Comparison of binary encoding schemes for electron-beam fabrication of computer generated holograms

Comparison of binary encoding schemes for electron-beam

fabrication of computer generated holograms

Hamid Farhoosh, Michael R. Feldman, Sing H. Lee, Clark C. Guest, Yeshaiahu Fainman, and Reiner

Eschbach

The suitability of various binary encoding methods for electron-beam recording of computer generated

holograms is systematically evaluated. Subjected to the limitations of computing resources, a set of criteria is

established according to which these encoding schemes are evaluated and compared. This comparison can be

used to determine the optimum encoding method for desired wavefront properties.

1. Introduction

The many attractive features and capabilities ofelectron-beam (e-beam) lithography systems, such as alarge space bandwidth product (SBWP), direct writingat submicron resolution, and small distortion errors,have produced a growing interest in e-beam fabrica-tion of computer generated holograms (CGHs). Withthe advancement of e-beam technology in recent yearsit appears that e-beam fabrication techniques mayplay an important role in further advancement of digi-tal holography.

Electron-beam fabrication of computer generatedholograms has been investigated in depth by Arnoldl 2

and used in a few applications by others.3-5 The pur-pose of this paper is to systematically evaluate thesuitability of various encoding schemes for e-beamrecording of CGHs. We will also try to establish a setof criteria according to which, and depending on theparticular application at hand, an optimum encodingmethod can be determined.

For most types of CGH fabrication procedure, theSBWP is limited by the capabilities of the recordingdevice. General comparisons of CGH encoding meth-ods have been provided in several papers.6 -8 Thesecomparisons are inherently based on the limitedSBWP of the recorder. However, when e-beam lithog-

The authors are with University of California at San Diego, De-

partment of Electrical Engineering & Computer Sciences, La Jolla,California 92093.

Received 15 February 1987.

0003-6935/87/204361-12$02.00/0.© 1987 Optical Society of America.

raphy is employed, the SBWP of the hologram is limit-ed, not by the recording device, but by computer mem-ory, computer time restrictions, and data storagecapacities. These constraints impose a different basisfor comparison of encoding methods. The evaluationpresented in this paper is based on the ability of encod-ing methods to achieve high-quality holograms (interms of diffraction efficiency, SNR, and image wave-front bandwidth) while subjected to the above limita-tions. In this way the suitability of these methods for e-beam recording of CGHs can be compared.

The characteristics of an e-beam fabricated CGHwill depend to some extent on the type of e-beamlithography system employed. However, for the pur-pose of this paper it is sufficient to describe the partic-ular e-beam recorder by four parameters: the SBWP;minimum feature size df; minimum distance betweenaddressable points dr; and maximum placement errorover a distance dr (6dr). Typical values of these pa-rameters are SBWP = 1010, df = 0.5 Mm, dr = 0.1 Am, 5dr= 1/32 ,um.9 10 These parameters allow fabrication oflarge CGHs with submicron resolution and small dis-tortion errors. However, if the whole available area isused to record a single Fourier transform CGH on apoint by point basis, a two-dimensional FFT operationof the order of 105 X 105 is required. This is beyond thepractical limits of data storage systems. Therefore,the information content of an e-beam fabricated CGHis limited, not by the recorder capabilities, but by thelarge amounts of computation and data storage in-volved in the process.

The paper is organized as follows: Section II con-tains a brief review of six encoding methods which areconsidered in this paper. In Sec. III we discuss a set ofcriteria by which these encoding schems can be evalu-ated. Section IV contains tabulated results of theevaluation applied to the six encoding schemes and

15 October 1987 / Vol. 26, No. 20 / APPLIED OPTICS 4361

x

1 I -11

X 1 X

x

a) image plane

ax,

b) CGH c) CGH cell

Fig. 1 (a) Coordinate system and notation in the image plane. (b)Coordinate system in the CGH plane. (c) The CGH cell structure.

comparison among them. Final conclusions are pre-sented in Sec. V.

1. Review of Encoding Schemes

Because of the (currently) binary nature of e-beamrecording, we shall only consider here those encodingstrategies that transform the computed complex wave-front into a binary pattern. There are many well-known binary encoding methods that could be suitablefor e-beam writing. In this paper we discuss in detailsix binary encoding techniques which are representa-tives of the existing methods. These have beengrouped, after Allebach," into two categories: (1)those that sample the desired holographic wavefront atthe center of each cell, including the original Lohmannmethod 12 ' 1 3 , Lee's delayed phase method 1 4 as it is com-monly implemented (to be subsequently referred to asthe Lee(1) method); and (2) those that sample thewavefront at the center of each aperture, including thebinary version of Burch's method,15 the revised Loh-mann method,16-18 the Lee's delayed phase method asoriginally proposed (to be subsequently referred to asthe Lee(2) method), and Arnold's staircase method.1Since each encoding method has several variations, webriefly describe the form of each method to be consid-ered here.

Figure 1 illustrates the coordinate notation used inthis paper. In the image plane, x and y are the coordi-nates, X and Yare the image dimensions, and x and byare the image sampling distances. In the hologramplane, x' and y' are the coordinates, X' and Y' are theCGH dimensions, and Ax' and by' are the samplingdistances. Resolution elements in the hologram planeare df X df square areas. The normalized offset of thereconstructed image with respect to the optic axis isq(q = X/X, where X0 is the actual image offset).Figure 2 shows how different encoding schemes sampleand represent the complex wavefront in the hologramplane. In Lohmann's method [Fig. 2(a)] the hologramis divided into many cells, and a single aperture is

(a) Cb)

(c) 8x (d)

ads 0:0 A::; 4 4 :tf 00 ~local t

C L 0 ~~~~~~~~step Size

(e) (f)

Fig. 2. Sampling periods and representation of different encodingschemes. (a) Lohmann method in which the height and location ofthe apertures are determined by the amplitude and phase, respec-tively, of the wavefront at the center of the cell. (b) Revised Loh-mann method, where the wavefront is evaluated at each samplingpoint, and a single aperture is placed at the sampling point whichmost nearly satisfies Eq. (1). The aperture height is determined bythe amplitude of the wavefront at the sampling point. (c) In theLee(1) method the wavefront is sampled at the center of each cell,and the height of each aperture is determined by the component ofthe sample along the appropriate basis vectors [0,7r/2,7r,(37r)/2]. (d)Lee(2) method is similar to Lee(1) method except that the wavefrontis sampled at the center of each of the four apertures. (e) Burch'smethod in which the hologram transmittance function given in Eq.(2) is sampled at the center of each cell and a square aperture of areaproportional to the sampled value is placed at the center of the cell.(f) In Arnold's method the hologram is drawn fringe by fringe. Foreach fringe, the location of the next sampling point is determined bybeginning at a center line point and moving in the y direction by anamount equal to the local step size that is updated at each sampling

point.

placed in each cell. The height and location of theapertures in the cells are determined by the amplitudeand phase, respectively, of the wavefront at the centerof the cell. The revised Lohmann method [Fig. 2(b)]referred to here is method 1 of Ref. 17, which is some-times termed Lee's interferogram method. Thismethod is similar to Lohmann's original method ex-cept that at each sampling point the wavefront is eval-uated and a single aperture is placed at the samplingpoint which most nearly satisfies the following equa-tion:

2r + (x',y') = 27rn,tc

(1)

where 0(x',y') is the wavefront's phase and t is thecarrier period. The amplitude of the wavefront at thesampling point determines the height of the aperture.

