Maria Bayard Dühring - Solid Mechanics

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FIG. 4: Results for the finite model at resonance with h/2p = 1. (a) and (b): The displacements u3 for mode 3 (SH2) with fp = 1004 ms −1 . (c) and (d): The displacements u2 for mode 4 (VP2) with fp = 1510 ms −1 . FIG. 5: Results for the optical model with h/2p = 1. (a): The refractive index variation in the waveguide. (b): The distribution of the x3-component of the time averaged power flow for the fundamental mode in the waveguide with arbitrary scale when no strain is applied. 6
TABLE II: The difference in effective refractive index for the two first order optical modes for the six acoustic modes with h/2p = 1 as well as for the thin electrodes with h/2p = 0.01. ∆neff,1/ √ P [W −1/2 ] ∆neff,2/ √ P [W −1/2 ] mode 1 (SH1) 1.55 · 10 −3 8.97 · 10 −4 mode 3 (SH2) 9.08 · 10 −5 6.62 · 10 −5 mode 5 (SH3) 3.08 · 10 −6 2.50 · 10 −6 mode 2 (VP1) 7.35 · 10 −5 7.21 · 10 −4 mode 4 (VP2) 1.18 · 10 −5 6.35 · 10 −5 mode 6 (VP3) 6.49 · 10 −6 1.38 · 10 −5 thin electrodes 2.51 · 10 −6 1.71 · 10 −6 the imaginary part in the refractive index in the Ni electrodes that absorb energy from the system. The fundamental mode is mainly polarized in the x1-direction. The waveguide supports several modes and the next mode is also a first order mode with a similar mode shape. It is mainly polarized in the x2-direction with the effective refractive index neff,2 = 2.147 + 6.090 · 10 −4 i. C. Improving the Acousto-Optical Interaction The coupled acousto-optical model is now solved for each of the six acoustic modes for the case where a wave crest is at the waveguide and a wave trough is at the waveguide, respectively. In the following, only the real part of the effective refractive index is considered and the change between the two extreme cases is given as ∆neff,j = |n c eff,j − n t eff,j|, (19) where j is 1 for the mode that is polarized in the x1direction and 2 for the mode polarized in the x2-direction. The modes will be denoted optical mode 1 and 2. The superscript c in (19) indicates the value at a wave crest and t the value at a trough. The difference ∆neff,j is normalized to the square root of the time averaged applied electric power P calculated by the following expression found in [15] V (iωD2m2) P = ℜ ∗ dL, (20) 2 Lel where Lel corresponds to the interfaces between the electrodes and the substrate and m2 is the normal vector to these interfaces. The star indicates the complex conjugate. The results are reported in Table II for the six acoustic modes and for thin electrodes with h/2p = 0.01 for both optical mode 1 and 2. The tendency is that the SH modes interact most efficiently with optical mode 1 and the VP modes interact best with optical mode 2. The acousto-optical interaction decreases with increasing mode number within the two different categories of acoustic modes. The six acoustic modes with high aspect ratio electrodes interact better in general compared to the acoustic mode for the conventional thin electrodes. The interaction is a result of how efficiently the acoustic mode is excited with the applied electrical power and how well the acoustic and optical modes match each other. The calculations show that the acoustic modes of lower order are more efficiently excited by a certain amount of electrical power than the modes of higher order such that larger values of the strain and hence bigger change in refractive index for lower order modes are obtained. A reason for this tendency is that for higher aspect ratios the movement of the structure tend to clean cantilever vibrations with the mechanical energy concentrated in the electrodes and just below them where they are attached to the substrate. The graphs in Fig. 3 show that the lower order acoustic modes are closer to this limit for h/2p = 1 and therefore more strain is concentrated in the area just below the electrodes. As the structure with high aspect ratio electrodes is more compliant than the almost plane surface with the thin electrodes more power has to be applied to the thin electrode in order to get the same strain. It can therefore be expected that the change in refractive index is biggest for the low order acoustic modes with high aspect ratio electrodes. The other important question is how well the acoustic modes match with the optical modes, in other words how well the changes in refractive index and the optical modes overlap. The optical modes will mainly detect the changes in refractive index in their polarization direction. So for optical mode 1, it is the SAW induced changes of the refractive index in the horizontal direction ∆n11 that are important. Optical mode 2 must overlap with ∆n22. The acoustic modes of the same polarization type all have a similar pattern of changes in refractive index around the optical waveguide. As an example ∆n11/ √ P is plotted for SH1 in Fig. 6(a) with the power flow of optical mode 1 indicated with contour lines. ∆n22/ √ P is plotted with optical mode 2 in Fig. 6(b). When these two figures are compared it is seen that ∆n11/ √ P has bigger values than ∆n22/ √ P , but the distribution of these parameters have a similar pattern that both match the optical modes well. This is a tendency for all the SH modes. However, for SH modes of higher order the wave patterns tend to go deeper into the substrate and therefor the strain concentrations are not as big around the waveguide. The same is plotted for VP1 in Fig. 6(c) and 6(d), note the different color scales. In this case ∆n22/ √ P has the biggest values and the distribution overlap well with the optical mode. ∆n11/ √ P does not overlap so well, as it has a pattern where ∆n11/ √ P changes sign into the depth. A part of optical mode 1 is therefore overlapping with ∆n11/ √ P close to zero. This is also a tendency for all the VP modes. Thus, the mode patterns explain that the interaction is different for the SH and VP modes with respect to the polarization direction of the optical mode. The interaction is most efficient for SH1 with optical mode 1 and it is more than 600 times bigger than for the case with the thin electrodes. In Fig. 7(a) and 7(b) the change in refractive index ∆n11/ √ P and ∆n22/ √ P are plotted for 7
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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
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vi Contents 6 Design of acousto-opt
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2 Chapter 1 Introduction waves. The
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4 Chapter 2 Time-harmonic propagati
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10 Chapter 2 Time-harmonic propagat
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12 Chapter 3 Topology optimization
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22 Chapter 4 Design of sound barrie
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28 Chapter 5 Design of photonic-cry
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34 Chapter 6 Design of acousto-opti
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58 Chapter 7 Concluding remarks
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60 References [14] E. Yablonovitch,
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62 References [42] A. A. Larsen, B.
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64 References [67] K. Svanberg, “
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Publication [P1] Acoustic design by
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558 In Ref. [3] it is studied how t
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560 2.2. Design variables and mater
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562 When the mesh size is decreased
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564 objective function, Φ [dB] 130
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566 ARTICLE IN PRESS Fig. 6. The di
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568 ~kðxÞ 1 ¼ k1 ARTICLE IN PRES
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570 ARTICLE IN PRESS M.B. Dühring
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572 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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Page 115 and 116: Surface acoustic wave driven light
Page 117 and 118: IDT GaAs AlGaAs 0.10 0.05 0.00 Ampl
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Page 136 and 137: where ˜pijkl are the rotated strai
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Page 160 and 161: FIG. 6: Results for the acousto-opt
Page 162: Suzuki, A. Onoe, T. Adachi and K. F
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Maria Bayard Dühring - Solid Mechanics

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