1. INTRODUCTIONThe pioneering work of Eli Yablonovitch1 and SajeevJohn2 has evolved into what are known today as photoniccrystals, or photonic bandgap materials. Many novelphenomena have been observed that promise significantimpact on future photonic systems. Photonic bandgapscan improve efficiency of optoelectronic devices by sup-pressing emission of photons within the bandgap.1,3
Minibands4 within the bandgap may offer even higher de-grees of spectral selectivity for spectroscopy and dense-wavelength-division multiplexing. Wave guiding aroundtight bends5–7 can accommodate denser waveguide rout-ing and lead to a host of new and novel devices based onthis property. Highly dispersive photonic crystals8,9 cancompensate for large amounts of channel dispersion, slowthe propagation of light, or perform difficult phase-transfer functions. Negative refraction9,10 may lead todramatically improved imaging optics or improve thequality of micro-optic devices such as micromirrors andmicrolenses.
One-dimensional (1D) and two-dimensional (2D) photo-nic crystals have received considerable attention in theliterature as a result primarily of their ease of simulation,analysis, and fabrication compared with their three-dimensional (3D) counterparts. While 1D and 2D crys-tals have many applications, 3D crystals have at least twoimportant advantages. First, they are capable of havinga complete bandgap where propagation is forbidden atany angle. Second, they are capable of localizing photonsat a point defect in the lattice.
Various techniques have been employed to build 3Dphotonic crystals.11–23 A particularly promising methodis holographic lithography.13–23 It has been shown that
all fourteen Bravais lattices for 3D crystals can be formedby interfering just four noncoplanar beams.23 In a singleprocess step, high-resolution photoresist materials suchas Epon® SU-8 can be exposed by UV, electron beam, orx-ray radiation to form photonic crystals.24 Holographiclithography may become a fabrication method of choicebecause this method and associated materials are flexibleand easily integrated with other devices.
Simulation tools and design methods for photonic crys-tals are still evolving. Analysis has been performed onideal 3D lattices,25 but virtually no work has been devotedto modeling the fabrication of these structures. Through-out the literature, the geometry of photonic crystalsformed by holographic lithography is predicted using asimple intensity-threshold method. For negative resistssuch as SU-8, this method assumes all portions of resistexposed at an intensity above some threshold will remainafter developing to form the lattice. While very fast andsimple to implement, this technique ignores all the phys-ics of lithography and all phenomena that produce distor-tions. It is accurate only for very-high-contrast expo-sures and crystals with only large features relative to thelattice distortions.
As dimensions of photonic crystals are reduced to oper-ate at shorter wavelengths, distortion artifacts from fab-rication become more pronounced. For holographic li-thography, beam quality, reflections, standing waves,reaction kinetics, acid diffusion, resist shrinkage, and de-veloping effects are just some potential sources of latticedistortion. For one class of photonic crystals, structuralfluctuations in excess of 15% have a large effect on the po-sition and width of the bandgap.26,27 It is reasonable toassume the dispersive properties and bandgaps of other
2004 Optical Society of America
1704 J. Opt. Soc. Am. A/Vol. 21, No. 9 /September 2004 R. C. Rumpf and E. G. Johnson
crystal structures are more sensitive to lattice distortion,especially as dimensions are reduced to operate at shorterwavelengths.
This paper describes a comprehensive model of the fab-rication and behavior of photonic crystals formed by holo-graphic lithography. It includes exposure, postexposurebaking, developing, and simulation of transmission spec-tra. This is the first effort known to the authors to incor-porate the physics of lithography into models of photoniccrystals. By simulating the phenomena that cause dis-tortion in crystal lattices, more accurate predictions oftheir final geometry and optical behavior can be obtained.Properties of the crystal may be more rigorously tunedand optimized. While the conclusions of this paper aregenerally applicable to other photoresists, the negativephotoresist SU-8 was selected for modeling in this workbecause of its large array of additional applications in mi-crophotonic systems.28
To demonstrate comprehensive modeling, the face-centered-cube (FCC) photonic crystal fabricated andtested by Miklyaev et al.13 was accurately duplicatedthrough simulation. This crystal symmetry was selectedsince it has a convenient top-side-only exposure configu-ration and was shown to have a complete photonic band-gap when there is sufficient index contrast betweenmaterials.29 This can be achieved by using the SU-8 lat-tice as a template13 or by backfilling the lattice with ahigh-index material. Numerical results from each stageof the model are presented and compared with results ob-tained by thresholding. The model is used to study theeffect of optical absorption, reflections, and standingwaves during the exposure process. Numerical resultsare presented showing that optical absorption during ex-posure leads to chirped lattices, while reflections can re-duce exposure contrast and introduce strong lattice dis-tortions. Finally, hybrid photonic crystals are formed bysimulating successive holographic exposures in the same
resist. A diamond lattice and three different hybridtrigonal–FCC lattices are presented as numerical ex-amples. The technique of performing multiple exposurescan also be used to introduce defects into a lattice,30
modify lattice symmetry,14 form photonic crystals withless than four beams,31 or form other novel micro-opticstructures.