In the Lee(1) method [Fig. 2(c)] each cell is com-posed of four apertures. The wavefront is sampled atthe center of each cell, and the height of each apertureis determined by the component of the sample alongthe appropriate basis vectors [0, 7r/2, ,r and 3(7r/2) rad].The Lee(2) method [Fig. 2(d)] is similar to the Lee(1)method except that the wavefront is sampled at thecenter of each of the four apertures. The height of the

4362 APPLIED OPTICS / Vol. 26, No. 20 / 15 October 1987

Table 1. Bandwidths, SNR, Diffraction Efficiency, NPS, and the Interpolation Formulas for Various Encoding Methods

Encoding min Ouantiztion Representation Distortion SNR Actual efficlency NPSMethod q mmT ~ o uSNR SNR(Minimuml t, a V, 'I. .

2 I r S 2,lLohmatnn Mu I npdrMK (n-l~df~u [4n2n ] [(M) + k4 ) n d (n-l)d X(O1M1) X2(O1 N(10) 2

Leell) M0 I 4 dTMK 1n4-)d 1M9 n [T4n +y 64M1 ] l M d(51) 2 o M M N

Burch 2 2 ]dS TI;7d1MU 42 960 (M~ 2+rMf2 VA|6 /d, X~qM ) I( U (0-3) Mx 112~

Rev~sed I np d, II In (nil) dM [. 4n 1n 1344 (-l n -l) d X~qM M) X (O M ) loll l) 111 MN2

Lee(2) M1 4 dM, (nl d M 2n2 1344 M 4d (na I M" (0-3) -2M MN2

al a 2 1 XMq.MM)XyO M -r o

Burch 2 qn~=l df VT5.--7 (M 302 _120 lM2 +M2 0- .T df\ l d X(qM)X(0, MIII3I .4MM NArol 2 4l rTM - pr d1 r

2n d 'U

aperture at each sampling point is proportional to thecomponent of the wavefront along the correspondingbasis vector.

Burch's encoding method [Fig. 2(e)] is achieved bysampling the function

T(x',y') = 0.5[1 + IG(x',Y')I cos(2r t-o Y)] (2)

where IG(x',y')I and O(x',y') are the amplitude andphase of the desired wavefront at the hologram plane,respectively, and t, is the carrier period. Sampling ofT(x',y'), which is always a positive-real function, isperformed at the center of each cell, and a squareaperture of area proportional to this sample is placedat each sampling point.

Arnold's staircase method [Fig. 2(f)], used to encodeconstant amplitude wavefronts, is similar to the re-vised Lohmann method, except the sampling is done inan aperiodic manner. The hologram is drawn fringeby fringe, and sampling is performed near fringe loca-tions to find center line points or points which satisfyEq. (1). For each fringe, the location of the nextsampling point is determined by beginning at a centerline point and moving in the y direction by an amountequal to the local step size (the distance betweenneighboring sampling points which is updated at eachstep).1 At each sampling point 0(x',y') and (8k)/(Oy')are evaluated, and the local step size is adjusted so thatEq. (1) is accurate to within a maximum phase error of&k. The center line points are connected by placing arectangle of width of one-half of the fringe separationat each center line point. Note that two wavefrontevaluations are required for each rectangle, one ofwhich is approximately at the center of the aperture.

III. Evaluation Criteria

In this section we present and discuss a set of criteriaaccording to which the encoding schemes discussed inthe previous section can be evaluated. These criteriaare divided into two groups as follows: (1) hologramqualities, which include image wavefront properties(e.g., bandwidth), SNR, and diffraction efficiency; (2)computer limitations, which include computation tolland size of graphical data, i.e., the amount of data

needed by the e-beam system for the actual CGHfabrication. The second group contains constraintsthat determine practical limitations in hologram syn-thesis.

A. Hologram Qualities

1. Image Wavefront PropertiesParameters that determine the characteristics of the

reconstructed image are the image SBWP (No) and itsbandwidths [Box = 1/(x),Boy = l/(Y)5 . The SBWP ofthe CGH(NxNy) can be larger than IV\7, e.g., due to theaddition of a carrier wave or by interpolation, which isachieved either directly by (linear, cubic, sinc, etc.)interpolation between sampling points or, in the caseof Fourier transform holograms, indirectly by embed-ding the object in a larger field before the FFT opera-tion. In general one has

NxNY = MXMyN, (3)

where Mx and My are interpolation factors that relatethe number of points in the hologram plane to thenumber of points in the image plane. Note that inter-polation is sometimes required by the encoding meth-od, and sometimes it is introduced to improve thequality of the CGH. The interpolation factor Mx (as-suming a carrier wave in the x direction) also dependson the normalized offset of the reconstructed image qas Mx = qM8, where M, is the number of wavefrontsamples per carrier period. The minimum number ofinterpolations required by the encoding schemes areshown in Table I. These minimum values are based onq = 1 for all encoding methods except the Burch'smethod for which q = 0.5.19

Because of the mismatch between the SBWP of theCGH and that of the image, not all the informationthat is encoded in the CGH is translated into imageinformation; part of the SBWP of the CGH is used toseparate multiple images and to enhance the quality ofthe image. To evaluate the efficiency of the encodingmethods in terms of their usage of the available CGHbandwidths [Bhx = 1/(5x)Bhy = 1I(by')] we define thebandwidths of the reconstructed wavefront (B',B,) as


N = BB 0,XY = BB'X'Y. (4)

The reconstructed wavefront is defined as the particu-lar wavefront in the hologram plane, which, whenpropagated to the image plane, will form the image.From the above arguments it follows that

B' .X Bh. ' BOYY BhyX' M., " Y' M,(5

The wavefront bandwidth depends on the number ofamplitude and phase quantization levels (na,fnp) usedby the encoding method and the interpolation factors(M.,My) as presented in columns 3 and 4 of Table I.Generally, the dependency is inversely proportional tothe number of amplitude and phase quantization lev-els and the interpolation factors.

2. Signal-to-Noise RatioThe definition we adopt for the SNR is 7' 8

SNR average intensity in the desired imagemean square error

XY J J Ig(xty)l 2dxdy= t . (6)f J Ih(xy) - g(x,y)l2 dxdy

where g(x,y) is the desired image, h(x,y) is the actualreconstructed image. The desired image is obtained ifthe hologram has recorded the complex wavefront inan ideal manner with no errors. If we represent theideal (discrete) wavefront at the hologram plane withAmn exp(hkmn), the desired image is then given by

Nx-1 N-1 2rg(x,y) = 3 E Ain exp(mn) exp n (xmx + ) (7)

m=O n=OL

The SNR as well as the diffraction efficiency of aCGH is reduced by the amount of error introduced invarious stages of hologram production. Assumingthat the size of the hologram is large enough that highfrequencies are not cut off, other major sources of errorin all CGHs are (a) sampling errors (spatial quantiza-tion), (b) quantization error (modulation quantiza-tion); (c) representation related errors (aperture size +detour phase errors); (d) distortion errors caused bythe nonideal behavior of the recording device. Thefirst three errors in this list are inherent to all CGHs.The other two depend on the encoding scheme andrecording device, respectively.

