computer-originated planar holographic optical elements

Computer-originatedplanar holographic optical elements

Silviu Reinhorn, Yaakov Amitai, and Albert A. Friesem

We present novel, to our knowledge, methods for the analytical design and recording of planar holo-graphic optical elements in thick materials. The recording of each planar holographic element is doneby interference of two aspherical waves that are derived from appropriately designed computer-generatedholograms such that the element has the desired grating function for minimizing aberrations and closelyfulfills the Bragg condition over its entire area. The design and recording methods are described, alongwith calculated results of representative elements. © 1998 Optical Society of America

OCIS codes: 090.2890, 090.1760, 090.2900.

1. Introduction

In recent years much effort has been directed to-ward miniaturization of optical systems to dimen-sions comparable with those that use electronics.1–4

One promising approach involves planar opticsconfigurations.5–8 A general planar configurationcomprises a planar transparent substrate that hasseveral holographic optical elements ~HOE’s! on ei-ther one or both sides of the substrate. The holo-graphic elements can function as conventional opticalelements such as lenses, beam splitters, polarizers,and mirrors. The light between the HOE’s propa-gates inside the substrate and exploits either totalinternal reflection or reflective coatings on the sub-strate surfaces. The alignment of several HOE’sthat are integrated on one substrate can be done withrelatively high accuracy during the fabrication stage.The final planar configuration is rigid, and there is noneed for further alignment of the individual HOE’s,as is usually necessary with free-space optical config-urations.

There are two main approaches to recording HOE’supon planar substrates. One involves direct record-ing and lithographic techniques, and the other, holo-graphic interferometric recording. With directrecording and lithographic techniques a computer-generated mask is transferred to form elements as

The authors are with the Department of Physics of ComplexSystems, Weizmann Institute of Science, Rehovot 76100, Israel.

Received 20 August 1997; revised manuscript received 6 Janu-ary 1998.

0003-6935y98y143031-07$15.00y0© 1998 Optical Society of America

surface-relief gratings that can perform almost anyarbitrary wave-front transformations. Unfortu-nately, for planar configurations the required gratingperiod of the relief grating is quite small. For exam-ple, grating period L, which couples a normally inci-dent wave into a transparent substrate so that thediffracted beam will be trapped inside the substrateby total internal reflection, is L 5 ly~n sin u!, wherel is the wavelength of the incident wave, n is theindex of refraction of the substrate, and u is the dif-fraction angle inside the substrate. When u is equalto the critical angle of the substrate, then L 5 l,which leads to a grating with line widths of the orderof ly2. As a result, for visible radiation only grat-ings that have binary profiles can be recorded bycurrently available methods for electron-beam directwriting. Such gratings have a relatively low diffrac-tion efficiency, typically below 50%.

One way to improve the diffraction efficiency is todesign planar elements with a moderate off-axis an-gle inside the substrate ~10°! so the needed gratingshave periods of the order of 4l. Such gratings can berecorded with multilevel lithography to yield blazedprofiles, thereby improving the diffraction efficiencysignificantly ~more than 90% for eight levels! over arelatively wide range of incident angles. When suchplanar elements are used the substrate surfaces mustbe coated with reflective materials to trap the lightinside the substrate. Unfortunately, the reflectivematerials are lossy, so after the many internal reflec-tions the overall light efficiency is again relativelylow.

For planar optical configurations in which the ele-ments are recorded as surface-profile gratings wemust impose certain geometrical conditions to ensure

10 May 1998 y Vol. 37, No. 14 y APPLIED OPTICS 3031

that undesirable interaction between internally re-flected rays and the planar elements does not occur.Specifically, we must choose the proper dimensionsand locations for the planar HOE’s as well as theproper thicknesses for the substrates. The lightrays that are diffracted by the planar element intothe substrate must propagate a lateral distance thatis greater than the aperture D of the element to avoidundesired outward diffraction from the same ele-ment. The ratio between the aperture size of theplanar element and the substrate thickness T is thuslimited by DyT # 2 tan u, where u is greater than thecritical angle.

