Photonic crystals with anomalous dispersion: Unconventional propagating modes
in the photonic band gap
Daniel Hermann, Marcus Diem,
*
and Sergei F. Mingaleev
†
Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany
and Institut für Theoretische Festkörperphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany
Antonio García-Martín
‡
and Peter Wölfle
Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany
Kurt Busch
Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany;
Department of Physics and College of Optics and Photonics: CREOL and FPCE,
University of Central Florida, Orlando, Florida 32816, USA;
and Institut für Theoretische Festkörperphysik, Universität Karlsruhe, 76128 Karlsruhe, Germany
共Received 24 September 2007; published 11 January 2008
兲
We present an investigation of the optical properties of photonic crystals whose constituent materials exhibit
anomalous dispersive behavior. In particular, the anomalous dispersion near resonances may lead to additional
propagating modes in the gap of the undoped system for a localized region of wave-vector space. Such a
system may be realized by infiltrating quantum dots in polymer suspensions into the pores of two-dimensional
high-index photonic crystals. An evaluation of the absorption lengths associated with these unconventional
modes and corresponding transmission calculations demonstrate that this effect can be observed in currently
accessible structures.
DOI: 10.1103/PhysRevB.77.035112 PACS number共s兲: 42.70.Qs, 42.25.Bs
I. INTRODUCTION
Progress in photonics is closely related to development of
optical materials with tailor-made properties. Photonic crys-
tals 共PCs兲
1–3
represent a novel class of man-made optical
materials. The judicious design of these two-dimensional
共2D兲 or three-dimensional 共3D兲 periodic dielectric arrays al-
lows one to tailor the photonic dispersion relation and the
corresponding mode structure to almost any need. In particu-
lar, the flexibility associated with material composition, lat-
tice periodicity, and symmetry together with the deliberate
creation of defect structures makes PCs the optical analog of
an electronic semiconductor.
The usefulness of PCs and defect structures embedded in
them may be substantially enhanced if the structures exhibit
one or more forms of tunability. This has led a number of
authors to propose PC structures whose constituent materials
exhibit tunable properties such as temperature-dependent re-
fractive indices,
4
strongly dispersive behavior,
5
as well as
electro-
4–6
and magnetooptically
7
controllable anisotropies.
In this paper, we consider photonic band structures and
absorption lengths of PCs whose constituent materials ex-
hibit anomalous dispersion. By embedding these materials
into a PC, one may expect to modify the photonic band
structure considerably, if the resonance frequency is tuned to
lie close to a photonic band edge. As we will show, the
hybridization of the electromagnetic wave mode with the
local dielectric modes leads to a splitting of the wave mode
into three submodes in a limited region of wave-vector
space. Of these three modes, two are stable and form a
bubble shape in the band structure. However, as any
frequency dependence of the effective dielectric constant is
necessarily accompanied by a dissipative component, the
question arises as to whether these bubble modes are
overdamped. To answer this question, it requires a full treat-
ment of the propagating and absorptive properties of the
electromagnetic wave. As a consequence, we describe a
highly efficient on-shell methodology based on photonic
Wannier functions,
8
which allows us to solve for the photo-
nic band structure and absorption lengths associated with
complex-valued and frequency-dependent dielectric con-
stants. We present a careful study of this problem and con-
clude that the bubble modes should, indeed, be observable in
currently accessible materials.
The paper is organized as follows. In Sec. II, we define
the model system used in the following sections. The nu-
merical methods for band structure and attenuation length
calculations of PCs with dispersive components are de-
scribed and discussed in Sec. III. In Sec. IV, we present
results for the model system, discuss the effects of the
anomalous dispersion of the constituent materials on the
photonic band structure, and investigate the attenuation
length resulting from the nonzero imaginary part of the
dielectric constant. Finally, we summarize our results in
Sec. V.
II. MODEL SYSTEM
In the following sections, we discuss in detail the case of
TM-polarized radiation in 2D macroporous silicon PCs
9
共square lattice of pores with lattice constant a in silicon,
dielectric constant of silicon
Si
=12.0, and electric field po-
larized parallel to the pore axis兲, where the pores 共radius
r/ a=0.475兲 have been infiltrated with a polymer 共typical di-
PHYSICAL REVIEW B 77, 035112 共2008兲
1098-0121/2008/77共3兲/035112共10兲 ©2008 The American Physical Society035112-1
