February 1, 1996 / Vol. 21, No. 3 / OPTICS LETTERS 177

Diffractive lens fabricated with mostly zeroth-order gratings

Frederick T. Chen and Harold G. Craighead

School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853-2501

Received October 6, 1995

We demonstrate a spherical diffractive lens fabricated in fused quartz for use at the 632.8-nm wavelength.The lens is constructed by use of a modulated two-dimensional binary grating with a high transmitted zeroth-order efficiency. Rigorous eigenmode analysis is used to correlate the desired phase modulation with the fillfactor. Fabrication requires only one lithography step. Using the lens, we were able to image a focal spotwith a diffraction-limited spot size (FWHM). 1996 Optical Society of America

Carefully arranged subwavelength microstructuresconstitute artif icial dielectric optical components.This was f irst demonstrated with microwaves byKock.1 The concept was applied to diffractive opti-cal elements independently by Stork et al.2 and byFarn.3 Recently we demonstrated a transmissiveblazed grating fabricated in fused quartz at a visiblewavelength (l ­ 632.8 nm) that was designed byuse of an approximate effective-medium theory fortwo-dimensional zero-order dielectric grating.4 Kipferet al. demonstrated diffractive spherical mirrors basedon a similar principle at the 10.6-mm wavelength.5 Inprevious research the periods of the gratings to bemodulated were chosen so that only one transmit-ted and one ref lected order were propagating. Forlow-index dielectric substrates this ensures a hightransmitted zeroth-order eff iciency. As we show inthis Letter, however, larger periods allowing moreorders to propagate can be used, as long as thetransmitted zeroth-order eff iciency is suff icientlyhigh. Such gratings can be called mostly zeroth-ordergratings (MZOG’s). The main advantage of usinglarger periods is that fabrication is easier, at the costof somewhat lower eff iciencies. In this Letter wedemonstrate what we believe to be the f irst diffractivelens designed by use of MZOG’s and fabricated in afused-quartz substrate for operation at the 632.8-nmwavelength.

The basic strategy for designing transmissive bi-nary diffractive optical elements with high efficiencyhas been described.2 – 4,6 The function of the elementis determined by the macrostructure, i.e., by thelarge-scale phase profile. For paraxial applicationsscalar theory is adequate for f inding the optimumphase profile for operation of the element. The phaseprofile is generated not directly by the surface reliefbut by the local microstructure, which consists of ahigh-spatial-frequency dielectric grating that allowsmost of the incident light to pass into the trans-mitted zeroth order. The phase of the transmittedzeroth-order light is determined by the fill factor ofthe grating as well as by the period/wavelength ratiodyl and the grating depth h. Hence, by modula-ting the fill factor while keeping the period and thedepth fixed, one obtains arbitrary phase modulationover the entire aperture of the diffractive element.The high-spatial-frequency grating is effectivelyhomogenized into an artificial dielectric layer. In

0146-9592/96/030177-03$6.00/0

contrast to that for the macrostructure, a rigorousvectorial approach is required for describing thehigh-spatial-frequency grating microstructure. Rig-orous eigenmode theory6,7 is a natural and eff icientmethod for analyzing surface-relief dielectric gratings.Noponen and Turunen6 have suggested the use of high-spatial-frequency lamellar dielectric gratings withperiods as large as 0.8l. If two-dimensional symmet-ric gratings are used, calculations based on rigorouseigenmode analysis7 show that, at normal incidence,transmitted zeroth-order efficiencies of $ 80% can beachieved for all fill factors with periods as large as1.1l for grating depths that permit sufficient phasemodulation sfmax , 3py2 2 2pd. Surprisingly, this isnot true for TE-polarized light normally incident ontoa lamellar grating. Moreover, at normal incidence, asymmetric two-dimensional grating exhibits no polari-zation sensitivity.

For the purpose of demonstrating the basic principlewe modeled a square array of square pillars 1.032 mmtall with a period of 700 nm. Normal incidence from aquartz substrate (n ­ 1.46) to air was assumed. Thir-teen Rayleigh orders along each axis were retained inour calculations. This height was chosen to accommo-date a phase range of 0–3 py2. According to scalartheory, this phase range yields a first-order diffrac-tion eff iciency of 81% for a four-level staircase phaseprofile.8 By using pillars one can avoid reactive ionetching lag effects during the fabrication process, be-cause reactant and product transport are not hinderedsignificantly.9,10 In Fig. 1 we plot the eff iciency andthe phase of the transmitted zeroth order as a functionof pillar width. Only the component polarized paral-lel to that of the incident light is shown. The cal-culated amount of depolarization in the zeroth orderactually never exceeds several percent. For volumefractions sufficiently near zero or unity, one may ex-pect near-homogeneous behavior (less scattering). Forpillar widths above 350 and below 600 nm there is in-creased scattering into higher diffraction orders, re-sulting in the dip in the transmitted zeroth-orderefficiency.

Using the data of Fig. 1 as a lookup table, we de-signed and fabricated a diffractive lens in fused quartzfor focusing normally incident 632.8-nm light. Thelens had a square aperture 1 mm on a side and a focallength of 2 cm. A fused-quartz substrate was coatedwith 100 nm of thermally evaporated aluminum.

1996 Optical Society of America

178 OPTICS LETTERS / Vol. 21, No. 3 / February 1, 1996

Fig. 1. Eff iciency and phase of transmitted zeroth orderfor a 700-nm-period square pillar array as a function ofpillar width. The pillar height is 1.032 mm, and the wave-length used is 0.6328 mm. Only the component polarizedparallel to the normally incident light is shown.

