## Differential uncertainty analysis for evaluating the accuracy of S-parameter retrieval methods for electromagnetic properties of metamaterial slabs |

Optics Express, Vol. 20, Issue 27, pp. 29002-29022 (2012)

http://dx.doi.org/10.1364/OE.20.029002

Acrobat PDF (2859 KB)

### Abstract

We apply a complete uncertainty analysis, not studied in the literature, to investigate the dependences of retrieved electromagnetic properties of two MM slabs (the first one with only split-ring resonators (SRRs) and the second with SRRs and a continuous wire) with single-band and dual-band resonating properties on the measured/simulated scattering parameters, the slab length, and the operating frequency. Such an analysis is necessary for the selection of a suitable retrieval method together with the correct examination of exotic properties of MM slabs especially in their resonance regions. For this analysis, a differential uncertainty model is developed to monitor minute changes in the dependent variables (electromagnetic properties of MM slabs) in functions of independent variables (scattering (S-) parameters, the slab length, and the operating frequency). Two complementary approaches (the analytical approach and the dispersion model approach) each with different strengths are utilized to retrieve the electromagnetic properties of various MM slabs, which are needed for the application of the uncertainty analysis. We note the following important results from our investigation. First, uncertainties in the retrieved electromagnetic properties of the analyzed MM slabs drastically increase when values of electromagnetic properties shrink to zero or near resonance regions where S-parameters exhibit rapid changes. Second, any low-loss or medium-loss inside the MM slabs due to an imperfect dielectric substrate or a finite conductivity of metals can decrease these uncertainties near resonance regions because these losses hinder abrupt changes in S-parameters. Finally, we note that precise information of especially the slab length and the operating frequency is a prerequisite for accurate analysis of exotic electromagnetic properties of MM slabs (especially multiband MM slabs) near resonance regions.

© 2012 OSA

**OCIS Codes**

(290.3030) Scattering : Index measurements

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: August 17, 2012

Revised Manuscript: October 15, 2012

Manuscript Accepted: October 22, 2012

Published: December 13, 2012

**Citation**

Ugur Cem Hasar, Joaquim J. Barroso, Cumali Sabah, Yunus Kaya, and Mehmet Ertugrul, "Differential uncertainty analysis for evaluating the accuracy of S-parameter retrieval methods for electromagnetic properties of metamaterial slabs," Opt. Express **20**, 29002-29022 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-29002

