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Photonic band-gap formation by optical-phase-mask lithography

Timothy Y. M. Chan, Ovidiu Toader, and Sajeev John

Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, M5S 1A7, Canada

?Received 2 November 2005; published 26 April 2006?

We demonstrate an approach for fabricating photonic crystals with large three-dimensional photonic band

gaps ?PBG’s? using single-exposure, single-beam, optical interference lithography based on diffraction of light

through an optical phase mask. The optical phase mask ?OPM? consists of two orthogonally oriented binary

gratings joined by a thin, solid layer of homogeneous material. Illuminating the phase mask with a normally

incident beam produces a five-beam diffraction pattern which can be used to expose a suitable photoresist and

produce a photonic crystal template. Optical-phase-mask Lithography ?OPML? is a major simplification from

the previously considered multibeam holographic lithography of photonic crystals. The diffracted five-beam

intensity pattern exhibits isointensity surfaces corresponding to a diamondlike ?face-centered-cubic? structure,

with high intensity contrast. When the isointensity surfaces in the interference patterns define a silicon-air

boundary in the resulting photonic crystal, with dielectric contrast 11.9 to 1, the optimized PBG is approxi-

mately 24% of the gap center frequency. The ideal index contrast for the OPM is in the range of 1.7–2.3. Below

this range, the intensity contrast of the diffraction pattern becomes too weak. Above this range, the diffraction

pattern may become too sensitive to structural imperfections of the OPM. When combined with recently

demonstrated polymer-to-silicon replication methods, OPML provides a highly efficient approach, of unprec-

edented simplicity, for the mass production of large-scale three-dimensional photonic band-gap materials.

DOI: 10.1103/PhysRevE.73.046610PACS number?s?: 42.70.Qs

I. INTRODUCTION

Photonic band-gap ?PBG? materials ?1,2? are periodically

ordered dielectric microstructures which forbid electromag-

netic waves of a certain spectral region from propagating in

the crystal. The most profound properties of these artificial

materials arise from their ability to trap or localize light ?3?.

These photonic crystals provide a robust platform for inte-

grating active and passive devices in an all-optical microchip

?4?. In order to realize an optical microchip, capable of lo-

calizing and micromanipulating light, it is necessary to have

high-quality, three-dimensional ?3D? PBG materials. Effi-

cient, large-scale microfabrication of PBG materials, with

high accuracy and low cost, has been a major scientific and

technological challenge over the past decade. The difficulties

in large-scale microfabrication of 3D architectures have led

to extensive studies of alternative 2D photonic crystal mem-

brane architectures. While 2D photonic crystals are more

amenable to conventional methods of semiconductor mi-

crolithography, they lack the most profound properties of the

photonic band gap: namely, complete localization of light

and control over the electromagnetic density of states. In this

paper, we suggest that the large-scale microfabrication of 3D

photonic band-gap materials is considerably simplified using

optical-phase-mask lithography ?OPML?. We describe the

design of optical phase masks ?OPM’s? that reduce the task

of large-scale and repetitive synthesis of PBG materials, with

photonic band gaps as large as 24% relative to center fre-

quency, to two simpler tasks. The first is the illumination

?single exposure? of a photoresist material with a single laser

beam at normal incidence to the phase mask and sample

surface. The second is the replication of the “developed”

photoresist with a high-refractive-index semiconductor, such

as silicon, using previously established methods ?5?.

The diamond lattice structure has been shown theoreti-

cally ?6? to be the quintessential architecture for creation of a

large 3D PBG. This discovery has spurred several theoretical

blueprints and subsequent fabrication attempts for photonic

crystals based on “diamondlike” structures employing non-

spherical bases on an fcc lattice. One diamondlike architec-

ture is the layer-by-layer “woodpile” structure comprised of

stacked two-dimensional photonic crystals ?7,8?, which can

have a PBG approximately 18% of the gap center frequency.

Techniques such as repetitive deposition and etching of sili-

con ?9,10?, wafer-fusion and laser-assisted alignment ?11?,

and nanofabrication of the two-dimensional layers followed

by microassembly of the layers ?12? have been used to pro-

duce high-quality woodpile structures with PBG’s in the op-

tical regime. Unfortunately, these samples are only a few

periods deep in the stacking direction. Recently, “direct laser

writing” processes involving two-photon absorption ?causing

polymerization? in resins have been used to produce wood-

pile structures as a proof of concept ?13–15?. An alternative

approach towards fabricating diamondlike structures uses

glancing-angle deposition methods ?16? to form silicon

square spiral posts on a silicon substrate. A PBG as large as

24% of the gap center frequency has been predicted for suit-

ably architectured spiral structures ?17,18?. Optical reflectiv-

ity in a weakly disordered version of these silicon square

spiral crystals has revealed a 3D PBG of roughly 10% rela-

tive to the center frequency ?19?. An fcc lattice of crisscross-

ing pores is another diamondlike architecture which has been

shown to exhibit a 3D PBG ?20?. Attempts to fabricate this

structure on a submicron-scale have included electron beam

lithography followed by reactive ion etching ?21,22?, deep

x-ray lithography ?LIGA? patterning of an x-ray sensitive

resist ?23?, and photoelectrochemical etching followed by

focused-ion-beam etching ?24?. However, the first method

produced only a few periods of the structure, with severe

imperfections at the pore crossing points, while samples cre-

ated by the latter two methods have had feature sizes too

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large for a PBG in the optical regime. Recently, new “slanted

pore” architectures have been introduced ?25? whose simpler

geometries may facilitate their fabrication by various pore

etching methods. Another approach towards the fabrication

of 3D photonic crystals relies on the colloidal self-assembly

of silica spheres into an fcc opal lattice. The silica spheres

are used as a template which is inverted by chemical vapor

deposition of silicon, followed by selective etching of the

silica template ?26–28?, in a process which can be performed

at large scales. In contrast to the approaches already men-

tioned, this “inverse opal” architecture cannot be character-

ized as diamondlike, and as a result, the PBG is only 9% of

the gap center frequency ?29? and vulnerable to disorder

?30?, necessitating that the fabrication methods yield very-

high-quality structures.

