Maria Bayard Dühring - Solid Mechanics

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083529-4 M. B. <strong>Dühring</strong> and O. Sigmund J. Appl. Phys. 105, 083529 2009 TABLE I. The elastic stiffness constants 10 11 Nm −2 and the density kg m −3 for the materials used in the piezoelectric model. E Material c11 C ˜ ijkl = M ijmnM klpqC mnpq. 20 For a given optical mode of order and with the out of plane propagation constant , the effective refractive index neff, is defined as neff, = /k0, 21 where k 0 is the free space propagation constant. It is assumed that the propagating optical modes have harmonic solutions on the form H p,x 1,x 2,x 3 = H p,x 1,x 2e −i x 3, 22 where Hp, is the magnetic field of the optical wave. The governing equation for the magnetic field is thus the timeharmonic wave equation Hp 2 eijk klblmemnp k0Hp =0, 23 x jb xn− where e ijk here is the alternating symbol and b ikn kj= ij. For a given value of k 0, the propagation constant for the possible modes is found by solving the wave equation as an eigenvalue problem, whereby the effective refractive indices can be obtained. In this work the set of equations are reduced such that the model is only solved for the transverse components of the magnetic fields H 1 and H 2. As the energy of the lower order optical modes is concentrated in the waveguides, the calculation domain, where the eigenvalue problem is solved, can be reduced to a smaller area around each of the waveguides. At the boundary of such a domain it is assumed that the magnetic field is zero as the energy density quickly decays outside the waveguide, thus the boundary condition is e ijkH jm k =0, 24 where e ijk again is the alternating symbol and m k is the normal unit vector pointing out of the surface. III. RESULTS E c12 In this section results are presented for the SAW propagation, the optical waves in the waveguides, and their acousto-optical interaction. E c13 E c22 GaAs 1.183 0.532 0.532 1.183 0.532 1.183 0.595 0.595 0.595 5316.5 Si 1.660 0.639 0.639 1.660 0.639 1.660 0.796 0.796 0.796 2330 SiO 2 0.785 0.161 0.161 0.785 0.161 0.785 0.312 0.312 0.312 2200 ZnO 2.090 1.205 1.046 2.096 1.046 2.106 0.423 0.423 0.4455 5665 E c23 E c33 E c44 A. Simulation of SAWs The piezoelectric model is first employed to solve the propagation of a SAW in a GaAs sample with straight surface to show that the model performs as expected. The SAW is generated by ten double electrode finger pairs. Each of the electrodes has a width equal to 0.7 m and is placed 0.7 m apart such that the wavelength of the generated SAW is 5.6 m. GaAs is a piezoelectric material with cubic crystal structure. The material constants used in the piezoelectric model are given in Tables I and II and are here given in the usual matrix form for compactness reasons. The frequency is f SAW=510 MHz where f SAW= SAW/2 and the applied electric potential is V p=1 V. The constant j controlling the damping in the PMLs is set to 10 10 . GaAs is piezoelectric in the 110 direction, so for the SAW to propagate in this direction the material tensors have to be rotated by the angle 2=/4. Second order Lagrange elements are used for all the calculations. In Fig. 2a the displacement u 2 in the x 2-direction is plotted in the domain with a displacment scaling factor set to 1000, and it is seen that the SAW is generated at the electrodes and propagates away from them to the right and the left. In the PMLs the wave is gradually absorbed. The same results are illustrated at Fig. 2b where the absolute amplitude absu 2 is plotted along the material surface. It is seen how a standing wave is generated at the electrodes because the SAW here travels in both directions. Away from the electrodes the SAW is a traveling wave and therefore the amplitude is nearly constant. It is also seen how the SAW is absorbed in the PMLs. In Fig. 3a the displacements u 1 and u 2 are plotted as function of the depth for x 1=85 m. It is seen that the displacements are concentrated within one SAW wavelength of the surface as expected. Below the surface they have smaller oscillations. This is due to the finite depth and the use of the PMLs that are not exactly acting as an infinite domain. On Fig. 3b the absolute amplitude absu 2 normalized with the square root of the applied electric power P is plotted as function of the frequency f SAW at a point on the surface to the right of the TABLE II. The piezoelectric stress constants and the permittivity constants for the materials used in the piezoelectric model. Material e 14 C m −2 e 15 C m −2 e 24 C m −2 e 25 C m −2 e 31 C m −2 e 32 C m −2 e 33 C m −2 e 36 C m −2 E c55 S 11 10−11 Fm E c66 S 22 10−11 Fm S 33 10−11 Fm GaAs 0.160 0 0 0.160 0 0 0 0.160 9.735 9.735 9.735 Si 0 0 0 0 0 0 0 0 11.5 11.5 11.5 SiO 2 0 0 0 0 0 0 0 0 2.37 2.37 2.37 ZnO 0 0.480 0.480 0 0.573 0.573 1.321 0 7.38 7.38 7.83 Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