2. MODELING CRYSTAL FABRICATIONSimulating the holographic lithography process was ac-complished by constructing separate models for exposure,postexposure baking (PEB), and developing that succes-sively operated on a common 3D materials mesh. Depo-sition techniques such as spin coating were not consid-ered and an even film of resist was assumed. Figure 1shows a block diagram of the overall model and factors ac-counted for in each step.
The exposure model determined the distribution of ab-sorbed energy within the resist, often called the aerial im-age. The PEB model simulated the blurring effects ofacid diffusion that occur during the bake to form the la-tent image. It also accounted for shrinkage during thebake and computed the dissolution rate based on energyin the latent image. The development model simulatedhow the resist would dissolve into a developer solutionbased on this dissolution rate information. The results ofeach modeling step depended heavily on the results of theprevious steps. With this approach, crystal geometrywas determined with a higher degree of fidelity than hadyet been achieved in the literature.
For periodic crystals, the size of the modeling problemwas reduced by storing and processing just a single pe-riod, or unit cell, of the lattice. Virtually all structuresformed by holographic lithography are periodic becausethey are formed by interfering separate beams at discreteangles. It was not accurate to assume that the lattice
Fig. 1. Block diagram of a comprehensive model to simulate the fabrication and behavior of photonic crystals formed by holographiclithography. Factors accounted for in each step are listed next to the blocks.
R. C. Rumpf and E. G. Johnson Vol. 21, No. 9 /September 2004 /J. Opt. Soc. Am. A 1705
was periodic in the direction normal to the surface of theresist, because it was found that optical absorption al-tered the geometry of the crystal with depth.
A. ExposureFor a more rigorous simulation of the exposure process,the finite-difference time-domain (FDTD) method was se-lected. This technique can account for refraction at theresist surface, index matching materials, beam coherence,intensity, polarization, wavelength, absorption, disper-sion, and nonlinear material properties, as well as reflec-tions and standing-wave effects. Thorough treatment ofthe FDTD method can be found in the literature.32,33
This technique has been successfully used to model theexposure process in previous work.34–36
The total-field–scattered-field formulation37,38 wasused to provide the holographic source. This method waschosen because it was simple to implement and allowedbackward-reflected waves to pass back through the sourceregion without being artificially reflected.
It was not possible to run a FDTD simulation over theentire duration of the exposure because this would re-quire an extraordinary number of iterations. Instead,exposure dose was extrapolated from the energy absorbedduring a limited number of wave cycles after steady statehad been reached. This approach can be applied only toresists that behave linearly during exposure. The refrac-tive index and absorption of SU-8 does not change signifi-cantly for wavelengths at and below 365 nm.39
The instantaneous power density (W/m3) absorbed in alossy material is40
Pd 5 sE2 5 sh02E2 (1)
where s is the conductivity (loss) of the resist, E is theelectric field magnitude, h0 is the free-space impedance,and E is the normalized electric field employed in manyFDTD implementations, where E 5 h0E. After steadystate had been reached, the energy absorbed by the resistduring each time step was summed over N wave cycles byusing the following equation with field values interpo-lated at common points in the mesh:
j8~i, j, k; t ! 5 j8~i, j, k; t 2 Dt ! 1 @Ex2~t ! 1 Ey
2~t !