Recently, the holographic lithography method ?31–33? has

been suggested as an alternative approach to large-scale syn-

thesis of 3D photonic crystals with large PBG’s in the optical

regime. In this approach, a 3D intensity pattern formed by

the interference of four or more laser beams exposes a pho-

topolymerizable material such as a photoresist. The photore-

sist undergoes a chemical alteration when the total light in-

tensity at position r ? due to the interference pattern, I?r ??, is

maintained over a time ?? such that the “exposure” I?r ????

exceeds a specified threshold T. For negative photoresists,

the “underexposed” regions can then be selectively removed

using a developer substance which leaves the “overexposed”

regions intact. ?For positive photoresists, the overexposed

regions are removed and the underexposed regions remain

after developing.? The developed material can then be infil-

trated at room temperature with SiO2?34? and burned away,

leaving behind a daughter “inverse” template. Finally, the

daughter template is inverted by high-temperature infiltration

with silicon ?27,35? and selective chemical etching of the

SiO2. As a result, a 3D silicon photonic crystal is formed, in

which the silicon-air boundary is defined by the original,

optical isointensity surface I?r ????=T. Most previous theoret-

ical reports have discussed the formation of 3D photonic

crystals by holographic lithography based on single exposure

of the photoresist by the interference pattern of four laser

beams. It has been shown ?36–39? that using configurations

of this form, it is possible to produce a diamondlike structure

with a PBG approximately 25% of the center frequency

when synthesized with a material with a dielectric constant

of 11.9, corresponding to Si. Despite the promise of multi-

beam interference lithography, the precise alignment of four

laser beams from different directions is experimentally in-

convenient. Restricting all four beams to be launched from

the same half-space ?umbrella setup? reduces the PBG sig-

nificantly ?40,41?.

In this paper, we circumvent these complications and

drawbacks using an interference pattern generated by a

single beam diffracting through a carefully designed phase

mask ?42?. The use of diffractive interference patterns from

phase masks has previously been shown to produce dia-

mondlike photonic crystals provided that two independent

optical exposures are performed with two separate positions

of the phase mask ?43?. However, realignment of the second

exposure with the first exposure is a daunting experimental

challenge. Simple fcc structures that do not lead to a large

PBG can be achieved with a single exposure ?44?. Here, we

introduce a novel approach to create a diamondlike structure

by OPML based on single exposure of a photoresist by a

laser beam leading to a five-beam interference pattern. We

present a phase mask design that yields a diamondlike struc-

ture with a PBG of 24% of the gap center frequency when

synthesized with a material with a dielectric constant of 11.9.

In Sec. II we introduce the target five-beam intensity pattern

and its relation to the intensity pattern emerging from the

diffraction of a single beam through an OPM. In Sec. III, we

describe the proposed OPM architecture. In Sec. IV we show

several phase mask geometries that can produce photonic

crystals with large PBG’s and we study the effects on the

resulting PBG when the phase mask parameters are varied

and the polarization of the incident beam is varied. In par-

ticular, we find that when the photoresist has a refractive

index corresponding to 1.67 ?undeveloped SU-8 at 355-nm

wavelength ?45??, the ideal index contrast for the OPM is in

the range of 1.7–2.3.

II. DIFFRACTION OF LIGHT BY OPTICAL PHASE

MASKS

Consider a single, monochromatic beam with vacuum

wavelength ?0and wave vector k?0=−2?/?0z ˆ, normally inci-

dent onto an optical phase mask and exposing a region with

refractive index n?−?below the mask. The phase mask is as-

sumed to have a square Bravais lattice symmetry, with lattice

constant a, finite thickness in the z direction, and mirror

planes normal to x ˆ and y ˆ. For a uniform incident beam with

infinite extent in the x and y directions, the electric field in

the exposure region consists of the unscattered beam with

wave vector G?00=−2?n?−?/?0z ˆ and diffracted beams with

wave-vectors G?mn=2?/a?m,n,−?mn?, where m and n are ar-

bitrary integers. These diffracted beams have wave-vector

components in the xy plane, 2?/a?m,n,0?, corresponding to

the Fourier components of the OPM dielectric profile. The z

component of the wave vector of mode ?m,n? is determined

by the condition of energy conservation ?k?0?=?G?mn?/n?−?:

?mn= ± ??2− m2− n2?1/2,

? ? an?−?/?0.

?1?

In general, ?mncan be a complex number, describing either

propagating or evanescent beams in the region of the photo-

resist ?z?0?. From earlier studies of multibeam optical in-

terference lithography ?38,46?, it is desireable to have five

and only five propagating modes in the exposure region:

namely, the ?0,0?, ?0,±1?, and ?±1,0? modes, with corre-

sponding wave vectors

G?00=2?

a?0,0,− q0?,

G?ij=2?

a?i,j,− q1?,

?2?

where i¯?−i and j¯?−j. Here we have defined

qj???2− j.

