www.elsevier.com/locate/optmat

Optical Materials 27 (2005) 1250–1254

Advanced photonic crystal architectures from colloidalself-assembly techniques

Tushar Prasad a,*, Rajesh Rengarajan b, Daniel M. Mittleman b, Vicki L. Colvin a

a Department of Chemistry, Rice University, Houston, TX 77005, USAb Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005, USA

Received 19 October 2004; accepted 9 November 2004

Available online 22 January 2005

Abstract

We describe two different types of novel architectures based on photonic crystals of sub-micron colloids. The first involves the

formation of photonic superlattices from colloidal photonic crystals. The superlattice periodicity induces the formation of mini-

bands due to folding of the photonic band structure. This represents a way by which mid-gap states can be incorporated into a col-

loidal photonic crystal via a specifically engineered structural modification. The second idea involves applying the superprism

concept to three-dimensionally periodic structures. Near a photonic band edge, the diffraction angle is strongly dependent on wave-

length. We analyze this effect in the context of macroporous polymer thin films formed from colloidal crystal templates.

2004 Elsevier B.V. All rights reserved.

Keywords: Colloidal photonic crystal; Optical superlattice; Superprism

1. Introduction

The development of three-dimensional photonic crys-

tals with stop bands in the visible and near-IR has at-

tracted much attention recently, in part because oftheir potential value in the fabrication of photonic inte-

grated circuits [1]. Photonic crystals with three-dimen-

sional periodicity have been fabricated using a variety

of lithographic and selective etching techniques [2–4],

holographic methods [5], and colloidal self-assembly

[6,7]. Unlike most other techniques, this latter method

relies on either entropic or chemical forces to drive the

self-assembly of micron-sized particles. The resultingcrystals have a number of appealing features. They are

readily fabricated in a large area planar thin film format

with controllable thickness up to hundreds of repeating

layers. In addition, self-assembly and templating tech-

0925-3467/$ - see front matter 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.optmat.2004.11.019

* Corresponding author. Tel.: +1 713 348 3489; fax: +1 713 348

2578.

E-mail address: tushar@rice.edu (T. Prasad).

niques offer an impressive versatility with respect to

the materials used in fabricating the crystals [8–13], as

well as their structural morphology [14,15].

There has been much work on the incorporation of

specific types of structural defects into three-dimen-sional colloidal crystals [16,17]. Such defects give rise

to propagating modes lying within the forbidden gap

in the photonic density of states. These modes are a cru-

cial element in the development of photonic crystals as

waveguides, resonant cavities for low-threshold lasers,

or as other photonic devices [1,18,19]. Here, we present

the observed miniband formation in a colloidal crystal

superlattice.Other strategies in the application of photonic crys-

tals avoid the need for structural defects, and instead ex-

ploit the complex band structure of the periodic

medium. An excellent example is the superprism phe-

nomenon which uses the band structure anisotropy to

determine the light propagation direction. This anisot-

ropy can be large in the vicinity of a stop band. As a re-

sult, one can find in this spectral range an extraordinary

T. Prasad et al. / Optical Materials 27 (2005) 1250–1254 1251

sensitivity of the outgoing angle to the angle of the

incoming beam, and to its wavelength. Such an effect

could be extremely valuable in the construction of com-

ponents for WDM applications, as well as for optical-

based sensing devices [20,21]. In this work, we also de-

scribe the computation of the superprism effect in a mac-roporous polymer formed from a colloidal crystal

template.

Fig. 1. SEM cross-sectional image of a three-layer (ABA) colloidal

superlattice. The silica colloidal crystal has been converted to a

macroporous polymer, to facilitate imaging of the cross-section. Both

the AB and the BA interface are clearly evident. These interfaces are

smooth, despite of the 20% difference in the sphere diameters.

2. Experimental

Monodisperse silica colloids with diameters ranging

from 200 to 500 nm are synthesized following the Sto-ber–Fink–Bohn method [22]. The as-synthesized silica

sols are purified and redispersed in 200 proof ethanol

by at least six centrifugation/redispersion cycles. Previ-

ously described technique is used to fabricate three-

dimensionally ordered planar colloidal crystals with

thickness ranging from one monolayer to 50 monolayers

[6]. In short, a glass slide is immersed vertically into

15 ml purified silica sol (1% particle volume fraction)contained in a glass scintillation vial. After ethanol

slowly evaporates, an iridescent film is formed on top

of the glass slide. A large area (1 cm · 3 cm) sample

can be made over 3–5 days. After each single coating

is deposited, the film is taken out of the silica sol and

air-dried for 10 min and then dipped again into another

purified silica sol with differing particle size. This coat-

ing-drying-coating cycle can be repeated many timesand each time the particle size can be arbitrary selected.

The thickness of each crystalline sub-unit can be easily

tuned by changing the concentration of the silica sol.