With holographic recording it is possible to recordvolume phase gratings of high spatial frequencies, sothe incident beam will be diffracted into the substratewith high diffraction efficiency at an internal anglelarger than the critical angle. Here there is no needfor any reflective coatings, so the substrate remainstransparent and there is little, if any, reflection loss.Moreover, a planar holographic lens in volume phasematerials has a variable grating period, in which thelocal Bragg condition varies along the area of thegrating. In such a case, unlike for surface-reliefgratings, the internally propagating light rays willdiffract outward from the substrate only at desiredlocations where they satisfy the local Bragg condi-tions.9

For ensuring that geometric and chromatic aber-rations are minimized and the diffraction efficienciesare maximized, several optimization design methodsfor the grating function of HOE’s were developed,both for low diffraction angles10 as well as for HOE’swith high diffraction angles.11 In all these methodsit is usually necessary to resort to aspherical wavefronts to record the desired grating functions. Suchwave fronts can be derived from interim hologramsdesigned in accordance with a recursive design tech-nique12 as well as from computer-generated holo-grams ~CGH’s!.13

With the recursive technique the final element isrecorded with predistorted aspherical waves thatoriginate from interim holograms that were recordedand read out with either spherical or even predis-torted aspherical waves. The technique can be ex-tended to include predistorted waves that willmaximize the diffraction efficiency as well.14 Yet forpractical considerations the number of iterationsmust be limited, so the recursive design technique isusually confined to elements with relatively simplegrating functions.

When the aspherical wave fronts for recording aHOE are derived from CGH’s,15 it is possible to obtainmore-general grating functions for which the geomet-ric and chromatic aberrations are minimized. Thediffraction efficiency for such a computer-originatedholographic optical element, however, is typicallyrather low. Some attempts to increase the diffrac-tion efficiency were reported for relatively simple andspecific elements.14

In this paper we present a new design method andsimple configuration for recording generalized

3032 APPLIED OPTICS y Vol. 37, No. 14 y 10 May 1998

computer-generated holographic optical elementsthat have both minimum aberrations and maximumdiffraction efficiencies. The design method exploitsa design in which the grating vector of the HOE isoptimized and a recording configuration in which theCGH’s are both on the same substrate. It leads to afinal HOE that has the small grating periods neededfor planar configurations and arbitrary three-dimensional grating functions. Such characteristicsare difficult, if not impossible, to obtain with the ear-lier design methods.

2. Design Considerations

The recording configuration of our device, in whichthe aspherical wave fronts are derived from twoCGH’s that are on one substrate, is shown in Fig. 1.When both CGH’s are formed on the same substrateit is possible to control the distance between themwith great accuracy. This substrate can be alignedparallel to the final HOE plane so the wave frontsimpinging upon the holographic recording mediumcan be controlled accurately. For planar optical el-ements it is usually required that one of the recordingbeams impinge upon the holographic recording me-dium at a large off-axis angle. For example, if therequired off-axis angle of the reference beam is 60°and the recording wavelength is 0.5 mm, the gratingperiod of the CGH is L 5 0.5y~sin 60! 5 0.577 mm.Consequently, the CGH’s must be recorded withelectron-beam direct writing. With today’s avail-able technology it is possible to record only binarygratings with such fringe density. Fortunately, thediffraction efficiencies of the CGH’s do not influencethe diffraction efficiency of the final element, so theCGH’s can be recorded as binary gratings, usually amuch simpler procedure.

When one is dealing with HOE’s it is mainly thephases of the recording waves that establish the rel-evant diffraction relations, given by

fi~x, y! 5 fc~x, y! 6 fh~x, y!, (1)

Fig. 1. Geometry for recording the final HOE with recording wavefronts derived from CGH’s.

where fi and fc denote the phases of the diffractedimage and reconstructing waves, respectively, theplus or minus sign refers to the positive and negativefirst diffraction orders from the hologram, and fhdenotes the holographic grating function, given by

fh~x, y! 5 fo~x, y! 2 fr~x, y!, (2)

where fo and fr denote the phases of the object andthe reference recording waves, respectively. Equa-tion ~1! is the basic holographic equation, indicatingthat a HOE acts as a wave-front transforming ele-ment, i.e., it transforms the incoming wave front fcinto the output wave front fi. Usually, when thereis a range of possible input and output wave frontsthe grating function fh~x, y! must perform the de-sired transformation with minimum aberrations16–18

by exploitation of some optimization method.12,19,20

Such methods can exploit the fact that fh~x, y! can bean arbitrary two-dimensional function, so there is aninfinite set of two-dimensional functions fo~x, y! andfr~x, y! that can generate it.

