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In a recent paper, Zhao et al. [Opt. Express 19(12), 11605 (2011)], proposed the use of differential evolution technique to optimize figure of merit of a negative index metamaterial (NIM) for the visible spectrum. In this comment, we argue that certain ambiguities associated with the effective parameter retrieval should be also addressed in the paper for the accurate implementation of the technique for NIMs. Furthermore, the figure of merit reported in the paper is unrealistically large.
©2014 Optical Society of America
In a recent paper, Zhao et al. [1], proposed that geometry of a negative index metamaterial (NIM) can be optimized to obtain high figure of merit (FOM = −n′/n″) by combining differential evolution (DE) algorithm with an electromagnetic simulation software. They presented an example by iteratively optimizing FOM of a fishnet metamaterial for the visible spectrum. Each iteration consists of s-parameter calculation, retrieval of effective parameters, and calculation of geometric dimensions for improved FOM. However, authors do not consider the effect of multiple retrieved branches for n′given by [2],
Effective parameters of a metamaterial can easily be misinterpreted by using the wrong branch for n′ especially when there are discontinuities in the retrieved index [3]. Various rigorous approaches have been proposed to select the correct branch. See, for example [3,4]. Such an approach needs to be also implemented in the DE algorithm to avoid erroneous results in DE-optimized NIM. Errors in FOM due to the use of the wrong branch may occur at any iteration of the optimization process making the final result questionable. Figure 1 shows retrieved refractive index for optimized fishnet structure with the same boundary conditions and assumptions as presented in [1], using transient as well as frequency domain solver of CST Microwave Studio (MWS). Similar discontinuities were also observed in the retrieved parameters (not shown) for intermediate generations in [1]. Observed discontinuities in n′ branches make the selection of the correct branch difficult, increasing doubts on results of each iteration. An appropriate implementation of DE for optimizing the FOM of an NIM should incorporate an efficient automated process to resolve this branch ambiguity.
Fig. 1 Retrieved effective index of optimized fishnet metamaterial using: (a) transient solver (b) frequency domain solver of CST MWS. In both cases, the discontinuity in n′ occurs at frequency close to possible negative index band, increasing the possibility of wrong interpretation of the DE-algorithm.
It should be noted that CST MWS gives more realistic results for FOM if collision frequency (fc) for the Drude model in CST is defined using its angular frequency value rather than its linear frequency i.e. by taking fc = 85THz in the material properties for the Drude model of silver rather than taking fc = 85THz/(2π) = 13.5THz. As shown in Fig. 2, the highest FOMs (1.7 and 0.6, respectively) obtained using CST with fc = 85THz and COMSOL with fc = 13.5THz closely approximate the FOM ( = 1.1) obtained using the experimental data for silver [5]. On the other hand, FOM obtained by CST with fc = 13.5THz is noticeably larger. Similar discrepancy can be also observed in [6, Fig. 5], where a comparison of CST simulation with an experimental result is presented. Nevertheless, if the CST simulation with fc = 13.5THz is considered, the highest FOM obtained would be 6.4 as opposed to 15.2, reported in [1]. Therefore, even if the ambiguity due to multiple branches is resolved, the FOM of the optimized fishnet structure reported in [1], is still unrealistically large that it cannot be achieved by any means (even numerically, using the smallest possible collision frequency).
Fig. 2 Simulated FOM (considering m = 0 branch only) for the optimized fishnet metamaterial using: (a) CST transient solver with fc = 13.5THz (b) CST frequency domain solver with fc = 13.5THz (c) CST frequency domain solver with fc = 85THz (d) COMSOL Multiphysics with fc = 13.5THz (e) CST with 2nd order fit of experimental values for permittivity of silver. All the FOMs are significantly smaller than what is reported in [1].
As a concluding remark, the use of DE-algorithm for NIMs is associated with important ambiguities due to the existence of multiple branches for n′ in the retrieval procedure. The lack of considering these ambiguities leads to significant error and deficiencies in the arrived conclusions of the paper. Therefore, the proposed implementation of the DE algorithm for NIMs is wrong and cannot be used for NIMs.
References and links
1. Y. Zhao, F. Chen, Q. Shen, Q. Liu, and L. Zhang, “Optimizing low loss negative index metamaterial for visible spectrum using differential evolution,” Opt. Express 19(12), 11605–11614 (2011). [CrossRef] [PubMed]
2. 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. B 65(19), 195104 (2002). [CrossRef]
3. S. Arslanagic, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antenn. Propag. M. 55(2), 91–106 (2013). [CrossRef]
4. J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B 80(3), 035109 (2009). [CrossRef]
5. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
6. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]
