Photolithographic fabrication method of computer-generatedholographic interferograms

Markus Kajanto,Arto Salin

Eero Byckling, Juha Fagerholm, Jussi Heikonen, Jari Turunen, Antti Vasara, and

We consider the fabrication of high-quality interferogram-type diffractive optical elements with conventionalphotolithographic techniques and compare the results with those achievable with electron-beam lithography.The fringes associated with the phase transfer function of the binary phase holographic interferogram areapproximated with rectangles, which can be realized at submicron accuracy using a pattern generator and

step-and-repeat camera. The effects of the rectangle quantization are analyzed both numerically andexperimentally with the aid of diffraction patterns produced by simple focusing elements. Both resolutionand diffraction efficiency of the best holograms approach their theoretical values.

1. Introduction

Computer-generated holograms (CGHs) with opti-mized complex-amplitude transmission functions canperform rather general wavefront transformations.'Consequently, these diffractive optical elements havefound use in such diverse areas as display holography,optical information processing, aberration correctionin optical systems, optical interconnection, multiple-beam generation, angular spectrum shaping, laserscanning, optical testing, and pattern recognition (seeRefs. 1-7 and references cited therein). Traditionally,computer-generated holograms have been fabricatedusing either a CRT output or computer-controlled penand ink plotters (or, more recently, laser scanners8'9)for patterning, followed by photoreduction with anSLR camera lens. Problems associated with the limit-ed accuracy of those commonly used CGH fabricationmethods have largely been overcome by the use ofelectron-beam lithography,10 -'2 but the equipment re-quired is expensive and hence available at a few labora-tories only. A considerably lower cost alternative isprovided by the conventional photolithographic ICmask production techniques. We have applied these

Markus Kajanto is with California Institute of Technology, De-

partment of Applied Physics, Pasadena, California 91125; Arto Salin

is with Technical Research Center of Finland, Semiconductor Lab-

oratory, SF-02150 Espoo, Finland; and the other authors are withHelsinki University of Technology, Department of Technical Phys-ics, SF-02150 Espoo, Finland.

Received 11 January 1988.0003-6935/89/040778-07$02.00/0.© 1989 Optical Society of America.

techniques to fabrication of various types of diffractiveoptical element, including holographic lenslet andgrating arrays, computer-optimized aspheric lenses,13

and multiple-beam generators.14 In this paper, weconcentrate on the effects of the limitations of thephotolithographic apparatus in production of binaryinterferometric-type holograms.

A description of our photolithographic process, theapparatus used, and its accuracy is found in Sec. II. InSec. III, an efficient method is presented for findingthe fringes of a holographic interferogram defined byits phase transfer function, and for approximating thefringe structure by a network of optimally sized andlocated rectangles. The rectangle approximation in-herently leads to quantization; a universal measureA, termed the relative local phase error, is establishedfor the severeness of the quantization errors of thistype.

A numerical analysis of the effects of quantization isperformed in Sec. IV, using a simple off-axis focusinghologram as an example. Making use of the Fresneldiffraction theory, we estimate the dependence on A4Pof the spot size, diffraction efficiency, and the requirednumber of approximating rectangles. It turns out thatholograms exhibiting diffraction-limited resolutionand nearly theoretical diffraction efficiency can beproduced using a moderate number of rectangles, andhence at a comparatively low cost. The analysis ofSec. IV may, in a way, be viewed as a 2-D generalizationof the approach presented in Ref. 15 for estimation ofthe quantization errors occurring in electron-beamlithography.

In Sec. V, an experimental verification is provided ofthe ability of our fabrication method to produce high-quality diffractive optical elements. In the final sec-

778 APPLIED OPTICS / Vol. 28, No. 4 / 15 February 1989

tion, the merits and drawbacks of the photolithograph-ic technique, compared with electron-beam lithog-raphy, are discussed. It will be concluded that, exceptin applications requiring ultimate resolution, the pho-tolithographic method is entirely adequate.

II. Fabrication Method

A holographic interferogram is completely charac-terized by its phase transfer function d(x,y), the argu-ment of the complex-amplitude transmittance t(x,y)of the element. For a binary interferogram, the re-gions to be recorded if maximum diffraction efficiencyis desired, are the fringes of the hologram, i.e., thebands 2w(n - 1/4) < (x,y) < 27r(n + 1/4), where n is aninteger.