?3?

In order for these modes to propagate in the exposure region,

we require ?an?−?/?0?2?1, or ?0/a?n?−?. The next lowest-

order modes are the ?±1,±1? modes, which have wave vec-

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tors of the form G??11?=2?/a?±1,±1,−q2?. For these ?and all

higher-order? modes to be evanescent in the exposure region,

we require ?an?−?/?0?2?2, or ?0/a?n?−?/?2. Therefore, in

order to retain only the desired modes, the incident beam

wavelength is constrained by

n?−?/?2 ? ?0/a ? n?−?.

Provided that the constraint in Eq. ?4? is satisfied, then an

intensity pattern is produced corresponding to the interfer-

ence pattern of five monochromatic plane waves of fre-

quency ?, wave vectors G?lcorresponding to Eq. ?2?, polar-

ization vectors E?l, and phases ?l, where l is in the set L

=?00,10,1¯0,01,01¯?. The electric field associated with this

interference pattern is given by

E??r ?,t? = e−i?tei?G?00·r ?+?00??E?00+?

?4?

l?00

E?lei?K?l·r ?+?l??,

?5?

where K?land ?lare defined as

K?l? G?l− G?00,

?l? ?l− ?00.

?6?

The corresponding intensity pattern is given by

I?r ?? = I0+ 2?

l?00

+ 2?

l??l

?00·lcos?K?l· r ? + ?00·l+ ?l?

l?00?

?l?·lcos??K?l− K?l?? · r ? + ?l?·l+ ?l− ?l??,

?7?

where

I0??

l

?E?l?2

?8?

and

?l?·l? ?E?

l?

*· E?l?,

?l?·l? arg?E?

l?

*· E?l?.

?9?

The spatial modulation of this intensity pattern is periodic

with a Bravais lattice whose reciprocal lattice vectors can be

represented byanythree

=?K?10,K?1¯0,K?01,K?01¯?:

K?ij=?2?i

a

vectorsfrom theset

K

,2?j

a

,2?

c?,

?10?

where i¯?−i and j¯?−j. Here we have defined

c ? a?q0− q1?−1= a?? −??2− 1?−1.

?11?

It is easy to verify that the fourth vector from K can then be

written as a linear combination, with integral coefficients, of

the other three vectors in the set. This reveals that the five-

beam intensity pattern resulting from diffraction of light

through the OPM has tetragonal Bravais lattice symmetry,

with aspect ratio c/a.

The form of the intensity pattern can be simplified by

symmetry arguments. For a normally incident beam and a

phase mask with reflection symmetry in both x and y, as

considered here, the electric-field components and phases

satisfy the following symmetries:

E10

x= E1¯0

x,

E10

y= E1¯0

y,

E10

z= − E1¯0

z,

E01

x= E01¯

x,

E01

y= E01¯

y,

E01

z= − E01¯

z,

?10= − ?1¯0,

?01= − ?01¯.

?12?

The relations involving the phases ? imply that they can be

eliminated by a translation of the origin, r ?→r ?−?1?, where

?1?=a/2???10,?01,0?. Using the fact that the unscattered

?central? beam is transverse ?E00

volving the E-field components imply that

z=0?, the relations ?12? in-

?00·10= ?00·1¯0? c1,

?00·10= ?00·1¯0? ?1,

?00·01= ?00·01¯? c2,

?00·01= ?00·01¯? ?2,

?10·01= ?1¯0·01¯? c3,

?10·01= ?1¯0·01¯? ?3,

?10·01¯= ?1¯0·01? c4,

?10·01¯= ?1¯0·01? ?4.

?13?

Here the relations involving ?iare modulo 2?. Defining c5

??1¯0·10, ?5??1¯0·10, c6??01¯·01, and ?6??01¯·01and noting

that ?K?01¯−K?1¯0?=?K?01−K?10? and ?K?01−K?1¯0?=?K?01¯−K?10?, we

can write the intensity pattern in the photoresist as

I?r ?? = I0+ 2?c1cos?K?10· r ? + ?1? + c1cos?K?1¯0· r ? + ?1?

+ c2cos?K?01· r ? + ?2? + c2cos?K?01¯· r ? + ?2?

+ 2c3cos ?3cos??K?01− K?10? · r ??

+ 2c4cos ?4cos??K?01¯− K?1¯0? · r ?? + c5cos??K?10− K?1¯0? · r ?

+ ?5? + c6cos??K?01− K?01¯? · r ? + ?6??.

?14?

It has been previously shown ?46? that structures with large

PBG’s can be created by intensity patterns of the form

I?r ?? = I0+ C?cos?K?10· r ?? + cos?K?1¯0· r ?? + cos?K?01· r ??

− cos?K?01¯· r ???,

?15?

where C is a real number.1In order to write Eq. ?14? in this

form, we make another translation r ?→r ?−?2?, such that ?2?

satisfies K?10·?2?=?1, K?1¯0·?2?=?1, and K?01·?2?=?2. It is easy

to verify that the translation ?2?=(0,??2−?1?a/2?,?1c/2?)

provides the above properties. Under this change of coordi-

nates, the intensity pattern becomes

1In Ref. ?46? the vectors are written in terms of a different coor-

dinate system.

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I?r ?? = I0+ 2?c1cos?K?10· r ?? + c1cos?K?1¯0· r ??

+ c2cos?K?01· r ?? + c2cos?K?01¯· r ? + 2??2− ?1??

+ 2c3cos ?3cos??K?01− K?10? · r ? + ??1− ?2??