In this way, a layered structure or an optical superlattice

with an arbitrary pattern of sphere sizes can be assem-

bled [23]. Transmission spectra are obtained by using

an Ocean Optics ST2000 fiber optic UV-near-IR spec-

trometer. An Oriel model 6000 UV lamp with 68806 ba-sic power supply is used to polymerize styrene.

3. Results and discussion

3.1. Optical superlattices

Fig. 1 shows scanning electron microscope (SEM)images of cross-sectional views of a typical sample.

These show macroporous polymer superlattices, formed

by infiltrating the original silica superlattice structure

with polystyrene, and then removing the silica spheres

by etching to produce an inverted structure [11]. This

process replicates the structure of the original silica

films, in a form amenable to cross-sectional imaging.

These data clarify the morphology of the samples andpermit us to accurately determine the radii of the spheres

and the number of layers in each superlattice period.Also, they permit a careful evaluation of the quality of

the interface between crystalline layers with different lat-

tice constants. Fig. 1 shows a cross-sectional view of an

ABA structure, with three successive depositions. The

two A sections consist of NA = 11 111 lattice planes

of a close-packed face-centered cubic (fcc) colloidal crys-

tal composed of 451 nm diameter spheres. The middle

B section is NB = 17 111 planes of an fcc crystal,with sphere diameter of 381 nm. The figure shows that

both the A and B layers are planar and of uniform thick-

ness throughout the structure, and that the preferred

vertical orientation of the 111 crystalline axis is pre-

served. It is interesting to note that both interfaces are

flat and well defined, indicating ordered growth of larger

spheres on smaller ones and also of smaller spheres on

larger ones. Structures as thick as six layers (ABABAB)were fabricated.

The solid curves in Fig. 2 show the evolution of the

normal-incidence transmission spectra as the number

of repeat layers is increased. The lower two traces are

single crystals of the A and B layers with numbers of lat-

tice planes NA = 11 and NB = 17, respectively. With only

one layer (bottom spectra) the transmission shows a

broad photonic stop band consistent with that observedfor traditional colloidal crystals. When one size is lay-

ered on top of another, resulting in an AB structure,

two distinct stop bands with widths comparable to the

individual layers are observed (third spectrum from

the bottom). As the layering is repeated, however, each

additional layer reinforces the long-range periodicity of

the superlattice, resulting in significant modifications to

the observed stop bands.The superlattice periodicity in these films has the ef-

fect of modifying the original photonic band structure.

This results in the folding of the band structure along

the 111 direction in reciprocal space, leading to the

formation of minibands. The optical density spectra is

calculated using the scalar wave approximation [24],

700 800 900 1000 1100

ABABAB

ABABA

ABAB

ABA

AB

B

A

Opt

ical

Den

sity

(arb

. uni

ts)

Wavelength (nm)

Fig. 2. The solid curves show normal-incidence optical density spectra

of a series of films, in arbitrary units, vertically displaced for clarity.

The lower two represent the spectra of an 11-layer film of the Aspheres (451 nm) and a 17-layer film of the B spheres (381 nm),

respectively. The remaining spectra are measured on samples with

additional crystalline layers added alternately (AB, ABA, ABAB, etc.)

as labeled. The dashed curves show the simulated spectra, calculated

using the scalar wave approximation.

1252 T. Prasad et al. / Optical Materials 27 (2005) 1250–1254

and are shown in Fig. 2 (dotted curves). For the thicker

samples, neither the modulation depth nor the spectral

positions of the stop bands agree well with the simula-tions. This could be because of the accumulated effects

of disorder in the fcc lattices, or more likely because of

small variations in the thicknesses of the uppermost lay-

ers relative to the underlying layers. Despite these dis-

crepancies, the qualitative features of the experimental

spectra are reproduced in these simulations. The qualita-

tive correspondence between this simple theory and the

experimental results provides convincing evidence thatthe observed structure does indeed arise from superlat-

tice effects.

3.2. Three-dimensional superprisms

The basis of the superprism phenomena is anisotropy

in the photonic band structure, which is strongly present

near the photonic band gap. Accurate theoretical mod-eling is required in order to design and orient samples

for optimized sensitivity at a given wavelength. Earlier

theoretical predictions involving superprism effect have

simulated auto-cloned 3-D photonic crystals and other

2-D structures [21,25]. In this paper, we present a theo-

retical method for computing the expected response,

based on a full calculation of the photonic band struc-

ture. We base our models on three-dimensional macro-

porous polymer photonic crystals, formed by using

colloidal crystals as templates. Macroporous polymer

templates can be prepared from high quality silica col-

loidal crystals [11], with controlled film thickness [6].

