A Tutorial on the Classical Theories of Electromagnetic Scattering and Diffraction

TL;DR

This paper provides a comprehensive tutorial on classical electromagnetic scattering and diffraction theories, deriving fundamental results from Maxwell's equations and extending them to vector fields, with applications to obstacles, inhomogeneities, and neutron scattering.

Contribution

It offers a detailed derivation of classical diffraction and scattering theories, including the Sommerfeld solution and optical cross-section theorem, extending scalar results to vector electromagnetic fields.

Findings

Derivation of Huygens-Fresnel-Kirchhoff and Rayleigh-Sommerfeld theories

Extension to vector electromagnetic fields

Relation between scattering cross-section and forward scattering amplitude

Abstract

Starting with Maxwell's equations, we derive the fundamental results of the Huygens-Fresnel-Kirchhoff and Rayleigh-Sommerfeld theories of scalar diffraction and scattering. These results are then extended to cover the case of vector electromagnetic fields. The famous Sommerfeld solution to the problem of diffraction from a perfectly conducting half-plane is elaborated. Far-field scattering of plane waves from obstacles is treated in some detail, and the well-known optical cross-section theorem, which relates the scattering cross-section of an obstacle to its forward scattering amplitude, is derived. Also examined is the case of scattering from mild inhomogeneities within an otherwise homogeneous medium, where, in the first Born approximation, a fairly simple formula is found to relate the far-field scattering amplitude to the host medium's optical properties. The related problem of neutron scattering from ferromagnetic materials is treated in the final section of the paper.