083529-5 M. B. <strong>Dühring</strong> and O. Sigmund J. Appl. Phys. 105, 083529 2009 FIG. 2. Color online Generation of a SAW in a GaAs sample by the piezoelectric model. The position of the IDT is indicated. a The color indicates the displacement u 2 and the shape of the surface is deformed with the unified displacements u 1 and u 2. b The graph shows the absolute amplitude absu 2 along the material surface as function of x 1 IDT for the frequency interval of 210–810 MHz for 10 and 20 electrode pairs, respectively. The power P is calculated by the expression 16 P = R Lel ViD2m2 dL, 25 2 where L el corresponds to boundaries with the electrodes and m 2 is the normal vector to the upper surface. Figure 3b shows that the highest value of absu 2/P appears at f SAW =510 MHz, which corresponds to the frequency found by experiments for a GaAs sample, see Ref. 18. Hence, these results show that the piezoelectric model performs as ex- (a) (b) FIG. 3. Color online Results for the SAW in the GaAs substrate. a Displacements u 1 and u 2 as function of depth at position x 1=85 m. b The absolute amplitude absu 2 normalized with the square root of the electrical power P as function of the frequency f saw for 10 and 20 electrode pairs. FIG. 4. Color online Generation of SAWs in a SOI sample by the piezoelectric model. The results are given to the right of the IDT. a The color indicates the displacement u 2 and the shape of the surface is deformed with the unified displacements u 1 and u 2. b The graph shows the absolute amplitude absu 2 along the material surface as function of x 1. pected for generation of SAWs. It can also be observed from Fig. 3b that when the number of electrode pairs increases, the bandwidth decreases as more of the applied energy is confined to the SAW and consequently the amplitude increases. For twice as many fingers the bandwidth is halved, which is expected as the bandwidth is inversely proportional to the number of finger pairs see Ref. 15. In practical constructions several hundred finger pairs are used in order to generate a strong SAW. For computational reasons only a fraction of the finger pairs can be used in these calculations. This means that a SAW with a smaller amplitude as in practice is used and the acousto-optical interaction is therefore also smaller than in experiments. However, the same physical effects are expected, which is discussed further in Sec. IIIC1. In Refs. 8 and 9 results for the acousto-optical modulation in a MZI with GaAs waveguides on a Al 0.2Ga 0.8As layer were presented as well as results for a MZI with Si waveguides on a SiO 2 layer. In the first case a relative modulation of 40% was obtained with straight waveguides, but for the SOI sample the modulation was only between 0% and 8%. In this work attention is given to the SOI sample in order to understand the unsatisfactory performance and what can be done to improve the modulation. As Si is not piezoelectric the SAW is generated in a ZnO layer on top of the Si from which the SAW propagates toward the rest of the sample with the optical waveguides. The geometry is indicated on Fig. 1b. The IDT consists of double double finger electrode pairs where each of the electrodes again has a width of 0.7 m such that the wavelength of the generated SAW is 5.6 m. The material constants for Si, SiO 2, and ZnO used in the piezoelectric model are given in Tables I and II. The applied potential is V p=1 V and it is driven with the frequency f SAW=630 MHz. After numerical experiments the constant j in the PMLs is set to 10 10 . All the materials are rotated with 2=/4 such that the SAW propagates along the 110 direction. In Fig. 4a a zoom of the displacements u 2 around the end of the ZnO layer is plotted and it is seen how the wave is generated at the electrodes to the left and propagates from the ZnO layer to the rest of the sample with the waveguides to the right. In Fig. 4b the absolute ampli- Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
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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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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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34 Chapter 6 Design of acousto-opti
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60 References [14] E. Yablonovitch,
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62 References [42] A. A. Larsen, B.
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Maria Bayard Dühring - Solid Mechanics

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