+ 2c4cos ?4cos??K?01¯− K?1¯0? · r ? − ??1− ?2??

+ c5cos??K?10− K?1¯0? · r ? + ?5? + c6cos??K?01− K?01¯? · r ?

+ ?6+ 2??1− ?2???.

?16?

Finally, in order to achieve an intensity pattern correspond-

ing to Eq. ?15?, we desire to find a phase mask which pro-

duces modes whose corresponding E fields satisfy

c1= c2,

?2− ?1= ±?

2,

mod 2?,

2cjcos ?j= 0,

j = 3,4,

c5= c6= 0.

?17?

Equation ?17? provides a target pattern against which actual

intensity patterns from various phase masks can be evalu-

ated. For a given intensity pattern, we search for the largest

achievable PBG by calculating the photonic bands for struc-

tures whose solid-air boundaries are defined by several isoin-

tensity surfaces of the pattern. The optimal intensity thresh-

old is then defined as one whose isointensity surface yields

the largest PBG when the developed photoresist is replaced

with silicon. However, by comparing the coefficients of in-

tensity patterns to those in Eq. ?17?, one can save computa-

tional effort by discarding those intensity patterns which dif-

fer greatly from the target intensity pattern.

III. OPTICAL-PHASE-MASK ARCHITECTURE

We demonstrate the ability of a three-layer phase mask, as

shown in Fig. 1, to achieve the target intensity pattern de-

scribed in Sec. II. The phase mask consists of two identical,

orthogonally arranged, one-dimensional binary gratings

separated by a homogeneous slab of thickness t with refrac-

tive index na. The motivation for choosing such a design

comes from its simplicity and its flexibility through variation

of design to deliver target diffraction patterns.

The ideal intensity pattern described by Eqs. ?16? and

?17?, shown in Fig. 2, consists of four repeating intensity

slices along the tetragonal direction of the unit cell. This is

characteristic of diamondlike structures. The four slices are

labeled as A, B, C, and D in Fig. 2. Slices A and C differ by

an in-plane translation, and slices B and D are a 90° rotation

from slices A and C. In a very crude picture, the top layer of

the proposed phase mask can be thought of as creating a

two-dimensional diffraction pattern which generates the A

and C slices of the desired shape, while the bottom layer

generates a similar diffraction pattern that is rotated by 90°

and translated in the vertical direction. The homogeneous

OPM layer separating the binary gratings is introduced as a

mechanism to control the translation between the diffraction

patterns created by the two orthogonal, one-dimensional

gratings so that the spacing between the slicesA, B, C, and D

is appropriate. However, this simple picture provides only a

rough guide to the overall diffraction pattern. It does not

account for the effects of reflections at layer boundaries and

interference between the two orthogonal diffraction patterns

of the separated grating layers of the OPM. Therefore the

true intensity pattern must be calculated carefully.

Here, the diffraction-interference pattern created by the

phase mask is calculated using the Fourier modal method

?47? on a 1024?1024 grid and truncation order 441. In each

layer of the OPM and in the homogeneous regions above and

below, the electromagnetic field is expanded in terms of

modes whose wave-vector x and y components correspond to

the Fourier components of the OPM dielectric profile in the

xy plane. In regions that are homogeneous in x and y, the z

component of the wave vector of each mode is given by Eq.

?1?. In regions where the dielectric profile is periodic in the

xy plane, the z component of the wave vector is calculated by

Fourier expansion of the field in Maxwell’s equations. The

FIG. 1. Schematic representation of a three-layer phase mask.

The top and bottom layers are orthogonally oriented binary grat-

ings, and the middle layer is a homogeneous slab.

FIG. 2. ?Color online? Four unit cells ?one unit cell in the ver-

tical direction? of an isointensity surface in the target intensity pat-

tern, I?r ??=I0+C?cos?K?10·r ??+cos?K?1¯0·r ??+cos?K?01·r ??−cos?K?01¯·r ???.

The three planes indicate positions at c/4, 2c/4, and 3c/4 along the

tetragonal direction.

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resulting diffraction-interference pattern is obtained by

matching boundary conditions at the interfaces between the

layers. Photonic band structures are calculated with the

plane-wave expansion method ?6? using over 1440 plane

waves, while Fourier coefficients of the dielectric structure

are calculated using a discrete Fourier transform with 512

points per direction.

The binary grating layers of the phase mask, each of

thickness d, consist of alternating regions of refractive index

naand nbwith periodicity a. The naregions have width w

and the nbregions have width ?a−w?. We characterize a

given phase mask geometry by the set of adjustable param-

eters ?w,d,t?. All lengths are expressed in units of a, the

lattice constant of the phase mask and of the resulting pho-

tonic crystal. For concreteness, the refractive indices of the

regions above the phase mask ?from which the incident beam

is launched? and below the phase mask ?the region to be

exposed? are n?+?=1, corresponding to air, and n?−?=1.67,

corresponding to undeveloped SU-8 ?45?, respectively. We

restrict our discussions to configurations that produce an in-

tensity pattern with an aspect ratio c/a=?2, corresponding to

a fcc Bravais lattice. This ratio has been shown ?46? to maxi-

mize the PBG. Accordingly, we take the vacuum wavelength

of the incident beam to be ?0=?2?2n?−?a?/3. The polariza-

tion vector of the incident beam is characterized by

E?inc= cos???x ˆ − ei?sin???y ˆ,

?18?

where x ˆ and y ˆ are unit vectors in the x and y directions,

respectively, ? is the linear polarization angle from x ˆ as mea-

sured looking along the incident beam, and ? is an “elliptic-

ity” angle which indicates the phase delay between the x and

y polarization components. In order to simplify the problem,

we choose a linearly polarized incident beam ??=0?. The

symmetry of the target diamond structure suggests that we

choose ?=45°, so that the diffracted beams in the x and y

directions have equal intensities. Intensities are given in units

of the incident beam intensity.