1 Ez2~t !#ui, j,k . (2)
At the completion of the simulation, the material prop-erties were incorporated and the exposure extrapolatedover the total developing time Texp . The aerial imagewas the total energy absorbed during exposure and wascomputed from the result of Eq. (2) as
jA~i, j, k ! 5 Texp
csh02Dt
Nl0j8~i, j, k !, (3)
where c is the speed of light in free space, Dt is the dura-tion of the FDTD time steps, and l0 is the free-spacewavelength of the illumination.
B. Postexposure BakeThe PEB model simulated diffusion of mobile acids, incor-porated resist shrinkage due to cross linking, and com-puted the dissolution rate at each point throughout the
exposed resist. Dissolution rate R describes how quicklyresist dissolves into developer solution and was givenunits of millimeters per second. With this definition, thedistance d that a resist surface progresses after develop-ing for time t is
d 5 Rt. (4)
A number of simultaneous chemical reactions occurduring the PEB that are quantified by a system of coupledpartial differential equations.41 These equations de-scribe the concentration of active compounds as they areproduced by reactions during the bake. At the sametime, mobile acids diffuse into surrounding resist. Thecombined effects of reaction kinetics and diffusion blurthe aerial image to form the latent image. It is by thismechanism that standing waves and other quickly vary-ing artifacts are usually eliminated. Contrast of theaerial image is slightly reduced in the latent image as aresult of acid diffusion.
The general equation describing diffusion of mobile ac-ids is41
]M/]t 5 ¹~D • ¹M !, (5)
where M represents chemical concentration and D speci-fies the diffusion coefficient that can vary spatially or de-pend on concentration, temperature, and more. Chemi-cally amplified resists can exhibit different properties onthe surfaces because acids are not able to diffuse outsideof the resist and tend to concentrate here. For regions inthe resist away from edges, this effect is negligible andEq. (5) reduces to
]M/]t 5 D~¹2M !. (6)
Three types of diffusion were identified by Erdmannet al.41 that differed in how the parameter D was defined.All of these produced diffusion profiles that were close toGaussian and could be described by
g~r ! 51
~2preff2 !1/2
expS 2r2
2reff2 D . (7)
The parameter r was the distance from the origin of dif-fusion and reff was the effective diffusion length. The la-tent image jL incorporated the blurring effect of acid dif-fusion and was computed by convolving Eq. (3) with Eq.(7). Efficient numerical methods based on fast Fouriertransforms were used to perform this convolution42:
jL 5 jA ^ g. (8)
The dissolution rate R is a strong function of the con-centration of chemicals in the resist and of the strength ofthe developer. Process variables such as temperatureand pressure also affect dissolution rate, but were ignoredhere. An empirical equation called the enhanced-notchmodel was formulated to map dissolution rate of novolak–DNQ (diazonaphthoquinone) resists to relative photoin-hibitor concentration M used in Eq. (6).43 This modelwas selected because SU-8 is a derivative of novolak.24
The dissolution rate at each point in the resist wascomputed from the latent image energy density by using amodified formulation of the enhanced-notch model thatmapped absorbed energy directly to dissolution rate.
1706 J. Opt. Soc. Am. A/Vol. 21, No. 9 /September 2004 R. C. Rumpf and E. G. Johnson
Circumventing reaction kinetics in this manner requiresa separate chemical simulation or laboratory character-ization to generate the required mapping.43 Theenhanced-notch model reformulated to map energy di-rectly to dissolution rate for negative resists was
Rn~E ! 5 Rmax~1 2 E !NF ~an 1 1 !~1 2 E !Nnotch
an 1 ~1 2 E !NnotchG
1 RminFRminE21
RmaxE21G F1 2
~an 1 1 !~1 2 E !Nnotch
an 1 ~1 2 E !NnotchG ,
(9)
where
an 5Nnotch 1 1
Nnotch 2 1~1 2 Eth!Nnotch. (10)
In these equations, E was a normalized energy param-eter in the range 0 < E < 1. E 5 0 represented no ab-sorbed energy while E 5 1 represented the energy levelabove which it had negligible effect on solubility. Nor-malized energy was computed from latent image energyby dividing by the level of energy that would completelycross link the SU-8, jmax :
E 5 jL /jmax . (11)
Dissolution rate at each point in the exposed resist wascomputed by Eqs. (9) and (10). While not implementedin this work, a similar formulation for positive resistsshould be
Rp~E ! 5 RmaxENF ~ap 1 1 !ENnotch
ap 1 ENnotchG
1 RminF RminE
RmaxE G F1 2
~ap 1 1 !ENnotch
ap 1 ENnotchG ,
(12)
where
ap 5Nnotch 1 1
Nnotch 2 1Eth
Nnotch. (13)
An advantage of the enhanced-notch model is that theparameters characterizing the numerical fit have physicalmeaning. Rmax defines the highest dissolution rate thatoccurs where no energy was absorbed (i.e., E 5 0). Rmindefines the lowest dissolution rate in saturated regions(i.e., E 5 1). The parameter Eth defines the energythreshold where dissolution rate falls rapidly. The slopeof the dissolution rate curve in the region of Rmax is con-trolled through parameter N. Higher values yieldsteeper slopes. In a similar manner, Nnotch controls theslope in the central, or notch, region, where dissolutionrate falls rapidly.
