Maria Bayard Dühring - Solid Mechanics
Maria Bayard Dühring - Solid Mechanics Maria Bayard Dühring - Solid Mechanics
34 Chapter 6 Design of acousto-optical interaction [P3]-[P7] the electrodes increases the SAW properties change significantly, and in [82, 81, 83] numerical and experimental results show a multi-mode SAW propagation and a tenfold slowing of the phase velocity. The high aspect ratio (HAR) electrodes measured in the experiments are seen in figure 6.1(b) and 6.1(c). Recently, a Rayleigh wave generated by thin electrodes has been employed to create an optical switch by modulation of optical waves in the two waveguide arms of a Mach-Zehnder interferometer (MZI). A MZI is an optical device consisting of a waveguide that is split into two waveguide arms and after a distance the arms are joined again as illustrated by figure 6.2(a). By sending a Rayleigh wave perpendicular towards the waveguide arms a periodic modulation of the light is obtained. The distance between the waveguide arms is an unequal number of half Rayleigh wavelengths such that when a Rayleigh wave node is present at each waveguide the output light will not change. When a wave crest is located at one waveguide a trough will appear at the other waveguide, and the applied stresses with opposite sign will introduce a refractive index difference in the arms that causes an opposite phase change in them. By constructive and destructive interference the light will be turned on and off periodically with a multiple of the SAW frequency depending on the static phase difference in the two arms, which can be introduced by a length difference between them. Experimental results are reported in [23] and [P3]. For straight electrode fingers a 40% relative modulation was obtained for a GaAs/AlGaAs sample, but only a 0-8% relative modulation was found for a silicon on insulator (SOI) sample. In this chapter a numerical model is presented in order to study the acoustooptical interaction in a MZI and explain the difference in the experimental results. To model the SAW propagation and the optical modes in the waveguides, it is sufficient to consider a 2D model of a cross-section through the two waveguides and the geometry is seen on figure 6.2(b). Perfectly matched layers (PML) are employed at the domain borders to prevent reflections of the mechanical and electrical disturbances from the SAW. PMLs were introduced for time-harmonic elastic problems in (a) (b) Figure 6.2 Light modulation in a Mach-Zehnder interferometer. (a): 3D geometry of a MZI with a propagating surface acoustic wave (M. van der Poel [P3]). (b): The 2D cross-section through the waveguide arms of the MZI that is used in the simulations.
6.2 The acousto-optical model 35 [26] and are here extended to piezoelectric materials. The Rayleigh wave is generated at the IDT to the left in figure 6.2(b) and propagates in both the left and the right horizontal direction and is absorbed in the PMLs. First the acousto-optical interaction is investigated for the two material cases and a parameter study of the geometry is performed for the SOI case in order to improve the interaction. Then topology optimization, which has successfully been applied to optimize SAW filters and waveguides in elastic materials in [45], is employed to optimize the acoustooptical interaction based on the GaAs/AlGaAs sample. Finally SAW generation by HAR electrodes is studied. First the electrodes are modeled by a periodic model in order to find the mode shapes and the mechanical energy confinement to the electrodes. Then it is investigated if the acousto-optical interaction can be improved with SAWs generated by HAR electrodes compared to those generated by conventional thin electrodes. 6.2 The acousto-optical model In a piezoelectric material SAWs are generated by applying an electrical potential to electrode fingers on a material surface. This introduces mechanical deformations in the solid by the inverse piezoelectric effect and the behavior of the piezoelectric material is described by the following model found in [3]. A time-harmonic electrical potential V (xj, t) = V (xj)e iωt , (6.1) with the angular frequency ω is applied to the electrode. The mechanical strain Sij (assumed small) and the electric field Ej are given by the expressions Sij = 1 1 ∂ui 2 γj ∂xj + 1 ∂uj γi ∂xi and Ej = − 1 γj ∂V , (6.2) ∂xj where ui are the displacements. In order to prevent reflections from the sides and bottom of the material, PMLs are employed and the parameter γj is an artificial damping at position xj in the PML given by the expression γj(xj) = 1 − iσj(xj − xl) 2 . (6.3) Here xl is the coordinate at the interface between the regular domain and the PML and σj is a suitable constant. There is no damping outside the PMLs and here γj = 1. A suitable thickness of the PMLs as well as the value of σj must be found by calculations such that both the mechanical and the electrical disturbances are absorbed before reaching the outer boundaries. However, the absorption must also be sufficiently slow as reflections will occur at the interface between the regular domain and the PML if their material properties are not comparable. The mechanical stresses Tjk and the electric displacement Di both depend on the strain and the
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6.2 The acousto-optical model 35<br />
[26] and are here extended to piezoelectric materials. The Rayleigh wave is generated<br />
at the IDT to the left in figure 6.2(b) and propagates in both the left and the<br />
right horizontal direction and is absorbed in the PMLs. First the acousto-optical<br />
interaction is investigated for the two material cases and a parameter study of the<br />
geometry is performed for the SOI case in order to improve the interaction. Then<br />
topology optimization, which has successfully been applied to optimize SAW filters<br />
and waveguides in elastic materials in [45], is employed to optimize the acoustooptical<br />
interaction based on the GaAs/AlGaAs sample.<br />
Finally SAW generation by HAR electrodes is studied. First the electrodes are<br />
modeled by a periodic model in order to find the mode shapes and the mechanical<br />
energy confinement to the electrodes. Then it is investigated if the acousto-optical<br />
interaction can be improved with SAWs generated by HAR electrodes compared to<br />
those generated by conventional thin electrodes.<br />
6.2 The acousto-optical model<br />
In a piezoelectric material SAWs are generated by applying an electrical potential<br />
to electrode fingers on a material surface. This introduces mechanical deformations<br />
in the solid by the inverse piezoelectric effect and the behavior of the piezoelectric<br />
material is described by the following model found in [3]. A time-harmonic electrical<br />
potential<br />
V (xj, t) = V (xj)e iωt , (6.1)<br />
with the angular frequency ω is applied to the electrode. The mechanical strain Sij<br />
(assumed small) and the electric field Ej are given by the expressions<br />
Sij = 1<br />
<br />
1 ∂ui<br />
2 γj ∂xj<br />
+ 1<br />
<br />
∂uj<br />
γi ∂xi<br />
and Ej = − 1<br />
γj<br />
∂V<br />
, (6.2)<br />
∂xj<br />
where ui are the displacements. In order to prevent reflections from the sides and<br />
bottom of the material, PMLs are employed and the parameter γj is an artificial<br />
damping at position xj in the PML given by the expression<br />
γj(xj) = 1 − iσj(xj − xl) 2 . (6.3)<br />
Here xl is the coordinate at the interface between the regular domain and the PML<br />
and σj is a suitable constant. There is no damping outside the PMLs and here<br />
γj = 1. A suitable thickness of the PMLs as well as the value of σj must be found<br />
by calculations such that both the mechanical and the electrical disturbances are<br />
absorbed before reaching the outer boundaries. However, the absorption must also<br />
be sufficiently slow as reflections will occur at the interface between the regular domain<br />
and the PML if their material properties are not comparable. The mechanical<br />
stresses Tjk and the electric displacement Di both depend on the strain and the
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