Maria Bayard Dühring - Solid Mechanics
Maria Bayard Dühring - Solid Mechanics Maria Bayard Dühring - Solid Mechanics
Figure 3: The distribution of the objective function Φ normalized with √ P close to the optical waveguide. (a): Initial design. (b): Optimized design where air holes are represented with white color. Figure 4: The color bar indicates the distribution of the change in the refractive index component ∆n11 normalized with √ P close to the optical waveguide. The fundamental mode in the waveguide is indicated with contour lines. (a): Initial design. (b): Optimized design. the piezoelectric model. However, due to the objective function, which is only dependent on S2, some debatable mechanisms can appear in the design as the air hole close to the waveguide. This can on one hand create strain concentrations that can increase the objective function, but on the other hand air holes are areas where there is no periodic change in the refractive index components and these can also guide the optical wave, which is clearly unwanted. So it might not always be an advantage to have air holes close to the waveguide as the optical mode can extend out of the output domain. Holes close to the waveguide can be prevented by moving the design domain further away from the waveguide or try out other objective functions that both depend on the strain components and the strain-optical constants such that n11 is optimized more directly. The optimized design almost consists of solid material and air, but it will still be difficult to fabricate it with modern fabrication technics. To make designs that are easier to fabricate the optimization could be done with the constraint that the design variables are assigned to an entire column that starts at the surface as in [14]. 6. Conclusion and future work It was here shown that the method of topology optimization can be employed to improve the acoustooptical interaction between a surface acoustic wave and an optical wave in a waveguide. The SAW generation is modeled by a 2D piezoelectric model, which is coupled to an optical model for the optical 8
mode in the waveguide. The topology optimization approach is based on the piezoelectric model and the objective function is the squared absolute value of the normal strain in the vertical direction. It vas shown that the objective function could be increased more than 2 orders of magnitude by the optimization where several smaller air holes were created in the design domain that initially consisted of solid material. The effect of the air holes is to trap the SAW in the design domain close to the waveguide and to make strain concentrations around the holes that extend into the waveguide. The increased strain concentration in the waveguide changes the refractive index components such that the periodic difference in effective refractive index for the fundamental guided mode, as the SAW passes the waveguide, is increased 10 times compared to the initial design. So the topology optimization approach is a promising method to improve acousto-optical interaction. Further work includes testing of other objective functions that depend both on the strain components and the strain-optical constants such that the refractive index component in the horizontal direction is optimized directly. Finally, the constraint, that the design variables are assigned to entire columns starting at the surface, can be introduced in order to get designs that are easier to fabricate. 7. Acknowledgements This work received support from the Eurohorcs/ESF European Young Investigator Award (EURYI, www.esf.org/euryi) through the grant ”Synthesis and topology optimization of optomechanical systems” and from the Danish Center for Scientific Computing (DCSC). Also support from the EU Network of Excellence ePIXnet is gratefully acknowledged. The author is thankful to Ole Sigmund and Jakob S. Jensen from the Department of Mechanical Engineering, Technical University of Denmark, for helpful discussions related to the presented work. 8. References [1] R.M. White and F.W. Voltmer, Applied Physics Letters, 17, 314-316, 1965. [2] K.-Y. Hashimoto, Surface acoustic wave devices in telecommunications, modeling and simulation, Springer, Berlin, 2000. [3] M.M. de Lima Jr. and P.V. Santos, Modulation of photonic structures by surface acoustic waves. Reports on Progress in Physics, 68, 1639-1701, 2005. [4] M.M. de Lima Jr., M. Beck, R. Hey and P.V. Santos, Compact Mach-Zehnder acousto-optic modulator, Applied Physics Letters, 89, 121104, 2006. [5] M. van der Poel, M. Beck, M.B. Dühring, M.M. de Lima, Jr., L.H. Frandsen, C. Peucheret, O. Sigmund, U. Jahn, J.M. Hwam and P.V. Santos, proceedings of European Conference on Integrated Optics and Technical Exhibition, Copenhagen, Denmark, April 25-27, 2007. [6] M.B. Dühring and O. Sigmund, Improving the acousto-optical interaction in a Mach-Zehnder interferometer, Journal of Applied Physics, in press, 2009. [7] M.P. Bendsøe and O. Sigmund, Topology optimization, theory, methods and applications, Springer Verlag Berlin, 2003, ISBN 3-540-42992-1. [8] E. Wadbro and M. Berggren, Topology optimization of an acoustic horn, Computer Methods in Applied Mechanics and Engineering 196, 420-436, 2006. [9] G.H. Yoon, J.S. Jensen and O. Sigmund, Topology optimization of acoustic-structure interaction problems using a mixed finite element formulation, International Journal for Numerical Methods in Engineering 70, 1049-1075, 2007. [10] J. Du and N. Olhoff, Minimization of sound radiation from vibrating bi-material structures using topology optimization, Structural and Multidisciplinary Optimization, 33, 305-321, 2007. [11] M.B. Dühring, J.S. Jensen and O. Sigmund, Acoustic design by topology optimization, Journal of Sound and Vibration, 317, 557-575, 2008. 9
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Figure 3: The distribution of the objective function Φ normalized with √ P close to the optical waveguide.<br />
(a): Initial design. (b): Optimized design where air holes are represented with white color.<br />
Figure 4: The color bar indicates the distribution of the change in the refractive index component ∆n11<br />
normalized with √ P close to the optical waveguide. The fundamental mode in the waveguide is indicated<br />
with contour lines. (a): Initial design. (b): Optimized design.<br />
the piezoelectric model. However, due to the objective function, which is only dependent on S2, some<br />
debatable mechanisms can appear in the design as the air hole close to the waveguide. This can on one<br />
hand create strain concentrations that can increase the objective function, but on the other hand air<br />
holes are areas where there is no periodic change in the refractive index components and these can also<br />
guide the optical wave, which is clearly unwanted. So it might not always be an advantage to have air<br />
holes close to the waveguide as the optical mode can extend out of the output domain. Holes close to the<br />
waveguide can be prevented by moving the design domain further away from the waveguide or try out<br />
other objective functions that both depend on the strain components and the strain-optical constants<br />
such that n11 is optimized more directly.<br />
The optimized design almost consists of solid material and air, but it will still be difficult to fabricate<br />
it with modern fabrication technics. To make designs that are easier to fabricate the optimization could<br />
be done with the constraint that the design variables are assigned to an entire column that starts at the<br />
surface as in [14].<br />
6. Conclusion and future work<br />
It was here shown that the method of topology optimization can be employed to improve the acoustooptical<br />
interaction between a surface acoustic wave and an optical wave in a waveguide. The SAW<br />
generation is modeled by a 2D piezoelectric model, which is coupled to an optical model for the optical<br />
8
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