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Equivalent Homogeneous Metamaterial Model

In order to model a horn antenna lined with a metasurface, it is convenient to employ a homogeneous material model with effective permittivity and permeability values, rather than an infinitesimally thin boundary with the corresponding anisotropic surface impedances. Few, if any, simulation packages support anisotropic surface impedances, but many support a large range of effective material properties. This process begins by equating the composite metasurface structure to a homogeneous material backed by a conducting plane. The homogeneous material is assumed to have an effective permittivity e_eff and an effective permeability m_eff. The thickness of the homogeneous slab is set to be t, that of the composite metasurface. The effective parameters are chosen such that the homogeneous material gives rise to identical boundary conditions (i.e., surface impedances) as the actual metasurface.

For a homogeneous slab with thickness t, backed by a conducting ground plane, the anisotropic surface impedances can be expressed as:

where k₀ and ц₀ are the wave number and wave impedance of free space, respectively. Calculating the effective refractive index n_eff and wave impedance z_eff yields:

from which we can calculate the permittivity and permeability as:

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