IV. PHOTONIC BAND-GAP ARCHITECTURES

A. Direct structures

We first discuss photonic crystal structures consisting of

solid material in the regions of high light intensities ?above

the threshold of the photoresist? and air in the regions of low

light intensity ?below the threshold?. This, for example, cor-

responds to the case when a double-inversion process ?5? is

used with a negative photoresist or a single-inversion process

is used with a positive photoresist. For illustration, we con-

sider the case in which the grooves of the phase mask consist

of air, so that nb=1.

Figure 3 shows an iso-intensity surface in the interference

pattern createdbya phase

=?0.50,0.50,0.90? and na=2.00. The diamondlike character-

istics of the interference pattern are apparent in the dielectric

“nodes” connected to their nearest neighbors by tetrahedral

“bonds.” The five beams created in the photoresist by illu-

mination of the phase mask from above have the wave vec-

tors given in Eq. ?2?. The polarization vectors of these beams

maskwith

?w,d,t?

are determined by the Fourier modal method and are given

by

E?00= ?0.176+ i0.283,− 0.145− i0.303,0?,

E?10= ?− 0.110+ i0.002,0.218− i0.147,− 0.310+ i0.004?,

E?1¯0= ?− 0.110+ i0.002,0.218− i0.147,0.310− i0.004?,

E?01= ?0.129+ i0.202,− 0.004− i0.083,− 0.011− i0.235?,

E?01¯= ?0.129+ i0.202,− 0.004− i0.083,0.011+ i0.235?.

?19?

The symmetry of the intensity pattern, Eq. ?19?, compares

favorably with the target symmetry in Eq. ?17?. Quantita-

tively,

?c1,c2,cos??1− ?2?,2c3cos ?3,2c4cos ?4,c5,c6?

= ?0.119,0.106,0.044,0.000,0.010,0.015,0.009?.

?20?

High-quality materials synthesis requires that the contrast in

the optical diffraction pattern, between the highest-light-

intensity regions and the lowest-light-intensity regions, be

maximum. This makes the process less vulnerable to random

fluctuations causing unwanted disorder in the developed pho-

toresist. A dynamic range of the intensity pattern which is the

FIG. 3. ?Color online? The isointensity surface at Ithr=1.10 in

the intensity pattern created by a phase mask with ?w,d,t?

=?0.50,0.50,0.90?, na=2.00, and nb=1. The volume fraction of the

region inside the surface is ?24%. When the high-intensity regions

are replicated with silicon in an air background, the resulting struc-

ture displays a 24% 3D PBG.

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largest possible fraction of the background intensity I0leads

to the most effective exposure and development of the pho-

toresist. Accordingly, we define the intensity contrast as

C ? ?Imax− Imin?/?2I0?.

?21?

The intensity pattern in Fig. 3 has a background intensity

I0=0.818, calculated from the polarization vectors according

to Eq. ?8?, and reaches a minimum value Imin=0.204 and a

maximum value Imax=1.500. The intensity contrast is there-

fore C=0.762. For comparison, the maximum intensity con-

trast that can be achieved using counterpropagating four-

beam interference lithography to generate a diamondlike

photonic crystal is 0.816 when elliptical polarizations are

allowed and 0.472 when only linear polarizations are used

?46?.

Figure 4 shows the solid volume fraction and the PBG of

the silicon replica of the developed photoresist ?with a cor-

responding dielectric constant of 11.9? as functions of the

threshold isointensity Ithrfor photopolymerization. The map-

ping of the percentage volume fraction f to the isointensity

value can be approximated as f=−93Ithr+126. The optimized

structure has a solid volume fraction of 24% and yields a 3D

PBG of spectral width ??/?0=24%.

The photonic band structure corresponding to the opti-

mized structure is shown in Fig. 5. The 3D PBG opens be-

tween bands 4 and 5, and is centered at a/?0=0.38. The gap

is bounded on the upper edge at the R point and on the lower

edge somewhere on the ?-X segment.

It is important to consider the robustness and sensitivity

of the PBG to changes in the structural phase mask param-

eters. The lattice constant of the phase mask grooves is equal

to the lattice constant of the photonic crystal in the xy plane.

It follows that to produce a structure with a PBG in the

optical regime, the phase mask must have submicron fea-

tures. Typically, a 3D PBG centered at 1.5 ?m made from a

silicon photonic crystal requires a lattice constant a

?600 nm.

Figure 6 shows the change in the photonic band-gap

edges of the resulting structure when the phase mask geom-

etry deviates from ?w,d,t?=?0.50,0.50,0.90?, for a fixed

FIG. 4. Plot of the solid volume fraction and the relative PBG as

functions of the threshold iso-intensity for photopolymerization, for

the intensity pattern in Fig. 3. The high-intensity regions are as-

sumed to map to a dielectric constant of 11.9 ?corresponding to

silicon? while the low-intensity regions correspond to air.

FIG. 5. ?Color online? Photo-

nic band structure diagram for the

optimized structure created by the

intensity pattern in Fig. 3 charac-

terized by 11.9:1 dielectric con-

trast. The inset shows the posi-

tions of the high-symmetry points

in the reciprocal lattice. A 3D

PBG ofwidth

opens between bands 4 and 5.

