Maria Bayard Dühring - Solid Mechanics
Maria Bayard Dühring - Solid Mechanics Maria Bayard Dühring - Solid Mechanics
48 Chapter 6 Design of acousto-optical interaction [P3]-[P7] increase in Δ eff,1 [%] (b) 70 60 50 40 30 20 10 0 0 0.2 0.4 0.6 0.8 1 2r/w [−] Figure 6.12 Study of the influence of an air hole below the waveguide with radius r. (a): The color bar indicates ∆n11/ √ P and the time average power flow in the x3-direction is shown by the contour lines with an arbitrary scale. (b): Increase in ∆neff,1 as function of 2r/w. a simpler design change can be introduced. The optimization suggests that it as an advantage to have an air hole underneath the output domain. To explore this tendency further an air hole is introduced in the form of a half circle with the radius r, see figure 6.12(a). The increase in ∆neff,1 as function of 2r/w, where w is the width of the waveguide, is illustrated by figure 6.12(b). An optimum is obtained around 2r/w = 0.8 where the interaction has increased with 65% compared to the case without the air hole. When r is increasing two effects will influence the interaction. First, the hole will trap the Rayleigh wave in the waveguide and introduce additional strain concentrations that increase ∆n11/ √ P , see figure 6.12(a). As r grows, strain is confined more and more to the waveguide above the hole until finally the hole reaches a size, where the Rayleigh wave will be reflected instead. This explains the sudden dip in the graph. The other effect is that the optical mode gets more confined to the waveguide due to the refractive index contrast in the air. The total increase in interaction is small compared to the increase obtained by topology optimization, so the other air holes in the optimized design have an important influence on the performance. To determine if the increased interaction is mainly due to the strains around the hole or due to the better confinement of the optical wave, the interaction ∆neff,1 is compared for the four cases seen in table 6.2. The value 1 Table 6.2 Increase in acousto-optical interaction ∆neff,1 compared to the original GaAs/AlGaAs structure without the air hole. The original case corresponds to the value 1. optical mode without hole optical mode with hole strain without hole 1 1.03 strain with hole 3.29 1.65
6.5 High aspect ratio electrodes 49 corresponds to the original GaAs/AlGaAs structure in figure 6.5(a) where the strain and the optical mode are calculated without the hole. When the strain and the optical mode is found for 2r/w = 0.8, 1.65 is obtained. Then the strain is calculated without the hole and then the hole is introduced and the optical mode is found. This only gives an increase of a few percent. Finally, the opposite is done where the strain is computed with the hole and the optical mode is calculated with the hole filled. Now the interaction increases more than 2 times, so this indicates that the effect from the strain concentrations is dominant. However, it also indicates that the effect from the strain concentrations and the confinement of the optical mode are counteracting each other. This shows that the presented optimization method can be employed to improve the acousto-optical interaction and even though the designs are complicated to fabricate, they can be used as inspiration to introduce simpler changes of the initial design. A way to obtain designs that are easier to fabricate is to introduce a constraint that assign the design variables to an entire column that starts at the surface as in [45]. 6.5 High aspect ratio electrodes In the previous sections the acousto-optical model has been employed to study the interaction between a Rayleigh wave and optical waves in channel waveguides. This model is now utilized to investigate SAWs generated by high aspect ratio electrodes. First their mode shapes and the mechanical energy confinement to the electrodes are studied by a periodic model of a unit cell. The obtained results are compared to a model with a finite number of electrodes. Finally, the acousto-optical interaction between the new types of SAWs and an optical wave in a buried channel waveguide is investigated. The section is a summary of the results presented in publication [P6] and [P7]. 6.5.1 Periodic structure First a unit cell is studied with periodic boundary conditions connecting the left and the right boundaries and a PML at the bottom, see figure 6.13. The electrode consists of nickel (Ni) and the substrate is lithium niobate (LiNbO3). The height of the electrode is h and the period of the cell is p. When the aspect ratio of the electrode increases it is possible to excite more mode types than the two that exist for thinner electrodes. Six different modes exist for the aspect ratio h/2p = 1 and the mode shapes are plotted in figure 6.13 normalized to the applied electric potential equal to 1 V. Modes with unequal numbers are mainly polarized in the shear horizontal (SH) direction and modes with equal numbers are mainly vertically polarized (VP). All the modes are combinations of a vibration in the electrode and a surface acoustic wave in the substrate. Half of the phase velocity f · p, which will be denoted vp, for increasing aspect ratio is given in figure 6.14(a) for the six modes,
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6.5 High aspect ratio electrodes 49<br />
corresponds to the original GaAs/AlGaAs structure in figure 6.5(a) where the strain<br />
and the optical mode are calculated without the hole. When the strain and the optical<br />
mode is found for 2r/w = 0.8, 1.65 is obtained. Then the strain is calculated<br />
without the hole and then the hole is introduced and the optical mode is found.<br />
This only gives an increase of a few percent. Finally, the opposite is done where the<br />
strain is computed with the hole and the optical mode is calculated with the hole<br />
filled. Now the interaction increases more than 2 times, so this indicates that the<br />
effect from the strain concentrations is dominant. However, it also indicates that<br />
the effect from the strain concentrations and the confinement of the optical mode<br />
are counteracting each other.<br />
This shows that the presented optimization method can be employed to improve<br />
the acousto-optical interaction and even though the designs are complicated to fabricate,<br />
they can be used as inspiration to introduce simpler changes of the initial<br />
design. A way to obtain designs that are easier to fabricate is to introduce a constraint<br />
that assign the design variables to an entire column that starts at the surface<br />
as in [45].<br />
6.5 High aspect ratio electrodes<br />
In the previous sections the acousto-optical model has been employed to study the<br />
interaction between a Rayleigh wave and optical waves in channel waveguides. This<br />
model is now utilized to investigate SAWs generated by high aspect ratio electrodes.<br />
First their mode shapes and the mechanical energy confinement to the electrodes<br />
are studied by a periodic model of a unit cell. The obtained results are compared to<br />
a model with a finite number of electrodes. Finally, the acousto-optical interaction<br />
between the new types of SAWs and an optical wave in a buried channel waveguide<br />
is investigated. The section is a summary of the results presented in publication<br />
[P6] and [P7].<br />
6.5.1 Periodic structure<br />
First a unit cell is studied with periodic boundary conditions connecting the left<br />
and the right boundaries and a PML at the bottom, see figure 6.13. The electrode<br />
consists of nickel (Ni) and the substrate is lithium niobate (LiNbO3). The height<br />
of the electrode is h and the period of the cell is p. When the aspect ratio of<br />
the electrode increases it is possible to excite more mode types than the two that<br />
exist for thinner electrodes. Six different modes exist for the aspect ratio h/2p = 1<br />
and the mode shapes are plotted in figure 6.13 normalized to the applied electric<br />
potential equal to 1 V. Modes with unequal numbers are mainly polarized in the<br />
shear horizontal (SH) direction and modes with equal numbers are mainly vertically<br />
polarized (VP). All the modes are combinations of a vibration in the electrode and<br />
a surface acoustic wave in the substrate. Half of the phase velocity f · p, which will<br />
be denoted vp, for increasing aspect ratio is given in figure 6.14(a) for the six modes,
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