WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

4.1 Waveguide bends and junctions 33
incident
wave
PML
y
x
PML
dielectric
columns
air
absorbing
boundaries
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a) b)
Figure 4.1 Topology optimization of the material distribution in photonic crystal waveguides,
a) 90-degree bend, b) T-junction. Black: dielectric material with ε = 11.56, light
gray: air. The design domain for the 90-degree bend is indicated in dark gray and the
energy transmission in the structures is evaluated in the gray regions close to the output
ports. Unwanted reflections from the waveguide boundaries are eliminated by using PML
(Perfectly Matched Layers). From papers [10] and [11].
that otherwise occurs due to reflections at the waveguide discontinuities is reduced.
Fig. 4.2b shows the optimized material distribution for the 90-degree bend. The
design is obtained by optimizing the sum of the transmitted power for the three
frequencies ω = 0.34, ω = 0.38 and ω = 0.42. The resulting structure has a nonintuitive
appearance very different from structures reported in precedent studies
(such as the one shown in Fig. 4.2a taken from Mekis et al. (1996)). More importantly,
the structure also outperforms previously reported structures significantly
by having a very low loss in a broad frequency range. Fig. 4.2c shows the performance
of the structures. A bend loss of less than 0.3% in the frequency range from
ω = 0.325−0.440 is obtained for the optimized structure. This can be compared to
the structure in Fig. 4.2a that has a similar low loss only in narrow frequency band
around ω ≈ 0.35. The circular dots in Fig. 4.2c correspond to transmission losses
for a post-processed optimized design in which the (few) elements with intermediate
values of the design variable that appear in Fig. 4.2b are forced to either 0 or 1. The
plot shows negligible discrepancy between the performance of two structures.
Fig. 4.1b illustrates the related design problem of a T-junction. A set of optimized
material distributions is presented in Fig. 4.3a-c plotted together with the
electric field amplitude for ω = 0.38. The structures are all optimized for maxi-
y
x
a
10a
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