Under usual circumstances, the hologram fringesmay locally be viewed as arcs of circles, whose radii ofcurvature are large compared with the local fringeperiod. In this case, the fringes can be satisfactorilyapproximated by trains of rectangles, which are rela-tively long compared with their widths. Such rectan-gle trains can be economically recorded on a chromemask plate using a photolithographic pattern genera-tor. Since the minimum feature size achievable withconventional pattern generators is typically a few mi-crometers, and the fringe widths are usually of theorder of a few wavelengths of light, the mask platemust be photoreduced by a high-quality wafer step-per.16 A step-and-repeat camera also makes it possi-ble to repeat the same pattern to conveniently produceaccurate periodic structures.14 Furthermore, it allowsthe combination of several pattern generator masksinto a single large hologram.

The photoreduction may be done on a chrome plate(covered by photoresist) to yield an amplitude-modu-lated binary interferogram, with maximum diffractionefficiency of 10.1% into the first diffraction order. Toimprove this efficiency four times, the amplitude-typehologram can be converted into a phase-only surface-relief structure by employing suitable etching tech-niques, such as ion-beam milling. Alternatively, thephotoreduction can be done on a plate covered only bya layer of photoresist with thickness suitably chosen toresult in a phase delay of rad. This technique isextremely simple, but the resulting hologram must behandled with care since photoresist is not one of themost durable hologram recording materials. More re-sistant phase-only holograms can be made, for exam-ple, on thin silicon nitride films.

The GCA 3600 pattern generator, which we haveused, can record rectangles with dimensions rangingfrom 4m to 3 mm. The accuracy of these dimensionsis better than 0.5 gm, and the positioning error isbelow 0.3 gAm over an area of 5 X 5 cm2 . Photoreduc-tion (five times) with a high-quality step-and-repeatcamera PAS 2000A thus gives a submicrometer featuresize (0.8 gim). By careful exposure, the dimensionalerrors of the rectangles can be kept at about 0.1 gAm.Due to the very low distortion of the imaging lens, andthe air-bearing stage with laser interferometer servocontrol, the absolute positioning errors over a 12.5- X

12.5-cm2 plate are well below 0.3 gAm. System perfor-mance is hence adequate for most practical applica-tions, provided that the errors due to quantizationeffects caused by the rectangle approximation of thefringe pattern may be considered negligible.

Ill. Rectangle Approximation of the Fringes

The number of approximating rectangles in the ho-logram is an important factor, since it affects the timeneeded for recording the hologram by a pattern gener-ator, and thus also the fabrication cost. Cost increaseis not severe when the total number of rectangles V inthe quantized hologram structure is of the order ofthousands. In large holograms, however, NV can beconsiderably larger than in typical VLSI applicationsand become a serious consideration. On the otherhand, it is clear that the increase of the number ofrectangles yields a closer approximation of the fringepattern and thus, presumably, better performance. Itis not perfectly clear how severely the quantizationerrors affect the performance of the hologram, al-though estimates have been presented for the 1-Dcase.15 To show (in the next section) that surprisinglylarge quantization errors can be tolerated in most ap-plications, we now proceed by considering the calcula-tion of the optimal locations of the rectangles and byestablishing a suitable quality criterion for this kind ofa rectangle approximation.

The boundaries of the fringes, the lines c(Px,y) =27r(n 1/4), as well as the fringe center lines 4(x,y) =27rn, can be tracked by several different algorithms.The problem may, for example, be formulated as aninitial-value problem associated with an ordinary first-order differential equation. Due to the high numeri-cal precision required, we have chosen a more directprocedure outlined briefly below.

If the phase function gradient VI(x,y) has zeros inthe hologram area (such as the center of an on-axisFresnel zone plate), the hologram must be divided intosubregions Z that do not contain any of these zeros.Once this is done, all fringes that appear in any of the(sub)regions 1D must cross the boundary 13 of Z) at leasttwice. By iteration along 3 it is a simple matter tofind, for some n, the points where the lines 4(x,y) =27rn and 4(x,y) = 2(n 1/4) intersect 3. Startingfrom the intersection points, these lines can be fol-lowed step by step inside 1.) until the boundary 3 isagain reached. The next point on the line is searchediteratively by scanning directions close to that of thetangent of the line at the found point. In practice, itsuffices to trace points on the center line 4(x,y) = 2rnonly. Points on the other lines will then be found indirections V(x,y)(x0 y0), at distances approximatelyequal to d(xo,y0)/4, where the local fringe periodd(xo,yo) in the neighborhood of the point (x0 ,y0 ) isdefined as

d(xo,yo) = 211V4(xy)(x 0Y,)h1 . (1)

The above outlined tracking procedure can be repeat-ed for all fringes (i.e., values of n) in the region L),finding the starting point of the next fringe by travers-

15 February 1989 / Vol. 28, No. 4 / APPLIED OPTICS 779

(xy) = 27r(n+1)

S

Fig. 1. Criterion for determination of lengthL of a rectangle: /d is

equal to a predetermined constant.

ing the boundary 13 to either direction. Since a fringecorresponding to a given n may in general cross theboundary 13 more than twice, the end point of everytracked fringe should be stored to avoid confusion.