These are inverted face-centered cubic (FCC) structures,

i.e. interconnected air holes in an FCC configuration ina polymer background. Though these inverted struc-

tures do not provide a high enough dielectric contrast

for the formation of a full band gap, they do possess

substantial stop bands indicative of a partial gap along

the (111) crystalline axis. This is an indication of strong

band structure anisotropy, which is sufficient for the

observation of the superprism effect.

We use an available software package [26] to simulatethe structure of macroporous polymer templates and de-

fine the lattice geometry for FCC and set the ratio of

sphere radius to the unit cell length to be slightly more

than 0.5. This corresponds to placing the air spheres

slightly closer together than their diameters, leading to

small windows which interconnect the internal air net-

work. This accurately reproduces the morphology of

the samples [11]. The dielectric constant of the polymerbackground is taken as 2.69 which corresponds to

poly(allyl)methacrylite (PAMA) and the typical band

structure is computed. From the complete photonic

band structure, all possible values of wave vectors in

the three-dimensional space for a particular energy or

frequency can be calculated. The plot of all these wave

vectors in the k-space for a particular energy gives an

equal-energy surface known as a dispersion surface. Itis analogous to the index ellipsoid in conventional crys-

talline optics, or to the Fermi surface in electronic

crystals.

The shape of dispersion surface depends on the cho-

sen value of the energy, specified by the frequency of the

incident light. For small values far from the stop band,

the band structure is isotropic in nature. In this case, the

dispersion surface is spherical with a radius given by c/nave, where nave is the average (homogenized) refractive

index. At higher energy values, near the photonic band

gap, the band structure anisotropy is strong. As a result,

the shape of dispersion surface deviates from spherical,

although it retains the symmetry of the Brillouin zone.

This distorted shape leads to the superprism phenome-

non. In Fig. 3, examples of iso-energy surfaces are

shown for the sample under consideration.The propagation direction can be obtained from the

dispersion surface. The propagation direction is ob-

tained as normal to the dispersion surface at the end

point of the propagation wave vector, since the group

velocity vG = $kx(k). If the dispersion surface is spheri-

cal, then this gradient points radially, and the wave

propagates in a direction parallel to its wave vector.

As a result, the propagation angle does not change dras-tically for small changes in the incident orientation or

incident wavelength. However, if the dispersion surface

Fig. 3. Typical dispersion surfaces computed for the three-dimensional photonic crystal described in the text. From left to right, these correspond to:

band #2 (X = 0.5), band #3 (X = 0.8), and band #4 (X = 0.73). Frequencies are normalized and are in dimensionless units.

T. Prasad et al. / Optical Materials 27 (2005) 1250–1254 1253

is distorted, then the gradient can be a sensitive functionof the incident angle.

Since we have the dispersion surface, it is possible to

calculate incident and propagation angles with respect

to any set of planes in the crystal. As an example, we

compute the sensitivity of the output propagation angle

on the input propagation angle, at fixed photon energy.

We chose an energy of 0.73 (in normalized units), since

that is the energy for which the dispersion surface ismost severely distorted (see Fig. 3). We chose an inci-

dent direction such that the light is incident on the

(001) face of the crystal, and is in the plane perpendic-

ular to the (001) set of planes. In this case, the scattered

light propagating inside the crystal is also confined in

this plane, and may be parameterized by a single angle.

Fig. 4 shows the computed relation between the angle of

the input beam and the internal propagation angle. Thesudden jump of the propagation angle from negative to

-100 -80 -60 -40 -20 0 20 40 60 80 100-80

-60

-40

-20

0

20

40

60

80 (001) geometryφin =45 o, Ω =0.73

θ p (d

egre

es)

θin (degrees)

Fig. 4. Computed dependence of the internal propagation angle as a

function of the input angle, at fixed photon energy of X = 0.73 (in

dimensionless units). Both angles are measured with respect to the

direction normal to the (001) lattice planes. The abrupt jump is a

manifestation of the superprism effect. With a 4 change in the input

angle, the internal angle varies by more than 60.

positive values is due to the propagation wave vectorcrossing through one of the high symmetry portions of

the dispersion surface. The curvature of the dispersion

surface across the high symmetry line changes drasti-

cally, resulting in a large change in the propagation an-

gle. This property of angle-sensitive propagation can be

applied to beam steering and waveguiding in integrated

optics [27].

4. Conclusion

To adapt photonic crystals for a variety of optoelec-

tronic functions, there is a need to create elements of

optical circuit architectures within them. We have pre-

sented two kinds of photonic crystal device structures

which can be easily fabricated through colloidal self-assembly techniques. Optical superlattices and superp-

risms will help in the development towards designing ad-

vanced photonic architectures which can combine light

sources and routing functions for applications pertain-

ing to integrated optics.

Acknowledgments

This work has been partially supported by the Na-

tional Science Foundation (CHE-967020) and the R.A.

Welch foundation (C-1342).

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