Sort: Year | Journal | Reset

### References

- V. G. Veselago, “The electrodynamics of substances with simultaneously negative values ofε and μ, ” Sov. Phys. Uspekhi10, 509–514 (1968). [CrossRef]
- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292(5514), 77–79 (2001). [CrossRef] [PubMed]
- M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K. Y. Kang, Y. H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470(7334), 369–373 (2011). [CrossRef] [PubMed]
- W. H. Wee and J. B. Pendry, “Universal evolution of perfect lenses,” Phys. Rev. Lett.106(16), 165503 (2011). [CrossRef] [PubMed]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84(18), 4184–4187 (2000). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
- R. Melik, E. Unal, N. K. Perkgoz, C. Puttlitz, and H. V. Demir, “Metamaterial-based wireless strain sensors,” Appl. Phys. Lett.95(1), 011106 (2009). [CrossRef]
- L. Jelinek, R. Marques, and M. J. Freire, “Accurate modeling of split ring metamaterial lenses for magnetic resonance imaging applications,” J. Appl. Phys.105(2), 024907 (2009). [CrossRef]
- C. Helgert, C. Rockstuhl, C. Etrich, C. Menzel, E.-B. Kley, A. Tunnermann, F. Lederer, and T. Pertsch, “effective properties of amorphous metamaterials,” Phys. Rev. B79(23), 233107 (2009). [CrossRef]
- D. A. Pawlak, S. Turczynski, M. Gajc, K. Kolodziejak, R. Diduszko, K. Rozniatowski, J. Smalc, and I. Vendik, “How far are we from making metamaterials by self-organization? The microstructure of highly anisotropic particles with an SRR-like geometry,” Adv. Funct. Mater.20(7), 1116–1124 (2010). [CrossRef]
- R. E. Collin, Field Theory of Guided Waves (Wiley-IEEE Press, 1990).
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter10(22), 4785–4809 (1998). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech.47(11), 2075–2084 (1999). [CrossRef]
- K. Aydin, Z. Li, M. Hudlicka, S. A. Tretyakov, and E. Ozbay, “Transmission characteristics of bianisotropic metamaterials based on omega shaped metallic inclusions,” New J. Phys.9(9), 326 (2007). [CrossRef]
- C. Sabah, “Multiband planar metamaterials,” Microw. Opt. Technol. Lett.53(10), 2255–2258 (2011). [CrossRef]
- Q. Zhao, L. Kang, B. Du, B. Li, J. Zhou, H. Tang, X. Liang, and B. Zhang, “Electrically tunable negative permeability metamaterials based on nematic liquid crystals,” Appl. Phys. Lett.90(1), 011112 (2007). [CrossRef]
- H. Nemec, P. Kuzel, F. Kadlec, C. Kadlec, R. Yahiaoui, and P. Mounaix, “Tunable terahertz metamaterials with negative permeability,” Phys. Rev. B79, 241108(R) (2009).
- C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B77(19), 195328 (2008). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
- D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65(19), 195104 (2002). [CrossRef]
- M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B62(16), 10696–10705 (2000). [CrossRef]
- T. Paul, C. Menzel, W. Smigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B84(11), 115142 (2011). [CrossRef]
- Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.79(2), 026610 (2009). [CrossRef] [PubMed]
- D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(33 Pt 2B), 036617 (2005). [CrossRef] [PubMed]
- P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express11(7), 649–661 (2003). [CrossRef] [PubMed]
- M. Bozzi, L. Perregrini, J. Weinzierl, and C. Winnewisser, “Efficient analysis of quasi-optical filters by a hybrid MoM/Bi-RME method,” IEEE Trans. Antenn. Propag.49(7), 1054–1064 (2001). [CrossRef]
- Z. H. Jiang, J. A. Bossard, X. Wang, and D. H. Werner, “Synthesizing metamaterials with angularly independent effective medium properties based on an anisotropic parameter retrieval technique coupled with a genetic algorithm,” J. Appl. Phys.109(1), 013515 (2011). [CrossRef]
- T. Driscoll, D. N. Basov, W. J. Padilla, J. J. Mock, and D. R. Smith, “Electromagnetic characterization of planar metamaterials by oblique angle spectroscopic measurements,” Phys. Rev. B75(11), 115114 (2007). [CrossRef]
- D. R. Smith, D. Schurig, and J. J. Mock, “Characterization of a planar artificial magnetic metamaterial surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.74(3), 036604 (2006). [CrossRef] [PubMed]
- K. B. Alici and E. Ozbay, “Oblique response of a split-ring-resonator-based left-handed metamaterial slab,” Opt. Lett.34(15), 2294–2296 (2009). [CrossRef] [PubMed]
- X. Chen, B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(4), 046610 (2005). [CrossRef] [PubMed]
