Maria Bayard Dühring - Solid Mechanics

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44 Chapter 6 Design of acousto-optical interaction [P3]-[P7] Figure 6.9 Results for the buried waveguide where the color bar indicates ∆n11/ √ P close to the optical waveguide. The time averaged power flow in the x3-direction of the fundamental mode is indicated by the contour lines with an arbitrary scale. in the SiO2 layer such that their upper surface is leveled with the substrate surface. The waveguides still support the two first order modes and for the first one polarized in the x1-direction the interaction is ∆neff,1 = 7.16 · 10 −5 W −1/2 . This is 8 times bigger than for the original waveguides on top of the surface. The power flow of this optical mode is plotted together with ∆n11/ √ P in figure 6.9 and the entire waveguide is now influenced by the change of the refractive index. For the optical mode polarized in the x2-direction the interaction is ∆neff,1 = 3.89 · 10 −5 W −1/2 . From an experimental point of view it is difficult to couple the optical mode to a waveguide if the height is small, and it is more complicated to fabricate a buried waveguide than a ridge waveguide. An alternative design is therefore to introduce a layer of Si next to the waveguide with the height l as seen in figure 6.10(a). This can simply be fabricated by stopping the etching of the Si on the surface before reaching the SiO2 layer. In this way the optical mode comes closer to the stresses at the surface, but the original height of the Si layer is kept such that the optical mode can be coupled to the waveguide. The interaction ∆neff,ν as function of l is illustrated in figure 6.10(b) for the two first order modes that the waveguide supports. The interaction is increasing for increasing l and it is bigger for the mode polarized in the x1-direction with ∆neff,1 = 1.11 · 10 −4 W −1/2 for l = 2.5 µm. This is 12 times bigger than for the original geometry and the increase has the same size as for the improved height and width. For increasing l the mode that is polarized in the x2-direction is not supported anymore. The power flow of the optical mode and ∆n11/ √ P are plotted in figure 6.10(a) for l = 0.2 µm. The mode is still confined to the waveguide area and the Si layer, but now the center is below the waveguide where the stresses are big. However, the drawback of this design is, that as the mode has a bigger extension for increasing l the dimensions of other parts in the geometry, as the input waveguides, taperings and splitters, must be correspondingly bigger.
6.4 Acousto-optical interaction in a Mach-Zehnder interferometer 45 Δn eff,ν [W −1/2 ] (b) x 10−4 1.2 1 0.8 0.6 0.4 0.2 mode 1 mode 2 0 0 0.05 0.1 0.15 0.2 0.25 height of Si layer, l [μm] Figure 6.10 Results for the waveguide surrounded by a Si layer with height l. (a): The color bar indicates ∆n11/ √ P and the time averaged power flow in the x3-direction of the fundamental mode is indicated by the contour lines with an arbitrary scale. (b): The acousto-optical interaction ∆neff,ν as function of l for the two first order modes. The study of the waveguide geometry shows that the acousto-optical interaction in the MZI can be increased significantly compared to the original chosen geometry. Both the Rayleigh wave and the optical wave must be adjusted in order to match well and it is important that the center of the optical mode is close to the substrate surface such that it overlaps the mechanical stresses. When choosing the most suitable geometry different aspects must be taken into account such as the size of the interaction, the fabrication possibilities, the size of the structure and single- and multi-modedness. Other effects that are not considered in the presented model can have an influence on the interaction. The confinement of the optical mode versus the loss has to be taken into account. When the optical mode gets less confined to the waveguide, as in the case where there is a Si layer next to the waveguide, more energy will be lost to the bulk material. On the other hand, the bigger the dimensions of the waveguide are above the substrate, the more energy can be lost to the air due to irregularities at the surface. The numerical model could also be extended to explore the influence of the mechanical deformations and the temperature on the interaction. Finally, it is important to examine if the polarization direction of the optical mode is unchanged when it propagates along the waveguide. If it changes it can have a significant influence on the interaction. In order to explore this effect a three-dimensional model is required. The presented results for the SOI sample were calculated with the values of C44, C55 and C66 taken into account, see table 6.1. This has not influenced the general design conclusions compared to the results in [P4], but the acousto-optical interaction has in general increased. When the stress-optical constants are rotated, the values of C44, C55 and C66 give a contribution to some of the other components.
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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,
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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
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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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