Highly Corrected Submicrometer Grid Patterning on Curved Surfaces

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Highly corrected submicrometergrid patterning on curved surfaces

Kenneth M. Baker

A compact holographic projector system was built and tested. This projection system offers a practicalapproach for making a highly corrected mesh or grid pattern on curved surfaces. The pattern can rangein size from multimicrometer to submicrometer dimensions and be recorded in either positive or negativephotoresist. Standing-wave interference patterns in the form of a diverging close-packed lattice of eitherhexagonal or square rodlike intensity maxima extending outward from a point or a locus of points areproduced by multiple-beam holography that involves the combination of a holographic diffraction gratingand a hypercomatic focusing objective. © 1999 Optical Society of America

OCIS codes: 110.3960, 090.2880.

1. Introduction

Grid patterning has been used widely in optics re-search and applications to make Bragg diffractiongratings, fan-out diffraction gratings,1 computer-generated holograms,2–4 kinoforms,5,6 phase-shiftmasks,7–9 and both bandpass and band-stop inter-ference filters. Grid patterning is also found in bi-nary phase gratings, which are used to generatebeam arrays for holographic interconnections in free-space photonic switching systems that optically in-terconnect large arrays of processing elements10 andto couple light coherently from a laser array into asingle on-axis beam.11 Specifically, submicrometer

atterning, when it is below the wavelength of thelluminating light, has been implemented asubwavelength-structured12 ~SWS! surfaces. SWSurfaces, also known as zero-order gratings, are usedor antireflection, fabrication of polarization compo-ents, narrow-band filters ~including color filters!,nd phase plates. SWS surfaces are also used onire-grid polarizers,13 beam splitters,14 wave plates

and retarders,15,16 and polarizing mirrors.17 In na-ture, SWS surfaces are responsible for the colorfulreflective structures decorating the wings of somebutterflies,18 and antireflection- ~AR-! structured sur-faces are found on the corneas of certain night-flying

The author is with Optimerix Company ~818-782-8718!, 13659ictory Boulevard, Van Nuys, California 91401-1735.Received 18 May 1998; revised manuscript received 9 September

998.0003-6935y99y020339-13$15.00y0© 1999 Optical Society of America

moths to reduce surface reflections that would other-wise betray the moth to its predators.19 Impor-tantly, AR-structured surfaces can be used instead ofAR coatings either for use with wavelengths forwhich ideal coating materials may not be available orfor applications such as space optics for which adhe-sion and thermal-expansion mismatches would makeAR coatings a liability.

Within these and other applications there has beena growing critical need for grid patterning on design-specified curved surfaces. Most gratings are fabri-cated on flat surfaces because they are derived frome-beam or laser-writer lithographic techniques.This has posed a formidable design barrier becausepatterns made on flat e-beam or laser-written platesare not transferable to curved surfaces.

For microstructure fabrication on curved surfacesholographic methods offer perhaps the best approach.A holographic process in which a laser beam isdivided into either three or four parts and then re-combined has resulted in grid patterning of interfero-metric fringes, and the method is applicable forexposure on curved surfaces.20–24 This type of holo-graphic exposure system that uses the three-beampattern, in which the beams emit from a source con-figuration as the apices of an equilateral triangle, wasused in a classified U.S. Army project at BattelleMemorial Institute and was purported to be the onlymethod known to produce grid patterning for pat-terned induced-transmission filters as nose cones forlaser-guided missiles.25 The four-beam pattern, inwhich the beams emit from a source configuration asthe corners of a square, was demonstrated for use inthe semiconductor industry.26 The equiangular

10 January 1999 y Vol. 38, No. 2 y APPLIED OPTICS 339

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four-beam patterning method was also proposed forfield emission-display applications and was com-pared with several multibeam, multiexposure meth-ods and also a five-beam system, which is similar tothe equiangular four-beam system except than anextra beam is added in the center.27

Another system similar to these systems was usedto produce both the equiangular three-beam patternand a five-beam pattern in which the beams are con-figured as the apices of a regular pentagon for inves-tigation of the attraction forces of optical fields.28

However, in these endeavors the widely separatedcomponents and the long beam path lengths of theexposure system render it inherently unstable. Thesetups are tedious and time consuming. Moreover,the exposure system contains no correcting factors tooffset the stretching of the holographic pattern fieldwhen it is applied to a curved photosensitive surface.In this case the patterning becomes elongated in thedirection of curvature. A system for contouring theholographic pattern field around curved surfaces tocorrect for the stretching of the pattern is describedherein.

2. Pattern Generation

In previous publications,29,30 the author introduced aompact holographic projector system ~see Fig. 1!hat employs a holographic diffraction grating fol-owed by a focusing objective and a tail-endorrective-optics section ~see Fig. 2! that producedighly corrected hexagonal or square grid-pattern ar-ays on or in curved-substrate surfaces. The sys-em, which was illuminated by an UV argon-ion laserperating at 363.789 nm, was used to produce direc-ional light filters composed of high-aspect-ratio azo-ye-formed microhoneycomb images within opticalesins. The design not only allowed for changes inhe direction and the curvature of the pattern field,hich could be projected outward from either a singleoint or a locus of points, but also corrected for theoreshortening of the interferometric angles towardhe edges of the field, which would otherwise cause atretching or a pincushion distortion in the grid pat-ern. This distortion occurs because the smaller thenterferometric angles are, the bigger the patternets. However, the basic design limited the down-izing of the pattern periodicity to a dimensional sizef several micrometers at a close-in distance adjacento the outer surface of the last lens element because

Fig. 1. Previously developed hologra

40 APPLIED OPTICS y Vol. 38, No. 2 y 10 January 1999

he interferometric ray angles were restricted to onlyfew degrees. Scaling the lens prescription either

arger or smaller does not change the pattern sizeroduced at the outer surface of the last lens elementecause the interferometric angles always remainhe same. Scaling the lens prescription changesnly the radius of curvature of the holographic pat-ern field at any same proportionate distance out-ard from the projector system. To bring theattern periodicity to micrometer or submicrometerimensions, it is necessary to have relatively largenterferometric angles. At the laser wavelength of63.789 nm, as was used for this project, the inter-erometric angle must be larger than approximately0.96° for the fringe periodicity to be in the submi-rometer range.

