Maria Bayard Dühring - Solid Mechanics

36 Chapter 6 Design of acousto-optical interaction [P3]-[P7]
electric field according to the constitutive relations
Tjk = ˜c E jklmSlm − ˜e T ijkEi, (6.4)
Di = ˜eijkSjk + ˜ε S ijEj, (6.5)
where ˜c E jklm are the elastic stiffness constants, ˜eijk are the piezoelectric stress constants
and ˜ε S ij are the permittivity constants. The materials are in general anisotropic,
and as it is only possible to generate the SAW by the inverse piezoelectric effect in
certain directions the material tensors have to be rotated. This is indicated by the
tilde above the material tensors. The rotation is done according to Euler’s transformation
theory as explained in [2]. The two governing equations give the stresses by
Newton’s second law and the electric displacement from Gauss law
1
γj
∂Tij
∂xj
= −ρω 2 ui and
1
γj
∂Dj
∂xj
= 0, (6.6)
where ρ is the mass density. The piezoelectric model can be solved by a plane
formulation obtained by assuming that Si3, S3j and E3 as well as Ti3, T3j and D3
are equal to zero, which is suitable when simulating Rayleigh waves as they are
mainly polarized in the plane. In that case the governing equations are solved for
the three unknowns u1, u2 and V . When SAWs generated by HAR electrodes are
considered, the waves can also have a significant displacement u3 out of the plane. In
that case the model is solved by omitting all derivatives with respect to x3. Second
order Lagrange elements are used for all the unknowns. The implementation in
Comsol Multiphysics is done similar to the problems described in [84, 85].
After the mechanical wave is computed by the piezoelectric model described
above, the refractive index nij in the material can be calculated either according to
the strain-optical relation [13]
∆bimbmj = ˜pijklSkl, (6.7)
where bimnmj = δij and ˜pijkl are the rotated strain-optical constants, or the stressoptical
relation [86]
nij = n 0 ij − ˜ CijklTkl. (6.8)
n 0 ij is the refractive index in the stress free material and ˜ Cijkl are the rotated stressoptical
constants. It is assumed that the stress-optical effect is dominant compared
to the electro-optical effect, which will be neglected here (see [23]). It is furthermore
assumed that the SAW will affect the optical wave, but the optical wave will not
influence the SAW. After the SAW induced changes in the refractive index are
computed the optical modes in the waveguides are found by solving the eigenvalue
problem described in section 5.1. From this the effective refractive indexes of the
guided optical modes ν are computed by neff,ν = βν/k0. The measure of the acoustooptical
interaction is defined as
∆neff,ν = |n c eff,ν − n t eff,ν|/ √ P , (6.9)