??/?0=24%

FIG. 6. The photonic band edges as functions of the change in

the geometry of the phase mask. The deviation is measured a dif-

ference from a phase mask with ?w,d,t?=?0.50,0.50,0.90?. The

photopolymerization intensity threshold is held constant, and the

structure is characterized by a 11.9:1 dielectric contrast.

CHAN, TOADER, AND JOHNPHYSICAL REVIEW E 73, 046610 ?2006?

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Page 7

value of the photopolymerization intensity threshold. For

small deviations in d and in t the effects are similar, since

both cause changes in the path length of light passing

through the phase mask. However, changes in d have the

additional effect of changing the strength of the diffraction

by the phase mask. Therefore, at larger deviations, the effect

of a change in d differs from that of a change in t.

Figure 7 shows an isosurface plot of the magnitude of the

PBG in three-dimensional space of ?w,d,t?, for a direct sili-

con structure created from a phase mask with na=2.0. The

isosurface shown in Fig. 7 corresponds to a gap size

??/?0=20%. Regions inside the depicted surface represent

structures with PBGs larger than 20% relative to the gap

center frequency. At constant values of the groove width w,

when each grating thickness d is increased, the grating sepa-

ration t must be decreased in order for the geometry to re-

main in the large PBG region. The size of the PBG tends to

be more sensitive to changes in t than in d and w. The mini-

mum sensitivity to t is reached for values of w between 0.50

and 0.55.

The variation of the intensity pattern ?and PBG of the

resulting structure? with changes in the incident beam char-

acteristics is also of importance. Although we do not make

use of the incident beam as a design parameter here, it is

worthwhile, for practical purposes, to know the robustness of

the diffraction-interference pattern ?and resulting PBG?

against deviations in the incident beam parameters.

First, we consider the effect of changing the linear polar-

ization angle ? on the relative PBG size and band-edge po-

sitions. The results are shown in Fig. 8. The relative PBG

magnitude remains above 10% over a change in ? of ±5°.

Removing the linear polarization restriction on the inci-

dent beam provides further design flexibility by allowing the

ellipticity angle ? in Eq. ?18? to vary. For illustration, we

consider a simple path in the two-dimensional polarization

parameter space by fixing ?=45° and allowing ? to vary.

Figure 9 shows the relative PBG size and band edge posi-

tions as ? is varied from 0° ?corresponding to the linear

polarization? to ±90° ?corresponding to two orthogonal cir-

cular polarizations?. The PBG does not close at any value of

?, although the relative PBG size drops to just above 10%

when the incident beam is circularly polarized. This indicates

that the intensity pattern is more dependent on the relative

amplitudes of the polarization vector components in the x ˆ

and y ˆ directions than on the relative phase between the two

components.

The refractive index of the phase mask is a particularly

important design consideration.

Figure 10 shows the PBG size in ?w,d,t? space with

groove width w=0.50, using different phase masks with re-

fractive indices of na=1.95, na=2.00, and na=2.05. For each

of the refractive indices, we identify regions in the ?d,t?

parameter space where structures with large PBG’s can be

generated upon replication with silicon. As one decreases the

refractive index of the phase mask from na=2.00 to na

FIG. 7. ?Color online? Isosurface plot of the magnitude of the

PBG in three-dimensional space of ?w,d,t? for a direct silicon

OPML photonic crystal created from a phase mask with na=2.0.

The isosurface corresponds to a gap size of ??/?0=20%. The rela-

tive PBG is larger than 20% inside the volume.

FIG. 8. The relative PBG size and band-edge positions as func-

tions of the linear polarization angle ? of the incident beam. The

beam passes through a phase mask with geometry ?w,d,t?

=?0.50,0.50,0.90?, and the resulting intensity pattern is synthesized

with a 11.9:1 dielectric contrast at a photopolymerization intensity

threshold of 1.10. The PBG magnitude remains above 10% over a

change in ? of ±5°.

FIG. 9. The relative PBG size and band-edge positions as func-

tions of the incident beam polarization “ellipticity” ? for structures

createdusinga phasemask

=?0.50,0.50,0.90? and synthesized with a material with dielectric

constant 11.9. When ?=0°, the incident beam is linearly polarized,

with linear polarization angle ?=45°, while at ?=±90°, the inci-

dent beam is circularly polarized.

withgeometry

?w,d,t?

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Page 8

=1.95, a shallower grating depth d and larger separator layer

thickness t are required to compensate and produce struc-

tures with large PBG’s. On the other hand, as the refractive

index is increased from na=2.00 to na=2.05, not only does

the large-PBG region move to larger groove depth d and

smaller grating separation t, but the maximum PBG size also

drops to about 18%. It is important to note that for high

refractive indices, even with small changes in the refractive

index of the phase mask, the size of the PBG can vary

greatly—for example, a phase mask geometry with d=0.50,

t=0.90, and na=2.0, which produces a structure with a 24%

PBG at Ithr=1.10. The same mask structure produces only a

15% PBG for na=2.05 and does not even produce a PBG

when na=1.95. It is therefore important to choose phase

mask structure according to the particular composition of the

mask.

It is of considerable importance to define an ideal range of

refractive index contrasts for the optical phase mask. This is

governed by a trade-off between high intensity contrast in

the interference-diffraction pattern and robustness of the op-

tical interference pattern to random perturbations in the OPM

architecture. Both of these factors are important for the de-

velopment of high-quality PBG materials with minimal dis-

order.