It is noteworthy to mention here a method that wasecently developed at Cornell University31,32 for in-reasing the size of the interferometric angles fromiffraction gratings. The method is used to pro-uce high-resolution submicrometer patterns. Theethod, when used on flat surfaces, utilizes sub-

trates bearing diffraction gratings on the upper andhe lower surfaces that, when laser illuminated, re-pond with cascaded beam diffractions. The inter-erence patterns obtained from selected diffractiverders can be used for the fabrication of gratings

projector system, shown end to end.

Fig. 2. Tail-end section of the holographic projector system of Fig.1. Only two off-axis beams are shown.

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employed in distributed-feedback lasers, distributedBragg reflected lasers, grating-coupled surface-emitting lasers, wavelength-division multiplexers,and wavelength-division demultiplexers.

Rather than using the cascading approach pre-sented in Refs. 31 and 32, the author opted for analternative compact holographic projector system de-sign33 that preserves all the pattern-corrective fea-tures of the former curved-field design. In addition,it allows for patterning from multimicrometers downto submicrometer in size.

A. New Holographic Projector System

A Coherent Enterprise Model 652 UV argon-ion laserwas selected for this study. This model is unpolar-ized in that it lacks Brewster windows, having only ahigh reflector at one end of the plasma tube and anoutput coupler at the other end. This type of laserwas used to avoid having an output beam where therays have the same directional polarization orienta-tion, as is the case with lasers with Brewster win-dows. The amount of interference of two interferingrays of light whose polarization orientations lie in thesame plane of polarization greatly decreases as theinterferometric angle gets larger and larger, until, at90°, there is no interference ~cos 90° 5 0!. After 90°,s the interferometric angle becomes even larger, thenterference comes back until, at 180°, there is fullnterference ~cos 180° 5 1!. It is not necessary forhe intersecting angle of two light rays to be at 90° forhis lack of interference to occur: It will occur when-ver the electric-field vectors of the rays become or-hogonal to each other, regardless of the angle ofntersection. There is no way for an equiangularhree-beam or equiangular four-beam configurationo be oriented so that the beams all share the sameingular polarization orientation and still create uni-ormity of interference among the beams. As a gen-ral rule, with interferometric angles smaller thanpproximately 10° the systems can be treated in acalar sense, with polarization ignored.The laser beam, which first may be put through a

imer-actuated mechanical shutter, was directed into aeam-expander–beam-collimator containing a pinholepatial filter. The beam-expander–beam-collimator,s shown in Fig. 3, is an exact scale up ~see Table 1! ofNewport Corporation Tropel UV Model T27-100-150,xcept for the T27-6 two-element input objective thatas obtained from Newport Corporation.

Fig. 3. New holographic pro

The next element, a custom-made narrow band-ass filter ~Andover Corporation!, is used to filter outhe 351.112-nm line and the two weak lines at51.418 nm and 357.661 nm while allowing the63.789-nm line to pass through. This last line con-ains approximately half the power of the laser, orpproximately 50 mW. The output beam is thenassed through a pair of plano-aspheric calcium flu-ride lenses made by means of diamond turning.hese elements constitute a beam-profile re-haper34,35 and are used for transversely redistribut-

ing the nonuniform Gaussian ~TEM00! intensityprofile of the beam to achieve substantially even grid-pattern illumination at the exposure surface of thephotoresist-coated substrate, which is placed in anindex-matching fluid within the space immediatelyafter the hypercomatic objective. The beam exitingthe profile reshaper is approximately 150 mm in di-ameter and is put directly into a holographic diffrac-tion grating that sits at the aperture stop ~pupil

lane! of the hypercomatic objective.

. Holographic Diffraction Gratingsigures 4~a! and 4~b! show the three-beam and the

our-beam beam-splitting holographic diffractionrating designs,29 respectively, used in the holo-

graphic projector system. Both grating designs emitsets of spectral orders in which the first-order beamsare the most prominent. The first-order beams di-verge equiangularly and are of the same size andshape as the input beam but individually are ofequally divided lesser intensity. These types of ho-lographic diffraction gratings are also known as fan-out elements or Fourier phase-array generators.1The three-beam grating emits beams that have across-sectional configuration as the apices of an equi-lateral triangle, and the four-beam grating emitsbeams that have a cross-sectional configuration asthe corners of a square. The grating period is thewidth between any two parallel sides of the unit cellsshown in heavy outline in both designs. The rela-tion between the grating period and the emittedbeam angles is expressed by

d 5ml

sin um, (1)

um 5 sin21 ml

d, (2)

r system, shown end to end.
jecto

Table 1. Lens Data for the Illuminator Section of the Holographic Projector System when Configured End to End and Scaled to 150 mm Outa

11111

A

i

m

3

Current Lens Surface withRespect to the Object ~.OBJ!

Radius of the Surfacein the y–z Plane ~RDY!

Thickness of the Lensand Spacing ~THI! Type of Glass ~GLA! Lens Use

RefractiveIndexb

Object Infinity InfinitySTO ~STOP! Infinity 25.4000002 7.50000 3.220000 Silica ~SPECIALc! T27-6 Objective 1.4747243 226.30000 0.130000 T27-6 Objective4 3.25000 1.910000 Silica ~SPECIAL! T27-6 Objective 1.4747245 7.50000 3.544150 T27-6 Objective6 Infinity 673.6637057 22086.30721 20.662448 Silica ~SPECIAL! 1.4747248 1139.62211 14.0624649 1622.30214 28.124929 Silica ~SPECIAL! 1.4747240 2490.03001 1.8749951 Infinity 22.237444 Silica ~SPECIAL! 1.4747242 2674.97954 0.0000003 Infinity 47.6248794 Infinity 12.700000 BK7 ~Schott! Bandpass Filter 1.536489

15 Infinity 47.624879 Bandpass Filter16 Infinity 28.124929 CaF2 ~Special! 1.44501617 937.497625 937.49762518 625.40251 28.124929 CaF2 ~Special! 1.44501619 Infinity 25.400000Image ~IMG or LAST! Infinity 0.000000

dditional Data on Lens Surfaces 17 and 18

Lens Surface Type of Lens Conic Constant K K Control Code ~KC!d Intersect Code ~IC!eFresnel Curvature

Flag ~CUF!fFresnel Curvature Control

Code ~CCF!g

17 Asphere 20.788480 100 YES 0.000000 10018 Asphere 20.734690 100 YES 0.000000 100

Aspheric Coefficientsh A, B, C, and D and Codes AC, BC, CC, and DC for Lens Surfaces 17 and 18

Lens Surface

Coefficient Coefficient Code

A B C D AC BC CC DC

17 20.700502 3 1026 0.864616 3 10210 0.000000 3 100 0.000000 3 100 100 100 100 10018 0.682884 3 1028 20.353346 3 10211 0.000000 3 100 0.000000 3 100 100 100 100 100

Aperture Data: Cir, circular; Si, surface i.