It is often more convenient to deal with light raysinstead of wave fronts, especially when one is calcu-lating the diffraction efficiency from volume gratingsby the coupled-wave theory. The normalized prop-agating vectors, which can be regarded as the direc-tion cosines of the rays, can be written as

kq 5lp

2p=fq, (3)

where q 5 o, r, c, i and p 5 o, c, = is the gradientoperator with respect to the spatial coordinates x andy, lo denotes the recording wavelength, and lc de-notes the reconstructing wavelength. Equation ~3!represents a two-dimensional projection of the prop-agation vector upon a plane perpendicular to the di-rection of propagation. The grating vector of thehologram is defined as

kh 5lo

2p=fh~x, y! 5

lo

Lx~x, y!x 1

lo

Ly~x, y!y, (4)

where Lx~x, y! and Ly~x, y! are the grating spacings inthe x and the y directions, respectively.

When the HOE is recorded in thick materials, khhas a three-dimensional grating vector distribution.The lateral two-dimensional distribution influencesthe optical wave-front transformation properties ofthe HOE, whereas the third dimension z is related tothe mechanism of transferring energy from the inci-dent wave to the diffracted wave. The grating spac-ings Lx~x, y! and Ly~x, y! in the x and the y directionshave the forms

Lx~x, y!

5lo

sin bo~x, y!cos ao~x, y! 2 sin br~x, y!cos ar~x, y!,

Ly~x, y! 5lo

sin ao~x, y! 2 sin ar~x, y!, (5)

where bo and br are the angles between the projectionof the propagating vectors of the object and referencewaves, respectively, on meridional plane ~x, z! andthe z axis, and ao and ar are the angles between theactual propagation vectors and the meridional plane,respectively, as illustrated in Fig. 2. The diffractionrelations can now be written as

kxi5 kxc

6 mkxh,

kyi5 kyc

6 mkyh,

kzi5 6~1 2 kxi

2 2 kyi

2!1y2, (6)

where m 5 lcylo. The sign of kz is determined inaccordance with

sign~kzi! 5 sign~kzc

!, ~transmission hologram!,

sign~kzi! 5 2sign~kzc

!, ~reflection hologram!.

To obtain a high diffraction efficiency it is necessaryto satisfy the Bragg condition over the entire area ofthe HOE. This is achieved when the three-dimensional grating vector kh fulfills the condition

kh~x, y! 5 ki~x, y! 2 kc~x, y! 51m

@ko~x, y! 2 kr~x, y!#.

(7)

In general, it is difficult to record a HOE thatsimultaneously has low aberrations and high dif-fraction efficiency over its entire area. Specifi-cally, the aberrations and the image geometrydepend on the two-dimensional grating distributionacross the surface of the holographic element,16–18

whereas the diffraction efficiency is determined bythe three-dimensional grating distribution.9 Inother words, it is difficult to fulfill the condition setforth in Eq. ~7!, i.e., to obtain a set of conservingvectors from which we can find realizable phases forthe recording waves, fo and fr. There are twosituations when Eq. ~7! can be easily satisfied:first, when m 5 1, i.e., there is no wavelength dif-ference between recording and readout, and second,when kh is a one-dimensional function, i.e., kh~x!.

In planar optical elements with the usual high off-axis angle there is one direction in the hologramplane where the grating spacing is much smallerthan that in the orthogonal direction. Typically, the

Fig. 2. Coordinate system for recording and readout of the HOE.


off-axis angle is aligned with respect to the x axis, soLx , Ly, for example, in planar elements in which thelight is trapped inside the substrate by total internalreflection, where u is approximately 45°, according toEq. ~5!, LxyLy , 0.025. In such elements the angu-lar and wavelength selectivities because of the Braggcondition are greater in the x direction than in the ydirection.9 Thus we can exploit the advantage thatkh is almost a one-dimensional function. Accord-ingly, we develop a design procedure, assuming thatthe recording setup is that illustrated in Fig. 1. Ourdesign method is appropriate for a situation in whichthe final desired HOE is recorded as a volume gratingthat can have a two-dimensional grating function.In addition, its readout wavelength might be differ-ent from that of the recording. Our design proce-dure has two stages. First, we calculate the phasesof the object and the reference waves at the HOEplane that form the desired grating function, whichsimultaneously nearly fulfills the Bragg conditionover an area as large as possible. Second, we tracethese wave fronts to the CGH’s plane by an ordinaryray-tracing approach.21

To calculate the needed object and reference waveswe begin with the desired grating function fhd

~x, y! ofthe HOE and determine the desired grating vectorkhd

~x, y! according to

khd~x, y! 5

lc

2p=fhd

~x, y!. (8)