A 70-nm-thick layer of poly(methyl methacrylate)was then spin coated onto the aluminum layer.The pattern data were written in the poly(methylmethacrylate) resist by a Cambridge InstrumentsEBMF 10.5yCS electron beam lithography systemwith a 76-nm beam diameter and a 20-kV accelerationvoltage. The two intermediate phases were obtainedby use of square pillars 350 and 550 nm on a sidespaced 700 nm apart. The minimum phase level cor-responded to complete exposure, whereas the maxi-mum phase level corresponded to no exposure. Theouter radii of the regions of constant phase are givenby rm ­ fs f 1 mly4d2 2 f 2g1/2. Following exposure,the poly(methyl methacrylate) was developed in 1:1methyl isobutyl-ketone and isopropyl alcohol. Thesample was then etched in a BCl3yCl2yCH4 plasmauntil the aluminum layer was cleared in the exposedareas. An oxygen ash converted remaining aluminumchloride residues into aluminum oxide to prevent corro-sion of the aluminum mask on exposure to atmospherichumidity.11 A low-pressure, magnetically conf inedCF4 plasma was used to etch the fused quartz in the ar-eas where the aluminum had been cleared. The etchwas performed in steps, with the etch depth monitoredby an Alpha-Step profilometer scan of a wide patchnear the lens, until the desired etch depth was mea-sured. Afterward the aluminum layer was removedwith a 2:1 hydrochloric acid:nitric acid solution. Thelens was sputter coated with ,20-nm AuyPd and exam-ined with a scanning electron microscope. Figure 2(a)is a micrograph of the lens. The four quadrants werestitched together without f ine alignment with a com-mon vertex at the center. The stitching errors wereestimated to be 1 mm in the x direction and 0.3 mmin the y direction. These stitching errors are dueprimarily to the nonplanarity of the substrate. Thestitching is not expected to produce any degradation inthe efficiency because each quadrant has equal focus-ing power. However, the spot may suffer from slightaberrations owing to the stitching. Figure 2(b) is acloseup of a section of one Fresnel zone.

The focusing power of the lens was characterized bya measurement of the first-order diffraction eff iciency.An efficiency of 53% was measured with a knife-edge atthe focal plane and a detector. Because of scatteringinto higher orders, the use of a 700-nm period modu-lated grating generates amplitude modulation, whichlowers the expected diffraction eff iciency of the lensfrom 81% to 65%, in accordance with scalar diffractiontheory. Additional calculations that we performedshow that, if a continuous phase variation from 0 to2p is exploited, with a grating depth of 1.376 mm, thetheoretical efficiency would easily surpass 80%. Thesubsequent feature dimension requirements, however,

Fig. 2. Scanning electron micrograph of a diffractive lens.The lens aperture is a 1-mm square, and the focal length is2 cm. (b) Scanning electron micrograph of a section of oneFresnel zone, showing the four phase levels.

Fig. 3. Focal spot profile as measured by a CCD camera.

February 1, 1996 / Vol. 21, No. 3 / OPTICS LETTERS 179

are diff icult to achieve at the present time. The 12%measured discrepancy in efficiency arises for a com-bination of reasons. It is estimated that fabricationerrors themselves contribute no more than 2% reduc-tion. The breakdown of scalar theory, the breakdownof the homogenization approximation, and scatteringfrom the transition ambiguity of the phase boundariesare the main sources of eff iciency loss. In additionto eff iciency, focal spot quality is an important mea-sure of lens performance. A 203 microscope objectivewas used to magnify the focal spot, and an image ofthe spot was captured by a CCD camera system con-trolled by a microcomputer. Figure 3 is a plot of thepoint-spread function, i.e., the intensity profile at thefocal plane of the lens. The FWHM was measured tobe ,15 mm, in agreement with the theoretical value,1.22s fynumberdl ­ 15.44 mm. The fact that the spotappears to lack symmetry may be attributed to f ieldstitching in the electron-beam lithography and may beeliminated, if desired, by the use of alignment duringlithography. Because of the small number of Fresnelzones, local surface imperfections and nonuniformitiescan be expected to have a more drastic effect on the spotprofile.

The attractiveness of the MZOG design schemethat we have presented in this Letter is the simplif iedfabrication process compared with multilevel or analogsurface profiling techniques. Discrete, well-definedbinary patterns are etched into fused quartz, an idealsubstrate for most optical applications. The desireddiffraction pattern is achieved through artif icial indexmodulation rather than through explicit surface-relief blazing. Electron-beam lithography was usedfor demonstration at a visible wavelength. Phase-shifting i-line (365-nm) lithography would enablefeature sizes of 0.3 mm to be achieved, permit-ting demonstration at wavelengths of 1.3 mm andhigher.12 With the advent of KrF excimer laser (248-nm) lithography the use of phase-shifting technologywould push the resolution to 0.2 mm. These devel-

opments are encouraging for the mass production ofdiffractive micro-optical elements based on MZOG’s.

We performed numerical calculations by usingthe computing facility provided by the MaterialsScience Center at Cornell University. Fabricationwas performed at the Cornell Nanofabrication Fa-cility. We thank Jari Turunen at the University ofJoensuu, Finland, for helping to resolve an instabilityproblem early on in the calculations and Rick Bojkoof the Cornell Nanofabrication Facility for assistancewith electron-beam lithography. We also thank thereviewers for their helpful comments. This researchwas supported by the U.S. Air Force Photonics Labora-tory at Rome, New York.

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