Enhanced Diffraction from a Grating on the Surface of a Negative-Index Metamaterial

We show by numerical simulation as well as by measurements on negative-index metamaterial wedge samples, that the unavoidable stepping of the refraction interface—due to the finite unit-cell size inherent to metamaterials—can give rise to a well-defined diffracted beam in addition to the negatively refracted beam. The direction of the diffracted beam is consistent with elementary diffraction theory; however, the coupling to this higher order beam is much larger than would be the case for a positive index material. The results confirm recent theoretical predictions of enhanced diffraction for negative-index grating surfaces.

Figure 1

(a) Schematic diagram of the Shelby et al. metamaterial wedge used to demonstrate negative refraction. White lines in the diagram indicate planes on which SRRs and wires are patterned. In this case, the lines indicate SRRs and wires were patterned in two dimensions. (b) Schematic diagram of the 2.5 mm structure. This new structure has a cubic unit-cell 2.5 mm in size, and is patterned in only one dimension, as indicated by the lines. (c) Diagram of the SRR (black lines) and wire (in gray) for the Shelby et al. sample, with dimensions: $s=2.63\mathrm{m}\mathrm{m}$, $c=0.25\mathrm{m}\mathrm{m}$, $b=0.3\mathrm{m}\mathrm{m}$, $g=0.46\mathrm{m}\mathrm{m}$, $w=0.25\mathrm{m}\mathrm{m}$. (d) Diagram of the SRR (in black) and wire (in gray) for the 2.5 mm sample, with dimensions: $s=2.2\mathrm{m}\mathrm{m}$, $c=0.2\mathrm{m}\mathrm{m}$, $b=0.15\mathrm{m}\mathrm{m}$, $g=0.3\mathrm{m}\mathrm{m}$, $w=0.14\mathrm{m}\mathrm{m}$. The substrate used is 0.25 mm thick FR4 circuit board ($ϵ=3.8$), with a copper thickness of roughly 0.017 mm.

Figure 2

Field plot showing the refracted and diffracted beams at the stepped interface of a negative-index wedge. For the wedge in this simulation, $ϵ=-5.09$ and $\mu =-1.41$, so that $n=-2.68$. The frequency for the simulation is 11.5 GHz. The steps along the refraction surface of the wedge have dimensions 15 mm by 5 mm. In analogy to the experiments, a 1 cm high (in the direction perpendicular to the page) and 6 cm wide guided region is simulated, bounded by electric boundary conditions (parallel to the page). The refracted and diffracted beams exit the slab at angles of $-58°$ and $+30°$, respectively, in agreement with Eq. (1).

Figure 3

Simulated angular power spectra at a radius of 10 cm away from the surface of the stepped negative-index wedge of Fig. 3. Each curve corresponds to a different incident wavelength (frequency). All angles are relative to the refraction surface normal.

Figure 4

Map of the transmitted power as a function of frequency (vertical axis) and angle away from direct incidence (horizontal axis), for (a) a Teflon wedge with $15\mathrm{m}\mathrm{m}\times 5\mathrm{m}\mathrm{m}$ steps; (b) the Shelby et al. wedge; (c) the 2.5 mm unit-cell wedge. The vertical white lines indicate the angle of the surface normal.