In the following discussions we consider the SNRresulting from different sources of error, and in doingso we treat sources of error individually as indepen-dent processes. We also assume a uniform amplitudeand phase probability distribution over the CGHplane, which is a justified assumption when a diffuseris used to spread the information over the entire holo-gram. Such an assumption appears to be a commonone used in many image coding methods 2 0 ; however, aspointed out in Ref. 20, due to the fact that errors, ingeneral, are image-dependent and not simply additive,the SNR as defined in Eq. (6) with these simplified

assumptions may not be adequate for measuring imagequality. For this reason sometimes subjective criteriaare used to determine the CGH quality.'1 Here, how-ever, we have chosen the statistical approach that ap-proximately quantifies the SNR and hologram quality.In this error analysis we shall follow the sequentialorder of steps that are taken to fabricate a CGH.

a) Sampling:

Effects of sampling are well known. Sampling inthe object plane results in multiple Fourier spectra,and sampling in the hologram plane results in thereconstruction of multiple images. If the sampling inthe Fourier plane is done according to the Nyquistcriterion, the reconstructed images do not overlap anda clean image results. On the other hand, since a phasefunction is often introduced in the object plane toalleviate the dynamic range problems in the Fourierplane, the object plane sampling frequency is often lessthan the Nyquist rate. The consequence of this un-dersampling is speckle noise between sampling points,which can be reduced by replicating the Fourier spec-tra in the hologram plane. Since replicating patternswith an e-beam lithography system requires negligibleincreases in computation time and data storage re-quirements, we neglect this error source and evaluatethe SNR only at the sampling points.

b) Quantization:

Once the function to be recorded as a hologram isobtained, it must be quantized to discrete values. Ex-pressions for mean square error (MSE) due to quanti-zation effects were taken from Refs. 6-8, 19 and 21 andmodified to allow for interpolation. [MSE is definedas the denominator in Eq. (6).] In general, the result-ing SNR due to quantization effects SNRQ is depen-dent on the particular image to be encoded. However,by assuming a uniform probability distribution for thehologram transmittance (constant probability over the0-1 range), an image independent expression forSNRQ can be obtained. The SNRQ for the encodingmethods discussed in Sec. II are calculated in part 1 ofAppendix A, and the results are summarized in column5 of Table I.

c) Representation Related Errors:

After the hologram data are quantized, they must betransformed into geometrical shapes for plotting.Representation related errors are caused by the meth-od a particular encoding scheme uses groups of resolu-tion elements (smallest features) to encode an ampli-tude or phase value. They include aperture errors anddetour phase errors. Aperture errors are caused by thefact that the encoding schemes use a group of resolu-tion elements to encode the wavefront's amplitude at asingle sampling point. Detour phase errors are due tothe fact that the apertures represent the phase at thecenter of the cell and not the center of the aperture.Detour phase errors are eliminated in those methodswhich sample the object spectrum at the center of eachaperture. As shown in part 2 of Appendix A, the SNR


due to representation related errors SNRR depends onthe image coordinates. The minimum SNR, occurringat the edges of the image, is presented in column 6 ofTable I. Note that only the lowest-order terms inaperture error (the term proportional to My-4) anddetour phase error (the term proportional to MX-2) areincluded in these expressions. If the interpolationfactors are larger than one, this is sufficient in calculat-ing the SNRR. Otherwise, higher-order terms in Eqs.(A6) and (A9) of Appendix A must be included in theseexpressions.

d) Distortion Error:

The last stage in CGH production is the actual plot-ting of the hologram. Errors introduced in this stepare due to finite positioning accuracy of the plottingdevice; i.e., the error is caused by placing the aperturein a location which is slightly off its intended position.This displacement causes a phase error which can becalculated in exactly the same way as it was deter-mined for phase quantization. From the derivationsin part 3 of Appendix A it follows that the distortionrelated SNR (SNRD) is given by

5.1% efficiency. Note that these diffraction efficien-cies apply to amplitude holograms. The diffractionefficiency of the corresponding phase hologram (e.g., asurface relief hologram) may be found by multiplyingthese values by a factor of 4.1,2

The actual diffraction efficiency X is defined as

PsP.,

(11)

where P, is the total light power in the signal in thepresence of error. The actual diffraction efficiency isdetermined by the manner in which the object wave-front is encoded as well as by the sampling error. Asmentioned in Sec. III.A.2 sampling in the hologramplane results in the reconstruction of multiple images.This causes power to be diverted from the primaryimage, thereby reducing the diffraction efficiency.Other sources of error do not divert power from theprimary image by significant amount, and they will notbe considered in diffraction efficiency calculations.Therefore, the actual diffraction efficiency can be writ-ten as

77 = rt, (12)

(SNRD)min12

ir 2(bd,)2 [ + (IM)2]

where t is the carrier period, which is assumed to be inthe x direction. Although the expression for SNRD isthe same for all encoding methods, the two parameters,t, and by', appearing in this expression depend on theparticular method employed. These two parametersare listed in columns 7 and 8 of Table I.

Assuming that the errors discussed above are addi-tive, the total SNR can be written as

SNR-1 = [SNRi] i = Q,R,D. (9)

3. Diffraction EfficiencyThe theoretical diffraction efficiency is defined as

ideal

inc

(10)

where Pideal is the power in the signal for an idealhologram, and Pinc is the light power incident on thehologram. An ideal hologram is one in which a wave-front is encoded by means of a particular codingscheme in an otherwise error-free manner. In generalt is image dependent and has a maximum value when

a plane wave is encoded in the CGH. For comparisonpurposes it suffices to assume such a maximum for allencoding schemes. For example, on encoding a planewave with the Lohmann, revised Lohmann, and Ar-nold methods, a regular (periodic) binary diffractiongrating with 50% duty cycle results with a theoretical(first-order) diffraction efficiency of nt = 1/r 2 = 10.1%.Burch's method results in a binary grating with sinu-soidally varying aperture size with an efficiency of6.25%, and the Lee(1) and Lee(2) encoding methodsgive rise to a binary grating with 25% duty cycle with

where r is the power reduction factor due to sampling.This power reduction factor is calculated in AppendixB, where it is shown to be a function of interpolationparameters [Eqs. (B7) and (B9)]. The results in thisAppendix show that for the Lohmann and Lee(1)methods, Nyquist sampling of the wavefront gives r =60% [from Eq. (B9)]. For methods that sample thewavefront at the Nyquist rate at the center of aper-tures r = 48% for q = 1 and Mx = 3, My = 1. Theresulting actual diffraction efficiencies are given incolumn 9 of Table I, where the function X(q,Mx) isdefined in Eq. (B6) in Appendix B.