The step size used during the tracking proceduremust be relatively small compared with the sizes of thefinal rectangles. Each rectangle, can, however, be de-fined by giving the coordinates of three of its cornerpoints only. It thus saves computer resources to fit therectangles into each fringe immediately after its pathhas been tracked. Assuming that the fringes are arcsof circles, and that their separation remains constant,over the area occupied by one approximating rectan-gle, length L of any rectangle can be determined by thefollowing criterion, illustrated in Fig. 1. The deviation6, at a distance L, of the tangent S from the fringecenter line may not exceed a product ad, where a is apredetermined constant chosen small enough to en-sure that the assumptions made above are approxi-mately valid. From the geometry of Fig. 1, westraightforwardly obtain an approximate result:

L = [ad(ad + 2R)]" 2 . (2)

Here the fringe period d, defined by Eq. (1), can becalculated from the phase function, and the local radi-us of fringe curvature R can be estimated from threetracked points on the line d(x,y) = 27rn. In view of Eq.(2), a fine hologram structure (small R and/or d) im-plies a small value of L.

Once the length of a rectangle has been determined,an equation for its optimal location (lateral deviationD from the fringe center line) is obtained by straight-forward calculation from the geometry of Fig. 2. Theresult is

D = R[1 arctan L)]* (3)

The optimal width of the rectangle is naturally w = d/2. Using the geometry of Fig. 2, we introduce a quanti-ty Ai = 27rD/d, which describes the magnitude of therelative local phase error caused by replacement of acurved fringe section by a rectangle. To a good ap-proximation, AiD is equal to 27ra/12, and hence

a = 12A4/27r. (4)

If we choose a particular value of AcI for all rectanglesand use Eqs. (4), (2), and (3), sequentially, to calculatethe network of the approximating rectangles, the

L

/<

D

R

\ (x~y)=2(n-1/4p q(x.y)=27rn

\P(xy)= 27r(n+1/4)

Fig. 2. Determination of position S (and width w) of a rectangle;the shaded and dotted areas must be equal.

quantization error caused by each rectangle is practi-cally equal, and also independent of the particularform of the phase function. Thus, if we analyze theperformance of any hologram as a function of A-1, wemay confidently expect that the analysis is, to a goodapproximation, universally valid. For example, if acomplex aspheric holographic interferogram is used asan aberration compensator in an image-forming opti-cal system, the contributions of its quantization errorsto the image degradation can be estimated from theresults of the next section, where the effects of quanti-zation in simple focusing elements are analyzed.

IV. Numerical Error Analysis

The diffraction pattern in any transverse observa-tion plane behind an arbitrary hologram, which is com-posed of rectangles, can be straightforwardly calculat-ed by Fresnel diffraction theory. For each elementaryrectangle, complex field amplitudes are evaluated at agrid of points in the observation plane by the well-known formula 7 utilizing Fresnel integrals. The dif-ferent orientations of various rectangles are taken careof by simple coordinate transformations. The super-position principle of diffraction theory is applied tosum up the contributions from all the rectangles. Fi-nally, the total intensity distribution is found bysquaring the resulting field distribution at the obser-vation plane.

Although a NAG Library'8 subroutine evaluates aFresnel integral in just 10-gs CPU time in the IBM3090/150VF supercomputer, calculation of the diffrac-tion patterns of large holograms may be quite a formi-dable task. Holograms typically contain thousands ofrectangles, each of which requires 8MN evaluations ofa Fresnel integral, if we want MN resolution points inthe observation plane. However, with low focal ratiosthe number of rectangles and hence also the calcula-tion time are reasonable, and as discussed above, theresults obtained give good indication of the quality ofmore complicated holograms, which have the samevalue of the phase error A/1.