- U. C. Hasar and J. J. Barroso, “Retrieval approach for determination of forward and backward wave impedances of bianisotropic metamaterials,” Prog. Electromagn. Res.112, 109–124 (2011).
- R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B65(14), 144440 (2002). [CrossRef]
- A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011). [CrossRef]
- A. M. Nicolson and G. Ross, “Measurement of the intrinsic properties of materials by time–domain techniques,” IEEE Trans. Instrum. Meas.19(4), 377–382 (1970). [CrossRef]
- X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(1), 016608 (2004). [CrossRef] [PubMed]
- U. C. Hasar, “A microwave method for accurate and stable retrieval of constitutive parameters of low- and medium-loss materials,” IEEE Microw. Wirel. Compon. Lett.20(12), 696–698 (2010). [CrossRef]
- U. C. Hasar, “Procedure for accurate and stable constitutive parameters extraction of materials at microwave frequencies,” Prog. Electromagn. Res.109, 107–121 (2010). [CrossRef]
- J. Qi, H. Kettunen, H. Wallen, and A. Sihvola, “Compensation of Fabry-Perot resonances in homogenization of dielectric composites,” IEEE Antennas Wireless Propag. Lett.9, 1057–1060 (2010). [CrossRef]
- X.-X. Liu, D. A. Powell, and A. Alu, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B84(23), 235106 (2011). [CrossRef]
- W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE62(1), 33–36 (1974). [CrossRef]
- A. H. Muqaibel and A. Safaai-Jazi, “A new formulation for characterization of materials based on measured insertion transfer function,” IEEE Trans. Microw. Theory Tech.51(8), 1946–1951 (2003). [CrossRef]
- S. Xia, Z. Xu, and X. Wei, “Thickness-induced resonance-based complex permittivity measurement technique for barium strontium titanate ceramics at microwave frequency,” Rev. Sci. Instrum.80(11), 114703 (2009). [CrossRef] [PubMed]
- O. Büyüköztürk, T.-Y. Yu, and J. A. Ortega, “A methodology for determining complex permittivity of construction materials based on transmission-only coherent, wide-bandwidth free-space measurements,” Cement Concr. Compos.28(4), 349–359 (2006). [CrossRef]
- U. C. Hasar, “Unique permittivity determination of low-loss dielectric materials from transmission measurements at microwave frequencies,” Prog. Electromagn. Res.107, 31–46 (2010). [CrossRef]
- Z. Szabo, G.-H. Park, R. Hedge, and E.-P. Li, “Unique extraction of metamaterial parameters based on Kramers-Kronig relationship,” IEEE Trans. Microw. Theory Tech.58(10), 2646–2653 (2010). [CrossRef]
- V. V. Varadan and R. Ro, “Unique retrieval of complex permittivity and permeability of dispersive materials from reflection and transmitted fields by enforcing causality,” IEEE Trans. Microw. Theory Tech.55(10), 2224–2230 (2007). [CrossRef]
- J. J. Barroso and U. C. Hasar, “Resolving phase ambiguity in the inverse problem of transmission/reflection measurement methods,” Int. J. Infrared Millim. Waves32(6), 857–866 (2011). [CrossRef]
- U. C. Hasar, J. J. Barroso, C. Sabah, and Y. Kaya, “Resolving phase ambiguity in the inverse problem of reflection-only measurement methods,” Prog. Electromagn. Res.129, 405–420 (2012).
- O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A tespwise Nicolson-Ross-Weir-based material parameter extraction method,” IEEE Antennas Wirel. Propag. Lett.10, 1295–1298 (2011). [CrossRef]
- B. Kapilevih and B. Litvak, “THz characterization of high-dielectric constant materials using double-layer sample,” Microw. Opt. Technol. Lett.49(6), 1388–1391 (2007). [CrossRef]
- U. C. Hasar and I. Y. Ozbek, “Complex permittivity determination of lossy materials at millimeter and terahertz frequencies using free-space amplitude measurements,” J. Electromagn. Waves Appl.25(14-15), 2100–2109 (2011). [CrossRef]
- U. C. Hasar and A. Abusoglu, “Using millimeter and terahertz frequencies for complex permittivity retrieval of low-loss materials,” J. Electromagn. Waves Appl.25(17-18), 2389–2398 (2011). [CrossRef]
- B. Kapilevich, Y. Pinhasi, and B. Litvak, “Measurement of complex permittivity of lossy materials in free space using matched THz power meter,” Int. J. Infrared Millim. Waves32(12), 1446–1456 (2011). [CrossRef]
- J. Zhou, Th. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry, and C. M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett.95(22), 223902 (2005). [CrossRef] [PubMed]
- E. Pshenay-Severin, F. Setzpfandt, C. Helgert, U. Hubner, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tunnermann, F. Lederer, and T. Pertsch, “Experimental determination of the dispersion relation of light in metamaterials by white-light interferometry,” J. Opt. Soc. Am. B27(4), 660–666 (2010). [CrossRef]
