Effective Permittivity in Microwave Remote Sensing Problems: Media as Mixtures with Scatterers of Gaussian Packets

In this paper, the focus is on the dielectric properties of inhomogeneous media. The polarizability and dipole moment of different types of small inhomogeneous scatterers are solved. After knowing the dipole moment, the effective permittivity of a dielectric mixture that contains these kinds of scatterers can be solved. A series solution of the differential equation for the scattered potential will be given. For cases where a scatterer does not possess sharp boundaries, a variational approach appears most efficient. A stationary functional for the polarizability is derived which can be used to calculate accurate estimates for the dipole moment with simple trial functions. Results are shown for inhomogeneous scatterers with linear, parabolic, and Gaussian permittivity profiles.