Aperture Aperture Type Radius ~mm! Aperture Aperture Type Radius ~mm!

CIR S2 Clear aperture ~CA! 2.000000 CIR S14 Clear aperture ~CA! 100.000000CIR S3 Clear aperture ~CA! 2.000000 CIR S16 Clear aperture ~CA! 85.000000CIR S4 Clear aperture ~CA! 2.000000 CIR S17 Clear aperture ~CA! 85.000000CIR S5 Clear aperture ~CA! 2.000000 CIR S18 Clear aperture ~CA! 85.000000CIR S7 Clear aperture ~CA! 100.000000 CIR S19 Clear aperture ~CA! 85.000000CIR S8 Clear aperture ~CA! 100.000000 CIR S16 Clear aperture ~CA! 100.000000CIR S9 Clear aperture ~CA! 100.000000 CIR S17 Edge aperture ~EDG! 100.000000CIR S10 Clear aperture ~CA! 100.000000 CIR S18 Edge aperture ~EDG! 100.000000CIR S11 Clear aperture ~CA! 100.000000 CIR S19 Edge aperture ~EDG! 100.000000CIR S12 Clear aperture ~CA! 100.000000

Illuminator Specification DataEntrance pupil diameter ~EPD! 0.76676mm Wavelength ~WL! 363.789 nmInfinite Conjugates Paraxial Image

Effective focal length ~EFL! 223.5276 mm Height ~HT! 0.0000 mmBack focal length ~BFL! 285586.9006 mm Angle ~ANG! 0.0000°Front focal length ~FFL! 21.0190 mm Entrance Pupilf-Number ~FNO! 291.5224 Diameter ~DIA! 0.7668 mmImage distance ~IMG DIS! 25.4000 mm Thickness ~THI! 0.0000 mmOverall length ~OAL! 1896.5274 mm Exit Pupil

Diameter ~DIA! 8.1541 mmThickness ~THI! 283209.7851 mm

aThe labels used in this table correspond to those used in CODE V, e.g., STO refers to the first, or STOP, lens surface after the image, thenumbers in column one under object designate the front and back surfaces of each lens element, and LAST refers to the last surface or themage. When labels have been used instead of the CODE V abbreviation, the abbreviation is included in small capitals following the label.

bRefractive indices are given at the design wavelength of 363.789 nm.cSPECIAL indicates that the material used is not in a specific CODE V manufacturer’s catalog but is in the CODE V special catalog foriscellaneous glasses and other optical materials.dThe K control code refers to whether K is a variable when optimized; a value of 100 means that it is not and a value of 0 that it is.eFor the intersect code, a YES entry indicates that the standard intersect code, in which the intersection is the one closest to the surface

vertex, was used, a NO entry indicates that the other intersection was used.fFor the CUF, a nonzero value indicates a Fresnel surface.gThe CCF refers to whether the CUF is a variable when optimized; the values and their meanings are the same as those for the K control code.hThe coefficients A, B, C, and D are the fourth, sixth, eighth, and tenth polynomial aspheric sag coefficients, respectively, to the conic

constant. The codes AC–DC, like the K control code, indicate whether the coefficients are variables when optimized; the values and theirmeanings are the same as those for the K control code.


dl

ltdaicwotddsbe

s

where d equals the grating period, um equals thediffraction angle of the mth-order beam, m equals the

iffraction order, and l equals the wavelength of theight.

The three-beam design of Fig. 4~a! has four phaseevels and a diffraction efficiency of 0.2500 for each ofhe three first-spectral-order beams. Its zero-orderiffraction efficiency is 0.0625. The phase depthsre 0 5 0, 1 5 2py3, 2 5 p, and 3 5 5py3, which aren terms of radians, or, if divided by 2pyl, they areonverted to 0 5 0, 1 5 ly3, 2 5 ly2, and 3 5 5ly6,hich is in wavelength, where l is equal to one wavef retardation. This wavelength depth is known ashe optical depth. For calculation of the actual etchepth for a specified substrate, the optical depth isivided by n 2 1, where n is the refractive index of theubstrate at the operational wavelength. The four-eam design of Fig. 4~b! has two phase levels andliminates the zeroth order.35,36 It has a diffraction

efficiency of 0.164256 for each of the four first-spectral-order beams. The phase levels in radiansare 0 5 0 and 1 5 p.

For accommodating the 150-mm-diameter inputbeam the holographic diffraction gratings are madeon large synthetic-quartz plates ~Hoya, Type7.25R25!. The circularly shaped written gratingarea has very little clearance at four points becausethe limit on the industry standard e-beam machine is

Fig. 4. Holographic diffraction gratings that ~a! emit three equi-angular spectral order beams and ~b! emit four equiangular spec-tral order beams.

approximately 152 mm. The four-phase-level grat-ings are usually written with one e-beam step andone optical ~laser-writer! step for reasons that mostlyhave to do with the difficulty of putting e-beam resistback onto a plate after the first etch. The registra-tion generally is not degraded when writing a largeplate with a small address size. The e-beam widthfor these large plates should be at least 0.5 mm, whichleaves a practical feature-limitation size of approxi-mately 3 mm. Reactive-ion etching is probably bet-ter for smaller features, but either reactive-ionetching or ion milling can be used.