- C. Sabah, “Multiband metamaterials based on multiple concentric open ring resonators topology,” IEEE J. Sel. Topics Quantum Electron. 2012 (DOI#: 10.1109/JSTQE.2012.2193875).
- C. Sabah, “Multi-resonant metamaterial design based on concentric V -shaped magnetic resonators,” J. Electromagn. Waves Appl.26(8-9), 1105–1115 (2012). [CrossRef]
- D. M. Pozar, Microwave Engineering (Wiley, Hoboken, NJ, 2005).
- T. Weiland, R. Schuhmann, R. B. Greegor, C. G. Parazzoli, A. M. Vetter, D. R. Smith, D. C. Vier, and S. Schultz, “Ab initio numerical simulation of left-handed metamaterials: Comparison of calculations and experiments,” J. Appl. Phys.90(10), 5419–5424 (2001). [CrossRef]
- G. Lubkowski, B. Bandlow, R. Schuhmann, and T. Weiland, “Effective modeling of double negative metamaterial macrostructures,” IEEE Trans. Microw. Theory Tech.57(5), 1136–1146 (2009). [CrossRef]
- R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.64(5), 056625 (2001). [CrossRef] [PubMed]
- T. J. Cui and J. A. Kong, “Time-domain electromagnetic energy in a frequency-dispersive left-handed medium,” Phys. Rev. B70(20), 205106 (2004). [CrossRef]
- C. Sabah and S. Uckun, “Multilayer system of Lorentz/Drude type metamaterials with dielectric slabs and its application to electromagnetic filters,” Prog. Electromagn. Res.91, 349–364 (2009). [CrossRef]
- J. Baker-Jarvis, R. G. Geyer, and P. D. Domich, “A nonlinear least-squares solution with causality constrains applied to transmission line permittivity and permeability determination,” IEEE Trans. Instrum. Meas.41(5), 646–652 (1992). [CrossRef]
- S. Xu, L. Yang, L. Huang, and H. Chen, “Experimental measurement method to determine the permittivity of extra thin materials using resonating metamaterials,” Prog. Electromagn. Res.120, 327–337 (2011).
- C. Alexander and M. Sadiku, Fundamentals of Electric Circuits (McGraw-Hill, 2002).
- U. C. Hasar, I. Y. Ozbek, E. A. Oral, T. Karacali, and H. Efeoglu, “The effect of silicon loss and fabrication tolerance on spectral properties of porous silicon Fabry-Perot cavities in sensing applications,” Opt. Express20(20), 22208–22223 (2012). [CrossRef] [PubMed]
- G. Lubkowski, R. Schuhmann, and T. Weiland, “Extraction of effective metamaterial parameters by parameter fitting of dispersive models,” Microw. Opt. Technol. Lett.49(2), 285–288 (2007). [CrossRef]
- R. Storn and K. Price, “Differential evaluation–A simple and efficient heuristic for global optimization over continuous spaces,” J. Glob. Optim.11(4), 341–359 (1997). [CrossRef]
- K. Price, R. Storn, and J. Lampinen, Differential Evolution - A Practical Approach to Global Optimization (Springer, 2005).
- K. Price and R. Storn, “Differential evaluation (DE) for continuous function optimization,” http://www.icsi.berkeley.edu/~storn/code.html .
- The MathWorks, http://www.mathworks.com .
- J. Baker–Jarvis, E. J. Vanzura, and W. A. Kissick, “Improved technique for determining complex permittivity with the transmission/reflection method,” IEEE Trans. Microw. Theory Tech.38(8), 1096–1103 (1990). [CrossRef]
- A. H. Boughriet, C. Legrand, and A. Chapoton, “Noniterative stable transmission/reflection method for low-loss material complex permittivity determination,” IEEE Trans. Microw. Theory Tech.45(1), 52–57 (1997). [CrossRef]
- U. C. Hasar and C. R. Westgate, “A broadband and stable method for unique complex permittivity determination of low-loss materials,” IEEE Trans. Microw. Theory Tech.57(2), 471–477 (2009). [CrossRef]
- J. J. Barroso and A. L. de Paula, “Retrieval of permittivity and permeability of homogeneous materials from scattering parameters,” J. Electromagn. Waves Appl.24(11-12), 1563–1574 (2010). [CrossRef]
- K. Chalapat, K. Sarvala, J. Li, and G. S. Paraoanu, “Wideband reference-plane invariant method for measuring electromagnetic parameters of materials,” IEEE Trans. Microw. Theory Tech.57(9), 2257–2267 (2009). [CrossRef]
- S. J. Kline and F. A. McClintock, “Describing uncertainties in single−sample experiments,” Mech. Eng.75, 3 (1953).
- J. Baker–Jarvis, M. D. Janezic, J. H. Grosvenor, Jr., and R. G. Geyer, “Transmission/reflection and short–circuit line methods for measuring permittivity and permeability,” NIST, Boulder, CO, Tech. Note 1355, (1992).
- G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists: A Comprehensive Guide (Academic Press, 2005).
- E. Kreyszig, Advanced Engineering Mathematics (Wiley, 2006).
- H. J. Pain, The Physics of Vibrations and Waves (Wiley, 2008).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.