In Fig. 3 the quantized structure of a simple off-axisfocusing element with a paraxial phase transfer func-tion V(x,y) = k(x 2 + y2)/2F, where k = 27r/A, is dis-

780 APPLIED OPTICS / Vol. 28, No. 4 / 15 February 1989

Hotel

I

I----

4

.0

Fig. 4. Diffraction pattern of the structure shown in Fig. 3. Thescales are in meters.

Fig. 3. Structure of an F:50 (X = 633 nm) off-axis focusing elementwith allowable local phase error A4/2ir = 1/6.

played for A(/27r - 1/6. The focal length of the holo-gram is F = 100 mm, the size of the (square) aperture is2 X 2 mm, and the aperture is centered at a point (xoy o)= (1.5 mm,1.5 mm). Figure 4 shows the calculateddiffraction pattern corresponding to the hologram inFig. 3, evaluated at a distance F behind the hologramplane in 100 X 100 points. It is interesting to note that,although the quantization looks extremely rough, reso-lution (Rayleigh criterion) is, within the accuracy ofFig. 4, equal to the theoretical 62 gm (for X = 633 nm) ofan equivalent lens. This somewhat surprising resultshows that, if the rectangles are optimally located inthe sense discussed in the previous section, rather largelocal phase errors A can be tolerated without signifi-cant loss of resolution.

Rough quantization has, however, a noticeable ef-fect in diffraction efficiency: only 4.15% of the inci-dent light is diffracted inside the main lobe, while thetheoretical value is 8.23%. Had the calculation beenperformed for a phase-only hologram instead of anamplitude-type hologram, a fourfold increase of dif-fraction efficiency would of course have resulted.Clearly, much of the light is concentrated in the strongspurious spots located on both sides of the main lobe.Their peak intensity is 12% of the peak intensity ofthe main lobe. These ghosts, which look familiar tousers of ruled grating spectrometers, are caused by theclear vertical periodicity in the fringe structure. Thediffraction efficiency into the first diffraction order(the area covered by the 100 X 100 grid in Fig. 4) is9.32%, i.e., slightly lower than the theoretical efficiencyof 10.1%.

To avoid the unwanted periodicities, and hence theghost spots, we could use random reduction of therectangle lengths. However, it has proved better tointroduce randomness only in the length of the firstrectangle of every fringe, i.e., that touching the bound-

Fig. 5. Structure of an F:50 off-axis focusing element with random-ness in the length of the first rectangle of every fringe.

ary 13 of . In this way, the required number ofrectangles remains practically unchanged, and the pe-riodicities are in fact destroyed more completely. Theresult of randomness (the other parameters being un-changed) is shown in Fig. 5, and the correspondingdiffraction pattern is displayed in Fig. 6. Resolutionremains unaffected, as expected, and the diffractionefficiency inside the main lobe increases slightly, to4.46%. Most notable, of course, is the almost totaldisappearance of the ghost peaks. The diffractionefficiency into the grid area of Fig. 6 has been reducedto 7.39%. Randomness, which causes sharper holo-gram structure, thus tends to smooth the ghost spots,scattering light to form a weak widespread halolikebackground.

15 February 1989 / Vol. 28, No. 4 / APPLIED OPTICS 781

O 1.

IV

2 c

Fig. 6. Diffraction pattern of the structure shown in Fig. 5.

To increase diffraction efficiency into the main lobe,we clearly have to reduce the quantization error A4.In Fig. 7, we have chosen Ab/27r 1/40. Now therectangles approximate the fringes very well, and thediffraction pattern, shown in Fig. 8, looks like a puresinc2 distribution. Diffraction efficiency into themain lobe is as high as 8.16%, and in the grid area it is9.81%. The drawback accompanied with this im-provement is over a twofold increase in the number ofrectangles.

The structure in Fig. 3 (corresponding to A4P/27r = 1/6) clearly represents an upper practical bound for thetolerable quantization error. On the other hand, nosignificant improvement can be gained by reducing thephase error A4/27r below 1/40 (Fig. 7). By calculatingseveral hologram structures, diffraction patterns, andefficiencies, we obtain the following empirical depen-dence of the diffraction efficiency i7 on the quantiza-tion error, valid in the AV/27r < 1/6 range:

77 = 1 - 18.5(A(/2r) 2 ]%.

Here no = 10.1% for an amplitude-coded hologram, and1 = 40.5% for a binary-phase interferogram. Thecorresponding figures for the diffraction efficiency in-side the main lobe are no = 8.23% and no = 32.9%. Thelaw expressed in Eq. (5) is to a good approximationindependent of the form of the phase function. Itwould be useful to have a similar law for the depen-dence of the number of approximating rectangles JV onAd?, but On does depend on the form of the phasefunction.

