Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation

TLDR

Diffraction of light focused by a high‑NA lens across a planar interface between materials of mismatched refractive indices introduces spherical aberration. The authors solve the diffraction problem for a planar interface between two isotropic, homogeneous media perpendicular to the optical axis, deriving the electric and magnetic field vectors in the second material. They obtain a rigorous solution that satisfies the homogeneous wave equation, presents a simple form suitable for numerical computation, offers a physical interpretation, and derives its paraxial approximation.

Abstract

The diffraction of electromagnetic waves for light focused by a high numerical aperture lens from a first material into a second material is treated. The second material has a different refractive index from that of the first material and introduces spherical aberration. We solve the diffraction problem for the case of a planar interface between two isotropic and homogeneous materials with this interface perpendicular to the optical axis. The solution is obtained in a rigorous mathematical manner, and it satisfies the homogeneous wave equation. The electric and magnetic strength vectors are determined in the second material. The solution is in a simple form that can be readily used for numerical computation. A physical interpretation of the results is given, and the paraxial approximation of the solution is derived.