The components of the desired grating vector that ful-fill the relations for the first ~11! diffraction order are

kxhd~x, y! 5 kxi

~x, y! 2 kxc~x, y!,

kyhd~x, y! 5 kyi

~x, y! 2 kyc~x, y!,

kzhd~x, y! 5 @1 2 kxhd

2~x, y! 2 kxhd

2~x, y!#1y2. (9)

Equation ~7! can be solved to yield ko and kr in thepresence of a wavelength shift when the grating vectorof the HOE is one dimensional. To obtain such a one-dimensional vector we set kq~x, y! 5 kq~x, y 5 0!,where q 5 i, c, o, r. Now we let kr~x, y! be

kr~x, y! 5 kr~x, y 5 0!. (10)

Because kr~x, y! depends on only the x coordinate it isa conserving vector, so we can obtain a unique solu-tion for frd

~x, y! by using

frd~x, y! 5

2p

lo * kr~x, y!dx. (11)

Here the integration constant was set arbitrarily to 0.Then, in accordance with Eq. ~2!, we calculate thedesired phase of the object wave fod

~x, y! as

fod~x, y! 5 fhd

~x, y! 1 frd~x, y!. (12)

When the object and reference waves that have thephases given in Eqs. ~11! and ~12! interfere, the de-sired grating distribution is obtained, and the Bragg


condition is completely fulfilled along the line y 5 0.The deviation from the Bragg condition over the restof the area is quite small because of the reasons men-tioned above. Finally, these calculated wave frontsare transferred to the CGH plane by one of the wave-front transference methods.21

3. Calculated Results

To verify our design we calculated the diffractionefficiencies of several planar HOE’s that were re-corded with waves from CGH’s, i.e., computer-originated HOE’s. These calculations exploited theanalytical coupled-wave theory.9 Initially we con-sidered a planar holographic lens that operates in thegeometry depicted in Fig. 3. This lens transformsan on-axis spherical wave that originated from apoint source located at coordinates ~0, 0, z! into anoff-axis plane wave that is trapped inside the sub-strate by total internal reflection. The grating func-tion fhd

spherical of the lens has the form

fhd

spherical~x, y! 5 22p

lc@~x2 1 y2 1 z2!1y2 2 nx sin u#,

(13)

where z is the focal distance, n is the substrate’s indexof refraction, and u is the off-axis angle inside thesubstrate.

To illustrate our design procedure we chose thespecific example of the spherical grating functiongiven in Eq. ~13!. To record such a grating functionwe must solve Eq. ~7! explicitly to find frd

and fod.

We begin with a reconstruction spherical wave whosenormalized propagation vectors inside the substrateare

kxc5 2

xn~x2 1 y2 1 z2!1y2 ,

kyc5 2

yn~x2 1 y2 1 z2!1y2 ,

kzc5 ~1 2 kxc

2 2 kyc

2!1y2. (14)

The normalized propagation vectors of the desireddiffracted beam inside the substrate are

kxi5 sin u,

kyi5 0,

kzi5 ~1 2 kxi

2 2 kyi

2!1y2. (15)

Fig. 3. Computer-originated holographic optical element in a pla-nar configuration.

Now we insert Eqs. ~14! and ~15! into Eq. ~7! and letkyc

5 kyi5 0 to calculate ko~x, y 5 0! and kr~x, y 5 0!.

For convenience, let us define

S ; 2m~kxi2 kxc

!,

C ; 2m~kzi2 kzc

!. (16)

Thus Eq. ~7! simplifies to a set of two equations withfour coupled variables, given by

kxo2 kxr

5 S,

kzo2 kzr

5 C. (17)

Using the relations kzo5 =1 2 kxo

and kzr5 =1 2 kxr

and performing some mathematical manipulationsyield

kxr5

2b 1 Îb2 2 4ac2a

, (18)

where

a 5 4S2 1 4C2,

b 5 4S3 1 4C2S,

c 5 S4 1 C4 2 4C2 1 2C2S2.

By inserting the result obtained in Eq. ~18! into Eq.~11! we can find the phase of the reference recordingbeam Frd

, and then by using Eq. ~12! we find thephase of the object recording beam Fod

.We now design a CGH such that the holographic

lens will be recorded with lo 5 514.5 nm and recon-structed with lc 5 800 nm, the substrate index ofrefraction is n 5 1.5, and the off-axis angle is u 5 45°.We calculate the diffraction efficiency as a function ofthe coordinates x and y, as illustrated in Fig. 4. Inthe calculation we assume that the emulsion thick-ness of the recording material is 10 mm and that thefocal distance z of the lens is 10 mm. The resultsindicate that the relative diffraction efficiency is100% along the line y 5 0, as expected, and it dropsto approximately 95% at the edges. These results

Fig. 4. Calculated diffraction efficiency for a spherical lens thathas a grating function fhd

spherical.

are calculated for an aperture for which the lens hasan f-number of 1.66, which is quite small.