B. Computer Limitations

As mentioned in Sec. I, unlike conventional CGHgeneration, in an e-beam fabricated CGH the SBWP isessentially limited by computational and data storagecapabilities.

1. Computation TollA typical e-beam system is capable of writing a CGH

with a SBWP of -1010. Yet, if a computer such as aVAX is employed, the largest 2-D FFT that can beperformed in a reasonable amount of time is -2048 X2048 = 4 X 106 points. Because of the mismatch in theSBWP of the e-beam system and that of the computingresources, the SBWP of the image (N20) and the inter-polation parameters M. and My are under computa-tional constraints.

When considering the computation limitations, itmust be noted that there are basically two classes ofCGH. Wavefronts of the first class are obtained bysampling an image and then computing a Fresnel orFourier transform. In this case, interpolation is oftenachieved by embedding the object in a larger arraywith a frame of zeros before taking its Fourier trans-form. The resulting size of the FFT that must be


handled by the computing resources is MXMyN2o. If aless exact type interpolation (e.g., linear interpolation)is performed, the computational demands on the re-sources will be reduced to an FFT of size N20 plus theoverhead for interpolation. But the hologram recon-struction quality will be correspondingly degraded.The second class of CGH includes wavefronts that areknown in analytic form. Hence, no FFT is neces-sary-the analytic wavefront need only be sampled.For this type of hologram, a total of N2oMxMY samplesof the wavefront must be evaluated. In this case, thenumber of samples is not limited by the FFT operationbut by the wavefront complexity.

2. Size of Graphical DataAnother constraint on the encoded wavefront is the

amount of graphical data needed to generate the holo-gram. Arnold' calculated the area density of primitiveshapes generated by three encoding algorithms to de-termine the relative amount of graphical data generat-ed by each encoding method. In this section we calcu-late, for all encoding methods, the number of primitiveshapes (NPSs), defined as the number of rectangularapertures of arbitrary size oriented along the axes ofthe e-beam sytem.

For the Lohmann, revised Lohmann, Lee(1), Lee(2),and Burch methods, the NPS is given by

NPS = (number of hologram cells) X N =Xay NC_ = M.MMN:

(13)

where N, is the average number of primitive shapes ineach cell. For Arnold's encoding method, the requiredNPS differs from the above expression, since the de-sired hologram transmittance is sampled in an aperi-odic manner. The derivation of the NPS and compu-tation toll of this method can be found in Appendix C.The resulting NPS in this case is

NPS = PqMN = PMxMATN2, (14)np

where P, defined in Eq. (C3), is a reduction factorwhich is always less than unity. Hence the NPS issimilar to the NPS for the revised Lohmann method(see Table I) except for the factor Pr.

It must be noted that, aside from the NPS, whichdepends on the encoding method, the amount ofgraphical data depends on the format under whichgraphical data are coded as well. Currently, the inputdata format requirements of industrial e-beam lithog-raphy facilities are designed to be efficient for VLSI-type data. These data formats are not efficient forCGH fabrication. Substantial savings in the storagecan be achieved should more suitable formats be em-ployed for CGH fabrication.

IV. Comparison of Encoding Schemes

In Sec. III we presented a set of criteria that deter-mines the quality of a CGH when subjected to con-straints imposed by the computing resources. Thequality of CGH has been characterized by three crite-

ria: bandwidth of the reconstructed wavefront; SNR;and diffraction efficiency. Table I summarizes thedependence of these criteria on the e-beam writer pa-rameters (dftdrcdr), the interpolation factors (M,My),and the number of amplitude and phase quantizationlevels (na,np). This dependence is such that for fixede-beam writer parameters, the SNR and diffractionefficiency generally increase by increasing M, My, na,and np at the cost of decreasing the bandwidths, whichare inversely proportional to these parameters. (ForBurch's method bandwidths are inversely proportion-al to the square root of na.) For an image of fixedSBWP (N), the computation toll and the NPS in-crease as the interpolation factors increase. There-fore, for all encoding schemes there is a trade-off be-tween the SNR and diffraction efficiency of the CGH,on the one hand, and the bandwidths, computationtoll, and NPS, on the other hand. Depending on theparticular application of the CGH, the four parametersMx, My, na, and np, and the particular encoding schemeshould be chosen so that a compromise is reached thatbest satisfies the application's requirements within thecomputational constraints.

A. Qualitative Comparison of Encoding Schemes

Comparison of the bandwidths expressions in col-umns 3 and 4 of Table I shows that for fixed values ofM, and My, the Lohmann, revised Lohmann, and Ar-nold's methods achieve the highest value of Bx becausedr (= 0.125 ,im) is less than df (= 0.5 ,im). Amongmethods that encode amplitude and phase of the wave-front and for a constant value of na, Burch's methodachieves the highest value of B because it depends onthe square root of n compared to linear dependencefor the other encoding methods.

In comparing the SNRs (columns 5-8 in Table I),note that since the positioning accuracy of the e-beamwriter is extremely high, dr (= 1/32 Ai) is very small,the distortion errors are negligible for all encodingmethods. For quantization SNR (SNRQ), phasequantization errors are larger than amplitude quanti-zation errors because of an order of magnitude differ-ence in their multiplicative coefficients. For example,to obtain a SNRQ of 100 due to amplitude quantiza-tion, only five to seven amplitude levels are sufficient,while eighteen phase levels are necessary to obtain thesame value of SNRQ for the phase quantization. Inthe case of representation related SNR (SNRR), detourphase errors are much stronger than aperture errors,since detour phase errors are proportional to M 2 andaperture errors are proportional to M-4, and there is alarge difference in the multiplicative coefficients.From these arguments and the SNR expressions inTable I it follows that the Lee(2) and Burch methodshave the highest SNR because they have no phasequantization and detour phase errors, while the Loh-mann and Lee(1) methods suffer the most due to thefact that both types of phase error are present. Toreduce these phase errors both np and M must beincreased at the expense of the bandwidth B.

Data storage requirements of different encoding


Table II. Comparison of Encoding Methods for Amplitude and PhaseVarying Wavefronts with Minimum Interpolation, Constant NPS = 223 8.4

X 106, na = 5, n. = 12, d, = 0.5,um, d, = 0.125 Am, and 5d, = 1/32 Am

Encoding D x y e M meho . 0 R 0 total 6 x, NY

(min) (min) (min) 1mm I/mm xIO

Lohmann 6.1 30 0.6 610 0.6 670 500 8.4 1 I

Leem) 3.1 100 1.7 1000 1.7 500 500 4.2 1 I

Burch 3.7 300 7.9 250 7.4 500 1000 4.2 2, 1

RevisedLohmann 7.4 30 13.8 610 9.3 670 500 8.4 12,1

Lee(2) 3.9 100 13.8 1000 12.0 500 500 4.2 4,1

schemes are directly proportional to the NPS that theygenerate, and the NPS is proportional to MXMyN2o Acomparison of the NPS in the last column of Table Ishows that for a fixed value of MXMyNo the NPS gener-ated by the Lee(1), Lee(2), and Burch methods is atleast twice that generated by the Lohmann and revisedLohmann methods. The NPS generated by the Ar-nold method is less than that generated by the Loh-mann method by a factor of Pr (Pr < 1). This is mainlydue to the aperiodic nature of the sampling in theArnold method, which was designed to reduce theNPS.

