General, closed-form expressions for the time-domain surface impedances of a homogeneous, lossy half-space

We have demonstrated that the TE and TM time-domain surface impedances can be expressed in terms of modified Bessel functions and incomplete Lipschitz-Hankel integrals (ILHI). Use of the series expansions to compute the ILHI can greatly improve the computational efficiency. The closed-form solutions provide physical insight into the TE and TM time-domain surface impedances associated with different incident angles and different lossy half space parameters. During early times, the incident angle, the material properties, and the polarization of the incident wave play important roles. However, at late times, the response of the surface impedance is independent of the incident angle, material permittivity, and the polarization of the wave.