Generalized antireflection coatings for complex bulk metamaterials

We present the optimized design of an antireflection coating to efficiently couple an incident plane wave into a metamaterial with a complex field profile. We show that such an antireflection coating must enable spatial engineering of the field profiles at the coating/metamaterial interface to achieve high transmission, and therefore it is required to be inhomogeneous. As a demonstration, we investigate theoretically a waveguide-based negative-index metamaterial, which under normal incidence cannot be excited due to the antisymmetric propagating eigenmode. Through careful engineering of the field profile, lateral position, and thickness of the coating layer, we enhance the transmission under normal incidence from $0\%$ to $100\%$. This principle may generally be applied to overcome low coupling efficiency between incident plane waves and complex mode profiles in metamaterials.

Figure 1

(a) Sketch of metal-dielectric metamaterial geometry. (b) The negative-index waveguide mode field profile $h_y\left(x\right)$ ($d_m=45\mathrm{nm},{\varepsilon }_m=-3.5,d_d=20\mathrm{nm},{\varepsilon }_d=6.25$, and ${\lambda }_0=450\mathrm{nm}$). The field profile is antisymmetric across the dielectric core. (c) Calculated field profile due to a plane wave $\left({\lambda }_0=450\mathrm{nm}\right)$ incident on the metal dielectric waveguide array at normal incidence. The spatial field profile shown is composed of 15 repetitions of the unit cell. (d) Field profile due to a plane wave at an angle of incidence of ${70}^{\circ }$. Negative refraction is evident in the phase fronts, indicated by ${\theta }_i$ and ${\theta }_r$. The scale bar in (c), (d) is 250 nm.

Figure 2

(a) Sketch of the air/coupling layer/metal-dielectric metamaterial geometry. (b) Magnetic field profile of the propagating mode in the coupling layer with ${\varepsilon }_{\mathrm{Hi}}=20.25,{\varepsilon }_{\mathrm{Lo}}=1$, and ${\rho }_{\mathrm{Hi}}=0.87$, and a 65 nm unit cell size. The propagation constant of the eigenmode is ${\beta }_1^{\left(2\right)}=43.85\mu {\mathrm{m}}^{-1}$.

Figure 3

(a) Simulated reflectivity (at ${\lambda }_0=450\mathrm{nm}$) of the coupling layer/waveguide array interface as a function of displacement $\mathrm{\Delta }$. The coupling layer is defined by ${\varepsilon }_{\mathrm{Hi}}=20.25,{\varepsilon }_{\mathrm{Lo}}=1$, and ${\rho }_{\mathrm{Hi}}=0.87$, and the unit cell size is $a=65\mathrm{nm}$. The waveguide array is the same as in the above. (b) Simulated reflectivity of the combined coupling layer and waveguide array for $\mathrm{\Delta }=7.8\mathrm{nm}$ with changing coupling layer thickness $L$. Clear oscillations are visible, corresponding to a standing wave in the coupling layer. The two local maxima in reflection are not equal in amplitude due to the influence of evanescent waves in the coupling layer.

Figure 4

(a) Simulated field profile at normal incidence. As is clear, the incident plane wave now very effectively couples to the antisymmetric waveguide mode. (b) Field profiles for a plane wave at an angle of incidence of ${70}^{\circ }$. The scale bar in (a), (b) is 250 nm. (c) Reflectivity from the total system as a function of angle of incidence (red). As a reference, the reflectivity of the bare multilayer substrate is also shown (blue).