The computation toll of a CGH depends on whetherthe CGH function is known in analytic form or it needsto be computed by FFT operation. If the CGH func-tion is known in analytic form, the computation in-volves MXMyN2o evaluations of the analytic function.For Fourier transform holograms, if the interpolationis done by embedding the object array in a larger field,the computation toll is that of performing a two-di-mensional FFT of size MXMyN'. If the interpolation isdone in a separate step, the size of the FFT would bereduced to N2, but there would be an extra overheadfor performing the interpolation. Here, when we com-pare the computation time of different encodingschemes, we assume that either all methods encodewavefronts that are known in analytic form or they allencode wavefronts whose computation requires FFToperation. Therefore, for fixed values of MXMyN2, thecomputation toll is the same for all encoding schemesand proportional to MXMyNO2, except for the Arnoldmethod for which the computation toll is proportionalto 2(Pr/np) MXMyVo, which is about n-1 times the tollfor other encoding methods (assuming Pr = 0.5). Notethat if the interpolation factors are kept at their mini-mum values, a comparison of expressions in Table Ishows that the computation toll for Lohmann, Lee,'and Arnold methods, being proportional to N2, is halfof that for the Burch method, one-quarter that for theLee(2) method, and n-1 times that for the revisedLohmann method. (For Arnold's method Pr = 0.5 isassumed.)

B. Quantitative Comparison of Encoding Schemes

The qualitative analysis given above can be reexam-ined quantitatively in the numerical examples given in

Table Ill. Comparison of Encoding Methods for Amplitude and PhaseVarying Wavefronts with Constant M1M, = 24, Constant NPS = 223 8.4X 106, na = 5, np = 12, d, = 0.5jim, d, = 0.125 Am, and Ad, = 1/32 Am

SNR -

etodn P. 0 R D total 'Y oI M meodng S O (min) (min) (min) 1/mm 1/mm x10 Y

Lohmann 9.4 30 37 680 16 56 250 3.5 12, 2

LeeCI) 4.8 100 100 1200 48 42 250 1.8 12,2

Burch 6.0 300 1200 7200 170 170 250 3.5 6,4

Lohmann 93 30 220 680 25 670 250 42.0 12, 2

Lee(2) 5.0 100 220 1200 65 170 250 7.0 12,2

Table IV. Comparison of Encoding Methods forAmplitude and Phase Varying Wavefronts with Constant

MAY = 24, Constant N2= 106, n = 5, n = 12, d,

0.5 Am, d, = 0.125 Aim, and d, = 1/32 Am

Encoding r SNR Bx B NPS M, M

method % tota ) mm /mm x Y

Lohmann 9.8 8 110 125 24 6,4

Lee I) 4.9 31 83 125 48 6,4

Burch 5.8 170 170 250 24 6,4

Lohmann 93 25 670 250 2 12,2

Lee(2) 5.0 90 330 125 12 6,4

Table V. Comparison of Encoding Methods for Phase OnlyVarying Wavefronts with Constant MxMYN

2= 24 X 106,

Variable Number of Quantization Levels, d, = 0.5 Am, d, =0.125 Am, and 5d, = 1/32 jm

SNR B' B' NPS N2

% total 6 % method 1mn /mm 1/mm x10 I 0O

Lohmann 10 9.7 125 125 24 1.8 8,16np=8

Lee0i) 4.9 14.7 125 125 48 7.5 4,4na- 5

Arnoldnp= 16 1 0 15.6 500 500 12 60 16,4

RevisedLohmann 8.9 17.9 1000 1000 3 15 8,2

n p = B

Lee(2) 3.9 10.8 500 500 12 60 4,1n a= 5

Tables II-V. In these tables the expressions summa-rized in Table I were used to compare encodingschemes when subjected to computer limitations. InTables II and III the methods are compared based onlimited data storage capacity, while the comparison inTables IV and V is based on limited computation time.Thus in the first two tables the NPS is kept constant,while in the second two it is the product MMy and theimage SBWP (N2b) that are held constant for all encod-ing schemes. Typical e-beam writer parameters of df= 0.5 gm, dr = 0.125 gim, and 5dr = 1/32gm are assumedand used in all four tables.


In Tables II-IV methods that encode the amplitudeand phase of the wavefront are compared. In thesetables the number of amplitude and phase quantiza-tion levels are kept constant at na = 5, np = 12. Thesevalues were chosen to obtain a relatively high SNRQand at the same time to keep the numerical values ofthe bandwidths B'x and B about the same. In Table IIthe interpolation factors are set to their minimumvalues required by the encoding methods. This allowsthe encoding methods to achieve maximum values ofN2 (for a constant NPS) and bandwidth (for fixed naand np). The resulting efficiency and SNR, however,are small, and the SNR is dominated by SNRR. Notethat for the Lohmann and Lee(1) encoding methodswhen the interpolation factors are equal to one, higher-order terms in Eqs. (A6) and (A9) must be included inthe evaluation of SNRR. Also note that the minimumSNRR values, occurring at the spatial limits of thereconstructed image (see part 2 of Appendix A), areconsidered in our analysis; as one approaches the cen-ter of the image SNRR increases dramatically. Fromthe numerical values in Table II it follows that themethods that sample the wavefront at the center ofeach aperture (i.e., the Burch, revised Lohmann, andLee(2) methods) have considerably larger SNR thanthe Lohmann and Lee(1) methods, with the Lee(2)method yielding the highest SNR. However, the Loh-mann and revised Lohmann methods yield better dif-fraction efficiency than other encoding methods. Italso follows that Lee(1) and Lee(2) methods are not asefficient as the other encoding methods in use of theavailable bandwidths. Furthermore, it can be seen inthis table that for a given NPS, the SBWP of theimages reconstructed by the Lohmann and revisedLohmann methods is twice that reconstructed by otherencoding methods.

In going from Tables II to III interpolation factorsare increased so that the productMxMy = 24. In TableIII Mx and My are chosen in a manner to optimize theSNRR. It can be seen that the SNR values are im-proved considerably. Diffraction efficiencies have im-proved as well, but the bandwidths are significantlyreduced. Since the NPS is kept constant, the SBWPof the image is also reduced as a consequence of in-creasing the interpolation factors. In comparing theencoding schemes in Table III we see that the revisedLohmann method uses the bandwidth much more effi-ciently than other encoding methods; it also allows thehighest image SBWP and diffraction efficiency.Compared to other encoding schemes, Burch's methodis moderate in terms of bandwidth and diffractionefficiency, but it has the highest SNR.