Then we compare the results from two representa-tive planar holographic cylindrical lenses. The firsthas a grating function fhd

cylindrical,x of the form

fhd

cylindrical, x~x, y! 5 22p

lc~Îx2 1 z2 2 nx sin u!. (19)

The second lens has a grating function fhd

cylindrical,y ofthe form

fhd

cylindrical, y~x, y! 5 22p

lc~Îy2 1 z2 2 nx sin u!. (20)

The first cylindrical lens transforms an on-axis cylin-drical wave, emerging from a line source parallel tothe y axis along the line ~0, y, z 5 10!, into an off-axisplane wave that is coupled inside the substrate bymeans of total internal reflection. The second cylin-drical lens transforms an on-axis cylindrical wave,emerging from a line source parallel to the x axisalong the line ~x, 0, z 5 10!, into an off-axis planewave.

The diffraction efficiencies of the cylindrical lensesare calculated with the same parameters that were

Fig. 5. Calculated diffraction efficiency for cylindrical lenses thathave grating functions ~a! fhd

cylindrical,x and ~b! fhdcylindrical,y.


used for the spherical lens above. The results ofthese calculations are presented in Fig. 5. For thegrating function fhd

cylindrical,x, where all the opticalpower is in the x direction, the relative diffractionefficiency is 100% over the entire area of the lens, asshown in Fig. 5~a!. On the other hand, for the grat-ing function fhd

cylindrical,y, where the optical power isonly in the y direction, the relative diffraction effi-ciency is 100% along only the y 5 0 line, and it dropssymmetrically with respect to the y axis to approxi-mately 95% at the edges, as can be seen from Fig.5~b!. This is so because the design method requiresthat the final grating fulfill exactly the Bragg condi-tion on the y 5 0 line. Thus, for the first cylindricallens whose grating function and grating vector do notdepend on the y coordinate, the Bragg condition isfulfilled over the entire area, whereas for the secondlens whose grating function and grating vector de-pend on x and y coordinates, the deviation from theBragg condition increases as a function of the dis-tance from the y axis.

Now we consider a planar holographic imaging lenswhose grating function has the form

fhd

imaging~x, y! 5 22p

lc$~x2 1 y2 1 z2!1y2

2 n@~x 2 xo!2 1 y2 1 zo

2#1y2%. (21)

This lens transforms an on-axis spherical waveemerging from a point source located at coordinates~0, 0, z! into a spherical wave that converges to apoint inside the substrate at coordinates ~xo, 0, zo!.For the calculations we use xo 5 zo 5 10 mm, so theoff-axis angle of the central diffracted ray is also 45°.The calculated results of the relative diffraction effi-ciency as a function of the lens aperture are shown inFig. 6. As is evident, the relative diffraction effi-ciency along the y 5 0 line is 100%, and at the edgesit drops to 98.5%.

4. Concluding Remarks

We have presented a novel method for designingcomputer-originated HOE’s that can be recorded with

Fig. 6. Calculated diffraction efficiency for an imaging lens witha grating function fhd

imaging.


a wavelength that is different from that used forreadout and still obtain exactly the desired gratingfunction that closely fulfills the Bragg condition overthe entire area of the HOE. This method is espe-cially appropriate for recording HOE’s that must op-erate at a wavelength for which there are no suitablerecording materials. Also, it is appropriate for pla-nar HOE’s whose spatial frequencies are high andmust be recorded with a wavelength that it is shorterthan that of readout.

The CGH’s from which the recording waves for thefinal HOE’s are derived are fabricated with the aid ofelectron-beam direct writing and advanced litho-graphic technology. The CGH for the referencewave is fabricated as a one-dimensional grating, i.e.,it contains only straight lines that have high spatialfrequencies. The CGH for the object wave front isfabricated as a grating with curved lines and rela-tively low spatial frequencies. Thus one can fabri-cate CGH’s with relative ease by exploiting currentelectron-beam and lithographic technologies.

This project was supported in part by the IsraeliMinistry of Science and Technology.

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computer-originated planar holographic optical elements

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