V. Experiments

We have successfully fabricated a large number ofdifferent types of focusing elements, gratings, and ho-lographic interconnect elements. Experimental re-sults have been in excellent agreement with numericalpredictions; for example, the ghosts were clearly visi-ble in holograms of the type displayed in Fig. 3. To

Fig. 7. Structure of an F:50 off-axis focusing element with allowa-ble local phase error AP/27r = 1/40.

(5)

4o*10s4

Fig. 8. Diffraction pattern of the structure shown in Fig. 7.

establish the ability of the fabrication technique toproduce very high-quality diffractive optical elements,we present more detailed test results of the structureshown in Fig. 7.

Diffraction efficiencies of the fabricated hologramswere determined by illuminating the hologram with aplane wave and comparing the first-order diffractedintensity to the intensity of the beam transmitted byan equally sized fully transparent aperture, also fabri-cated with a pattern generator. Diffraction efficien-cies of holograms of the type shown in Fig. 7 agreed wellwith theory: the ratio 1 7measured - nnumericall/7theoreticalwas of the order of a few percent in all cases.

782 APPLIED OPTICS / Vol. 28, No. 4 / 15 February 1989

ol d

Fig. 9. Measured profile of the diffraction pattern of the structureshown in Fig. 7.

Profiles of diffraction patterns were measured byscanning a detector (placed behind a pinhole) over thepattern, which was first magnified with the aid of amicroscope. Perfectly diffraction-limited perfor-mance was observed; this is clearly evidenced by thescan shown in Fig. 9, in which deviations from the idealsinc2 distribution are below 2%. A slight ripple in thesidelobes is mostly due to coherent noise originatingfrom dust and minor imperfections in the testing sys-tem.

VI. Discussion and Conclusions

In this paper, we have aimed to demonstrate thatconventional photolithography can be applied to pro-duce high-quality computer-generated interferogram-type holograms at a relatively low cost. From bothnumerical and experimental results, we deduce thatthe inherent rectangle quantization does not apprecia-bly reduce resolution, provided that the approximat-ing rectangles are located optimally with respect to thefringes (Fig. 2); the quantization effects were seen onlyin the diffraction efficiency.

We emphasize once more that, although the numeri-cal and experimental results presented in this paperexclusively deal with focusing elements, the fabrica-tion and analysis techniques presented here are by nomeans restricted to this simple case. Rather generalcomputer-optimized phase transfer functions can berealized; one example appears in Ref. 13. Also, themethod is well suited for fabrication of holograms thatare inherently composed of rectangles, e.g., Lohmann-type display holograms and Dammann grating beamsplitters.' 4

Although electron-beam fabrication of computer-generated holograms has recently gained considerablepopularity due to the high precision available, it alsohas some drawbacks, and it is not yet clear what meth-od might be most generally useful for fabrication ofdifferent types of holographic element (see the discus-sion in Ref. 19). The clear advantage of E-beam lith-ography over our photolithographic technique is thatthe minimum feature size is an order of magnitudesmaller (0.1 gm vs 1 gm). This small feature sizepermits in principle fabrication of almost any holo-gram for visible and near-UV light, while our method isnot practical if the period is below 2-5 ,gm, dependingon the form of the hologram phase function. However,progress is still made in development of the photo-lithographic apparatus, and linewidths of 0.4 gim willbe feasible in the near future.2 0 An important consid-eration is also the distortion in patterns written by theE-beam system. This distortion is typically a consid-erable portion of the wavelength of light.10 The ef-fects of plotting distortion in the quality of the imageformed by the hologram have, to our knowledge, notbeen fully investigated, but it can be anticipated thatthese effects narrow the difference in performancebetween photolithographic and E-beam techniques.Finally, at submicron feature size levels, photoresistsare superior to electron resists in terms of the MTF.20

This work was partially funded by the Academy ofFinland and Tekniikan Edistamissaatio, Finland. In-quiries for reprints and further information should beaddressed to A. Vasara.

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319 (1987).

N. Streibl of the Physics Institute of the University of Erlangen-Nuremberg, photographed by W. J. Tomlinson, of Bellcore, at theInternational Conference on Optical Nonlinearity and Bistability of

Semiconductors, August 1988.

784 APPLIED OPTICS / Vol. 28, No. 4 / 15 February 1989