2. Hypercomatic ObjectiveFigure 5 shows the hypercomatic objective with a raytrace of two off-axis input beams at 61.0°. The sub-strate to be exposed is placed just to the right of theregion of ray convergence, where the rays diverge.The anterior surfaces of the second and the third lenselements are both aspheric, with the third elementbeing a conic section asphere with a convex ellipticalsurface ~see Table 2!. These elements are now fab-ricated easily from UBK7 optical glass by a relativelyrecent commercial process known as diamond grind-ing.37 In this process a diamond-turning machine isused but with a special diamond-grinding tool ratherthan a single diamond point. This tool causes a scal-loping effect on the grinding surface of the glass thatproduces microscopic pits that require careful post-polishing for removal. Because of the periodicitythat is due to the diamond-turning machine, even asurface roughness of 40-nm rms can cause visibleFresnel ring patterns to emanate from a finished lenswhen it is illuminated with blue laser light. An al-ternative optical glass aspheric-lens fabrication pro-cess38 that uses a different grinding system tends tobe zone free, which simplifys the postpolishing andthus may be preferable.

In the lens data listed in Table 2, the last lenselement appears to be a matching-sided triplet of lenselements all composed of K10. The middle elementis actually a refractive-index-matching fluid, and itsindex is matched to that of the 1.0-mm-thick window~which follows! and the index of the immersed sub-trate ~which is photoresist coated!. For the fabri-

Fig. 5. Hypercomatic objective with two beams at 61°.


Table 2. Lens Data for the Hypercomatic ~Immersion! Objectivea

3

cation of grid patterns on convex surfaces, the convexsurface of the substrate is coated with photoresist,and the substrate is exposed through its base. Be-cause of the design of the hypercomatic objective, a

Current Lens Surface withRespect to the Object ~.OBJ!

Radius of the Lens inthe y–z Plane ~RDY!

Th

Object InfinitySTO ~STOP! Infinity2 374.440103 2311.612614 63.845765 231.732576 58.248257 66.816908 19.561299 20.00000

10 InfinityImage ~IMG! Infinity

Additional Data on Lens Surfaces 4 and 6

Lens Surface Type of Lens Conic Constant K K Control Cod

4 Asphere 20.199721 1006 Conic section 20.133709 0

asphereAspheric Coefficients A, B, C, and D and Codes AC, BC, CC, and D

LensSurface

Coefficients

A B

4 0.000000 3 100 2313517 3 10211 0.00Illuminator Specification Data

Entrance pupil diameter ~EPD! 150.000Field angles in the x direction ~XAN! 0.000Field angles in the y direction ~YAN! 0.250Upper vignetting factor in the y direction ~VUY! 0.000Lower vignetting factor in the y direction ~VLY! 0.000Infinite Conjugates

Effective focal length ~EFL! 63.494Back focal length ~BFL! 1.000Front focal length ~FFL! 60.889f-Number ~FNO! 0.277Image distance ~IMG DIS! 1.000Overall length ~OAL! 155.419

aThe labels and names used in this table correspond to those uincluded in parentheses following some entries. See Table 1 foot

bRefractive indices are given at the design wavelength of 363.78


wide variaty of indices can be used for the index fluid,the window, and the photosensitive substrate withlittle appreciable effect on the pattern. Thus onecan use crown glass such as BK7 for the substrate

ess of the Lenspacing ~THI! Type of Glass ~GLA! Refractive Indexb

Infinity0.0000008.000000 UBK7 ~Schott! 1.5364459.8373720.015398 UBK7 ~Schott! 1.5364450.1000005.923614 UBK7 ~Schott! 1.5364450.1000001.443147 K10 ~Schott! 1.5237650.000000 K10 ~Schott! 1.5237651.000000 K10 ~Schott! 1.5237650.000000

! Intersect Code ~IC!Fresnel Curvature

Flag ~CUF!Fresnel CurvatureControl Code ~CCF!

YES 0.000000 100

Lens Surface 4

Coefficient Code

D AC BC CC DC

3 100 0.000000 3 100 100 0 100 100

m Wavelength ~WL! 363.789 nm0.00000° Paraxial Image

20.25000° Height ~HT! 0.1818 mm0.00000 Angle ~ANG! 0.2500°0.00000 Entrance Pupil

Diameter ~DIA! 150.0000 mmm Thickness ~THI! 0.0000 mmm Exit Pupilm Diameter ~DIA! 102.6521 mm

Thickness ~THI! 44.4526 mmmm

n CODE V; for clarity, abbreviations ~in small capitals! have beenfor particular definitions and explanations of values..

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and the window by matching the index fluid at the363.789-nm wavelength. At the index shown,Cargille Laser Liquid Code 561039 works very wellbecause it has a siloxane composition and does nottend to solvate a coating of the photoresist. It iscustom matched to the index of K10 at 363.789 nmand has approximately 96% transmission through a1.0-cm path at this wavelength. @K10 was chosenbecause many of the experimental substrates weremade from CR-307 plastic resin ~PPG Industries,Inc.!, which has a refractive index of 1.5223 6 0.0005at this wavelength.#

Depending on the photoresist used, an index mis-match between it and the substrate can occur, givingrise to reflections that result in an orthogonalstanding-wave pattern within the photoresist layer.40

This pattern, with planes of maximum and minimumintensity parallel to the substrate surface, oftencauses the photoresist to develop unevenly throughthe nodal planes, which leads to ragged line edges.The remedy for the uneveness is either to keep thethickness of the photoresist to less than half the pe-riod of the orthogonal standing wave or to apply aprecoat of an AR coating before applying the photore-sist.

The anterior side of the window is coated with op-tical blacking, except for a round hole in the coatingthat acts as an aperture for blocking out the un-wanted second- and higher-order diffractions fromthe holographic diffraction grating. On the window,in the center of this aperture, is a small dot of opticalblacking for blocking out the zeroth order. The dotis small enough that the rays of the caustic evenlysurround and bypass it. For direct surface exposureof a grid pattern, such as on the concave side of asubstrate coated with photoresist, the index fluid andthe window can be omitted altogether, and the sys-tem can be used in air, because again there is littleeffect on the pattern. In this case, a substrate ap-erture is mounted in place, and a thin protrudingpiece of opaque material can be used to block out theunwanted zeroth order.

