volume | PIER Journals

Abstract

Due to the increasing complexity of metamaterial geometric structures, direct optimisation of these designs using conventional approaches, such as Gradient-based and evolutionary algorithms, are often impractical and limited. This is in part due to the inherently high computational cost associated with running multiple expensive high-fidelity full-wave simulations, commonly required to optimise the constitutive parameters of a single metamaterial particle. In order to alleviate this issue, we propose an efficient optimisation approach which exploits the Co-Kriging methodology, such that we can successfully couple varying levels of discretisation and solver accuracy obtained from a 3d full wave numerical solver suite. In contrast to other optimisation strategies, we investigate the improvement in efficiency of optimisation through the use of the LOLA-Voronoi, in conjunction with Expected Improvement and the embedding of a trustregion framework within our optimisation algorithm, to accelerate the convergence of Co-Kriging. Finally, the effectiveness of the outlined algorithm will be demonstrated by a quantitative evaluation of the performance of optimised planar 2D negative index of refraction structures.

1. Pendry, J. B., A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. on Microwave Theory and Techniques, Vol. 47, No. 11, 2075-2084, Nov. 1999.
doi:10.1109/22.798002 Google Scholar

2. Smith, D. R., W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Physical Review Letters, Vol. 84, 4184-4187, May 2000.
doi:10.1103/PhysRevLett.84.4184 Google Scholar

3. Bradley, P. J., "Quasi-Newton model-trust region approach to surrogate-based optimisation of planar metamaterial structures," Progress In Electromagnetics Research B, Vol. 47, 1-17, 2013. Google Scholar

4. Huang, R., Z.-W. Li, L. B. Kong, L. Liu, and S. Matitsine, "Analysis and design of an ultra-thin metamaterial absorber," Progress In Electromagnetics Research B, Vol. 14, 407-429, 2009.
doi:10.2528/PIERB09040902 Google Scholar

5. Koziel, S., J. W. Bandler, and Q. S. Cheng, "Robust trust-region space-mapping algorithms for microwave design optimization," IEEE Trans. on Microwave Theory and Techniques, Vol. 58, No. 8, 2166-2174, Aug. 2010.
doi:10.1109/TMTT.2010.2052666 Google Scholar

6. Koziel, S. and D. Echeverra Ciaurri, "Reliable simulation-driven design optimization of microwave structures using manifold mapping," Progress In Electromagnetics Research B, Vol. 26, 361-382, 2010.
doi:10.2528/PIERB10090202 Google Scholar

7. Krige, D. G., "A statistical approach to some basic mine valuation problems on the witwatersrand," problems on the witwatersrand, Metallurgical and Mining Society of South Africa, Vol. 52, No. 6, 119-139, Dec. 1951. Google Scholar

8. Broomhead, D. S. and D. Lowe, "Multivariable functional interpolation and adaptive networks," Complex Systems, Vol. 2, 321-355, 1988. Google Scholar

9. Box, G. E. P. and N. R. Draper, "Empirical Model-building and Response Surface," John Wiley & Sons, Inc., 1986. Google Scholar

10. Jones, D. R., M. Schonlau, and W. J. Welch, "Effcient global optimization of expensive black-box functions," Journal of Global Optimization, Vol. 13, No. 4, 455-492, Dec. 1998.
doi:10.1023/A:1008306431147 Google Scholar

11. Jones, D. R., "A taxonomy of global optimization methods based on response surfaces," Journal of Global Optimization, Vol. 21, No. 4, 345-383, 2001.
doi:10.1023/A:1012771025575 Google Scholar

12. Forrester, A., A. Sobester, and A. Keane, Engineering Design via Surrogate Modelling: A Practical Guide, Wiley, 2008.
doi:10.1002/9780470770801

13. Forrester, A. I. J., A. Sbester, and A. J. Keane, "Multifidelity ptimization via surrogate modelling," Proceedings of the Royal Sociery A, Vol. 463, No. 2088, 3251-3269, Dec. 2007.
doi:10.1098/rspa.2007.1900 Google Scholar

14. Deschrijver, D., K. Crombecq, H. M. Nguyen, and T. Dhaene, "Adaptive sampling algorithm for macromodeling of parameterized S-parameter responses," IEEE Trans. on Microwave Theory and Techniques, Vol. 59, No. 1, 39-45, Jan. 2011.
doi:10.1109/TMTT.2010.2090407 Google Scholar

15. Koziel, S., L. Leifsson, I. Couckuyt, and T. Dhaene, "Robust variable-fidelity optimization of microwave filters using Co-Kriging and trust regions," Microwave and Optical Technology Letters, Vol. 55, No. 4, 765-769, 2013.
doi:10.1002/mop.27447 Google Scholar

