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Retrieval of the Frequency-Dependent Effective Permeability and Permittivity of the Inhomogeneous Material in a Layer

By Armand Wirgin
Progress In Electromagnetics Research B, Vol. 70, 131-147, 2016


This study is focused on how to obtain the effective or equivalent properties of inhomogeneous materials, which, contrary to the usual metamaterials, are assumed to possess only a sandwichlike form of heterogeneity. More specifically, the aim is to see how the method of inversion, and associated type and amount of data, condition the outcome of the inversion, notably as concerns the possibility or not of exotic features such as simultaneous negative permittivity and permeability in certain frequency intervals. Two inversion schemes are considered and compared: the Nicolson-Ross-Weir (NRW) scheme and an optimization scheme. The adopted form of the optimization scheme provides only numerical retrievals, but it applies to any number of far-field data couples, which fact makes it a useful tool for determining whether the retrieved properties of an inhomogeneous material really are independent of the angle of incidence as is required for effective properties. It is shown, via the optimization scheme, that the apparently infinite number of solutions predicted by the NRW scheme is reduced to a single solution-closest to the predictions of a mixture model-when the constraint of independence with respect to angle of incidence is invoked. Moreover, this solution exhibits none of the exotic features of the properties of the usual metamaterials except temporal dispersion and loss even when the component materials of the inhomogeneous layer are neither dispersive nor lossy.


Armand Wirgin, "Retrieval of the Frequency-Dependent Effective Permeability and Permittivity of the Inhomogeneous Material in a Layer," Progress In Electromagnetics Research B, Vol. 70, 131-147, 2016.


    1. Barroso, J. J. and A. L. De Paula, "Retrieval of permittivity and permeability of homogeneous materials from scattering parameters," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 11, 1563-1574, July 2010.

    2. Chambouleyron, I. and J. M. Martinez, "Optical properties of dielectric and semiconductor thin films," Handbook of Thin Films Materials, Vol. 3, Nalwa H. S. (ed.), Academic Press, New York, 2001.

    3. Chen, X., T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E, Vol. 70, 016608, 2004.

    4. De Paula, A. L., M. C. Rezende, and J. J. Barroso, "Experimental measurements and numerical simulation of permittivity and permeability of Teflon in X band," J. Aerosp. Technol. Manag., Vol. 3, 59-64, Sao Jose dos Campos, 2011.

    5. Hadamard, J., Lectures on Cauchy's Problem in Linear Partial Differential Equations, Yale University Press, New Haven, 1923.

    6. Liu, X.-X. and A. Alu, "Generalized retrieval method for metamaterial constitutive parameters based on a physically-driven homogenization approach," Phys. Rev. B, Vol. 87, 235136, 2013.

    7. Liu, X.-X., D. A. Powell, and A. Alu, "Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures," Phys. Rev. B, Vol. 84, 235106, 2011.

    8. Markos, P. and C. M. Soukoulis, "Transmission properties and effective electromagnetic parameters of double negative metamaterials," Optics Expr., Vol. 11, 649-661, 2003.

    9. Menzel, C., T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, "Validity of effective material parameters for optical fishnet metamaterials," Phys. Rev. B, Vol. 81, 035320, 2010.

    10. Nicolson, A. M. and G. Ross, "Measurement of the intrinsic properties of materials by time-domain techniques," IEEE Trans. Instrum. Meas., Vol. 19, 377-382, 1970.

    11. O'Brien, S. and J. B. Pendry, "Photonic band-gap effects and magnetic activity in dielectric composites," J. Phys. Condens. Matter, Vol. 14, 4035-4044, 2002.

    12. Ogam, E., Z. E. A. Fellah, and P. Baki, "The inverse problem of acoustic wave scattering by an air-saturated poroelastic cylinder," J. Acoust. Soc. Am., Vol. 133, No. 3, 1443-1457, 2013.

    13. Seal, M. D., M. W. Hyde, IV, and M. J. Havrilla, "Nondestructive complex permittivity and permeability extraction using a two-layer dual-waveguide probe measurement geometry," Progress In Electromagnetics Research, Vol. 123, 123-142, 2012.

    14. Sihvola, A., "Mixing models for heterogeneous and granular Media," Advances in Electromagnetics of Complex Media and Metamaterials, Zouhdi S., Sihvola A. and Arsalane M. (eds.), Kluwer, Amsterdam, 2002.

    15. Simovski, C. R. and S. A. Tretyakov, "On effective electromagnetic parameters of artificial nanostructured magnetic materials," Photonics Nanostruct. Fundamen., Vol. 8, No. 4, 254-263, 2010.

    16. Smith, D. R., S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B, Vol. 65, 195104, 2002.

    17. Smith, D. R., D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E, Vol. 71, 036617, 2005.

    18. Weir, W. B., "Automatic measurement of complex dielectric constant and permeability at microwave frequencies," Proc. IEEE, Vol. 62, 33-36, 1974.

    19. Wirgin, A., "Optical properties of a noble metal with a string-of-pearls insulator inhomogeneity," Physica A, Vol. 157, 382-387, 1989.

    20. Woodley, J. and M. Mojahedi, "On the signs of the imaginary parts of the effective permittivity and permeability in metamaterials," J. Opt. Soc. Am. B, Vol. 27, 1016-1021, 2010.