Maria Bayard Dühring - Solid Mechanics

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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)
6.3 Modeling surface acoustic waves 37 where nc eff,ν is the case where a surface acoustic wave crest is at the waveguide and nt eff,ν is the case where a trough is at the waveguide. The applied electrical power P can be calculated by the expression, see [2], P = ℜ V (iωD2m2) ∗ dL, 2 (6.10) Lel where Lel corresponds to boundaries with the electrodes and m2 is the normal vector to the upper surface. The star indicates the complex conjugate and ℜ is the real component. 6.3 Modeling surface acoustic waves To illustrate the performance of the piezoelectric model with the PMLs, a Rayleigh wave is generated by ten double electrode fingers in a GaAs sample with straight surface. The material is rotated 45 o around the x2-axis in order to get the appropriate piezoelectric properties in the propagation direction. An alternating electric potential with magnitude ±1 V is applied to the electrodes and the resonance frequency is found to be f = 510 MHz (where f = ω/2π), which is the same as found in experiments, see [87]. On figure 6.3(a) the color scale indicates the displacement u2 and the surface is deformed with the unified displacements u1 and u2 with a scaling factor equal to 1000. The Rayleigh wave is confined to the surface as expected and the correct wavelength defined by the electrodes is obtained. The wave propagates away from the electrodes in the middle and is gradually absorbed in the PMLs at the borders. On figure 6.3(b) the absolute value of u2 along the upper surface is plotted. A standing wave pattern is generated at the electrodes as the wave here travels in both the left and the right direction. Away from the electrodes the wave is only traveling and the amplitude is therefore nearly constant. It is not constant because the PMLs are not exactly acting as an infinite half space. The electrical potential V is plotted in figure 6.3(c). It is observed that the settings for the PMLs work such that both the mechanical displacements and the electrical potential are absorbed before they reach the outer boundaries. Thus, the results from the piezoelectric model have the expected properties. 6.4 Acousto-optical interaction in a Mach-Zehnder interferometer This section is an overview of publication [P3]-[P5] where the interaction between a Rayleigh wave and the optical waves in a MZI is studied. The acousto-optical interaction has been investigated experimentally for both a GaAs-/AlGaAs sample and an SOI sample, see [23] and [P3]. For the former case a good performance is obtained, but for the latter case almost no modulation of the optical wave is observed. The described acousto-optical model is here employed to simulate both
Page 1: Optimization of acoustic, optical a
Page 4 and 5: Title of the thesis: Optimization o
Page 6 and 7: Resumé (in Danish) Forskningsomr˚
Page 8 and 9: Publications The following publicat
Page 10 and 11: vi Contents 6 Design of acousto-opt
Page 12 and 13: 2 Chapter 1 Introduction waves. The
Page 14 and 15: 4 Chapter 2 Time-harmonic propagati
Page 20 and 21: 10 Chapter 2 Time-harmonic propagat
Page 22 and 23: 12 Chapter 3 Topology optimization
Page 32 and 33: 22 Chapter 4 Design of sound barrie
Page 38 and 39: 28 Chapter 5 Design of photonic-cry
Page 44 and 45: 34 Chapter 6 Design of acousto-opti
Page 66 and 67: 56 Chapter 7 Concluding remarks des
Page 68 and 69: 58 Chapter 7 Concluding remarks
Page 70 and 71: 60 References [14] E. Yablonovitch,
Page 72 and 73: 62 References [42] A. A. Larsen, B.
Page 74 and 75: 64 References [67] K. Svanberg, “
Page 77: Publication [P1] Acoustic design by
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Page 82 and 83: 560 2.2. Design variables and mater
Page 84 and 85: 562 When the mesh size is decreased
Page 86 and 87: 564 objective function, Φ [dB] 130
Page 88 and 89: 566 ARTICLE IN PRESS Fig. 6. The di
Page 90 and 91: 568 ~kðxÞ 1 ¼ k1 ARTICLE IN PRES
Page 92 and 93: 570 ARTICLE IN PRESS M.B. Dühring
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574 frequency or the frequency inte
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Publication [P2] Design of photonic
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photonic-crystal fibers used for me
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2.2 Design variables and material i
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two air holes, is Λ = 3.1 µm, the
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Figure 4: The geometry of the cente
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performance will decrease and some
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[23] J. Riishede and O. Sigmund, In
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Surface acoustic wave driven light
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IDT GaAs AlGaAs 0.10 0.05 0.00 Ampl
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Publication [P4] Improving the acou
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083529-2 M. B. Dühring and O. Sigm
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Publication [P5] Design of acousto-
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Figure 1: The geometry used in the
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where ˜pijkl are the rotated strai
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where ∂Φ ∂w R 2,n − i ∂Φ
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Figure 3: The distribution of the o
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[12] J.S. Jensen and O. Sigmund, To
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Energy storage and dispersion of su
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093504-3 Dühring, Laude, and Kheli
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093504-5 Dühring, Laude, and Kheli
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Publication [P7] Improving surface
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FIG. 1: The geometry of the acousto
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finer mesh. For the coupled model i
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FIG. 4: Results for the finite mode
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FIG. 6: Results for the acousto-opt
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Suzuki, A. Onoe, T. Adachi and K. F
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Maria Bayard Dühring - Solid Mechanics

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