Maria Bayard Dühring - Solid Mechanics

46 Chapter 6 Design of acousto-optical interaction [P3]-[P7]
The most important change is that the normal component C11 is increased around
four times. The Rayleigh wave introduces a stress pattern around the waveguide
where the normal stresses in the x1-direction in general have the biggest absolute
values. When the values of the interaction for the different waveguide geometries are
compared with the values reported in [P4], it shows that they are in general increased
around four times - the same as the increase of the C11 component. This indicates
that the normal stresses in the horizontal direction give the biggest contribution to
the acousto-optical interaction in the SOI case.
6.4.2 Topology optimization
Finally, the method of topology optimization is applied to increase the interaction
between the Rayleigh wave and the optical mode for the GaAs/AlGaAs sample.
Method
The problem is simplified such that only one waveguide is considered and the rest
of the problem setting is the same as described for the GaAs/AlGaAs sample in
the beginning of the chapter. During the optimization air and solid material is
distributed in a design domain Ωd below the waveguide such that the objective
function Φ is optimized in the output domain Ωo. The output domain consists of
the waveguide combined with an area of the same size just below it, as illustrated at
figure 6.11. The design variable ξ takes the value 0 for air and 1 for the solid material,
which is AlGaAs in the upper part of Ωd and GaAs in the lower part. All the material
parameters from the piezoelectric and the optical models are interpolated linearly
between the two material phases. The purpose of the optimization is to maximize
the acousto-optical interaction between the Rayleigh wave and the optical wave.
As the fundamental mode in the waveguide is polarized in the x1-direction it is
mainly interesting that the refractive index component n11 changes and the change
depends on the different strain components - the more the strain components change
the more n11 changes. The optimization problem is only stated for the piezoelectric
model and the aim is to maximize an expression, which depend on the normal strain
components. The squared absolute value of the normal strain in the vertical and
the horizontal direction, respectively, in the output domain Ωo have been tested as
objective function Φ, as well as their sum. The results obtained for the three cases
were similar and in the following the squared absolute value of the normal strain
in the vertical direction S22 is chosen as Φ. The formulation of the optimization
problem is thus given as
max log(Φ) = log
ξ
Ωo
|S22(r, ξ(r))| 2
dr , objective function, (6.11)
subject to 0 ≤ ξ(r) ≤ 1 ∀ r ∈ Ωd, design variable bounds. (6.12)
To check that the acousto-optical interaction has indeed improved by the optimization,
the optical model is solved for both the initial design and the optimized design.