WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1192 J. Opt. Soc. Am. B/Vol. 22, No. 6/June 2005 J. S. Jensen and O. Sigmund
range. To ensure a high bandwidth of the junction, we
maximize the transmission for a number of target frequencies
simultaneously and use an active-set strategy to
update these frequencies repeatedly. To identify the frequencies
with lowest transmission, we use fast frequency
sweeps based on Padé approximants. The performance of
the junction design obtained with this new method (Section
4) is compared with the performance of designs we
get by optimizing the junction for single frequencies (Section
3). In Section 2 we introduce the basic optimization
procedure along with the continuation method and a new
penalization scheme.
2. TOPOLOGY OPTIMIZATION OF
TWO-DIMENSIONAL PHOTONIC
CRYSTAL STRUCTURES
Our implementation of topology optimization is based on
the finite-element method. We use a frequency-domain
method based on a 2D model of plane polarization (2D
Helmholtz equation) in domain :
· Ax ux + 2 Bxux =0 in. 1
In Eq. (1), is the wave frequency, ux is the unknown
field in the plane x=x,y, and Ax and Bx are the
position-dependent material coefficients. For E polarization
A=1 and B= rxc −2 , and for H polarization A
= r −1 x and B=c −2 , where rx is the dielectric constant
and c is the speed of light in air. In the following we consider
E polarization only.
The 2D computational model of the T junction is shown
in Fig. 1. The domain consists of 1515 square unit
cells, each with a centrally placed circular rod with the dielectric
constant r=11.56 and diameter 0.36a, where a is
the distance between the individual rods. This configura-
Fig. 1. Computational model consisting of 1515 unit cells in
with centrally placed circular rods of diameter 0.36a. A wave input
is provided on inp, and absorbing boundaries are specified on
abs. Perfectly matching layers PML are added to avoid reflections
from the input and output waveguide ports. The objective is
to maximize the power transmission in the two subdomains near
the output ports J1 and J2 .
tion has been shown to have a large bandgap in E polarization
in the frequency range =0.302–0.443 2c/a,
and, with single rows of dielectric columns removed, a
guided mode is supported for =0.312–0.443 2c/a. 14
One creates T-junction waveguide by removing single
rows of rods as shown in Fig. 1.
Incident wave and absorbing boundary conditions are
specified on inp and abs, respectively (Fig. 1),
n · A u =2iABU on inp,
n · A u + iABu =0 on abs, 2
where U is a scaling factor for the wave amplitude (in the
following, set to unity). In Eqs. (2), n is the outwardpointing
normal vector at the boundary, and the material
coefficients are those of air in E polarization: A=1 and
B=B air=c −2 .
To eliminate reflections from the input and output
waveguide ports, we add more anisotropic perfectly
matching layers (PMLs) (denoted PML in Fig. 1). The
governing equation in these layers is
x sy A
sx u
+ x
y sx A
sy u
+ y 2sxsyBu =0 inPML, 3
where the factors s x and s y are complex functions of the
position and govern the damping properties of the layers.
A standard implementation for PhC waveguides is used
with the PhC structure retained in the PMLs. 15
A Galerkin finite-element procedure is applied, 16 leading
to the discretized set of equations:
− 2 M + iC + Ku = f, 4
where u is a vector of discretized nodal values of ux. On
the element level the nodal values are interpolated as
ux=Nxu, which leads to the following form of the
finite-element matrices:
M = BeM e
e + BesxesyeM ePML
e , Me =
C = Cabs =
eabs
AeBeCe , Ce = NTNdS, K = AeK e
e + Ae ePML
sye e
x + sxeK K e = K x e + Ky e =
NT
x
N T NdV,
s x e
s yeK y e,
N NT N
dV + x y y dV,
f =2i AeBef einp
e , fe = NTdS, 5
where A e, B e, s xe , and s ye are assumed element wise constants.
We discretize the domain by using standard quadratic
bilinear elements, 16 with 1919 elements for each unit