In Tables IV and V comparison of encoding schemes,based on limited computation time, are presented. Inthese tables both N and the product MXMy are heldconstant. Thus the computation time for each meth-od is approximately the same regardless of how theinterpolation is carried out. Table IV compares fiveamplitude and phase schemes while constrained to anobject SBWP of 106 elements with 24 interpolationsper element. In this table Mx and My are chosen so

thatB and By have similar values. The comparison ofCGH qualities for different encoding schemes is verysimilar to that given for Table III. It can be seen fromNPS entries in Table IV that for a given image SBWPand MXMy, the revised Lohmann method generates theleast NPS, while the highest NPS is generated by theLee(1) method.

Table V shows a comparison of methods used toencode wavefronts with constant amplitude values.(Note that Lee's methods still need to quantize compo-nents of the wavefront along the basis vectors.) Hereagain our comparison is based on limited computationtime, since in this case wavefronts typically are knownin analytic form. Therefore, the product MXMyN isheld constant for all encoding methods. The parame-ters M., My, and np (also n for Lee's methods) arechosen so that the total SNR of -10 is achieved by allencoding schemes, and the bandwidths are the same inboth directions. In these conditions, it can be seenfrom Table V that the Arnold (assuming Pr = 0.5) andLee(2) methods can encode more image points thanother methods. It also follows that methods that sam-ple the wavefront at the center of apertures generateless NPS per image SBWP than those that sample thewavefront at the center of the cell. In terms of imagequality, Arnold's and Lohmann's methods gives thebest diffraction efficiency, while the revised Lohmannmethod makes the best use of the available band-widths. The Arnold and Lee(2) methods have highbandwidths as well.

C. Comparison of Information Capacity

We conclude this section with a comparison basedon Shannon's criterion, which in communication sys-tems is a measure of information capacity of a channel(in bits per second) and is given by22

C = B log 2 (1 + SNR). (15)

It has been shown23 24 that the information capacity ofa coherent optical system is given by

C = 2X'Y'BhBhy log2 (1 + SNR). (16)

Here we adopt a variation of this equation by replacingthe hologram bandwidths by the bandwidths of thereconstructed wavefront. In this case the informationcapacity per unit area per polarization state becomes

C = BIBy log2 (1 + SNR). (17)

Table VI compares the relative information capacity ofvarious phase and amplitude encoding schemes forfixed NPS using the same values for the interpolationand quantization parameters as was used in Table II.By keeping the NPS fixed, we are able to determine theefficiency of various encoding schemes in storing holo-gram information in a fixed amount of data that is sendto the e-beam facility for fabrication. In this table theinformation capacity for all methods are measuredrelative to that of Lohmann's method. Burch's, Ar-nold's, and the revised Lohmann method seem to havea relatively large information capacity.


Table VI. Comparison of Information Capacityof Encoding Methods for Amplitude and Phase

Varying Wavefronts. The NPS is held Fixed andthe Interpolation Factors are kept at their

Minimum: na = 5, np = 12, d, = 0.5 m, dr =0.125 jm, and Ad, = 1/32 jm

Encoding B. B SNR Relativemethod x Y (Min) C (Min)

Lohmann 670 500 0.6 1

Lee( ) 500 500 1.7 1.6

Burch 500 1000 7.4 6.8

Revised 670 500 9.3 5.0Lohmann

Lee(2) 500 500 12.0 4.1

V. Conclusions

Because of the ability of electron-beam lithographysystems to record patterns of submicron resolution onlarge substrates, they appear to be a prime candidatefor the recording of computer generated holograms.However, there are two obstacles in the utilization oftheir full potential: (1) limited processing power ofcommonly available computers and (2) a large amountof graphical data needed to specify geometrical pat-terns. In this paper six encoding methods have beencompared in terms of their ability to encode high quali-ty wavefronts when subjected to these limitations.The hologram quality was measured in terms of SNR,diffraction efficiency, and wavefront bandwidth. Thecomparison was achieved by the development of equa-tions relating these measures of hologram quality tothe number of primitive shapes (NPSs), interpolationfactors (M~,My), and image SBWP (No), which areparameters that are constrained by the computing re-sources.

The comparisons that were carried out in the previ-ous section can serve as a guide to choosing the mostsuitable encoding scheme for a particular application.Different applications of CGH impose different re-quirements on the quality of the CGH. In general, it isdesirable to have a high diffraction efficiency, highSNR, and large SBWP images with high resolution forall types of application. However, as we saw in Sec. IV,achievement of all these requirements at once is notpossible, and one should be willing to compromise onerequirement for another, depending on the particularapplication on hand. For example, if a CGH is to beused for display purposes, perhaps reconstruction of alarge SBWP image with high diffraction efficiency isthe most important requirement. In this case oneshould use an encoding method that provides the high-est bandwidth and diffraction efficiency. From thetables presented in the previous section the revisedLohmann method gives the highest wavefront band-width and diffraction efficiency and, therefore, may beconsidered the most suitable method to encode this

type of hologram. As another example, if the CGH isto be used as a holographic optical element (HOE), inmost cases the HOE is required to have a high SNR.Burch's method appears to be mostly suitable for thistype of application. But, in general, depending on theminimum SNR requirements and available computingpowers, any one of the methods that samples the wave-front at the center of apertures might be used.

Finally, from Table V it can be deduced that a largeportion of the SBWP provided by the e-beam recordercan be utilized by means of large interpolation factorswhich can be carried out in a reasonable amount oftime. Additional utilization of the SBWP can beachieved by CGH replication to reduce the specklenoise. Of course, there are other cases which utilizethe additional SBWP of the CGH (e.g., space variantholograms). These cases are beyond the scope of thispaper, and they will be considered in future work onthis subject.

This work was supported by the Air Force Office ofScientific Research, grant AFOSR85-0371, and by theDefence Advanced Research Project Agency, grantNOSC N66001-84-C-24.

We would like to acknowledge A. W. Lohmann'ssuggestion to include Shannon's formula in our com-parative study.

Reiner Eschbach was supported by the DeutscheForschungsgemeinschaft.

Appendix A: Derivation of Expressions for SNR

1. Quantization ErrorsLee19 and Gabel and Liu7 have shown that for Loh-

mann's and the revised Lohmann's methods the meansquare error (MSE) is given by

MSE = 1N`N 1 - sc -N 5 -1 N5 1A,,L \"PJI m=On=(Al)

where l/(NxNy) 17 IAIn12 represents the average am-plitude transmittance of the desired CGH. The MSEgiven in Eq. (Al) has two parts. The first term is dueto amplitude quantization, and the second term is dueto phase quantization. It was assumed in deriving Eq.(Al) that the amplitude quantization error is uniform-ly distributed over the range (-Q/2,Q/2), where Q = 1/na, and the phase quantization error is uniformly dis-tributed over the range (-'/2,4/2), where b = (27r)/np.1 9,21 The maximum amplitude is taken to be 1.Substituting Eqs. (Al) and (7) into Eq. (6) results in

N.-1 Ny-1

3 3 IAmnI2

0 n=O(A2)

1 2_ _

12n+ NN[1 - sinc( )] 3: E IAmn212n2 Nx)FI (I m=0 n=O

If we assume a uniform probability distribution for thewavefront amplitude at the hologram plane (constantprobability over the 0-1 range), we can write

(A3a)N N5-1 N-1

3X~ 3 E tAmnl, = /3.0N n0


SNAR =

For future purposes we assume uniform probabilitydistribution for the phase also, and we can write

(A3b)N`-1 NY5-Z 1 =3 7r.Y m n=O

In this case Eq. (A2) becomes

1 1SNRQ= 1 2 F Ii 1 + r24n. I (P n21PL \ f l p / J ~~a p

(A4)

By dropping the phase quantization term in Eq.(A4), one arrives at the SNRQ for the Burch method,i.e., for Burch's method SNRQ = 4na.

