr - Ermax tr (15)

where B = 2(log Ermax - log Ermin) /(Drmax - Drmin) andis the density corresponding to the minimum exposure,Drmax

ErminUnfortunately, by adding amplitude information, we also

add some undesired phase effects. If we assume that thephase contributed by the red -absorbing layer is also linearwith its density, we can compensate for this phase by sub-tracting an appropriate amount from the computed phaseexposure by replacing Equation 11 with

log E = log Emin + (0/27r) log(Emax /Emin)

- ßlog (Er/Ermin). (16)

194 OPTICAL ENGINEERING- MAY /JUNE 1974 VOL. 13 -NO. 3

determined. To get consistent results, we measure the spot brightness directly with a light meter after the CRT is fully warmed up.

The ideal display is a grid of adjoining squares with uni form illumination over each of the squares and no gaps be tween the squares; thus, the X and Y scales on the CRT are adjusted so that adjoining lines just overlap. One of two addi tional methods may be employed to approximate the ideal: the electron beam may be slightly defocused so the spot attains a square shape; or four adjoining positions (in a 2X2 square) may be used to represent one coefficient. The latter method, however, reduces the number of available display positions by a factor of four.

The distance from the camera to the CRT is determined so that the finest detail is recorded at spatial frequencies lower than 50 lines/mm, in order not to exceed the resolution of the film. We have used only 20 lines/mm. To minimize exposure time, the widest lens opening should be chosen so that the finest detail can still be resolved. The CRT is shielded from room light with a cardboard tube.

Since both the blue-sensitive and the green-sensitive layers contribute after processing to the phase information, it is desirable to choose a color "phase" filter that exposes both layers about equally. Thus, both layers can be simultaneously exposed in the linear parts of their density vs. log of exposure (H&D) curves. In addition, ROACHs require the red-sensitive layer to be relatively unexposed by the phase-controlling exposure in order to avoid exposure cross talk. For type P31 phosphor, Wratten filter No. 38A satisfies these requirements. Similarly, a red filter for the amplitude-controlling exposure is required that leaves the blue and green-sensitive layers unexposed. Wratten filters No. 26 and 92 are found to be satisfactory. If the available CRT phosphor were to emit predominantly blue or green, then just the one layer can be used to control the phase. A "white" phosphor is most desirable. The relative exposures of the three layers can be determined by numerical integration over the product of the CRT phosphor's spectral curve, filter spectral transmittance curve, and film spectral sensitivity curve. A visual idea of the effect of a given filter can also be obtained simply by looking at the light from the CRT through it. Naturally, if the holo gram is to be illuminated by a color other than red, then a different set of filters is required.

For phase-only holograms, the processed red-absorbing layer should be as transparent as possible, implying that the red-sensitive layer must be heavily exposed selectively. We accomplish this by photographing a uniformly illuminated white piece of paper through a deep red (Wratten No. 92) filter, using eight times the exposure indicated by an unfilter- ed meter reading from the white paper.

ESTABLISHING THE PHASE-CONTROLLING EXPOSURE

A gray scale is displayed and photographed using the parameters and phase filter determined previously. Then the film is rewound in the camera and re-exposed to red light. The resulting densities to blue and green light are measured, and H&D curves are plotted to determine Emin , the minimum exposure at which both the blue and green H&D curves start to become linear.

Assuming that phase excursions are linear with density27 , we expose over the linear part of the H&D curve. The more heavily exposed parts of Kodachrome II are thinner, giving positive phase. If an exposure range Em jn to Emax gives 27T phase, then the exposure E for a phase 0 is given by

log E = log Emin + log (E max /Emin ) 0/27T (11)

or

(12)

Determination of the value of Emax corresponding to 27T phase, the "phase matching" condition, can be accomplished by displaying and photographing a series of patterns through the phase filter, with exposure ranges Emin to various Emax ; the film then is re-exposed to red light. The pattern displayed should be one with a known diffraction pattern, such as a sawtooth or sine wave pattern or a phase-only hologram. Emax is determined by observing which hologram's image is nearest to the desired result. For a sawtooth pattern, which is nothing more than a prism modulo 2?r, phase-matching occurs when the maximum intensity is diffracted into the first order. For phase-matching, l°g(Emax /Emin ) is found to be approximately 0.7 if both the blue and the green-absorbing layers are used. After this step, phase holograms can be made using the filters and the values of Emin and E max determined in the calibration procedure.

