## Abstract

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],

*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.

It should be noted that CST MWS gives more realistic results for FOM if collision frequency
(*f _{c}*) for the Drude model in CST is defined using its angular frequency value rather than
its linear frequency i.e. by taking

*f*= 85THz in the material properties for the Drude model of silver rather than taking

_{c}*f*= 85THz/(2

_{c}*π*) = 13.5THz. As shown in Fig. 2, the highest FOMs (1.7 and 0.6, respectively) obtained using CST with

*f*= 85THz and COMSOL with

_{c}*f*= 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

_{c}*f*= 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

_{c}*f*= 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).

_{c}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
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**4. **J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical
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**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]