The hypercomatic objective is designed so that anaberration of coma forms a somewhat circular enve-lope or boundary on the optical-axis side of the group-ing of ray fans in the vicinity of the image withinwhich there is no radiation. This cross section of thecaustic is shown in enlarged detail in Fig. 6~a!; it ishe result of two off-axis input beams at 61°. Figure

6~b! shows the cross section of the caustic formed bytwo input beams at 62°, and Fig. 6~c! shows the crossection of the caustic formed by two input beams at4.0°.The way the light rays form the near-circular cross

ection of the caustic with progressive tangentsround its curvature provides a system for correctinghe foreshortening of the interferometric angles atff-axis points toward the edges of a near-sphericalide-angle field. By modifying the shape of the

austic through a change in the design of the lenslements, one can create a wide variety of curvedattern fields, so for a given curved substrate there

will be substantially constant interferometric anglesat various points subtended from the substrate.

If the cross section of the caustic is optimized to acircular shape the bisectors of the interferometric an-gles will all intersect at a common point at the centerof the caustic. In the three-beam or the four-beamsystem the bisectors correspond to an imaginary linefollowing along the axial center of each three- or four-ray interferometric light bundle and passing throughthe common intersection point. However, if thecross section of the caustic is elliptical these imagi-nary lines, corresponding to the bisectors, will inter-sect at a locus of points that will distribute along theoptical axis.

For beam input angles starting at approximately61.0° and larger not all the extreme rays get throughthe hypercomatic objective. This is not due to a fail-ure of the optical design but is a result of beam over-lap. The parts of the beams that overlap theperiphery of the field and do not cross the otherbeams are noninterferometric and are thus not use-able. As the input beams go from 61.0° to largerangles, there is more and more of a trade-off tosmaller pattern size for less field angle.

As the angle of the off-axis input beams increases,the angular amount of peripheral noninterferometricbeam overlap increases, and the addition of this an-gular amount on opposite sides ~including the angleetween the input beams! subtracts from the angularmount of the interferometric or working field that isvailable. This is why it is useful to start with aide-angle optical design. In the three-beam sys-

em, because it does not have bilateral symmetry, theubtracted amount is equal to double the noninter-erometric beam overlap on one side along with dou-le the off-axis angle for a single input beam.The aperture, which is used to block out second-

nd higher-order diffractions, encircles the caustic atclose distance. With the dot of optical blacking in

ts center, it constitutes an annular aperture. Itsiameter must be changed according to the gratingeriod of the holographic diffraction grating, which inurn controls the size of the caustic. To accomplishhe blocking of the second-order diffractions, it is nec-ssary for the aperture to block out some first-orderxtreme rays. Generally, however, these rays areoninterferometric rays.The relation between the angle of convergence of

wo interfering light beams and the fringe spacing inhe resultant standing-wave interference pattern isiven by

C 5 sin21 l

2D, (3)

D 5 ly2 sin C, (4)

where C equals the half-angle of convergence be-tween the incident beams, D equals the fringe spac-ing, and l equals the wavelength of the light.

The use of a holographic diffraction grating of thedesign shown in Fig. 4~a! produces a hexagonal grid


3

pattern as a result of parallel fringe patterning run-ning in three directions, with each direction rotated120° from each of the other two @see Fig. 7~a!#. Theinterference combination of the three first-orderbeams causes periodic regions of destructive and con-structive interference along the fringes. With agrating period of 5.196 mm, the width between theparallel fringes is approximately 1 mm at a distanceoutward along the optical axis of 13.81 mm from thecenter of the caustic. This corresponds to a center-to-center distance of approximately 1.155 mm be-tween the regions of intensity maxima in the two-dimensional hexagonal grid-patterning plot of Fig.7~a!. Therefore the center-to-center distance isequal to 2Dy=3, where D 5 ly2 sin C and C is thehalf-angle between any two of the three beams. The13.81-mm distance was determined by a ray trace ona CODE V lens-design program ~Optical Research As-sociates!.

The use of a holographic diffraction grating of thedesign shown in Fig. 4~b! with a grating period of 6mm produces parallel fringes with a width betweenthe fringes of approximately 1 mm at the same dis-

Fig. 6. Caustic formed with ~a! two beams at


tance outward along the optical axis of 13.81 mmfrom the center of the caustic if only a diagonal pairof the four first-order beams is used. The inputbeam angles are approximately 63.47606°. Whenall four beams are exposed simultaneously, interfer-ence effects among the beams cause regions of inten-sity maxima to appear along only alternate rows andalternate columns of the fringes. However, becauseof interference from the diagonal beam pairs, a regionof intensity maxima appears in the center of eachsquare of four intensity-maxima regions. Because ofthis center intensity-maxima region, the rows andthe columns of the pattern line up in directions ro-tated by 45°. Therefore the four interfering beams,which emit from a configuration as the corners of asquare, produce a pattern whose geometry is rotated45° to this square. The center-to-center distance be-tween the regions of intensity maxima in this case is=2 mm, which is derived from =2 D, where D 5 ly2sin C and C is the half-angle between the diagonalbeams. The pattern is slightly larger in periodicitythan the pattern that would be obtained in aphotoresist-coated substrate if an exposure were

, ~b! two beams at 62°, ~c! two beams at 64°.
61°

d

3T6atlfptrsEfibsrtt

glmrmthtf

s

i

made with only a diagonal pair of beams followed bya second exposure with the alternate diagonal pair ofbeams. Here the center-to-center distance betweenthe regions of intensity maxima would be approxi-mately 1 mm. For a more complete discussion of this

ifference in center-to-center distances see Ref. 27.