ADDING AMPLITUDE CONTROL FOR ROACHs

The desired amplitude transmittance t r is obtained by dividing the calculated amplitude A by a normalizing constant Amax so that tr is less than 1.0 everywhere. By displaying and photographing gray scales through the red filter, the H&D curve for red light can be obtained. The exposure Er needed to get a desired amplitude transmittance tr is deter mined from the H&D curve using the relation

= (13)

where Drmin is the density to red for the maximum exposure Ermax . If, for simplicity, only the linear part of the H&D curve is used, then

(H)

or

F = F tnr rmax r r (15)

where B = 2(log Ermax - log Ermin )/(Drmax -Drmax is the density corresponding to the minimum exposure,Ermin .

Unfortunately, by adding amplitude information, we also add some undesired phase effects. If we assume that the phase contributed by the red-absorbing layer is also linear with its density, we can compensate for this phase by sub tracting an appropriate amount from the computed phase exposure by replacing Equation 11 with

log E = log Emin + (0/27T) log(Emax /Emin )

-/5log(Er/Ermin ). (16)

194 OPTICAL ENGINEERING-MAY/JUNE 1974 VOL. 13-NO. 3


If computed log E is less than log Emin, then log(Emax /Emin)is added to log E in order to preserve the phase modulo 27r.The factor ß is determined experimentally and was found tobe about 0.25.

Finally, since the calibration procedure we used is onlyapproximate, one parameter at a time was varied to convergeon the best set of exposure parameters. In particular,ROACHs should be made with half and double the exposuresindicated by Equation 14. Once a good set of exposureparameters is found, ROACHs can be made dependably sincehighly repeatable commercial processing is available.

CONCLUSION

In this paper we have discussed new ways to generatecomputer holograms that are highly efficient, both in the useof the display device in synthesis and in the use of lightin reconstruction. We have also described our synthesisprocedure and practical advice for making these computerholograms.

ACKNOWLEDGMENT

We are grateful for the guidance of J.W. Goodman ofStanford University where this work was performed.

REFERENCES

1. L.B. Lesern, P.M. Hirsch, and J.A. Jordan, Jr., "Computer Synthe-sis of Holograms for 3 -D Display ", Commun. ACM, Vol. 11,pp. 661 -674 (1968).

2. J.R. Fienup and J.W. Goodman, "Three New Ways to Make Com-puter Generated Color Holograms ", J. Opt. Soc. Am., Vol. 63,p. 1325A (1973).

3. K.G. Birch and Frances J. Green, "The Application of Computer -Generated Holograms to Testing Optical Elements ", J. Phys. D.Applied Physics, Vol. 5, pp. 1982 -1992 (1972).

4. B.R. Frieden, "On Arbitrarily Perfect Imagery with a FiniteAperture ", Optica Acta, Vol. 16, pp. 795 -807 (1969); B.R.Frieden, "The Extrapolating Pupil, Image Synthesis, and SomeThought Applications ",Appl. Opt., Vol. 9, pp. 2489 -2496 (1970).

5. A. Vander Lugt, "Signal Detection by Complex Spatial Filtering ",IEEE Trans. Inform. Theory, Vol. IT -10, pp. 139 -145 (1964).

6. A.W. Lohmann and D.P. Paris, "Computer Generated SpatialFilters for Coherent Optical Data Processing ", Appl Opt., Vol. 7,pp. 651 -655 (1968).

7. M. Severcan, "Computer Generation of Coherent Optical Filterswith High Light Efficiency and Large Dynamic Range ", Informa-tion Systems Lab. Tech. Report No. 6415 -6, Stanford University(1973).

8. B.R. Brown and A.W. Lohmann, "Complex Spatial Filtering withBinary Masks ", Appl. Opt., Vol. 5, pp. 967 -969 (1966).

9. A.W. Lohmann, D.P. Paris and H.W. Werlich, "A Computer Gen-erated Spatial Filter, Applied to Code Translation ", Appl Opt.,Vol. 6, pp. 1139 -1140 (1967).

10. A.W. Lohmann and H.W. Werlich, "Incoherent Matched Filteringwith Fourier Holograms ", Appl. Opt., Vol. 10, pp. 670 -672(1971).

11. E.N. Leith and J. Upatnieks, "Reconstructed Wavefronts andCommunication Theory ", J. Opt. Soc. Am., Vol. 52, pp. 1123-1130 (1962).

