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A reformulated integral equation is solved inside an elliptic disk particle for an electromagnetic field formulation bridging the quasi-static and the physical optics approximations. The scattering amplitude tensor elements associated with such a field formulation are derived and then used to formulate the extinction cross sections. It is shown that the extinction cross sections have a frequency dependence and an incidence angle dependence similar to those associated with the physical optics approximation, and they have a particle shape dependence similar to that associated with the quasi-static approximation. Furthermore, at the high-frequency limits, it is shown that those cross sections could reach the value known in the literature by the extinction paradox, namely, twice the particle geometric shadow area.

Theory is presented to provide insight into the observation that attenuation through vegetation is proportional to vegetation water content. In this analysis, the canopy is modeled as a sparse layer of randomly oriented particles (leaves, stalks, etc.) over a flat, homogeneous ground plane (soil) and an expression is obtained for the "optical depth". The formulas developed by Ulaby and El Rayes are used to relate this expression to the water content of the canopy. In the low frequency extreme (Rayleigh scatterers), the attenuation varies almost linearly with water content and inversely with wavelength. In contrast, in the high frequency limit, the attenuation is independent of both water content and frequency, in between, geometry dependent "resonances" occur even at the low frequency end of the microwave spectrum (e.g. L-band) making the dependence of attenuation on frequency and water content specific to canopy architecture.

This study accounts for the contribution of multiple scattering to microwave brightness temperature of sea ice and its snow cover. The application of the distorted Born approximation in conjunction with the strong fluctuation theory has led to calculated values for brightness temperatures in agreement with measurements from sea ice and its snow cover at frequencies below 37 GHz. However, at frequencies above 37 GHz, the distorted Born approximation overestimates the brightness temperature. This is due to the fact that the distorted Born approximation accounts only for the singly scattered field. Physical and mathematical arguments showed that inclusion of higher order multiple scattering would lead to more reasonable values for the brightness temperature at higher frequencies where the multiple scattering is expected to be dominant.

The classical optical theorem that describes the extinction of electromagnetic radiation by an arbitrary particle is reformulated by use of the diffusion properties of the mutual spectral densities. This reformulation yields three novel coherency tensors: an extinction-loss coherency tensor, a scattering-loss coherency tensor, and an absorption-loss coherency tensor. The first tensor is used to establish the polarimetric optical theorem and to derive the extinction-loss matrix. The second tensor is used to derive the scattering-loss matrix. The third tensor is used to develop the absorption-loss matrix and the absorption cross sections. Furthermore, the derived extinction- and scattering-loss matrices are found to be similar to the extinction coefficient and phase matrices of the vector radiative transfer equations.

The conventional integral equation governing the electric field inside dielectric particles is reformed to bridge and to provide mathematical foundations for analytic techniques widely used to estimate such a field. The solution of the reformed equation inside a dielectric slab explained how inner-field formulations based on the Rayleigh, the Rayleigh-Gans, the quasi-static, and the Shifrin approximations can be supported by the particles. It also confirmed the approach employed to reform the integral equation. The analysis performed uncovered the differences between the depolarization tensor characterizing electrostatic fields inside the particles and the source dyadic resulting from the extraction of the singularity of the integral equation kernel.

The problem of electromagnetic wave interaction with a stratified dielectric medium is solved in the framework of molecular optics, leading to a new derivation for the refraction and reflection laws at stratified media interfaces. In addition, the solution confirms the Lorentz–Lorenz refractive index formula characterizing the transverse propagation modes and the characteristic frequencies associated with the longitudinal mode. The analytic results presented in this study support the existence of another longitudinal mode propagating with the vacuum wave number within the medium, and they provide a new concept for the Ewald–Oseen extinction theorem. Other new results of this study are that (1) the relation between Fresnel horizontal and vertical reflection coefficients and the relation between the corresponding transmission coefficients are revealed and (2) a new concept is presented for the Brewster angle, and (3) the concept is introduced of multiple reflections in formulating the reflection coefficient at the interface between two different dielectric materials.