WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

J. S. Jensen and O. Sigmund Vol. 22, No. 6/June 2005/J. Opt. Soc. Am. B 1191
Topology optimization of photonic crystal
structures: a high-bandwidth
low-loss T-junction waveguide
Jakob S. Jensen and Ole Sigmund
Department of Mechanical Engineering, Technical University of Denmark, Nils Koppels Allé, Building 404,
2800 Kgs. Lyngby, Denmark
Received September 13, 2004; revised manuscript received December 8, 2004; accepted December 29, 2004
A T junction in a photonic crystal waveguide is designed with the topology-optimization method. The gradientbased
optimization tool is used to modify the material distribution in the junction area so that the power transmission
in the output ports is maximized. To obtain high transmission in a large frequency range, we use an
active-set strategy by using a number of target frequencies that are updated repeatedly in the optimization
procedure. We apply a continuation method based on artificial damping to avoid undesired local maxima and
also introduce artificial damping in a penalization scheme to avoid nondiscrete properties in the design
domain. © 2005 Optical Society of America
OCIS codes: 000.4430, 130.1750, 230.7390.
1. INTRODUCTION
Since the primary publications, 1,2 photonic crystals
(PhCs) have attracted much attention owing to the potential
applications in integrated optical circuits and communication
devices. PhCs rely on the bandgap phenomenon
that causes total prohibition of wave propagation for some
frequencies in certain periodic media. Future applications
of PhC components will inevitably rely on high-level performance,
such as ultra-low-loss and high-bandwidth operation.
Thus an increased interest has recently been devoted
to the possibility of using inverse design techniques
in the form of various optimization methods to design
such components, 3 e.g., to maximize bandgaps of PhCs
and to reduce losses in PhC waveguide bends, junctions,
inlets, and outlets. In this paper we demonstrate how topology
optimization can be used to design low-loss and
high-bandwidth two-dimensional (2D) waveguide T junctions.
In a recent paper 4 a 90-deg bend in a 2D PhC waveguide
was designed with topology optimization. The optimization
was performed by maximization of the power
transmission of E-polarized waves through the bend for
three wave frequencies simultaneously, and the resulting
design, which was rather unconventional, displayed a
good performance in a large frequency range. In the
present paper the theoretical details behind the optimization
algorithm are explained in detail, and a new optimization
strategy based on active frequency sets, as well as
a new penalization scheme, is introduced to treat the
more difficult case of a T-junction waveguide. Although we
here consider the case of E polarization, the methods described
can immediately be applied to H polarization with
a change of material parameters. In Ref. 5 a double 120
-deg bend designed for H polarization was fabricated and
showed a satisfactory performance of the component in
good agreement with numerical simulations.
0740-3224/05/061191-8/$15.00 © 2005 Optical Society of America
The T junction in a 2D PhC with circular dielectric rods
in a square configuration has previously been studied and
subjected to optimization. In Ref. 6, T junctions (and 90deg
bends) were studied by use of a 2D finite-difference
time domain simulations, and the performance of the
junction was improved by addition of extra rods of various
radii in the corner regions. In Ref. 7 the reflection at the T
junction was minimized with a combination of genetictype
algorithms and gradient-based optimization with the
position and size of a selected number of rods used as design
variables. In a related study, 8 optimization based on
simulated annealing was used to optimize a 60-deg junction
in a waveguide based on a triangular configuration of
holes. In that case the design variables were the radii of a
number of holes. Recently researchers performed a similar
optimization study 9 by allowing the radii of selected
holes in a 60-deg bend to be varied to maximize the transmission
through the bend.
Unlike the aforementioned studies, topology optimization
is based on free distribution of material in a design
domain and hence does not restrict the design to consist
of circular rods, or indeed rods at all. The method was
originally developed for use in structural mechanics with
the aim of obtaining the stiffest possible structures with a
restricted amount of material. 10 This has led to new designs
that significantly outperformed structures designed
with standard shape- or size-optimization techniques, led
to an increasing popularity in aero and automotive industries,
and resulted in implementation in commercial optimization
tools. 11 In the past decade, topology optimization
has also been successfully applied in other areas such as
fluid mechanics and heat conduction. 12 Recently the
method has been applied to optimization of phononic
bandgap materials and structures. 13
In this paper we use topology optimization to design T
junctions with high transmission over a large frequency