Arnold's method, which encodes wavefronts of con-stant amplitude, is similar to Lohmann's method with-out amplitude quantization. Thus in this case we haveSNRQ = 3(n2/7r2).

From the expressions given by Gabel6 [Eqs. (12) and(13)] it can be shown that for Lee(1) and Lee(2) encod-ing schemes MSE = (NxNy)/6n2. The SNR in thiscase becomes SNRQ = 2na-

2. Representation Related ErrorsLohmann and Paris'3 have shown that the represen-

tation related errors for the Lohmann's encodingmethod is given by

E=-sineiyby'Amn \exp (X - X)bX'q5mnnEmn 1 X ) exP[- XFM ]

(ry by'A mn)2 _ _ __ _ _( __F _ a') (x - X0)6X'0mn (A)

6 XE

where M is the number of 27r phase shifts in the carrierwavefront in one cell, Xo is the distance from the centerin the image plane where the image is projected. Theapproximation in this equation is valid for small valuesof the arguments of the sinc and exponential functions.This error depends on the image coordinates. Themaximum error results when x and y take their maxi-mum values X/2,Y/2. Substituting the maximum er-ror from Eq. (AS) along with Eq. (7) into Eq. (6) andusing Eqs. (3) and relations Mx = (XFIXbx') and My =(XF)/(Yby') results in a minimum SNRR given by

(SNRR)min

t~ ~ ~~~ (r )4 7r 217 _1*(6

1344 My 3 (M) 60 M, 120 (M AMy

For the Lee(1) method, using the transmittancefunction given by Bucklew and Gallagher,8 one canshow that the reconstructed image is given by

h a~y) .~L / 1 X6X' \ N,-1 N-1

h(xy) = 8 sie -+ xbx' expt 3r) E y' X5 \4 4XF/ i

0 =0

X exp[j 27r (nxbx' + my8y')J

X {(1 - E"n)Almn - (1 - En)A3 mn

+ j [-(1 - En)A 2mn + (1 - Em4n)A4 mn] }

where

Emn) = 1 - sinc(Y 6 k m) exp [ P 4F

(Ary' by'Akmn) -j (x-_O),x6_ - 16 ' 4XF

(A7)

(A)

In these equations Akmn are the components of thewavefront's amplitude along the basis vectors (0,Ar/2,7r,37r/2), and P = 3,1,-1,-3 for k = 1,2,3,4, respec-tively. Following the same procedure as presented inthe Lohmann's case, one can show that the minimumSNRQ in this case is given by

(SNRR)min

1 (ior\ 4 + 5 ( r I - 1 X \4 1 ( 7r2 \21344 I\My) 16 1200 M.) 512 MXMY)

(A9)

The representation related errors given in Eqs. (A9)and (A6) contain four terms. The first term is due toaperture error, and the second term is due to detourphase error. The other two terms are higher-orderterms that are negligible for relatively large values ofM. Chavel and Hugonin25 and Allebach" haveshown that detour phase error can be eliminated bysampling the wavefront at the center of each aperture.Thus, when the revised Lohmann and Lee(2) methodsare employed, all the terms in Eqs. (A6) and (A9) areeliminated but the first term, and they reduce to(SNRR)min 1344 (My/r) 4 .

For Burch's method the hologram can be encoded byintroducing an aperture of area proportional to themagnitude of the transmittance function given in Eq.(2) at the center of a cell. In this case the representa-tion related error is just that due to the aperture. Ifthe aperture opening is varied in two dimensions, theerror is

(Emn)R,aperture = 1 - sin V [(x - x) )x'fA] -

(A1O)

Taking the maximum values of x and y and expandingthe sinc functions, it can be shown that

(SNRR)min 960 1 J

L4 M My E(All)

For Arnold's method there is a negligible detourphase error since the wavefront is sampled (almost) at


the center of each aperture. Yet there is an additionalrepresentation related error term due to phase errorcaused by the finite size of the apertures. Although anaperture's position may correspond to the correctphase at the center of the aperture, near the top andbottom of the aperture the phase may differ from thatat the center. The encoding method is designed sothat most of the apertures will have phase errors neartheir edges close to, but not exceeding, (0)/2. Thus,we can approximate this error as a white noise, result-ing in a phase error uniformly distributed over therange [-(6k)/2 + (60)/2] in the Fourier domain. Thisresults in

12 2SNRR = = 12-My(85)2 W,2

(A12)

by the same reasoning used above for the derivations ofthe SNRQ terms.

3. Distortion ErrorsEvaluation of distortion related SNR (SNRD) is very

similar to that of the quantization case. In this caseEqs. (Al) and (A4) are modified to yield

SNR- 1 12 (A13)

2 [1 - sinc()]

where 4' is the range of the phase distortion error, andit is given by

, = ( + 42)112, ). = 2rbdr, 27rbdrtc (t y)av

(A14)

where t is the carrier's period (in the x direction), and(ty)av is the average distance between fringes in the ydirection. Since 2tyav = B = 1/(by'My), the maxi-mum value of by is (0y)m: = (7r5dr)/(y'My). Theresulting minimum SNRD is given in Eq. (8).

Appendix B: Effects of Sampling on DiffractionEfficiency

For the general case, let G(x',y') denote the desiredFourier plane transmittance function. Then the actu-al hologram transmittance can be approximated by

T(x',y) = a, b rect X' )rect * [G(x'Y')

X comb ax' comb , (Bi)

where * denotes convolution. The reconstructed im-age is given by the continuous inverse Fourier trans-form of T(x',y') as

t(x,y) = sinc(8X' X) sinc y,) E E

Xg(X - nX 'Y mX y, (B2)

The image t(xy) is a sum of shifted versions of thedesired image g(x,y), all attenuated by a sinc function.These multiple images are due to sampling, not to theuse of the carrier. The primary image [goo(x,y)] isgiven by

g0 0(x,y) = sinc(ax' A) sinc (by' A) g(x,y), (B3)

and the power reduction factor due to sampling isgiven by

xT 2sny /x

fJe'+ X2 f sinc2(8xSj) 2in& (by' ;t],) Ig0 (xYy)I2dxdyJ =f I |goo(Xy)12 dxdy

(B4)

In general, Eq. (B4) is image dependent. However, ifwe assume a constant image amplitude, Eq. (B4) be-comes

r = I Y J2 sinc2 (x ) sinc2

(y, Y) dxdyXY 2 FX

= f+ sinc2 - du I sinc2 dv.q,5 M. fo0.5 \M,

Defining

X(a,b) = sinc - dua-0.5 b

gives

r = X(q,Mx)X(O,M),

(B5)

(B6)

(B7)

whose limit as M. -X is 1. However, in some casesthere is a constraint on M,/q = M,. For these cases,

lim r = sinc2( ) = k M, = constant, (B8)

where k < 1. Since this limit is <1, k is not part of thereduction factor due to sampling. It is part of thetheoretical efficiency. Thus, for encoding schemessubject to the constraint, qIMx = constant, Eq. (B7) ismodified to

r = X(q,M5)X(0,My) sinc-2(M ). (B9)

To evaluate the power reduction factor for particu-lar encoding schemes, a model developed by AllebachLis adopted. This model allows one to calculate theFourier transform of an arbitrary binary hologram.Using this model, it was found that for encoding meth-ods that sample the wavefront at the center of cells, thepower reduction factor is that given by Eq. (B7) with q= 0. For other encoding schemes Eq. (B7) can be usedwith an arbitrary q.

