WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

4.2 Photonic crystal building blocks 35
computation of a transmission curve with high frequency resolution. The computational
burden associated with this identification is reduced significantly by using
Padé approximants 2
An extension of the optimization formulation is necessary in order to obtain
the well defined distribution of dielectric material near the T-junction in Fig. 4.3c.
The first attempts resulted in intermediate design variables near the corners of the
junction and thus a well defined structure with air and dielectric material was not
obtained. However, if a penalization of the intermediate design variables using
artificial damping is introduced, the problem is resolved. Linear viscous damping is
added in each element with the damping being proportional to ǫxe(1−xe) in which ǫ
is a positive constant and xe is the element design variable. Thus if xe = 0 or xe = 1
no damping is present, but if xe has an intermediate value, energy is dissipated and
thepower transmission reduced. Thus, toavoidunwanted dissipationofenergy, xe is
forced toward either 0 or 1. The effect is clear as a well defined structure is obtained
if ǫ is chosen sufficiently large. This method has later been applied successfully to
other optimization problems as well.
4.2 Photonic crystal building blocks
A collaboration with a photonic crystal research group allowed for fabrication and
experimental testingofoptimizedphotoniccrystalwaveguide components. However,
thecomponentsillustratedinSection4.1areoptimizedforE-polarizedwavesandare
not easily fabricated due their realization as dielectric pillars placed in air. Instead
a number of optimized designs for H-polarized waves were generated and fabricated.
These basically consist of air holes in the dielectric material placed in a triangular
pattern. This configuration is more amenable to fabrication and also less prone to
out-of-plane losses.
Fig. 4.4 shows a compound photonic crystal structure that integrates three of
the optimized building blocks (dark regions represent air and gray regions represent
silicon). These building blocks allow for control and manipulation of theflow of light
in the crystal. With optimized bends and splitters this can be done with minimum
lossofenergy. Specifically, thestructurecontainsa120-degree, a60-degreebendand
asplitterthatseparatesawaveintotwo. Allcomponentsareoptimizedseparatelyby
maximizing the power transmission through a simpler structure for given frequency
ranges. This is carried out using the procedure described in Section 4.1. It is seen
from the figure that holes of irregular sizes and shapes are obtained. The structures
are hence qualitatively different than what could have been created using intuitive
trial-and-error design methods and also regular parameter/size optimization. The
performance of the individual components in Fig. 4.4 is very satisfactory with low
loss over a broad frequency range. In all cases a significant improvement of the
2 This procedure was the inspiration for the new method of frequency-range optimization using
Padé approximants described in Section 5.1.