Maria Bayard Dühring - Solid Mechanics

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40 Chapter 6 Design of acousto-optical interaction [P3]-[P7] (n eff,1 −n no )/ P 1/2 [W −1/2 ] (a) 4 2 0 −2 −4 x 10−6 6 left waveguide right waveguide −6 0 0.5 1 1.5 2 phase of SAW, φ x π saw Figure 6.4 Results for the SOI sample. (a): Increase of refractive index (neff,1 −nno)/ √ P in the two waveguides as function of the SAW phase φsaw. φsaw = 0 corresponds to a wave crest in the left waveguide and a trough in the right. (b): von Mises stress with arbitrary scale. at the crests and the troughs. It is observed from figure 6.4(a) that losses due to reflections appear between the waveguides as (neff,1−nno)/ √ P in the right waveguide never reaches the extreme values achieved in the left waveguide. It is noted that the values of (neff,1 − nno)/ √ P are small because of the limited number of electrodes. In experiments several hundred electrode fingers are employed for the wave generation. In order to understand the acousto-optical interaction it is essential to know the polarization direction of the electric field of the propagating optical modes. An optical mode polarized in a certain direction mainly detects the refractive index and its change in this direction. When designing a device for acousto-optical interaction it is therefore important that the refractive index changes in the polarization direction. The fundamental optical modes for the two different samples are both polarized in the x1-direction. The waveguide in the GaAs/AlGaAs sample supports one mode and it is of first order. In figure 6.5(a) the increase in refractive index in the x1-direction, ∆n11/ √ P , is plotted around one of the waveguides where a Rayleigh wave crest induces stresses. The time averaged power flow in the x3-direction is indicated by the contour lines with an arbitrary scale. The optical mode has its center in the waveguide and is extending to the substrate below the waveguide. The Rayleigh wave crest is introducing stresses in the entire waveguide area except the upper corners, such that the optical mode is overlapping well with ∆n11/ √ P . The interaction is ∆neff,1 = 1.50 · 10 −5 W −1/2 . A similar plot is seen in figure 6.5(b) for the fundamental mode in the SOI sample. In this case the height of the waveguide is bigger compared to the width and therefore the Rayleigh wave only introduces stresses and deformations at the bottom and does not deform the upper part. For that reason ∆n11/ √ P only has a significant value at the bottom of the waveguide. Because of the bigger material contrast between the waveguide and the substrate
6.4 Acousto-optical interaction in a Mach-Zehnder interferometer 41 Figure 6.5 Study of acousto-optical interaction in an optical waveguide. The color bars show the refractive index change ∆n11/ √ P . The time averaged power flow in the x3direction of the fundamental mode is indicated by the contour lines with an arbitrary scale. (a): GaAs/AlGaAs sample. (b): SOI sample. the optical mode is fully confined to the waveguide and its center is placed where ∆n11/ √ P is zero. The interaction is ∆neff,1 = 8.84 · 10 −6 W −1/2 , which is less than for the GaAs/AlGaAs case. This shows that the geometry of the waveguide is important for the acousto-optical interaction, and the two examples indicate that it is efficient to have an aspect ratio of the waveguide with a bigger width and a lower height such that the entire waveguide gets deformed as in the GaAs/AlGaAs case. The waveguide of the SOI sample supports another first order mode as well, which is polarized in the x2-direction. The interaction of this mode is less than the first mode with ∆neff,2 = 5.15 · 10 −6 W −1/2 . 6.4.1 Parameter study of waveguide geometry The acousto-optical model is now employed to improve the interaction in the SOI sample by varying parameters in the geometry. First the height h of the waveguides is studied and figure 6.6(a) shows the interaction ∆neff,ν as function of h for the two first order modes that the waveguides support. An optimal height h = 0.19 µm of the waveguides is found for the mode polarized in the x1-direction with ∆neff,1 = 1.61 · 10 −5 W −1/2 , which is 1.8 times bigger than for the original height. In figure 6.7(a) ∆n11/ √ P is plotted for the optimal height together with the power flow of the fundamental optical mode. As the waveguide has become thinner, the center of the optical mode has moved closer to the surface and is overlapping better with the change in refractive index compared to the original case seen in figure 6.5(b). However, when the height is decreasing the optical wave will be less confined in the waveguide and will be increasingly influenced by the SiO2 and the air. The air will not contribute to an index difference in the two waveguides and the SiO2 will have a negative influence on the difference, as the stress-optical constants related
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
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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
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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
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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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