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A Two-Stage Approach for Frequency Response Modeling and Metamaterial Rapid Design

By Xiao Guo, Chunlin Ji, Ruo Liu, and Tao Tang
Progress In Electromagnetics Research C, Vol. 76, 11-22, 2017


We introduce a novel two-stage approach for rapid design of massive metamaterials (MTMs), where performances of thousands of microstructures require evaluation. In Stage I, an equivalent circuit model is synthesized via rational function modeling to represent the frequency response of MTMs microstructures. In Stage II, Gaussian process (GP) regression models are unitized to build the relation between the physical setting of the microstructure, including geometric design variables and incident angles of electromagnetic (EM) waves and the representing parameters of the equivalent circuit model. As a consequence, the mapping from the microstructure physical parameters to the frequency response is easy to achieve and with high accuracy. We offer two metamaterial prototypes to illustrate that the proposed approach allows high efficiency in facilitating the design of massive MTMs. The experimental results demonstrate that our method is no longer limited by the complexity of microstructures and the spatial dispersion, induced by the variation of incident angle. We compare the accuracy of predicted responses against the reference data, and both examples yield average RMSE less than 0.05, which meets the requirements for many MTMS engineering applications.


Xiao Guo, Chunlin Ji, Ruo Liu, and Tao Tang, "A Two-Stage Approach for Frequency Response Modeling and Metamaterial Rapid Design," Progress In Electromagnetics Research C, Vol. 76, 11-22, 2017.


    1. Shelby, R. A., D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science, Vol. 292, 77-9, Apr. 6, 2001.

    2. Eleftheriades, G. V. and K. G. Balmain, Negative-refraction Metamaterials: Fundamental Properties and Applications, J. Wiley, Hoboken, NJ, ISBN: 978-0471744757, 2005.

    3. Marques, R., F. Mart´ın, and M. Sorolla, Metamaterials with Negative Parameters: Theory, Design and Microwave Applications, John Wiley & Sons, ISBN: 978-0-471-74582-2, 2011.

    4. Cui, T. J., D. Smith, and R. Liu, Metamaterials: Theory, Design, and Applications, Springer, New York, NY, ISBN: 978-1441905734, 2014.

    5. Liu, R., C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, "Broadband ground-plane cloak," Science, Vol. 323, 366-9, 2009.

    6. Liu, B. and C. Ji, "Bayesian nonparametric modeling for rapid design of metamaterial microstructures," International Journal of Antennas & Propagation, Vol. 2014, 187-187, 2014.

    7. Smith, D. R., S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Physical Review B, Vol. 65, 195104, 2001.

    8. Chen, X., T. M. Grzegorczyk, B. I. Wu, P. J. Jr, and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Physical Review E Statistical Nonlinear & Soft Matter Physics, Vol. 70, 811-811, 2004.

    9. Menzel, C., C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, "Retrieving effective parameters for metamaterials at oblique incidence," Physical Review B, Vol. 77, No. 19, 195328-1-195328-8, 2008.

    10. Rahm, M., D. Roberts, J. Pendry, and D. Smith, "Transformation-optical design of adaptive beam bends and beam expanders," Optics Express, Vol. 16, 11555-11567, 2008.

    11. Rahm, M., D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, "Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations," Photonics and Nanostructures-fundamentals and Applications, Vol. 6, 87-95, 2008.

    12. Schurig, D., et al., "Metamaterial electromagnetic cloak at microwave frequencies," Science, Vol. 314, 977-980, 2006.

    13. Koschny, T., P. Marko, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Physical Review B, Vol. 71, 5105, 2005.

    14. Li, J. and J. Pendry, "Hiding under the carpet: A new strategy for cloaking," Physical Review Letters, Vol. 101, 203901, 2008.

    15. Jiang, W. X., J. Y. Chin, Z. Li, Q. Cheng, R. Liu, and T. J. Cui, "Analytical design of conformally invisible cloaks for arbitrarily shaped objects," Physical Review E, Vol. 77, 066607, 2008.

    16. Bilotti, F., A. Toscano, L. Vegni, and K. Aydin, "Equivalent-circuit models for the design of metamaterials based on artificial magnetic inclusions," IEEE Transactions on Microwave Theory & Techniques Mtt, Vol. 55, 2865-2873, 2007.

    17. Chen, H., L. Ran, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, "Equivalent circuit model for left-handed metamaterials," Journal of Applied Physics, Vol. 100, 024915-024915-6, 2006.

    18. Gil, I., J. Bonache, J. Garcia-Garcia, and F. Martin, "Tunable metamaterial transmission lines based on varactor-loaded split-ring resonators," IEEE Transactions on Microwave Theory & Techniques, Vol. 54, 2665-2674, 2006.

    19. Antonini, G., "SPICE equivalent circuits of frequency-domain responses," IEEE Transactions on Electromagnetic Compatibility, Vol. 45, 502-512, 2003.

    20. Majumdar, P., Z. Zhao, Y. Yue, C. Ji, and R. Liu, "Equivalent circuit model of cross and circular ring FSS using vector fitting," 2014 3rd Asia-Pacific Conference on Antennas and Propagation (APCAP), 1042-1045, 2014.

    21. Majumdar, P., Z. Zhao, Y. Yue, C. Ji, and R. Liu, "Equivalent circuit model of different configurations of loop elements using vector-fitting," PIERS Proceedings, 2395-2399, Guangzhou, Aug. 25–28, 2014.

    22. Williams, C. K. and C. E. Rasmussen, Gaussian Processes for Machine Learning, MIT Press, ISBN 0-262-18253-X, 2006.

    23. Semlyen, A. and B. Gustavsen, "Vector fitting by pole relocation for the state equation approximation of nonrational transfer matrices," Circuits, Systems and Signal Processing, Vol. 19, 549-566, 2000.

    24. Gustavsen, B. and A. Semlyen, "Rational approximation of frequency domain responses by vector fitting," IEEE Transactions on Power Delivery, Vol. 14, 1052-1061, 1999.

    25. Munk, B., Frequency Selective Surfaces: Theory and Design, J. Wiley, Hoboken, NJ, ISBN 978-0- 471-37047-5, 2005.