. Field-Curvature Calculationshe cross sections of the caustics shown in Figs. 6~a!–~c! have elliptical shapes with their minor axes lyinglong the optical or z axis. In three-dimensionalerms they form parts of oblate spheroids. The se-ected holographic pattern fields with constant inter-erometric angles, and thus evenly sized gridatterns, will, in three dimensions, have contourshat are also shaped like surface sections of ellipses ofotation. In this case they will be sections of prolatepheroids with their major axes lying along the z axis.ach shape of a prolate spheroid section can be de-ned in two dimensions as a section of a large ellipsey a rotating isosceles triangle ~see Fig. 8! that inter-ects the caustic ~smaller ellipse!. As the triangle isotated, the point of intersection changes, and theriangle must be shortened to maintain contact withhe ellipse. Because the nonbase angle of the trian-

Fig. 7. ~a! Two-dimensional plot of the equiangular three-beamnterference pattern. ~b! Inverted isometric view of the equian-

gular three-beam interference pattern.

le remains unchanged, the shorter triangle is simi-ar to the original one. By knowing the new

easure of the base of the triangle, one can find theatio of the original to the shorter triangle. The neweasure of the triangle’s base is twice the radius of

he smaller ellipse at the angle of interest. Theeight of the triangle, which is the trace of the ellipsehat is formed by the equiangular segments, is thenound.

The equation for an ellipse in polar coordinatesuch as the one shown in Fig. 8 is

r~u! 5 FScos2 u

a2 1sin2 u

b2 D21G1y2

. (5)

When the values for u range from 0° to 360°, theentire ellipse is defined. Twice the measure of r~u!at u 2 90° is the base of the smaller isosceles triangle,and the ratio of the old base length to the new baselength is defined as K, so

K 52r~u 2 90°!

s1, (6)

where s1 is the measure of the original base of theisosceles triangle. The ratio of the new length to theoriginal length will also be K, so, if d1 is the originallength, the new length R is

R~u! 5 Kd1. (7)

Substituting Eq. ~6! for K into Eq. ~7! results in

R~u! 52r~u 2 90°!

s1d1. (8)

Substituting Eq. ~5! for r into Eq. ~8! yields

R~u! 52d1

s1HFcos2~u 2 90°!

a2 1sin2~u290°!

b2 G21J1y2

. (9)

Fig. 8. Geometric construction for the ellipse calculations.


ls

pzssbiCtst

3

Because the value of s1 is the original major axisength of the smaller ellipse, 2b, by substitution andimplification the final function becomes

R~u! 5d1

b ( 1

HFcos2~u 2 90°!a2 1

sin2~u 2 90°!b2 GJ1y2) . (10)

3. Pattern Characterizations

Figures 7~a! and 9~a! show two-dimensional com-puter plots of the equiangular three-beam and equi-angular four-beam standing-wave interferencepatterns, respectively, each showing a regular arrayof antinodes. If viewed as topographical maps, eachtime a line is crossed, the intensity changes by oneunit.25 From the center of each bright periodic spotor intensity maxima to one of the surrounding areasof zero or low intensity, approximately 9 lines arecrossed in the 3-beam pattern of Fig. 7~a!, and ap-proximately 16 lines are crossed in the 4-beam pat-tern of Fig. 9~a!. This results in the constructive-interference areas at the center of each periodic spotbeing either 9 or 16 times more intense than in theareas where destructive interference predominates,

Fig. 9. ~a! Two-dimensional plot of the equiangular four-beaminterference pattern. ~b! Inverted isometric view of the equian-gular four-beams interference pattern.


which are near the edges of the areas surroundingeach periodic spot.20–22 However, the conclusionthat there are points with either 9 or 16 times theirradiance of a noninterfering incidence point of asingle input beam is a result of using scalar theory.The assumption is that there are points where all thebeams add with perfect coherence and possess themost favorable electric-field vector orientations formaximum constructive interference. The exact vec-tor treatment shows that the only case of this perfectcoherent addition is the trivial case. All nontrivialcases have peak irradiances less than these givennumbers.

Phase aberrations are also of concern in the holo-graphic projector. In the three-beam system, ifthere is a difference in phase in one or two of thebeams it merely causes a shifting of the points ofintensity maxima and can be thought of as automat-ically adjusting.20 In the four-beam system, verycareful alignment must be carried out to prevent theformation of moire fringes across the exposure region.

Figures 7~b! and 9~b! show inverted isometric viewsof the intensity patterns of Figs. 7~a! and 9~a!, respec-tively, which spatially illustrates the periodic spotsas antinodal intensity wells. Figure 10 shows a non-inverted isometric view of the three-beam pattern ofFig. 7~a!, showing a single antinodal intensity peaksurrounded by six partial intensity peaks. Figures7~b!, 9~b!, and 10 were profiled with the laser-analysis

rogram GLAD ~Applied Optics Research!. The sixero-intensity points surrounding each periodic spothown in Fig. 7~a! can be seen as small protuberancesurrounding each antinodal well shown in Fig. 7~b!ut are much more apparent as low points surround-ng the central intensity peak shown in Fig. 10.ompletely surrounding each periodic spot shown in

he four-beam plot of Fig. 9~a! are lines of zero inten-ity that can readily be seen to surround each an-inodal well depicted in Fig. 9~b!.

4. Experiment

Tests were performed with several types of opticalsubstrates of which all had varying curvatures. Fig-ure 11 shows a scanning electron microscope ~SEM!photomicrograph of the three-beam interference pat-

Fig. 10. Noninverted isometric view of the equiangular three-beam interference pattern.

aihibCa

trvcs

ric

tern recorded in positive photoresist. The substratewas cleaned and prebaked and then spin coated at6000 rpm on its 10.94-mm radius-of-curvature con-vex side with a Shipley Microposit Model S1805,41

resulting in a coating thickness of approximately 0.36mm. The substrate was then soft baked, holograph-ically exposed through its base under immersion con-ditions, developed with Shipley developer ~MF-321!,nd then hard baked. The center-to-center period-city here is approximately 0.8 mm. The size of theoles can be increased in relation to the pattern by an

ncrease in the exposure time. The holes will thenecome close packed and more hexagonal in shape.onversely, a shorter exposure time produces smallernd rounder holes in relation to the pattern.When a negative photoresist is used, posts, rather

han holes in the resist, composed of the resist willesult. Then, if a thin film of a metal is applied by aapor-deposition step and is followed by a lift-off pro-ess in which the posts are removed, a metallic meshtructure will remain, possessing near-round holes.25

The mesh structure acts as a resonant bandpass filterfor spectral transmission, and, in the case of a hex-agonal array of round holes in an aluminum thin film,the spectral-transmission parameters and the polar-ization effects were calculated.42

The four-beam pattern produces similar holes, ex-cept they are aligned in squared rows. The holestend to go from round to square on longer exposureand also become quite closely packed. One cangreatly increase the periodicity of the lesser-exposedfour-beam pattern by double-exposing the substrate.Intermediately, between the exposures, the substrateis moved a slight distance diagonally to the rows witha micropositioning stage. The added holes will thenform staggered rows. When the substrate is rotatedby 45°, all the holes then appear as a square grid ofrows and columns.