12. J.J. Burch, "A Computer Algorithm for the Synthesis of SpatialFrequency Filters ", Proc. IEEE, Vol. 55, pp. 599 -601 (1967).

13. L.B. Lesern, P.M. Hirsch, and J.A. Jordan, "Holographic Displayof Digital Images ", AFIPS Conference Proceedings, Fall JointComputer Conference, Vol. 41, (Spartan Books, Inc., New York,1967), pp. 41-49.

14. B.R. Brown and A.W. Lohmann, "Computer- Generated BinaryHolograms ", IBM J. Res. Develop. Vol. 13, pp. 160 -168 (1969).

15. R.E. Haskell, "Computer- Generated Holograms with MinimumQuantization ", J. Opt. Soc. Am., Vol. 63, p. 504, (1973).

VOL. 13 -NO. 3

16. W.H. Lee, "Sampled Fourier Transform Hologram Generated byComputer ", Appl. Opt., Vol. 9, pp. 639 -643 (1970).

17. C.B. Burckhardt, "A Simplification of Lee's Method of Generat-ing Holograms by Computer ", Appl. Opt., Vol. 9, p. 1949 (1970).

18. L.B. Lesern, P.M. Hirsch, and J.A. Jordan, Jr., "The Kinoform:A New Wavefront Reconstruction Device ", IBM J. Res. Develop.,Vol. 13, pp. 150 -155 (1969).

19. T.S. Huang, "Digital Holography ", Proc. IEEE, Vol. 59, pp. 1335-1346 (1971).

20. P.L. Ransom, "Synthesis of Complex Optical Wavefronts ", ApplOpt., Vol. 11, pp. 2554 -2561 (1972).

21. R.J. Collier, C.B. Burckhardt, and L.H. Lin, Optical Holography(Academic Press, New York, 1971), pp. 542 -563.

22. D.C. Chu, J.R. Fienup, and J.W. Goodman, "Multiemulsion On-Axis Computer Generated Hologram ", Appl. Opt., Vol. 12, pp.1386 -1388 (1973).

23. D.C. Chu and J.W. Goodman, "Spectrum Shaping with ParitySequences ", Appl. Opt., Vol. 11, pp. 1716 -1724 (1972).

24. D.C. Chu, "Three Ways to Obsolete the Kinoform", J. Opt. Soc.Am., Vol. 63, p. 1325B (1973).

25. N.C. Gallagher and B. Liu, "Method for Computing Kinoformsthat Reduces Reconstruction Error ", Appl. Opt., Vol. 12, pp.2328 -2335 (1973).

26. W.J. Dallas, "Deterministic Diffusers for Holography ", Appl. Opt.,Vol. 12, pp. 1179 -1187 (1973).

27. R.L. Lamberts, "Characterization of a Bleached PhotographicMaterial ",Appl. Opt., Vol. 11, pp. 33 -41 (1972).


If computed log E is less than log Emin , then log(E max /E min ) is added to log E in order to preserve the phase modulo 27T. The factor ]3 is determined experimentally and was found to be about 0.25.

Finally, since the calibration procedure we used is only approximate, one parameter at a time was varied to converge on the best set of exposure parameters. In particular, ROACHs should be made with half and double the exposures indicated by Equation 14. Once a good set of exposure parameters is found, ROACHs can be made dependably since highly repeatable commercial processing is available.

CONCLUSION

In this paper we have discussed new ways to generate computer holograms that are highly efficient, both in the use of the display device in synthesis and in the use of light in reconstruction. We have also described our synthesis procedure and practical advice for making these computer holograms.

ACKNOWLEDGMENT

We are grateful for the guidance of J.W. Goodman of Stanford University where this work was performed.

16. W.H. Lee, "Sampled Fourier Transform Hologram Generated by Computer", Appl Opt., Vol. 9, pp. 639-643 (1970).

17. C.B. Burckhardt, "A Simplification of Lee's Method of Generat ing Holograms by Computer", Appl. Opt., Vol. 9, p. 1949 (1970).

18. L.B. Lesem, P.M. Hirsch, and J.A. Jordan, Jr., "The Kinoform: A New Wavefront Reconstruction Device", IBM J. Res. Develop., Vol. 13, pp. 150-155 (1969).

