Maria Bayard Dühring - Solid Mechanics

where
∂Φ
∂w R 2,n
− i ∂Φ
∂wI
= 2
2,n Ωop
∂wR 2
∂x2
− i2 ∂wI
2 ∂φ2,n R
dr = 2S2 − i2S
∂x2 ∂x2 Ωop
I 2
∂φ2,n
∂x2
dr. (29)
The sensitivity analysis follows the standard adjoint sensitivity approach [20]. For further details of the
adjoint sensitivity method applied to wave propagation problems, the reader is referred to [12]. Eq.(27)
for the derivative of the objective function then reduces to
dΦ ∂Φ
= + Re λ
dξ ∂ξ T ∂Kj2
∂ξ w2
. (30)
The vectors ∂Φ/∂ξ and
R 2S Ωop 2 − i2SI ∂φ2,n
2 ∂x2 dr as well as the matrix ∂Kj2/∂ξ are assembled in Comsol
Multiphysics as described in [21].
4.6. Practical implementation
The optimization problem Eq.(23)-(24) is solved using the Method of Moving Asymptotes, MMA [22],
which is an algorithm that uses information from the previous iteration steps and gradient information.
When the mesh size is decreased the optimization will in general result in mesh-dependent solutions
with small details, which make the design inconvenient to manufacture. To avoid these problems a
morphology-based filter is employed. Such filters make the material properties of an element depend on a
function of the design variables in a fixed neighborhood around the element such that the finite design is
mesh-independent. Here a Heaviside close-type morphology-based filter is chosen [23], which has proven
efficient for wave-propagation type topology optimization problems, see for instance [11]. The method
results in designs where all holes below the size of the filter (radius rmin) have been eliminated. A further
advantage of these filter-types is that they help eliminating gray elements in the transition zone between
solid and air regions.
5. Results
Results are now presented for the problem shown in Figure 1. The SAW is generated by 6 double
electrode finger pairs. Each of the electrodes has a width equal to 0.7 µm and is placed 0.7 µm apart such
that the wavelength of the generated SAW is 5.6 µm. The waveguide is placed at the surface and has the
width 1.4 µm and the height 0.3 µm. The output domain Ωo consists of the waveguide and an area just
below the waveguide with the same size, in order to increase the possibility that the waveguide will still be
able to confine an optical mode after the optimization. The materials GaAs and AlGaAs are piezoelectric
with cubic crystal structure. The material constants used in the piezoelectric model are given in Table 1
and are here given in the usual matrix form for compactness reasons. For the SAW to propagate in the
piezoelectric direction the material tensors have to be rotated by the angle ϕ2 = π/4. The frequency is
fsaw = 518 MHz (where fsaw = ωsaw/2π), which is the resonance frequency. The constant σj controlling
the damping in the PMLs is set to 10 10 . Elements with maximum side length e1 = 0.6 µm are used in
the domain except in the design domain where it is e2 = 0.1 µm and in the output domain where it is
e3 = 0.02 µm. The optical model is solved both for the initial design, where the design domain is filled
with solid material, and for the optimized design in order to compare the acousto-optical interaction.
The free space wavelength of the optical wave is set to λ0 = 950 nm and the material constants for GaAs
and AlGaAs used in the optical model are given in Table 2. For air simply the refractive index n0 = 1
is used. The logarithm to the objective function for the initial design is found to be Φ = −19.99 and
the acousto-optical interaction measure is ∆neff,1/ (P ) = 1.500 · 10 −5 . Note that these values are low
do the the limited numbers of electrodes, in practice several hundred electrode fingers are employed to
generate a SAW.
Table 1: The elastic stiffness constants, the density, the piezoelectric stress constants and the permittivity
constants for the materials used in the piezoelectric model.
Material c E 11,c E 22,c E 33 c E 12,c E 13,c E 23 c E 44,c E 55,c E 66 ρ e14,e25,e36 ε S 11,ε S 22,ε S 33
[10 11 Nm −2 ] [10 11 Nm −2 ] [10 11 Nm −2 ] [kgm −3 ] [Cm −2 ] [10 −11 Fm]
GaAs 1.1830 0.5320 0.5950 5316.5 -0.160 9.735
AlGaAs 1.1868 0.5396 0.5938 5005.2 -0.152 9.446
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