volume | PIER Journals

Abstract

This paper discusses computation of shape derivatives of electromagnetic fields produced by complex 2-periodic structures. A dual set of forward and adjoint problems for Maxwell's equations are solved with the method of moments (MoM) to calculate the full gradient of the object function by the adjoint variable method (AVM). The periodic fast multipole method (pFMM) is used to accelerate the solution of integral equations for electromagnetic scattering problems with periodic boundary conditions (PBC). This technique is applied to shape optimization problems for negative-index metamaterials (NIM) with a double-fishnet structure. Numerical results demonstrate that the figure of merit (FOM) of metamaterials can reach a maximum value when the shape parameters are optimized iteratively by a gradient-based optimization method.

1. Pendry, J. B., "Negative refraction makes a perfect lens," Phys. Rev. Lett., Vol. 85, 3966-3969, 2000.
doi:10.1103/PhysRevLett.85.3966

2. Zhang, S., W. Fan, N. Panoiu, R. Osgood, and S. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett., Vol. 95, No. 13, 137404, 2005.
doi:10.1103/PhysRevLett.95.137404

3. Dolling, G., M. Wegener, C. M. Soukoulis, and S. Linden, "Negative-index metamaterial at 780nm wavelength," Opt. Lett., Vol. 32, No. 1, 53-55, 2007.
doi:10.1364/OL.32.000053

4. Valentine, J., S. Zhang, T. Zentgraf, and X. Zhang, "Development of bulk optical negative index fishnet metamaterials: Achieving a low-loss and broadband response through coupling," Proceedings of the IEEE, Vol. 99, No. 10, 1682-1690, 2011.
doi:10.1109/JPROC.2010.2094593

5. Dolling, G., C. Enkrich, C. Soukoulis, and S. Linden, "Design-related losses of double-fishnet negative-index photonic metamaterials," Opt. Express, Vol. 15, No. 18, 11536-11541, 2007.
doi:10.1364/OE.15.011536

6. Smith, D. R. and S. Schultz, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients ," Phys. Rev., Vol. B65, 195104, 2002.

7. Kildishev, A. V., U. K. Chettiar, Z. Liu, V. M. Shalaev, D.-H. Kwon, Z. Bayraktar, and D. A. Werner, "Stochastic optimization of low-loss optical negative-index metamaterial," J. Opt. Soc. Am. B, Vol. 24, A34-A39, 2007.
doi:10.1364/JOSAB.24.000A34

8. Bossard, J. A., S. Yun, D. H. Werner, and T. S. Mayer, "Synthesizing low loss negative index metamaterial stacks for the mid-infrared using genetic algorithm," Opt. Express, Vol. 17, 14771-14779, 2009.
doi:10.1364/OE.17.014771

9. Zhao, Y., F. Chen, Q. Shen, Q. Liu, and L. Zhang, "Optimizing low loss negative index metamaterial for visible spectrum using differential evolution," Opt. Express, Vol. 19, 11605-11614, 2011.
doi:10.1364/OE.19.011605

10. Otani, Y. and N. Nishimura, "A periodic FMM for Maxwell's equations in 3D and its applications to problems to photonic crystals," J. Comput. Phys., Vol. 227, No. 9, 4630-4652, 2008.
doi:10.1016/j.jcp.2008.01.029

11. Otani, Y. and N. Nishimura, "An FMM for orthotropic periodic boundary value problems for Maxwell's equations," Waves in Random and Complex Media, Vol. 19, No. 1, 80-104, 2009.
doi:10.1080/17455030802616863

12. Ergül, Ö. and L. Gürel, "Efficient solutions of metamaterial problems using a low-frequency multilevel fast multipole algorithm," Progress In Electromagnetics Research, Vol. 108, 81-99, 2010.
doi:10.2528/PIER10071104

13. Bondeson, A., Y. Yang, and P. Weinerfelt, "Shape optimization for radar cross sections by a gradient method," Int. J. Num. Meth. Eng., Vol. 61, 687-715, 2004.
doi:10.1002/nme.1088

14. Chew, W. C., J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, Boston, 2001.

15. Fraysse, V., L. Giraud, S. Gratton, and J. Langou, "Algorithm 842: A set of GMRES routines for real and complex arithmetics on high performance computers," ACM Trans. Math. Softw., Vol. 31, No. 2, 228-238, 2005.
doi:10.1145/1067967.1067970

16. Johnson, P. B. and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B, Vol. 6, 4370-4379, 1972.
doi:10.1103/PhysRevB.6.4370

17. Byrd, R. H., J. Nocedal, and R. B. Schnabel, "Representations of quasi-Newton matrices and their use in limited memory methods," Mathematical Programming, Vol. 63, No. 4, 129-156, 1994.
doi:10.1007/BF01582063

18. Veronis, G., R. W. Dutton, and S. Fan, "Method for sensitivity analysis of photonic crystal devices," Opt. Lett., Vol. 29, No. 19, 2288-2290, 2004.
doi:10.1364/OL.29.002288

19. Niino, K. and N. Nishimura, "Preconditioning based on Calderon's formulae for periodic fast multipole methods for Helmholtz' equation," J. Comput. Phys., Vol. 231, 66-81, 2012.
doi:10.1016/j.jcp.2011.08.019

Dr. Wu Wang

Prof. Naoshi Nishimura