Local thickness-dependent permittivity model for nonlocal bounded wire-medium structures

A local thickness-dependent permittivity is derived in closed form for bounded nonlocal wire-medium structures. The model takes into account spatial dispersion (as an average per length of the wires) and the effect of the boundary, and shows that an effective local model of a nonlocal medium is thickness dependent. The permittivity is comprised of a local bulk (Drude) term and a boundary- and thickness-dependent term, the latter including the effect of spatial nonlocality. Results are obtained for different wire-medium topologies which possess strong spatial dispersion, demonstrating good agreement with a nonlocal homogenization model and full-wave simulation results.

Figure 1

Illustration of the mushroom structure with the TM-polarized plane-wave incidence. (a) Side view. (b) Top view. The structure comprises a patch array over a dielectric slab perforated with metallic vias. The periodicity of the vias and patches $a$, the gap between the adjacent patches $g$, the radius of the vias $r_0$, and the relative permittivity of the host medium ${\varepsilon }_h$.

Figure 2

Drude component (local) and thickness-dependent permittivity for a mushroom structure with different gaps. The relevant structural parameters are $a=2\mathrm{mm},L=1\mathrm{mm},r_0=0.05\mathrm{mm}$, and ${\varepsilon }_h=10.2$.

Figure 3

Local thickness-dependent permittivity for a bed-of-nails with short wires. The relevant structural parameters are $a=2\mathrm{mm},L=1\mathrm{mm},r_0=0.05\mathrm{mm}$, and ${\varepsilon }_h=10.2$.

Figure 4

Schematics of a WM two-sided structure loaded with metallic or graphene patches: (a) cross-section view of the structure by considering a perfect electric conductor (PEC)/perfect magnetic conductor (PMC) symmetry plane, (b) top view.

Figure 5

Local homogenization model of the WM slab loaded with metallic or graphene patches.

Figure 6

Reflection phase calculated with the nonlocal, local (Drude)), and local thickness-dependent permittivity models for a mushroom surface with $a=2\mathrm{mm},L=1\mathrm{mm},r_0=0.05\mathrm{mm},g=0.6\mathrm{mm},{\theta }_i={30}^o$, and ${\varepsilon }_h=10.2$.

Figure 7

Reflection phase calculated with the nonlocal and local thickness-dependent permittivity models for a mushroom surface at different angles of incidence with $a=2\mathrm{mm},L=1\mathrm{mm},r_0=0.05\mathrm{mm},g=0.6\mathrm{mm}$, and ${\varepsilon }_h=10.2$.

Figure 8

Reflection phase calculated with the nonlocal, local (Drude), and local thickness-dependent permittivity models for a mushroom structure with $a=1\mathrm{mm},g=0.1\mathrm{mm},L=5\mathrm{mm},r_0=0.05\mathrm{mm},{\theta }_i={30}^o$, and ${\varepsilon }_h=1$.

Figure 9

Magnitude of the (a) reflection and (b) transmission coefficients calculated with nonlocal, local (Drude), and local thickness dependent permittivity models for a two-sided mushroom structure with metallic patches for the following parameters: $a=2\mathrm{mm},2L=2\mathrm{mm},r_0=0.05\mathrm{mm},{\theta }_i={30}^o,g=0.2\mathrm{mm}$, and ${\varepsilon }_h=10.2$.

Figure 10

Magnitude of the (a) reflection and (b) transmission coefficients calculated with nonlocal, local (Drude), and local thickness dependent permittivity models for a two-sided mushroom structure loaded with graphene patches for the following parameters: $a=2\mathrm{mm},g=0.2\mathrm{mm},2L=2\mathrm{mm},r_0=0.05\mathrm{mm},{\varepsilon }_h=10.2,{\theta }_i={30}^o,T=300\mathrm{K},\tau =0.35\mathrm{ps}$, and ${\mu }_c=0.5\mathrm{eV}$.

Figure 11

Magnitude of the (a) reflection and (b) transmission coefficients calculated with nonlocal, local (Drude), and local thickness dependent permittivity models for a WM slab for the following parameters: $a=2\mathrm{mm},2L=2\mathrm{mm},r_0=0.05\mathrm{mm},{\varepsilon }_h=10.2$, and ${\theta }_i={30}^{\circ }$.