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
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
PHYSICAL REVIEW E 73, 046610 ?2006?
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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-
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-
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
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.
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
where i¯?−i and j¯?−j. Here we have defined
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-
CHAN, TOADER, AND JOHNPHYSICAL REVIEW E 73, 046610 ?2006?
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+?
where K?land ?lare defined as
K?l? G?l− G?00,
?l? ?l− ?00.
The corresponding intensity pattern is given by
I?r ?? = I0+ 2?
?00·lcos?K?l· r ? + ?00·l+ ?l?
?l?·lcos??K?l− K?l?? · r ? + ?l?·l+ ?l− ?l??,
The spatial modulation of this intensity pattern is periodic
with a Bravais lattice whose reciprocal lattice vectors can be
where i¯?−i and j¯?−j. Here we have defined
c ? a?q0− q1?−1= a?? −??2− 1?−1.
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:
z= − E1¯0
z= − E01¯
?10= − ?1¯0,
?01= − ?01¯.
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.
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??.
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 ???,
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-
PHOTONIC BAND-GAP FORMATION BY OPTICAL-¼
PHYSICAL REVIEW E 73, 046610 ?2006?
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???.
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
?2− ?1= ±?
2cjcos ?j= 0,
j = 3,4,
c5= c6= 0.
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
CHAN, TOADER, AND JOHNPHYSICAL REVIEW E 73, 046610 ?2006?
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,
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
are determined by the Fourier modal method and are given
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?.
The symmetry of the intensity pattern, Eq. ?19?, compares
favorably with the target symmetry in Eq. ?17?. Quantita-
?c1,c2,cos??1− ?2?,2c3cos ?3,2c4cos ?4,c5,c6?
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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PHYSICAL REVIEW E 73, 046610 ?2006?