A rigorous treatment of vertical dipole impedance located above lossy DPS, MNG, ENG, and DNG half-space

In this paper, accurate analytical expressions for the impedance of vertical electric and magnetic dipoles which are located over the half-space materials of arbitrary permittivity and permeability are developed. In this regard, the impedance variations are expressed in integral forms. For metamaterial half-space, a proper expression for approximating the Fresnel reflection coefficient is proposed. Using this approximate expression, the impedance integrals are analytically solved, and exact formulas for impedance variations are obtained. The results for the metamaterial half-spaces are compared with the case of natural materials (positive permittivity and permeability), and key differences are explained. The influences of sign changing in permeability of the half-space material on the impedance of vertical dipole are studied, and the results are validated by comparison with those of numerical solution of integrals. It is shown that for elevated dipoles over materials with high and/or low conductivities, the results of both methods are in complete agreement. For vertical dipoles above low loss materials, the results are somewhat identical. However, a better agreement could be obtained using higher order approximations for the integrand.