Figure 12 shows a photomicrograph of an interfero-gram recorded in dichromated gelatin for the charac-terization of the four-beam interference pattern.The center-to-center periodicity here is approxi-

mately 67 mm. The pattern was exposed by use of aholographic diffraction grating with a relatively large~200-mm! grating period. The dichromated gelatinesponded quite well at 363.789 nm. Note the sim-larities between the interferogram of Fig. 12 and theomputer plot of Fig. 9~a!.

In a recent paper43 optical transmission throughsubwavelength hole arrays in thin films of silver,chromium, and gold that were vapor deposited onquartz substrates was analyzed. The holes wereround and configured in both square and hexagonalarrays. It was found that the amount of light trans-mitted through an array was equal to twice or moreof the total amount of light impinging directly on theholes, providing evidence of a surface plasmon effect.Because of the light-transmission efficiency and thelack of diffraction effects, these gratings might finduses in near-field scanning optical microscopes44 orsubwavelength photolithography.

5. Conclusions

The proposed holographic projector system has dem-onstrated an accurate and practical fabricationmethod for creating grid patterns on curved surfaces.Of practical interest for this grid patterning is theapplication of SWS curved-lens surfaces. Or thesystem can be optimized for a flat field. In this in-stance the shape of the caustic in the projector designwill comprise parts of a prolate spheroid with theoptical or z axis as its major axis.

With the current projector design grid patternssmaller than the wavelengths of visible light canreadily be produced. If, however, even smaller sub-wavelength patterns are desired, the holographicprojector can be modified in design to work atexcimer-laser wavelengths. This entails fabricatingall the lens elements and the holographic diffractiongrating from either calcium fluoride or lithium fluo-ride and installing the system in a chamber with apositive nitrogen gas pressure to prevent the forma-tion of ozone. Of practical interest for applicationhere is the use of either the line from an argon fluo-

Fig. 11. SEM photomicrograph of the three-beam interferencepattern recorded in positive photoresist.

Fig. 12. Photomicrograph of an interferogram of the four-beaminterference pattern recorded in dichromated gelatin.


mirrors and broadband antireflection surfaces,” Appl. Opt. 31,

3

ride laser, which has a center wavelength of 193.375nm, or one of the two lines which have center wave-lengths of 157.523 nm and 157.629 nm, from a mo-lecular fluorine laser. In the projector system eitherof the two lines from the molecular fluorine laser hasa sufficiently short center wavelength for use in fab-ricating gratings with pitch sizes of 0.1 mm or lessand microstructures with feature sizes of 0.05 mm orless.

The author thanks David L. Shealy, chairman ofthe Department of Physics of the University of Ala-bama at Birmingham, for his assistance in the edit-ing of the manuscript.

References and Notes1. U. Krackhardt, J. N. Mait, and N. Streibl, “Upper bound on the

diffraction efficiency of phase-only fanout elements,” Appl.Opt. 31, 27–37 ~1992!.

2. B. R. Brown and A. W. Lohmann, “Computer-generated binaryholograms,” IBM J. Res. Dev. 13, 160–168 ~1969!.

3. W. J. Dallas, “Computer-generated holograms,” in The Com-puter in Optical Research and Applications, B. R. Frieden, ed.~Springer-Verlag, New York, 1980! pp. 291–363.

4. S. M. Arnold, “Electron beam fabrication of computer-generated holograms,” Opt. Eng. 24, 803–807 ~1985!.

5. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, Jr., “The kinoform:a new wavefront reconstruction device,” IBM J. Res. Dev. 13,150–155 ~1969!.

6. J. C. Patau, L. B. Lesem, P. M. Hirsch, and J. A. Jordan, Jr.,“Incoherent filtering using kinoforms,” IBM J. Res. Dev. 14,485–491 ~1970!.

7. M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Im-proving resolution in photolithography with a phase-shiftingmask,” IEEE Trans. Electron Devices ED-29, 1812–1846~1982!.

8. Y. C. Pati and T. Kailath, “Phase-shifting masks for microli-thography: automated design and mask requirements,” J.Opt. Soc. Am. A 11, 2438–2452 ~1994!.

9. K. Ronse, M. Op de Beeck, L. Van den hove, and J. Engelen,“Fundamental principles of phase shifting masks by Fourieroptics: theory and experimental verification,” J. Vac. Sci.Technol. B 12, 589–600 ~1994!.

10. R. L. Morrison, S. L. Walker, and T. J. Cloonan, “Beam arraygeneration and holographic interconnections in a free-spaceoptical switching network,” Appl. Opt. 32, 2512–2518 ~1993!.

11. J. R. Leger, G. J. Swanson, and W. B. Veldkamp, “Coherentlaser addition using binary phase gratings,” Appl. Opt. 26,4391–4399 ~1987!.

12. D. H. Raguin, “Subwavelength structured surfaces and theirapplications,” in Diffractive and Miniaturized Optics, Vol.CR49 of SPIE Critical Reviews ~SPIE, Bellingham, Wash.,1993!, pp. 234–261.

13. P. Yeh, “A new optical model for wire grid polarizers,” Opt.Commun. 26, 289–292 ~1978!.

14. K. Shiraishi, T. Sato, and S. Kawakami, “Experimental veri-fication of a form-birefringent polarization splitter,” Appl.Phys. Lett. 58, 211–212 ~1991!.

15. L. H. Cescato, E. Gluch, and N. Streibl, “Holographic quarter-wave plates,” Appl. Opt. 29, 3286–3290 ~1991!.

16. C. W. Haggans, L. Li, T. Fujita, and R. K. Kostuk, “Lamellargratings as polarization components for specularly reflectedbeams,” J. Mod. Opt. 40, 675–686 ~1992!.