Appendix C: NPS and Computation Toll for the Arnold'sMethod

From Ref. 1, the area density of primitive shapes forArnold's staircase method is

1 o 2irb |y' ax'

where At is the maximum phase error. Therefore, theNPS is given by

(Cl)NPS = 1 ff f 0 OIYi~q I~ Jy' Iy Ox' dx'dy'.


If we assume that the phase varying function has aconstant spatial frequency in the x direction equiva-lent to a carrier frequency, one can write

- d = constant = t = 2qB' and

J | dy' = Y( )av (C2)

then Eq. (Cl) reduces to

NPS = Pq NN2, P. = (C3)

'I )max

For this case, redefining the interpolation parameter,

- 1 1Y By' minimum sampling period

gives

1NPS = PqMN2 = P.- MMINV. (C4)

As mentioned earlier, computer resources limit thetotal number of wavefront samples that can be evalu-ated for holograms encoding wavefronts known in ana-lytic form. Since only two wavefront evaluations areneeded for each primitive shape when using this en-coding method, the total number of samples is equal totwice the NPS. Thus, while for the other encodingmethods the product MxMyN2o is limited by computingresources, for Arnold's method the constraining factoris 2 X NPS. Therefore, the total number of wavefrontsamples in this case is

2Pq N2,

and one can write (for the same computation toll)

(MxMYNO)Arnold's method= P (MxMYN')conventional methods (CS)

In short, aperiodic sampling has produced the fol-lowing effects: (1) For the same number of wavefrontsamples, the effective CGH SBWP has been increasedby a factor of np/(2Pr). (2) For the same CGH SBWP,the NPS has been reduced by a factor of Pr. (3) Noisehas been introduced through an additional phase errorterm (see Appendix A). This phase error can be re-duced at the expense of increasing the NPS [Eq. (C4)].

References1. S. M. Arnold, "Electron Beam Fabrication of Computer Gener-

ated Holograms," Opt. Eng. 24, 803 (1985).2. S. M. Arnold, "E-Beam Written Computer Generated Holo-

grams," Final Technical Report E47069, Honeywell Corpora-tion Technology Center, Bloomington, MN (1983).

3. K. M. Leung, J. C. Lindquist, and L. T. Shepherd, "E-BeamComputer-Generated Holograms for Aspheric Testing," Proc.Soc. Photo-Opt. Instrum. Eng. 215, 70 (1980).

4. J. L. Freyer, R. J. Perlmutter, and J. W. Goodman, "DigitalHolography: Algorithms, E-Beam Lithography, and 3-D Dis-play," Proc. Soc. Photo-Opt. Instrum. Eng. 437, 38 (1983).

5. R. A. Athale, C. L. Giles, and J. A. Blodgett, "Use of E-beamWritten CGH in Optical Pattern Recognition," Proc. Soc. Pho-to-Opt. Instrum. Eng. 437, 48 (1983).

6. R. A. Gabel, "Reconstruction Errors in Computer GeneratedBinary Holograms: A Comparative Study," Appl. Opt. 14,2252(1975).

7. R. A. Gabel and B. Liu, "Minimization of Reconstruction Errorsin Computer Generated Binary Holograms," Appl. Opt. 9,1180(1970).

8. J. Bucklew and N. C. Gallagher, Jr., "Comprehensive ErrorModels and a Comparative Study of Some Detour-Phase Holo-grams," Appl. Opt. 18, 2861 (1979).

9. D. R. Herriott and G. R. Brewer, "Electron-Beam LithographyMachines," in Electron-Beam Technology in MicroelectronicFabrication, G. R. Brewer, Ed. (Academic, New York, 1980),Chap. 3.

10. E. E. Erny, J. Webster, and M. Salsali, "State-of-the-Art Elec-tron-beam Maskmaking Production Tools," Report 60, VarianAssociates, Inc., Lithography Products Division (1984).

11. J. P. Allebach, "Representation-Related Errors in Binary Digi-tal Holograms: A Unified Analysis," Appl. Opt. 20, 290 (1981).

12. B. R. Brown and A. W. Lohmann, "Complex Spatial Filteringwith Binary Masks," Appl. Opt. 5, 967 (1967).

13. A. W. Lohmann and D. P. Paris, "Binary Fraunhofer HologramsGenerated by Computers," Appl. Opt. 6, 1739 (1967).

14. W. H. Lee, "Sampled Fourier Transform Hologram Generatedby Computer," Appl. Opt. 9, 639 (1970).

15. J. J. Burch, "A Computer Algorithm for the Synthesis of SpatialFrequency Filters," Proc. IEEE 55, 599 (1967).

16. B. R. Brown and A. W. Lohmann, "Computer Generated BinaryHolograms," IBM J. Res. Dev. 13, 160 (1969).

17. W. H. Lee, "Binary Computer-Generated Holograms," Appl.Opt. 18, 3661 (1979).

18. W. H. Lee, "Binary Synthetic Holograms," Appl. Opt. 13, 1677(1974).

19. W. H. Lee, "Computer-Generated Holograms: Techniques andApplications," Prog. Opt. 16, 119 (1978).

20. N. S. Jayant and P. Noll, Digital Coding of Waveforms, Princi-ples and Applications to Speech and Video (Prentice-Hall,Englewood Cliffs, NJ, 1984), pp. 31-35.

21. J. W. Goodman and A. M. Silvestri, "Some Effects of Fourier-Domain Phase Quantization," IBM J. Res. Dev. 14, 478 (1970).

22. See, for example, S. Haykin, Communication Systems (Wiley,New York, 1978), pp. 492-496.

23. P. B. Fellgett and E. H. Linfoot, "On the Assessment of OpticalImages," Philos. Trans. R. Soc. London Ser. A 247, 369 (1954).

24. I. J. Cox and C. J. R. Sheppard, "Information Capacity andResolution in an Optical System," J. Opt. Soc. Am. A 3, 1152(1986).

25. P. Chavel and J. P. Hugonin, "High Quality Holograms: TheProblem of Phase Representation," J. Opt. Soc. Am. 66, 989(1976).


Documents

Comparison of binary encoding schemes for electron-beam fabrication of computer generated holograms