During cross linking, many resists shrink in size.13,44
To handle the general case of anisotropic shrinkage, themesh resolution parameters were modified using Eqs.(14)–(16) with shrinkage factors sx , sy , and sz in each di-rection:
Dxs 5 ~1 2 sx! • Dx, (14)
Dys 5 ~1 2 sy! • Dy, (15)
Dzs 5 ~1 2 sz! • Dz. (16)
C. DevelopingTwo main approaches to simulating developing are ray-tracing methods45–47 and cellular automata methods.48–51
Ray-tracing methods treat progression of the resist–developer interface analogous to an optical wave frontpropagating through a thick inhomogeneous medium.These methods are fast and efficient but become inaccu-rate and complicated when surfaces become disjointed orintersect. Cellular automata methods run somewhatslower and require more computer memory but avoid theabove difficulties and are able to simulate the arbitrarilycomplex and detached surfaces of photonic crystals. Acellular automata method was implemented in this work.
At a microscopic level, the fundamental mechanismsgoverning dissolution of polymers are not well under-stood, and a number of theories have been proposed.52
Factors affecting this process are surface roughness, mo-lecular weight, temperature, pressure, developerstrength, agitation, and others. In this work, it was as-sumed there was sufficient agitation during developingthat the strength of the developer solution was the samethroughout the resist and that it remained constant overthe entire duration of the developing process.
D. Bandgap SimulationThe FDTD method was used to simulate transmissionspectra of the photonic crystal structure predicted by thedevelopment model. This method was chosen because itwas more rigorous than other methods, required no as-sumptions of propagating modes, and enabled the timeevolution of the fields to be completely visualized. In ad-dition, the response could be obtained over a wide rangeof wavelengths from just one simulation. The method iseasily extended to nonperiodic structures and crystalswith defects. Properties such as dispersion, polarization,and diffraction efficiency of Floquet modes could also becomputed. The FDTD method has been successfullyused to model photonic crystals in previous work.53–57
To model arbitrary photonic crystals, the program hadto handle very small (one cell or less) material structures.Many such methods for FDTD can be found in theliterature.58–61 Of these, the Dey–Mittra technique fordielectric materials61 seemed most applicable as it was re-ported to reduce errors of coarse meshes by approxi-mately 60%.62 The method requires material propertiesapplied to a field component to be the average observedthroughout the immediately surrounding volume (i.e.,weighted volume average). This method was less accu-rate when photonic crystals with fine structures weresimulated. This seemed to be because averaging mate-rial properties over very fine structures reduced the mag-nitude of reflections by lowering the contrast between ma-terials. To circumvent this issue, a simple diagonal split-cell method63 typically used to model perfect electricconductor surfaces was used to model conformal dielectric
R. C. Rumpf and E. G. Johnson Vol. 21, No. 9 /September 2004 /J. Opt. Soc. Am. A 1707
surfaces. To prevent reflections from being artificiallysuppressed, no averaging was performed.
