Maria Bayard Dühring - Solid Mechanics

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50 Chapter 6 Design of acousto-optical interaction [P3]-[P7] h p Ni LiNbO 3 PML Figure 6.13 The displacements in the structure for each of the six modes at resonance with h/2p = 1. u1 and u2 are plotted as deformations with an arbitrary scaling factor and u3 is given by the color bar for all six modes. half of the phase velocity, v p [ms −1 ] (a) 2500 2000 1500 1000 500 SH2 SH1 VP1 VP2 SH3 VP bulk SH bulk VP3 mode 6 mode 5 mode 4 mode 3 mode 2 mode 1 0 0 0.5 1 1.5 2 2.5 3 aspect ratio, h/2p [−] E mech,elec /E mech,tot [−] 1 0.8 0.6 SH1 SH3 0.4 VP1 VP3 mode 6 mode 5 mode 4 0.2 VP2 mode 3 mode 2 SH2 mode 1 (b) 0 0 0.5 1 1.5 2 aspect ratio, h/2p [−] 2.5 3 Figure 6.14 Results from the periodic model for the six modes. (a): Half of the phase velocity vp as function of h/2p. The polarization types are indicated along with the limits for the bulk waves. (b): Energy fraction Emech,elec/Emech,tot as function of h/2p. The limit for the SH modes is 0.97 and the limit for the VP modes is 0.93. and their polarization direction and the bulk wave limits are indicated. The phase velocity decreases with increasing aspect ratio for all the modes and for mode 1 vp is decreased up to 15 times. The modes do not appear above their bulk wave limits as they are dissipated to the bulk material. The observed modal shapes in figure 6.13 explain why more and more modes appear in the structure as the aspect ratio increases. When considering the SH modes, which are mainly polarized in the x3-direction, the mode shapes in the electrode in this direction are of increasing order for increasing mode number. This means that SH1 has a mode shape of order 1, SH VP
6.5 High aspect ratio electrodes 51 SH2 has one of order 2 and SH3 has one of order 3. The same is found for the VP modes where the mode shapes have increasing order in the x1- and x2-direction. Thus, more modes can exist for larger aspect ratios as the electrode is allowed to vibrate with modes of higher order. The fraction of mechanical energy in the electrode Emech,elec with respect to the total mechanical energy in the structure Emech,tot is plotted in figure 6.14 (b) for the six modes. The energy is more confined to the electrode for increasing aspect ratio. The electrode thus acts as a mechanical resonator, which slows down the SAW velocity because of mechanical energy storage. For modes of the same type, the fraction of mechanical energy in the electrode tends to the same value for increasing aspect ratio. The SH modes tend to a value around 0.97, which is larger than the limit for the VP modes at 0.93. The fact that the limits do not reach 1 explains that the wave is still (slightly) propagating, or rather coupled from one electrode to the other by surface waves. If the energy was fully trapped the surface wave would not propagate at all. For increasing aspect ratio the mode shapes tend to clean cantilever vibrations of slender beams, and the only energy left in the substrate is what connects the cantilever modes to the substrate. Modes of the same polarization type tend to the same energy ratio because they deflect in the same direction where the stiffness in the substrate is the same. As the wavelength gets shorter for increasing mode order for a fixed aspect ratio, the aspect ratio for the higher order modes must be bigger before they reach the energy limits. 6.5.2 Finite structure and acousto-optical interaction The problem is now extended to a finite structure with twelve electrode pairs by applying PMLs at the vertical borders, see figure 6.15. The electrodes are excited with an alternating electrical potential. The values of the phase velocity and the Figure 6.15 The geometry of the acousto-optical problem with Ni electrodes on a LiNbO3 substrate. Perfectly matched layers absorb the waves at the boundaries and the thick square indicates the optical domain.
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Optimization of acoustic, optical a
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Title of the thesis: Optimization o
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Resumé (in Danish) Forskningsomr˚
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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,
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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
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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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