From Scattering Parameters to Snell’s Law: A Subwavelength Near-Infrared Negative-Index Metamaterial

A general relation is derived between the band structure of an arbitrary low-loss unit cell and its effective index of refraction. In addition, we determine the maximum unit cell size that defines the “metamaterial regime” [D. R. Smith et al., Phys. Rev. E 71, 036617 (2005)]. Furthermore, these general rules allow for the design of a subwavelength near-infrared negative-index material, where the negative refractive index is verified by band calculations to be a bulk property. Full-wavelength simulations of prisms consisting of these unit cells suggest behavior consistent with Snell’s law in the negative-index regime.

Figure 1

Bloch bands for zero (dashes) and nonzero dissipation (circles and squares) Au layers, and (a) imaginary and (b) real parts of the effective propagation constant $k_{x,\mathrm{eff}}$ from single-layer scattering simulations (continuous line). In (b), the part of lossless band structure with a positive group velocity $v_g>0$ is marked by arrows. Inset in (a): The unit cell. Lattice constants $L_x=L_y=150\mathrm{nm}$, Au strip and central layer thicknesses $t=t_c=20\mathrm{nm}$, strip width $W=100\mathrm{nm}$, and spacer thickness $h=15\mathrm{nm}$. Au strips and central layer are embedded in a polymer dielectric.

Figure 2

(a) A Bloch wave propagates in the positive $\widehat{\mathbf{x}}$ direction and is obliquely incident upon the semi-infinite metamaterial crystal interface with air. Only a small number of unit cells near the interface are shown. Here, $a$ is the lattice constant, and $d$ is the interface periodicity. $\overrightarrow{k}$ and ${\overrightarrow{v}}_g$ are the first Brillouin-zone wave vector and group velocity of the incident Bloch wave. ${\overrightarrow{k}}_0$ is the free space wave vector of the refracted light. $\theta$ and ${\theta }_0$ are incidence and refraction angles, respectively. (b) Graphical solutions to $ka+l\times 2\pi =k_{lt}d$. Radii of circles I and II are $ka+2\pi$ (for $-\pi <ka<0$) and $\pi$, respectively.

Figure 3

Comparison of effective indices extracted from numerical simulations and band calculations. Inset: Time-averaged power flow on a logarithmic scale for an $m=3$ wedge at wavelength $\lambda =950\mathrm{nm}$. A negatively refracted beam, a diffraction order, and two weak side lobes are observed.