A metamaterial is an artificial substance engineered to have unnatural properties, such as a negative refractive index. The possibility of creating optical negative-index metamaterials (NIMs) using metallic and dielectric (insulating) nanostructures has triggered intense basic and applied research over the past several years.^{1–4} In view of potential applications of these materials in a variety of areas ranging from subwavelength imaging^{1} to cloaking,^{5} it is crucial to grasp the underlying physics.

Our common understanding of NIMs is entirely based on the concept of homogenization. In other words, the typical approach to understanding light propagation in such complexly structured materials consists in looking at how electromagnetic waves propagate in a simpler, homogeneous substance with equivalent optical properties. At microwave frequencies, techniques based on a transmission-line model (a theory that assumes that the elements of an electric circuit are distributed throughout the material of the circuit) provide an analytical treatment for this problem.

The microwave community can therefore benefit from a reliable framework for understanding and designing the properties of NIMs. Unfortunately, the optical-frequencies range has not been given full attention in this respect. In addition, the design of NIMs in the visible and near-IR spectra relies entirely on rigorous computations that solve the light-propagation problem without using approximations. These numerical methods are intrinsically nonintuitive and drastically complicate the design process.

We have focused our study on the metamaterial geometry that has been shown to provide negative refractive indices in the visible and near-IR with the lowest attenuation, the so-called fishnet structure.^{3,4,6} It consists of a 2D subwavelength array of holes in a periodic metal-dielectric-metal thin-film stack: see Figure 1(a). Furthermore, we have abandoned classical homogenization techniques for describing light propagation in the structure. Instead, we adopted a ‘microscopic’ point of view by tracking the energy as it propagates and scatters through the fishnet mesh, like a fluid flowing in a multi-channel crossed system. A related approach recently enabled us to explain extraordinary optical transmission, the phenomenon of greatly enhanced light transmission through 2D tiny-hole arrays in metallic films.^{7}

**Figure 1. **Elementary light-scattering events in a fishnet metamaterial. (a) This material consists of a 2D array of holes etched into a metal-dielectric-metal periodic stack, with the metal being represented in yellow and the dielectric in blue. (The scanning electron microscope image was published elsewhere and is used with permission.

^{4}) (b) Scattering of the super-mode supported by every 1D hole chain of the fishnet. (c) Scattering of the gap-SP (surface plasmon) mode supported by a metal-dielectric-metal waveguide. The events in (b) and (c) define five elementary scattering coefficients, the reflection ρ and transmission τ of the super-mode, the reflection

*r*_{sp} and transmission

*t*_{sp} of the gap-SP, and the coefficient α of coupling between the two modes. (d) Real and (e) imaginary (attenuation) parts of the fishnet refractive index,

*n*_{eff}. The model predictions (red curve) are compared with more complex numerical calculations (circles).

Our model's dynamics involves the flow of surface plasmons (SPs, coherent photonic-electronic modes that exist at the interfaces of metallic and dielectric materials) through two intersecting subwavelength channels: see Figure 1(b) and (c). The vertical channel consists of a 1D air-hole chain in a metal film while the horizontal channel is formed by metal-dielectric-metal waveguides that support the propagation of SP modes called gap-SPs. The modes of the vertical channel are called 1D-hole-chain modes or super-modes. Through simple coupled equations that describe how both channels exchange energy, we have derived analytical formulas for the metamaterial refractive index.

The model has advantages in addition to providing such quantitative predictions for this index. It allows, for example, a step-by-step analysis that explicitly dissociates horizontal and vertical contributions to the appearance of negative values with low loss in the refractive-index spectrum. In particular, it provides a quantitative picture of the resonant excitation of gap-SPs in the dielectric layers in the origin of the fishnet's negative magnetic response (responsible for the appearance of its negative refractive index). With this model, we are also able to calculate the characteristic length of the gap-SP resonance, which is found to be delocalized over four periods.

Reducing the attenuation of NIMs operating in the visible and near-IR is crucial, and the issue of loss compensation with gain media has recently received much attention.^{8–10} We have evidenced that the decrease of the gap-SP attenuation constitutes the main ingredient of this problem. In particular, we have shown that gain mainly affects the attenuation of the fishnet, leaving the real part of the negative refractive index unchanged. At the transparency threshold, the horizontal SP resonance becomes delocalized over a few tens of unit cells.

In summary, we have analyzed light transport in fishnet metamaterials at optical frequencies with a comprehensive and accurate model. In addition to providing the first analytical treatment of NIMs at optical frequencies, the model shines new light on how a negative refractive index is formed and how mode attenuation can be compensated by incorporating optical gain. In contrast to previous efforts that rely on direct extraction of macroscopic quantities, the present model uses a ‘microscopic’ approach that tracks the local transport of electromagnetic fields in the structure. In the future, we will extend the model to tackle the important problem of the definition of impedance of optical metamaterials.

*We acknowledge collaboration with Haitao Liu of the Institute of Modern Optics at Nankai University, China, and the authors of a 2008 Nature paper*^{4} for granting permission for the use of an image.

Jianji Yang, Christophe Sauvan

Laboratoire Charles Fabry

Institut d'Optique, CNRS

University of Paris-Sud

Palaiseau, France

Christophe Sauvan is a CNRS researcher. He is currently involved in theoretical and numerical studies of metamaterials, plasmonic devices, photonic crystals, and microcavities.

Philippe Lalanne

Institut d'Optique, CNRS

University of Paris-Sud

Palaiseau, France

Philippe Lalanne is also a CNRS researcher. His interests include computational physics and applications of subwavelength optical structures for metamaterials, plasmonics, photonic crystals, slow light, and microcavities.

References:

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2. C. M. Soukoulis, S. Linden, M. Wegener, Negative refractive index at optical wavelengths, *Science* 315, pp. 47-49, 2007.

3. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, S. R. J. Brueck, Experimental demonstration of near-infrared negative-index metamaterials, *Phys. Rev. Lett*. 95, pp. 137404, 2005.

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