Figure 11 shows the relative gap size of structures result-

ing from phase masks with groove width w=0.50 and refrac-

tive indices na=1.7, na=2.3, and na=2.5, with corresponding

threshold intensities Ithr=0.78, Ithr=1.37, and Ithr=0.93.

When na=1.7 and na=2.3, there are large-PBG regions in

?w,d,t? space, comparable in size to the regions shown for

na=2.0 in Fig. 10. However, when the refractive index of the

phase mask increases to na=2.5, the large-PBG regions in

the ?w,d,t? space shrink considerably. This implies that the

interference-diffraction pattern is very sensitive to small per-

turbations in the phase mask structure. For phase masks with

FIG. 10. ?Color online? The relative PBG size of structures cre-

ated by different phase masks with refractive indices ?a? na=1.95,

?b? na=2.00, and ?c? na=2.05 in ?w,d,t? space with groove width

w=0.50. The shade in a rectangle corresponds to the PBG size at

the value of the grating depth d and separator layer thickness t at the

lower left corner of the rectangle, according to the bar on the right.

The photopolymerization intensity threshold is set at a value of

1.10. Regions exposed to intensities above the threshold are repli-

cated with a material with dielectric constant 11.9.

FIG. 11. ?Color online? PBG size of structures created by phase

masks with refractive indices ?a? na=1.7, ?b? na=2.3, and ?c? na

=2.5, with groove width w=0.50, and various values of grating

depth d and separator thickness t. The corresponding photopolymer-

ization threshold intensities are ?a? Ithr=0.78, ?b? Ithr=1.37, and ?c?

Ithr=0.93. Regions exposed to intensities larger than Ithrare repli-

cated with silicon ?dielectric constant 11.9?. The shade of a rect-

angle in the figure corresponds to the PBG size resulting from a

phase mask with values of d and t at the lower left corner of the

rectangle ?the rectangle edges are omitted for clarity?.

CHAN, TOADER, AND JOHN PHYSICAL REVIEW E 73, 046610 ?2006?

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Page 9

nagreater than 2.5, this undesirable sensitivity worsens and

large-PBG regions in ?w,d,t? space are no longer robust to

manufacturing error in the OPM. On the other hand, when

the refractive index of the phase mask is decreased below

na=1.7, the intensity contrast of the resulting interference

pattern is lowered. This in turn makes the photonic crystal

architecture more susceptible to disorder arising from inho-

mogeneities in the photoresist and small random variations

in the photopolymerization threshold.

B. Inverse structures

In the previous section, we considered photonic band-gap

architectures that were high-refractive-index replicas of the

optical diffraction-interference pattern where the optical in-

tensity exceeds a prescribed threshold value. Such “direct

structures” could be made by silicon double inversion ?5? of

an SU-8 polymer photoresist exposed by the optical interfer-

ence pattern or by a single step in certain chalcogenide

glasses that are amenable to direct photopolymerization ?48?.

On the other hand, the situation may arise wherein the final

photonic crystal structure will be defined by high-refractive-

index material in the low-intensity regions of OPML and

consist of air in the high-intensity regions of the OPM

diffraction-interference pattern. We refer to these as “inverse

structures.” For example, use of a negative photoresist and a

single-inversion process or a positive photoresist and a

double-inversion process will yield such results. For the tar-

get intensity pattern described in Eq. ?16? with the coeffi-

cients given in Eq. ?17?, the equivalent but inverted structure

?solid and air regions interchanged? is defined by the isoin-

tensity value Ithr obtained by the “transformation” Ithr

→−Ithr+2I0?to within a translation of ?a/2,a/2,c??: Con-

sider the shape functions Sdof the direct structure and Siof

the inverted structure, defined by

Sd?r ?,Ithr? = ?„I?r ?? − Ithr…,

Si?r ?,Ithr? = ?„Ithr− I?r ??….

Here ??x? is the Heaviside step function ???x?=1 for x?0

and 0 otherwise?. When I?r ?? corresponds to the idealized,

target intensity pattern in Eq. ?16? with coefficients Eq. ?17?,

taking r ??=r ?−?a/2,a/2,c? implies I?r ???=−I?r ??+2I0. We can

therefore write

Si?r ??,− Ithr+ 2I0? = ?„− Ithr+ I?r ??… = Sd?r ?,Ithr?.

For this specific, idealized intensity pattern, the PBG struc-

ture can be achieved using either fabrication algorithm,

yielding solid material in high-intensity regions ?direct struc-

ture? or yielding solid material in the “transformed” low-

intensity regions ?inverse structure?. However, this equiva-

lence is specific to the idealized, target intensity. When the

coefficients in the intensity pattern do not match the targets

in Eq. ?17?, then there is no simple “transformation” to

equivalent but inverted structures at different intensity

thresholds. Since OPML intensity patterns in this paper differ

slightly from the idealized, target pattern ?see Eq. ?20??, it is

necessary to consider separately the case of “inverse struc-

tures.”

?22?

?23?

With the understanding that the actual OPML intensity

pattern differs slightly from the target pattern, we revisit the

phase mask with ?w,d,t?=?0.50,0.50,0.90? and na=2.0. Ap-

plication of the algorithm Ithr→−Ithr+2I0with initial thresh-

old Ithr=1.10 and I0calculated from Eq. ?19? yields a trans-

formed threshold intensity Ithr=0.54. When the regions

illuminated by intensities lower than the threshold value are

infiltrated by a solid with dielectric constant 11.9, the result-

ing structure displays an 18% PBG at a solid volume fraction

of ?24%. Adjusting the intensity threshold to Ithr=0.58 pro-

duces a structure with a 19% PBG, several percent smaller

than the case where the overexposed regions consist of the

same solid material.

