Scattering by a Dielectric Sphere Buried in a Half-Space With a Slightly Rough Interface

Analytical expressions for the scattering coefficients of a dielectric sphere buried under a rough interface are presented. The proposed method combines the small perturbation method (SPM) and the Mie solution by using the expansion of plane waves in terms of vector spherical functions (VSFs) and vice versa. First, using SPM, the zeroth- and the first-order perturbative scattered fields of a rough interface for illuminations from above and below are derived. Using these solutions, the field transmitted to the lower half-space is determined as a spectrum of down-going plane waves. The scattered fields from the sphere are then calculated using the vector Mie solution. Subsequently, the VSFs are expanded in terms of up-going plane waves. These plane waves illuminate the interface, and using SPM, the scattered fields in the upper and lower regions are determined as infinite summations of plane waves. The reflected plane waves are once again scattered by the sphere and the scenario repeats. By inspecting the form of the fields resulting from the few first interactions of the sphere and the rough interface, a recursive form is obtained for the scattered fields. This recursive form is then used to rewrite the system of equations in a form containing all interactions in a single-step formulation. Accordingly, the zeroth- and the first-order closed-form scattered fields are obtained. The derived expressions are analytically and numerically validated. Finally, the numerical results for the case of the rough interface with sinusoidal profile are presented and briefly discussed.