Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps
S. Mahmoodian,
*
R. C. McPhedran, and C. Martijn de Sterke
CUDOS, School of Physics, University of Sydney, NSW 2006 Australia
K. B. Dossou, C. G. Poulton, and L. C. Botten
CUDOS, School of Mathematical Sciences, University of Technology Sydney, NSW 2007 Australia
Received 22 October 2008; published 15 January 2009
We investigate the formation and coupling of defect modes in two-dimensional photonic crystal PC band
gaps associated with degenerate edges. Using a method based on Green’s functions and perturbation theory, we
derive a condition for the degeneracy of a defect mode in a PC band gap. We show that the interaction between
multiple degenerate defects splits this degeneracy and provide a semi-analytic model for the splitting using a
4 4 tight-binding matrix. We observe that the structure of this tight-binding matrix is related to the overlap
and corresponding symmetry of the defect modes. We confirm our analysis by comparing with numerical
results.
DOI: 10.1103/PhysRevA.79.013814 PACS numbers: 42.70.Qs, 42.25.Fx, 42.25.Bs
I. INTRODUCTION
Photonic crystals PCs are constructed by arranging di-
electric material in a periodic fashion 1. This arrangement
often consists of circular inclusions forming hexagonal or
square lattices, thereby having planes of rotational and re-
flection symmetry. Highly symmetric two-dimensional 2D
structures often have modes that exist in degenerate pairs.
Degeneracy results from the frequency of modes being un-
changed after particular symmetry operations 1; that is, the
rotated or reflected mode “sees” the same lattice as the origi-
nal mode. The occurrence of these degeneracies in perfectly
periodic PCs is well understood 1,2.
Since the construction of the first PCs with band gaps,
much focus has been directed towards the study of localized
defect modes. Defects can be introduced into PCs by break-
ing the periodicity in some fashion, leading to the formation
of levels in band gaps associated with localized modes. De-
fect modes created at degenerate band edges come in pairs.
Whether these modes are degenerate depends on the nature
of the defect and of the band-edge mode. Band-edge degen-
eracies can occur in two forms: different bands having the
same frequency on the band edge or two points in a band
having the same frequency, while not separated by reciprocal
lattice vectors. In this paper we discuss the latter type as it is
much more common in 2D PCs.
Currently, there are many fully numerical studies on de-
fect modes that also deal implicitly with defect-mode degen-
eracy. For example, Sakoda and Shiroma 3 calculated the
frequency of defect modes in the first three band gaps of a
square lattice, classifying degenerate and nondegenerate de-
fect modes by their symmetry. Kuzmiak and Maradudin 4
used finite-difference time-domain simulations to model de-
fect frequencies in the first three band gaps of a hexagonal
lattice and were able to identify all degenerate and nonde-
generate modes belonging to the C
6v
mode class. There are,
however, no rigorous analytic studies determining under
what conditions defect modes are degenerate. The first part
of our paper accomplishes precisely this task; that is, we
derive a condition to determine whether a perturbation in a
periodic PC lattice leads to degenerate or nondegenerate de-
fect modes.
The PC lattices we are considering are assumed to be
made from nonmagnetic, lossless dielectric material forming
2D hexagonal or square lattices with circular inclusions. De-
fects are constructed by altering an inclusion such that its
symmetry and that of the lattice is preserved. This can be
achieved, for example, by changing the refractive index of an
inclusion or altering its radius.
We have previously shown that in all band gaps the cre-
ation of a shallow defect in a PC leads to the formation of
defect modes possessing the symmetry of the band-edge
Bloch mode 5. Hence, for band gaps with degenerate band-
edge modes, the creation of a defect leads to the formation of
two localized defect modes. If the defect is created in an
arbitrary fashion, in general, the degeneracy is broken and
the two defect modes have different frequencies. Painter et
al. 6 illustrated such a case, where creating a defect which
had a lower symmetry than that of the lattice split a degen-
eracy. In this paper we demonstrate that the converse is not
necessarily true. We show that defects created by preserving
the circular symmetry of an inclusion do not necessarily lead
to defects modes that are degenerate. In fact, we show that
the condition for degeneracy of a defect mode created at a
degenerate band edge is twofold: i The two degenerate
band-edge Bloch modes must have the same electric energy
inside the defect region. ii The electric energy overlap of
the two band-edge Bloch modes must vanish over the defect
region.
Photonic crystal components such as channel drop filters
7 and coupled cavity waveguides 8 rely on the coupling
of defect modes. A well-known method for treating modes of
any system, not necessarily optical, with weak coupling is
the tight-binding TB method 9. More recently, the TB
method has been applied to nondegenerate PC defects where
the defect modes perturb each other weakly 8,10. In these
cases the treatment assumed that each defect had only a
single nondegenerate defect mode. As mentioned, defect
*
sahand@physics.usyd.edu.au
PHYSICAL REVIEW A 79, 013814 2009
1050-2947/2009/791/01381412 ©2009 The American Physical Society 013814-1