19. T.S. Huang, "Digital Holography",Proc. IEEE, Vol. 59, pp. 1335- 1346(1971).

20. P.L. Ransom, "Synthesis of Complex Optical Wavefronts", Appl Opt., Vol. 11, pp. 2554-2561 (1972).

21. R.J. Collier, C.B. Burckhardt, and L.H. Lin, Optical Holography (Academic Press, New York, 1971), pp. 542-563.

22. D.C. Chu, J.R. Fienup, and J.W. Goodman, "Multiemulsion On- Axis Computer Generated Hologram", Appl Opt., Vol. 12, pp. 1386-1388 (1973).

23. D.C. Chu and J.W. Goodman, "Spectrum Shaping with Parity Sequences", Appl Opt., Vol. 11, pp. 1716-1724 (1972).

24. D.C. Chu, "Three Ways to Obsolete the Kinoform", J. Opt. Soc. Am., Vol. 63, p. 13256(1973).

25. N.C. Gallagher and B. Liu, "Method for Computing Kinoforms that Reduces Reconstruction Error", Appl. Opt., Vol. 12, pp. 2328-2335 (1973).

26. W.J. Dallas, "Deterministic Diffusers for Holography", Appl. Opt., Vol. 12, pp. 1179-1187 (1973).

27. R.L. Lamberts, "Characterization of a Bleached Photographic Material", Appl. Opt., Vol. 11, pp. 33-41 (1972).

REFERENCES

1. L.B. Lesem, P.M. Hirsch, and J.A. Jordan, Jr., "Computer Synthe sis of Holograms for 3-D Display", Commun. ACM, Vol. 11, pp. 661-674 (1968).

2. J.R. Fienup and J.W. Goodman, "Three New Ways to Make Com puter Generated Color Holograms", /. Opt. Soc. Am., Vol. 63, p. 1325A (1973).

3. K.G. Birch and Frances J. Green, "The Application of Computer- Generated Holograms to Testing Optical Elements", /. Phys. D. Applied Physics, Vol. 5, pp. 1982-1992 (1972).

4. B.R. Frieden, "On Arbitrarily Perfect Imagery with a Finite Aperture", Optica Acta, Vol. 16, pp. 795-807 (1969); B.R. Frieden, "The Extrapolating Pupil, Image Synthesis, and Some Thought Applications",,4pp/. Opt., Vol. 9, pp. 2489-2496(1970).

5. A. Vander Lugt, "Signal Detection by Complex Spatial Filtering", IEEE Tram. Inform. Theory, Vol. IT-10, pp. 139-145 (1964).

6. A.W. Lohmann and D.P. Paris, "Computer Generated Spatial Filters for Coherent Optical Data Processing", Appl Opt., Vol. 7, pp. 651-655 (1968).

7. M. Severcan, "Computer Generation of Coherent Optical Filters with High Light Efficiency and Large Dynamic Range", Informa tion Systems Lab. Tech. Report No. 6415-6, Stanford University (1973).

8. B.R. Brown and A.W. Lohmann, "Complex Spatial Filtering with Binary Masks", Appl. Opt., Vol. 5, pp. 967-969 (1966).

9. A.W. Lohmann, D.P. Paris and H.W. Werlich, "A Computer Gen erated Spatial Filter, Applied to Code Translation", Appl Opt., Vol. 6, pp. 1139-1140 (1967).

10. A.W. Lohmann and H.W. Werlich, "Incoherent Matched Filtering with Fourier Holograms", Appl. Opt., Vol. 10, pp. 670-672 (1971).

11. E.N. Leith and J. Upatnieks, "Reconstructed Wavefronts and Communication Theory", /. Opt. Soc. Am., Vol. 52, pp. 1123- 1130 (1962).

12. J.J. Burch, "A Computer Algorithm for the Synthesis of Spatial Frequency Filters", Proc. IEEE, Vol. 55, pp. 599-601 (1967).

13. L.B. Lesem, P.M. Hirsch, and J.A. Jordan, "Holographic Display of Digital Images", AFIPS Conference Proceedings, Fall Joint Computer Conference, Vol. 41, (Spartan Books, Inc., New York, 1967), pp. 41-49.

14. B.R. Brown and A.W. Lohmann, "Computer-Generated Binary Holograms", IBM J. Res. Develop. Vol. 13, pp. 160-168(1969).

15. R.E. HaskelL, "Computer-Generated Holograms with Minimum Quantization", /. Opt. Soc. Am., Vol. 63, p. 504, (1973).



recent approaches to computer— generated hologramsja ...€¦ · use in image enhancement6,...

Documents