3. NUMERICAL RESULTSTo demonstrate comprehensive modeling, experimentalwork performed by Miklyaev et al.13 on a FCC photoniccrystal in SU-8 was duplicated through simulation.Table 1 summarizes the parameters used in the simula-tion. Figure 2 shows the beam configuration used to ex-pose SU-8 to form the crystal. The exposure wavelengthwas 355 nm, where the absorption of SU-8 is approxi-mately 10 mm21. A central beam of circular polarizationwas normally incident. Three other beams of linear po-larization in their planes of incidence were equally dis-tributed around the central beam at an angle of 38.94°with respect to the normal inside the resist. Because ofadhesion between the resist and substrate, shrinkageparallel to the substrate was assumed to be negligible,and an angle of 36.22° may have provided better cubicsymmetry after shrinkage. To achieve these angles in-side the resist, the beams were combined through a prismspecially designed to limit refraction. The intensity ofthe central beam was 1.5 times that of the side beams.The diameter of the beams was 8 mm, and the beams pro-vided a total of 8 mJ from a single 6-ns pulse. This con-figuration was simulated for a 5.4-mm thick film of SU-8,and the resulting aerial image in a portion of resist isshown in Fig. 3.
The nominal resolution of SU-8 was reported to be bet-ter than 100 nm so the effective diffusion length was setto 75 nm. The nominal resolution can be interpreted as
Table 1. Summary of Parameter Values
Material properties355 nm: n . 1.67, a . 10,000 m21,
s 5 88.66 V21 m21
365 nm: n . 1.668, a . 2000 m21,s 5 17.71 V21 m21
600 to 900 nm: n . 1.58, a . 10 m21,s . 8.39 3 1022 V21 m21
Shrinkage: sx 5 0, sy 5 0, sz 5 7.5%Notch model: N 5 0.8, Nnotch 5 100, Eth 5 0.5,
jmax 5 6 mJ/mm3
Rmax 5 7.5 mm/sec, Rmin 5 5 3 10210 mm/secDevelop
Time: Tdev 5 2 min
the dimension below which sharp edges and fine struc-tures are no longer possible. A shrinkage factor of 7.5%was also reported. Owing to adhesion between the resistand substrate, it was assumed that shrinkage parallel tothe substrate was negligible. Therefore, shrinkage wasimplemented along the vertical direction only.
Figure 4 shows a small portion of resist at the surfaceat incremental times during simulated developing. Aftertwo minutes, very little change was observed. The finalphotonic crystal and resulting transmission spectra areshown in Fig. 5, where they are compared with results ob-tained by thresholding. Both crystal geometry andtransmission spectra of the comprehensive model com-pare remarkably well with the measured results obtainedby Miklyaev et al.13 Because of low index contrast be-tween SU-8 and air, only a partial bandgap wasobserved.29
In general, it was found that threshold methods pre-dicted crystals with higher fill factors and failed to ad-equately represent the shape of atoms in the lattice. Forcrystals formed by holographic lithography, atoms are thelarger globs of resist formed at intensity peaks during ex-posure. Acid diffusion works to connect atoms in the lat-tice to form standing structures at exposure doses well be-low what thresholding predicts is necessary. Thebandgaps of crystals predicted by thresholding wereshifted to longer wavelengths because of the falsely highfill factor. Furthermore, the atoms were poorly repre-
Fig. 2. Beam configuration for holographic exposure of a photo-nic crystal in SU-8. LP, linear polarization; CP, circular polar-ization.
Fig. 3. Aerial image of a FCC lattice exposed in a small portionof resist.
1708 J. Opt. Soc. Am. A/Vol. 21, No. 9 /September 2004 R. C. Rumpf and E. G. Johnson
Fig. 4. Small portion of resist at the surface at incremental times during the development model.
sented by thresholding. This seemed to manifest itselfmostly at the bandgap edges.
A. Optical AbsorptionOptical absorption during the exposure process causedthe geometry of formed crystals to vary with depth or be-come chirped. While the period remained constant, thefill factor and shape of the crystal varied with depth. Thefill factor is the fraction of resist that remains after devel-oping. For top-side exposures, lower portions of the re-sist absorbed less energy than resist near the surface.
Fig. 5. Comparison of FCC photonic crystal in SU-8 and trans-mission spectra generated by comprehensive modeling and asimple threshold scheme. Position and width of bandgap aresignificantly different.
Top portions became more fully cross linked, developedmore slowly, and remained bulkier in the finished crystalthan portions near the bottom. Figure 6 shows photoniccrystals with varying degrees of chirp.
The effect of a chirped photonic crystal is a broadenedbandgap as shown in Fig. 7. This can be understood con-ceptually through effective-medium theory.64 For normalincidence, Bragg reflection occurs at a free-space wave-length of
Fig. 6. Cross section of three photonic crystals with varying de-grees of chirp. Higher absorption during exposure producesstronger lattice chirp.
