Steering light by a sub-wavelength metallic grating from transformation optics

Abstract

Transformation optics has shown great ability in designing devices with novel functionalities, such as invisibility cloaking. A recent work shows that it can also be used to design metasurfaces which usually come from the concept of phase discontinuities. However, metasurfaces from transformation optics have very complicated material parameters. Here in this work, we propose a practical design, a sub-wavelength metallic grating with discrete and gradient index materials. Such a design not only inherits some functionalities of metasurfaces from phase discontinuities, but also shows richer physics. Our work will also provide a guidance to recent activities of acoustic metasurfaces, especially for those made of extremely anisotropic metamaterials.

Figure 1. Periodical metallic grating with gradient indexes.

(a) The schematic of the metallic grating, where only one supercell is displayed. (b) The relationship of relative phase shift and refractive index. The red line is calculated approximately by φ = βd, while the black curve is the numerical result for a normally incident TM wave. In simulations, the parameters w, a and d are set as 1 μm, 0.75 μm and 2 μm, respectively. The impedances of all filled media are matched to the air.

Figure 2. Iso-frequency contours for different additional momentums ξ.

(a–d) are corresponding to four situations that 0 < ξ < k₀, ξ = k₀, k₀ <ξ < 2k₀ and 2k₀ ≤ ξ, respectively. In all cases, the solid and dashed black circles represent the iso-frequency contours of air corresponding to incident side and refracted side, respectively. Both black circles, with their centers located at k_x axis, have the same radius. The green circles indicate the iso-frequency contours for the cases of introducing phase shift ξ with n = 0 in equation (4), i.e., moving the dashed black one down with ξ, while the red and blue circles are iso-frequency contours for the cases of introducing reciprocal lattice corresponding to n = 2 and n = 3 in equation (4), respectively. When the incident wave is illuminating to the metallic grating from air, the green (or red) dashed arrows in (a–d) denote the incident directions. After refraction, the directions of refractive wave are marked by the green (or red) solid arrows. Two paralleled dashed lines indicate the tangential momentum conservation.

Figure 3. The case of metallic grating with ξ = 0.8k₀.

(a–c) are simulated magnetic field patterns for incident wave with different angles with θ_i = −30°, 30° and 50° respectively. The upper parts are the corresponding iso-frequency contours, while the corresponding patterns of plane wave incident on metallic grating are placed in the nether parts, where the patterns for Gaussian beams bumping on the metallic grating with 6 supercells are inserted in bottom, which are marked by the red dashed frames. (d) is relationship between transmittance and incident angle. The points a, b and c in (d) denote the corresponding transmittance for the above cases of (a–c), respectively.

Figure 4. The case of metallic grating with ξ = k₀.

(a–c) are simulated magnetic field patterns for incident wave with different angles with θ_i = −30°, 0° and 30°, respectively. The upper parts are the corresponding iso-frequency contours, while the corresponding patterns of plane wave incident on metallic grating are placed in the nether parts, where the patterns for Gaussian beams bumping on the metallic grating with 6 supercells are inserted in bottom, which are marked by the red dashed frames. (d) is relationship between transmittance and incident angle. The points a, b and c in (d) denote the corresponding transmittance for the above cases of (a–c), respectively.

Figure 5. The case of metallic grating with ξ = 1.2k₀.

(a–c) are simulated magnetic field patterns for incident wave with different angles with θ_i = −45°, 10° and 45°, respectively. The upper parts are the corresponding iso-frequency contours, while the corresponding patterns of plane wave incident on metallic grating are placed in the nether parts, where the patterns for Gaussian beams bumping on the metallic grating with 6 supercells are inserted in bottom, which are marked by the red dashed frames. (d) is relationship between transmittance and incident angle. The points a, b and c in (d) denote the corresponding transmittance for the above cases of (a–c), respectively.

Figure 6. The case of metallic grating with ξ = 2k₀.

(a–c) are simulated magnetic field patterns for incident wave with different angles with θ_i = −30°, 0° and 50°, respectively. The upper parts are the corresponding iso-frequency contours, while the corresponding patterns of plane wave incident on metallic grating are placed in the nether parts, where the patterns for Gaussian beams bumping on the metallic grating with 6 supercells are inserted in bottom, which are marked by the red dashed frames. (d) is relationship between transmittance and incident angle. The points a, b and c in (d) denote the corresponding transmittance for the above cases of (a–c), respectively.