WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

5. Photonic Wire Waveguides
Figure 4 shows a schematic setup of a PhW waveguide T-splitter. A straight waveguide with width w
and of dielectric constant ε = n 2 , carries an incoming wave that propagates from left towards right. The
input signal is to be split into two equal waves so that 50% of the incoming wave energy is transmitted
through both the upper and lower output ports. We use a two-dimensional model of plane polarized
light that for time-harmonic waves is governed by the Helmholtz equation:
∇ · (A∇u) + Bω 2 u = 0, (1)
where A = ε −1 and B = c −2 for TE-polarized waves, and A = 1 and B = εc −2 for TM-polarized waves.
The frequency of the wave is denoted ω and the unknown magnetic or electric field is u.
The boundary condition specifies an incoming wave at the left waveguide port:
n · (A∇u) = 2iω √ ABu0, (2)
where u0 is the specified amplitude of the incoming wave. In addition to the wave source, the computational
domain in Figure 4 is embedded in a perfectly matched layer (PML) [15] that ensures a minimized
reflection of outgoing waves. The governing equation in the PML is:
∂
∂x (sy A
sx
∂u ∂
) +
∂x
∂y (sx
sy
A ∂u
∂y ) + Bω2 sxsyu = 0, (3)
where the two complex functions sx and sy specify the absorbing properties in the x- and y−direction,
respectively. At the interface to the computational domain, sx = sy = 1 which transforms Eq. (3) into
the original Eq. (1), thus creating a perfectly matching interface.
5.1. Topology optimization
We use the method of topology optimization [4] to find a suitable material distribution in the design
domain (the area indicated with a question mark in Figure 4).
The optimization problem is formulated as:
⎧
⎨
⎩
max min Φ1, Φ2
subject to : S(̺, ω)p = f(ω)
0 ≤ ̺e ≤ 1 e = 1, . . . , Nd.
w
n=1
Figure 4: Design problem for a photonic wire waveguide T-splitter. The material distribution in the area
indicated with a question mark is found using topology optimization. The objective is to distribute 50%
of the input energy into the top and bottom output ports and thus eliminate reflection and radiation at
the waveguide junction.
4
?
n
(4)