R. C. Rumpf and E. G. Johnson Vol. 21, No. 9 /September 2004 /J. Opt. Soc. Am. A 1709
lBragg 5 2neffLz , (17)
where Lz is the period of the crystal in the direction ofpropagation and neff is the effective index. Effective-medium theory defines the effective index in terms of lo-cal fill factor and material properties of the crystal. Afirst-order approximation for the effective index is
neff2 ' fnres
2 1 ~1 2 f !nfill2 , (18)
where f is the local fill factor of the crystal, nres is the re-fractive index of the resist material, and nfill is the refrac-tive index of a material backfilled into the crystal, if any.From relation (18), it follows that for crystals that varywith depth, the effective index also varies with depth. Asthe effective index changes with depth, so does the peakreflected wavelength, lBragg . The bandgap is broadenedbecause different portions of the crystal have a slightlydifferent peak reflected wavelength. As observed in the
Fig. 7. Effect of chirped lattice on transmission spectra. Lat-tices with stronger chirp have broader bandgaps.
Fig. 8. Effect of reflections during exposure on transmissionspectra. Position and shape of bandgap are altered or the band-gap is completely eliminated.
transmission spectra of Fig. 8, crystals with higher de-grees of chirp showed broader bandgaps.
The position of the bandgap was different for eachchirped crystal that was simulated. The exposure dosewas selected such that the lattice structure would remainintact at the weakest part of the lattice. Given that thethickness, and therefore the effective index, at the bottomof the lattice was bounded, crystals with higher degrees ofchirp were bulkier at the surface and had higher averageeffective indices. This shifted the bandgaps to longerwavelengths.
Some control over the width of the bandgap wasachieved by selecting an exposure wavelength with appro-priate absorption to produce the desired amount of chirp.Shifting the exposure wavelength, however, also changedthe geometry and period of the crystal. For SU-8, the ab-sorption changes very abruptly below 365 nm, thus pro-viding a large adjustment of chirp with minimal changein geometry. The position of the bandgap can be con-trolled through the average fill factor by adjusting the ex-posure dose and exposure bias.
Practical limits exist for the extent to which a crystalcan be chirped. The fill factor at the top must be smallenough for developer solution to pass through. The fillfactor at the bottom must be sufficient to maintain struc-tural integrity and prevent release from the substrate.Simple thresholding schemes cannot accurately modelthis process, but a comprehensive model of the fabricationprocess can be used to detect and prevent these cases.
B. Reflections and Standing WavesDuring exposure, reflections at the air–resist and resist–substrate interfaces produce scattered and standingwaves in the resist. In simulation, this reduced exposurecontrast and distorted the lattice. Contrast was reducedwhen dark areas of the incident hologram became par-tially illuminated by scattered waves and bright areaswere dimmed by destructively interfering with scatteredwaves. The lattice was distorted when scattered wavesacted as additional beams in the holographic exposure.The effects of reflections and standing waves dependedheavily on the dissolution rate properties of the resist.SU-8 is somewhat robust to low-contrast exposures be-cause the low contrast is compensated for by a high-contrast dissolution rate curve that will be discussedbelow.
To investigate the effect of reflections and standingwaves, the refractive index of the substrate material wasmodified to control the reflection coefficient. It was foundthat reflections of '30% and above formed structures thatno longer resembled the original crystal. This does notmean that useful structures cannot be fabricated withstrong reflections. In fact, it is reasonable to assumethat many useful structures can be designed that actuallyrequire reflections, standing waves, or other distorting ef-fects to be present. In this case, comprehensive modelingis critical to the design of crystals.
C. Dissolution Rate PropertiesThree regions of the dissolution rate curve can be defined.In the first region, absorbed energy is insufficient to affectdissolution rate. The resist is fully soluble and can becompletely removed during developing. In the second, or
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notch, region dissolution rate decreases rapidly with ab-sorbed energy. High-contrast resists have very steepslopes. A form of gray-scale lithography is possible inthis region because solubility can be varied in an ‘‘analog’’manner to produce smooth structures. In the third re-gion, resist is completely cross linked, is insoluble, and re-mains completely intact after developing.
