WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

32 Chapter 4 Optimization of photonic waveguides
Paper [16] presents an example of an optimized, fabricated and experimentally
tested photonic crystal component with a more advanced functionality. The
λ−splitter splits an incoming wave by channeling shorter waves in one output direction
and longer waves in another output direction. The component is fabricated
with a nano-imprint lithography technique that facilitates an accurate reproduction
of fine structural details.
Inpaper [17]thedesignmethodologyisappliedtoasimilar problemofthedesign
of a T-junction in a photonic ridge waveguide. Numerical results are presented for
E- and H-polarized waves and the importance of the size of the design domain as
well as the chosen frequency range for the optimization procedure are discussed.
Main references
The contributions in this thesis are believed to be the first that use a numerical material
distribution optimization method to design photonic crystal waveguide bends
and splitters.
Earlier optimization studies for photonic crystal waveguides were mostly based
on simple geometrical variations such as the movement of one or several holes in the
structure, as found in Mekis et al. (1996), Moosburger et al. (2001), Chutinan et al.
(2002), and Olivier et al. (2002). Smajic et al. (2003) and Jiang et al. (2003) used
genetic algorithmswithrelatively fewdesignvariablestooptimize similar structures.
Geremia et al. (2002) marked an exception by optimizing the material distribution
in photonic crystal structures using a mathematical inversion technique. Optimization
of ridge waveguide splitters was considered previously using a combination of
numerical analysis and analytical considerations by Manolatou et al. (1999) and by
Sakai et al. (2002).
Closely related is the application of topology optimization to the design of electromagnetic
antenna structures as was demonstrated in the work by Kiziltas et al.
(2003).
4.1 Waveguide bends and junctions
Fig. 4.1 shows two models used for the fundamental study of the optimization of
bends and junctions in photonic crystal waveguides. Both structures are built
around a 2D photonic bandgap material for E-polarized optical waves. The unit
cell is squared with a circular inclusion of a dielectric material placed in air (light
gray indicates air and black indicates dielectric). The bandgap is located in the
(non-dimensional) frequency range from ω = 0.302−0.443. A single line of dielectric
inclusions is removed which creates a waveguide that allows a wave to propagate
with frequencies in the range from ω = 0.312−0.443. The aim of the optimization
study is to redistribute air and dielectric in the vicinity of the bend or junction so
that the power transmission through the components is maximized. Hereby the loss