Whilethephasemask

=?0.50,0.50,0.90? is better suited to “direct” PBG architec-

tures, other phase mask geometries exist for which the oppo-

site is true. For example, a phase mask with geometry

?w,d,t?=?0.50,0.10,0.50?

generates

interference pattern shown in Fig. 12, with an intensity con-

trast C=0.53. This leads to a structure with a 24% PBG after

Si infiltration of the regions with light intensity below Ithr

=0.43 in the template. On the other hand, the “direct” silicon

structure ?filling high-intensity regions? yields, at best, only a

21% PBG when the threshold intensity is optimized to Ithr

=0.70.

Figure 13 shows the PBG map of “inverse structures”

created by phase masks with w=0.50 and various values of d

and t. Here, the phase mask index of refraction is na=2.00

and the photopolymerization intensity threshold is Ithr=0.43.

There are several geometries in the region surrounding

?w,d,t?=?0.50,0.10,0.50? which yield “inverse structures”

exhibiting a large PBG. The other large-PBG regions in the

figure point to the possibility of generating large-PBG struc-

tures with other phase mask geometries. For each particular

region, the actual PBG can be made larger than those shown

in Fig. 13 when Ithris further optimized.

withgeometry

?w,d,t?

thediffraction-

FIG. 12. ?Color online? The isointensity surface at photopoly-

merization intensity threshold Ithr=0.43 in the intensity pattern gen-

erated by a phase mask with ?w,d,t?=?0.50,0.10,0.50?, na=2.0,

and nb=1.0. The volume fraction of the region inside the surface is

?22%. When the low-intensity regions ?regions inside the surface?

are replicated with a material with dielectric constant 11.9 in an air

background, the resulting structure displays a 24% 3D PBG.

PHOTONIC BAND-GAP FORMATION BY OPTICAL-¼

PHYSICAL REVIEW E 73, 046610 ?2006?

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Page 10

V. CONCLUSIONS

We have shown that single-exposure, optical-phase-mask

lithography based on diffraction and interference patterns

from a single beam normally incident on suitably designed

phase masks can be used to create diamondlike photonic

crystals with large PBG’s. The phase mask design was facili-

tated by introducing a target five-beam interference pattern in

an umbrella configuration, identical to the four-beam coun-

terpropagating interference pattern proposed previously for

generating a diamondlike photonic crystal ?46?. Using the

OPM to generate five interfering beams from a single inci-

dent beam places the role of controlling the relative phases,

polarizations, amplitudes, and wave vectors of the interfering

beams on the phase mask itself. Unlike multibeam holo-

graphic lithography ?32,37?, in OPML only the parameters of

the single incident beam need to be precisely controlled. The

relative PBG sizes of final silicon-based “direct” structures

remain over 10% even when the linear polarization angle

changes by ±5°. Since the OPML intensity pattern is deter-

mined by the design of the phase mask, it is only possible to

realize a close approximation to the target intensity pattern of

more general five-beam interference. The resulting reduction

of symmetry in the OPML pattern, however, leads only to a

slight decrease in the PBG size ?from 25% to 24% in the

cases discussed here?. The reduction in symmetry also leads

to a distinction between “direct” and “inverse” photonic

crystal architectures. We have investigated phase mask ge-

ometries for the situation when the incident beam enters the

mask from air ?n?+?=1? and the resulting interference pattern

illuminates a region below the mask with dielectric constant

n?−?=1.67. We have demonstrated OPM geometries for phase

masks with refractive indices navarying between 1.7 and 2.3

that produce intensity patterns to yield a variety of photonic

crystal structures with large PBG’s. The intensity contrast in

OPML is comparable to that which can be achieved using

counterpropagating four-beam holography. The optimal

phase mask geometry is sensitive to the refractive indices of

the phase mask and the photoresist. Therefore, the phase

mask structure must be tailored specifically to the chosen

constituent materials. Moreover, there may be several re-

gions in the parameter space of phase mask geometries that

can produce a suitable intensity pattern. We provided an il-

lustration of a robust geometry, suitable to fabricate a “di-

rect” silicon PBG material. The PBG in this “direct” silicon

structure persists despite 8% variations in the thickness of

the homogeneous middle layer of the phase mask and despite

more than a 10% variation in either the width or thickness of

the phase mask grooves. Our paper provides illustrative ex-

amples of OPML with specific, simple choices of phase

masks. By expanding the parameter space of the phase mask

design, other blueprints for fabricating photonic crystals by

single-exposure phase mask lithography may be found. For

instance, the depth of the two binary gratings ?woodpile lay-

ers? can be allowed to vary independently. Also, the regions

between the grooves of the phase mask can be infiltrated by

a material with refractive index nb, to allow for further flex-

ibility in the phase mask design.

The fabrication of the phase mask itself may require high

precision lithographic processes widely used for two-

dimensional microstructures. The lattice constant of the

phase mask grooves equals the lattice constant of the result-

ing photonic crystal itself, with corresponding submicron

feature sizes. However, after the phase mask is fabricated, it

can be used repeatedly to expose many photoresists and for

the simple and efficient mass production of 3D PBG materi-

als.

ACKNOWLEDGMENTS

This work was supported in part by the National Sciences

and Engineering Research Council ?NSERC? of Canada and

the Ontario Premier’s Platinum Research Prize for Science.

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