The dissolution rate curve had dramatic effects on thegeometry of formed crystals. The slope and width of thenotch region had the most significant impact. The otherparameters could be compensated for by tailoring the ex-posure time, exposure bias, and developing time. Inpractice, control of the dissolution rate curve can beachieved by selection of developer, selection of resist, andchemical modification of the developer and resist.
Rmax characterized the dissolution rate of unexposedportions of resist where developing was quickest. Thisparameter had the greatest effect on developing time.When this value was lowered, structures required moretime to develop. Crystal geometry was relatively unaf-fected if developing occurred for a sufficient period oftime. Rmin characterized the dissolution rate of fully ex-posed portions of resist where developing was slowest.The crystal lattice and processing variables were very ro-bust to changes in this parameter because values for itwere extremely small. It had to be modified by orders ofmagnitude before it had a noticeable effect. In such acase, it was very difficult to form crystals because the re-sist was much more soluble and tended to dissolve com-pletely. Eth characterized the position of the notch regionwhere dissolution rate fell rapidly. Crystal fill factor wasobserved to be quite sensitive to this parameter and itwas necessary to design the exposure such that the imagecontrast was modulated around the notch region. Othergeometry attributes were affected little by this parameter.Nnotch characterized the slope of the dissolution rate curve
Fig. 9. Relation of lattice geometry to image contrast and disso-lution curve of the resist. The alignment between exposure dosemodulation and the notch region determine the local fill factor ofthe crystal.
in the notch region and proved to be the most critical pa-rameter affecting crystal geometry. It had a dramatic ef-fect on both fill factor and chirp and can be more clearlyunderstood from the following discussion.
The combined effects of the dissolution rate curve andexposure dose can be understood and visualized throughFig. 9. The top graph in this figure shows the energy ab-sorbed in the resist as a function of depth along a verticalline. The bottom graph is the dissolution rate curve usedin the simulation. At the right of this figure is a simu-lated crystal illustrating how the final lattice geometrydepends on the relation between exposure dose, exposurecontrast, and the position and width of the notch region ofthe dissolution rate curve.
For holographic lithography, exposure dose varied con-siderably throughout the resist. For a crystal to formproperly, the modulation in exposure dose had to passthrough the notch region. When this was not the case,the resist would either dissolve completely or be com-pletely insoluble. The fill factor of the crystal dependedon the position of exposure modulation with respect to thenotch region. When the modulation was at the high sideof the notch region, a bulkier crystal (i.e., higher fill fac-tor) was formed because it developed slower. When themodulation was at the low side of the notch region, a crys-tal with finer features (i.e., lower fill factor) was formedbecause it developed faster. An exposure bias was thedominant control mechanism to align the notch regionwith exposure modulation to achieve a desired fill factor.
The formation of a chirped crystal can also be under-stood through Fig. 9. Optical absorption tapered thealignment between the notch region and center of expo-sure modulation with depth. This produced a crystal inwhich the fill factor tapered with depth in the same man-ner.
D. Multiple ExposuresHybrid lattices may be formed by successively exposing aresist with different patterns, essentially superimposingthe aerial images of each exposure. Predicting the finalstructure is not straightforward because the process ishighly nonlinear and each exposure serves as a nonuni-form bias for the others (i.e., cross-biasing).
A diamond lattice is formed by superimposing two FCClattices offset by a quarter lattice constant along eachcrystal axis.22 Synthesizing a diamond lattice in SU-8 bymultiple exposures was simulated with the comprehen-sive model. The resulting crystal and transmission spec-tra are shown in Fig. 10 and compared to correspondingresults obtained by the threshold method. The dose ofeach exposure was reduced by 48% to compensate forcross-biasing. Since the surface of the resist corre-sponded to the [111] plane of the lattice, the net offset be-tween exposures was 239.4 nm in the vertical direction.As a result of acid diffusion during PEB and close prox-imity of atoms, the diamond lattice resembled a FCC lat-tice elongated along its diagonal.