17. E. N. Glytsis and T. K. Gaylord, “High-spatial-frequency bi-nary and multilevel stairstep gratings: polarization-selective


4459–4470 ~1992!.18. H. Haidner, P. Kipfer, W. Stork, and N. Streibl, “Zero-order

gratings used as artificial distributed index medium,” Optik~Stuttgart! 89, 107–112 ~1992!.

19. C. G. Bernhard, “Structural and functional adaptation in avisual system,” Endeavor 26, 79–84 ~1967!.

20. J. J. Cowan, “The recording and large scale replication ofcrossed holographic grating arrays using multiple beam inter-ferometry,” in Application, Theory, and Fabrication of PeriodicStructures, Diffraction Gratings, and Moire Phenomena II,J. M. Lerner, ed., Proc. SPIE 503, 120–129 ~1984!.

21. J. J. Cowan, “The holographic honeycomb microlens,” in Ap-plications of Holography, L. Huff, ed., Proc. SPIE 523, 251–259~1985!.

22. J. J. Cowan, “Method and apparatus for exposing photosensi-tive material,” U.S. Patent 4,496,216 ~29 January 1985!.

23. Canon Kabushiki Kaisha, “Verfahren zur Herstellung einesTeil mit einer Anordnung von Mikrostrukturelementen aufdemselben,” Ger. Offen. DE 2,952,607 ~class G02B5y02! ~pub-lication date: 10 July 1980; application date: 28 December1979; issue date: 14 April 1994!. This patent is the oldestreference the author has found regarding the equiangularthree-beam and equiangular four-beam interference patterns.

24. T. Suzuki, K. Iizuka, K. Ohtaka, and H. Mizutani, “Focusingplate,” U.S. patent 4,421,398 ~20 December 1983!.

25. W. T. Pawlewicz, P. M. Martin, R. W. Knoll, B. T. Smith, andW. M. Myers, “Transparent conductive coatings for electro-optic windows,” Tech. Rep. MMT A3 1134 ~U.S. Army MissileCommand, Redstone Arsenal, Ala., 1987!.

26. S. H. Zaidi, S. R. J. Brueck, F. M. Schellenberg, R. S. Mackay, K.Uekert, and J. J. Persoff, “Interferometric lithography exposuretool for 180-nm structures,” in Emerging Lithographic Technol-ogies, D. E. Seeger, ed., Proc. SPIE 3048, 248–254 ~1997!.

27. X. Chen, S. H. Zaidi, S. R. J. Brueck, and D. J. Devine, “Inter-ferometric lithography of sub-micrometer sparse hole arraysfor field-emission display applications,” J. Vac. Sci. Technol. B14, 3339–3349 ~1996!.

28. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, “Opticalmatter: crystallization and binding in intense optical fields,”Science 249, 749–754 ~1990!.

29. K. M. Baker, D. L. Shealy, and W. Jiang, “Directional lightfilters: three-dimensional azo dye formed images within op-tical resins,” in Diffractive and Holographic Optics TechnologyII, I. Cindrich and S. H. Lee, eds., Proc. SPIE 2404, 144–158~1995!.

30. K. M. Baker, “Directional light filter and holographic projectorsystem for its production,” U.S. Patent 5,642,209 ~24 June1997!.

31. C. H. Lin, Z. H. Zhu, and Y. H. Lo, “New grating fabricationtechnology for optoelectronic devices: cascaded self-inducedholography,” Appl. Phys. Lett. 67, 3072–3074 ~1995!.

32. C. H. Lin, Z. H. Zhu, Y. Qian, and Y. H. Lo, “Cascade self-induced holography: a new grating fabrication technology forDFByDBR lasers and WDM laser arrays,” IEEE J. QuantumElectron. 32, 1752–1759 ~1996!.

33. K. M. Baker, “Extreme depth-of-field optical lens and holo-graphic projector system for its production,” U.S. patent5,822,091 ~13 October 1998!.

34. W. Jiang, D. L. Shealy, and J. C. Martin, “Design and testingof a refractive reshaping system,” in Current Developments inOptical Engineering III, R. E. Fischer and W. J. Smith, eds.,Proc. SPIE 2000, 64–75 ~1993!.

35. W. Jiang, D. L. Shealy, and K. M. Baker, “Optical design andtesting of a holographic projection system,” in Diffractive andHolographic Optics Technology, I. Cindrich and S. H. Lee, eds.,Proc. SPIE 2152, 244–252 ~1994!.

36. W. Jiang, D. L. Shealy, and K. M. Baker, “Physical optical

3

3

ries photoresists are available from Shipley Company, 455
analysis of the performance of a holographic projection sys-tem,” in Diffractive and Holographic Optics Technology II, I.Cindrich and S. H. Lee, eds., Proc. SPIE 2404, 227–234 ~1995!.
7. Applied Physics Specialties, Ltd., 17 Prince Andrew Place,Dow Mills, Ontario M3C 2H2, Canada.

8. R. Mathews, Optical Works, Inc., 26280 Olhava Road, NW,#A, Poulsbo, WA 98370–9435 ~personal communication, 6November 1997!.

39. R. P. Cargille Laboratories, Inc., 55 Commerce Road, CedarGrove, NJ 07009–1289.

40. E. H. Anderson, C. M. Horwitz, and H. I. Smith, “Holographiclithography with thick photoresist,” Appl. Phys. Lett. 43, 874–875 ~1983!.

41. Processing instructions for the Shipley Microposit S1800 se-

Forest Street, Marlboro, MA 01752.42. J. M. Kurmer, J. I. Halman, K. A. Ramsey, D. L. Jones, and J.

McManigal, “Polarization effects of resonant mesh structuresfabricated on IR transmitting windows,” in Window and DomeTechnologies and Materials II, P. Klocek, ed., Proc. SPIE 1326,165–175 ~1990!.

43. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A.Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 ~1998!. Seealso the review and commentary of this paper by R. Sambles,“More than transparent,” Nature 391, 641–642 ~1998!.

44. E. Betzig and J. K. Trautman, “Near-field optics: microscopy,spectroscopy, and surface modification beyond the diffractionlimit,” Science 257, 189–194 ~1992!.


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Highly Corrected Submicrometer Grid Patterning on Curved Surfaces