Maria Bayard Dühring - Solid Mechanics

8 th World Congress on Structural and Multidisciplinary Optimization
June 1-5, 2009, Lisbon, Portugal
Design of acousto-optical devices by topology optimization
Maria B. Dühring
Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Nils Koppels Allé,
Building 404, DK-2800 Kgs. Lyngby, Denmark, mbd@mek.dtu.dk
1. Abstract
In resent years it has been shown experimentally that it is possible to modulate optical waves in waveguides
by surface acoustic waves and a theoretical study shows that the modulation can be improved by
changing the waveguide geometry. Here a method to improve the acousto-optical interaction by means of
topology optimization is presented. The surface acoustic waves are generated by interdigital transducers
in a 2D piezoelectric model, which is coupled to an optical model where the optical mode in the waveguide
is found by solving the time-harmonic wave equation for the magnetic field. Only the piezoelectric model
is used in the optimization and the objective function is the squared absolute value of the strain in the
vertical direction in the waveguide. The objective function is maximized by distributing air and solid
material in an area below the waveguide. The optical model is solved for the initial and the optimized
design and the measure of the acousto-optical interaction is the difference in effective refractive index
for the fundamental mode between the two cases where a surface acoustic wave crest is in the waveguide
and a trough is in the waveguide, which is then normalized with the squared applied electric power. It
is here shown that the acousto-optical interaction can be increased almost 10 times by redistribution of
solid material and air in the design domain.
2. Keywords: Rayleigh waves, piezoelectricity, perfectly matched layers, optical waveguide, strainoptical
relation.
3. Introduction
This work elaborates on how to improve the acousto-optical interaction between a surface acoustic wave
(SAW), also known as a Rayleigh wave, and an optical wave in a waveguide by topology optimization.
SAWs are elastic vibrations that propagate along a material surface with most of their energy density
concentrated at the surface. In piezoelectric materials it is possible to generate the SAW by the inverse
piezoelectric effect by applying an electric potential to interdigital transducers at the material surface [1].
SAWs are now used in filters and resonators in telecommunication [2]. A new application is modulation
of optical waves for signal generation in semiconductor structures [3]. In [4, 5] experimental results are
reported for a compact and monolithic modulator consisting of a SAW driven Mach-Zehnder interferometer
and a relative modulation up to 40 % was obtained with straight interdigital transducer fingers.
A finite element model where a piezoelectric model is coupled to a optical model for the mode in the
waveguide was introduced in [6] in order to simulate the acousto-optical interaction is such a device. By
varying some of the geometry parameters as the width and the height of the waveguide it is shown that it
is possible to improve the acousto-optical interaction. A potential new way of improving the interaction
with a high freedom in the design is to use topology optimization. This is a gradient based optimization
method that has proven efficient in optimizing a wide range of static and dynamic problems in engineering
[7]. Recently, this method has been extended to problems in acoustic wave propagation problems
[8, 9, 10, 11] as well as to photonic problems [12, 13]. Topology optimization has been applied in [14] to
design surface acoustic wave filters and waveguides in 2 and 3 dimensions in elastic materials. Here the
coupled piezoelectric and optical model from [6] is employed in a topology optimization approach where
the optimization is employed to the piezoelectric model only, and the optical model is solved for the initial
design and the optimized design in order to check that the acousto-optical interaction indeed has improved.
In section 4 the acousto-optical model is described as well as the topology optimization approach.
In section 5 results are presented for interaction between a SAW and an optical wave in a ridge waveguide.
4. Topology optimization method for acousto-optical problems
The problem studied in this article is illustrated in Figure. 1. The size of the waveguide and the material
layers are the same as in the experiments presented in [4, 5] with a waveguide of GaAs on top of a
Al0.2Ga0.8As layer, which will further on be referred to as AlGaAs for simplicity. Both materials are
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