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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 by anythree

=?K?10,K?1¯0,K?01,K?01¯?:

K?ij=?2?i

a

vectorsfromtheset

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 created bya 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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