Next, a hybrid trigonal–FCC photonic crystal wasformed through simulation by successive exposures ofFCC and trigonal symmetries. The resulting crystal andtransmission spectra are shown in Fig. 11 and comparedwith results obtained by the threshold method. A trigo-
R. C. Rumpf and E. G. Johnson Vol. 21, No. 9 /September 2004 /J. Opt. Soc. Am. A 1711
nal lattice was chosen because it differed substantiallyfrom the FCC lattice in both period and geometry, butcould be exposed from the top side only in a similar expo-sure configuration. Here the angle between the centraland side beams was 70.5° so a prism and index matchingmaterials would again be necessary to fabricate thisstructure. The exposure dose was increased by 15% to
Fig. 10. Comparison of diamond photonic crystal in SU-8 andtransmission spectra generated by comprehensive modeling anda simple threshold scheme. Position and width of bandgap aresignificantly different.
Fig. 11. Comparison of trigonal photonic crystal in SU-8 andtransmission spectra generated by comprehensive modeling anda simple threshold scheme. Position and width of bandgap aresignificantly different. Exposure dose in comprehensive modelwas increased by 15% to increase fill factor and move bandgapinto the 600–900-nm range for comparison.
increase the fill factor of the trigonal lattice and move thebandgap to between 600 and 900 nm for easier compari-son.
Tunability in the process was demonstrated by simulat-ing three varieties of superimposed crystals. These wereformed by varying the relative doses of each exposure andare shown in Fig. 12. To form a lattice that was approxi-mately 75% FCC and 25% trigonal, an 81% dose was usedin the FCC configuration and a 27% dose in the trigonalconfiguration. These differed from 75% and 25% to com-pensate for cross-biasing. To form a lattice that was ap-proximately 50% FCC and 50% trigonal, a 54% dose wasused in each configuration. This crystal was verystrange in appearance and showed a double bandgap inthe transmission spectra resembling both the FCC andtrigonal crystals. The transmission spectra of the 50:50hybrid lattice is shown in Fig. 13 and is a confirmation
Fig. 12. Hybrid lattices formed by successive exposures of FCCand trigonal symmetries. Different hybrid lattices were formedby changing the relative doses of the two exposures. Percent-ages indicate dose relative to recipe for pure crystal.
Fig. 13. Hybrid photonic crystal (50% FCC, 50% trigonal)formed in SU-8 and resulting transmission spectra. Transmis-sion spectrum has a double bandgap showing properties of bothcomponent lattices.
1712 J. Opt. Soc. Am. A/Vol. 21, No. 9 /September 2004 R. C. Rumpf and E. G. Johnson
that the resulting crystal was a true combination of thetwo lattice types. The third crystal was exposed with a27% dose in the FCC configuration and 81% dose in thetrigonal configuration. It more closely resembled thetrigonal lattice.
4. CONCLUSIONThere exists a strong need to model the fabrication of pho-tonic crystals to more rigorously understand and predicttheir optical properties. Conventional intensity-threshold methods fail to predict lattice geometry ad-equately, especially when the dimensions of crystals aresmall. This is due to the dramatic effects many processparameters have on final lattice geometry and subse-quent behavior of the crystal. The first comprehensivemodel of holographic lithography for photonic crystalsknown to the authors was developed in this work to ad-dress this need. With this approach, photonic crystalscan be more rigorously designed and analyzed. Distort-ing artifacts may be better controlled and even exploitedto develop novel photonic crystals and other micro-opticstructures.
Experimental results were accurately duplicatedthrough simulation of a FCC photonic crystal formed inSU-8. The impact of optical absorption and scatteringduring exposure was investigated and discussed. It wasshown that holographic lithography inherently formschirped photonic crystals with broadened bandgaps. Re-flections and standing waves can lower the image con-trast and distort the lattice. It was shown that under theright conditions, hybrid photonic crystals can be formedby performing multiple exposures in the same resist.This method was simulated to form a diamond lattice andthree hybrid trigonal–FCC lattices. The transmissionspectra of a 50:50 trigonal–FCC lattice showed a doublebandgap with properties of both component lattices.
Comprehensive modeling of fabrication is seen as anenabling technology for the realization of optimized pho-tonic crystals formed by holographic lithography. If dis-torting effects are to be controlled and exploited to formnovel photonic crystals, rigorous process modeling is criti-cal. It was shown that simple threshold schemes ignorethe physics of lithography, tend to predict bulkier crys-tals, and fail to predict adequately the shape of atoms inthe lattice. This falsely shifts the bandgap to longerwavelengths and alters the shape of the bandgap edges.Similar treatment of other fabrication methods may proveto be equally valuable.
Corresponding author E. G. Johnson’s e-mail address [email protected].
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