16. Liu, R., A. Degiron, J. J. Mock, and D. R. Smith, "Negative index material composed of electric and magnetic resonators," Applied Physics Letters, Vol. 90, No. 26, 263504-263504-3, 2007.
doi:10.1063/1.2752120 Google Scholar

17. Zhou, J., E. N. Economon, T. Koschny, and C. M. Soukoulis, "Unifying approach to left-handed material design," Optics Letters, Vol. 31, No. 24, 3620-3622, Dec. 2006.
doi:10.1364/OL.31.003620 Google Scholar

18. Withayachumnankul, W., C. Fumeaux, and D. Abbott, "Near-field interactions in electric inductive-capacitive resonators for metamaterials," Journal of Physics D: Applied Physics, Vol. 45, No. 48, 485101, 2012.
doi:10.1088/0022-3727/45/48/485101 Google Scholar

19. Liu, H., T. Li, S. M. Wang, and S. N. Zhu, "Magnetic plasmon modes introduced by the coupling effect in metamaterials," 2008 International Workshop on Metamaterials, 50-52, 2008.
doi:10.1109/META.2008.4723529 Google Scholar

20. Kante, B., S. N. Burokur, A. Sellier, A. de Lustrac, and J.-M. Lourtioz, "Controlling plasmon hybridization for negative refraction metamaterials," Phys. Rev. B, Vol. 79, 075121, Feb. 2009.
doi:10.1103/PhysRevB.79.075121 Google Scholar

21. Zhou, J., T. Koschny, L. Zhang, G. Tuttle, and C. M. Soukoulis, "Experimental demonstration of negative index of refraction," Applied Physics Letters, Vol. 88, No. 22, 221103-221103-3, 2006.
doi:10.1063/1.2208264 Google Scholar

22. Zhou, J., L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, "Negative index materials using simple short wire pairs," Phys. Rev. B, Vol. 73, 041101, Jan. 2006.
doi:10.1103/PhysRevB.73.041101 Google Scholar

23. Zaoui, W. S., K. Chen, W. Vogel, and M. Berroth, "Low loss broadband polarization independent fishnet negative index metamaterial at 40 GHz," Photonics and Nanostructures --- Fundamentals and Applications, Vol. 10, No. 3, 245-250, 2012.
doi:10.1016/j.photonics.2011.02.003 Google Scholar

24. Zhang, S., W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, "Experimental demonstration of near-infrared negative-index metamaterials," Phys. Rev. Lett., Vol. 95, 137404, Sep. 2005.
doi:10.1103/PhysRevLett.95.137404 Google Scholar

25. Ding, P., E. J. Liang, W. Q. Hu, L. Zhang, Q. Zhou, and Q. Z. Xue, "Numerical simulations of terahertz double-negative metamaterial with isotropic-like fishnet structure," Photonics and Nanostructures --- Fundamentals and Applications, Vol. 7, No. 2, 92-100, 2009.
doi:10.1016/j.photonics.2008.12.005 Google Scholar

26. Shen, N.-H., L. Zhang, T. Koschny, B. Dastmalchi, M. Kafesaki, and C. M. Soukoulisand C. M. Soukoulis, "Discontinuous design of negative index metamaterials based on mode hybridization," Applied Physics Letters, Vol. 101, 081913, 2012.
doi:10.1063/1.4748361 Google Scholar

27. Smith, D. R., D. C. Vier, Th. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E, Vol. 71, 036617, Mar. 2005.
doi:10.1103/PhysRevE.71.036617 Google Scholar

28. Nicolson, A. M. and G. F. Ross, "Measurement of the intrinsic properties of materials by time-domain techniques," IEEE Trans. on Instrumentation and Measurement, Vol. 19, No. 4, 377-382, 1970.
doi:10.1109/TIM.1970.4313932 Google Scholar

29. Varadan, V. V. and R. Ro, "Unique retrieval of complex permittivity and permeability of dispersive materials from re°ection and transmitted ¯elds by enforcing causality," IEEE Trans. on Microwave Theory and Techniques, Vol. 55, No. 10, 2224-2230, 2007.
doi:10.1109/TMTT.2007.906473 Google Scholar

30. Barroso, J. J. and U. C. Hasar, "Constitutive parameters of a ," Journal of Infrared, Millimeter, and Terahertz Waves, Vol. 33, No. 2, 237-244, 2012.
doi:10.1007/s10762-011-9869-3 Google Scholar

31. Hsieh, F.-J. and W.-C. Wang, "Full extraction methods to retrieve e®ective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models," Journal of Applied Physics, Vol. 112, No. 6, 064907, 2012.
doi:10.1063/1.4752753 Google Scholar

32. Alu, A., "Restoring the physical meaning of metamaterial constitutive parameters," Phys. Rev. B, Vol. 83, 081102, Feb. 2011.
doi:10.1103/PhysRevB.83.081102 Google Scholar

Dr. Patrick J. Bradley