A Tutorial on the Classical Theories of Electromagnetic Scattering and Diffraction

Masud Mansuripur

doi:10.1515/nanoph-2020-0348

Abstract

AbstractStarting 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.

Highlights

A thorough appreciation of these theories requires an understanding of the Maxwell-Lorentz electrodynamics[4,5,6,7,8,9,10,11] and a working knowledge of vector calculus, differential equations, Fourier transformation, and complex-plane integration techniques.[12]
We described some of the fundamental theories of EM scattering and diffraction using the electrodynamics of Maxwell and Lorentz in conjunction with standard mathematical methods of the vector calculus, complex analysis, differential equations, and Fourier transform theory
The scalar Huygens-Fresnel-Kirchhoff and Rayleigh-Sommerfeld theories were presented at first, followed by their extensions that cover the case of vector diffraction of EM waves

Summary

Introduction

The classical theories of electromagnetic (EM) scattering and diffraction were developed throughout the nineteenth century by the likes of Augustine Jean Fresnel (1788-1827), Gustav Kirchhoff (1824-1887), John William Strutt (Lord Rayleigh, 1842-1919), and Arnold Sommerfeld (1868-1951).[1,2,3] A thorough appreciation of these theories requires an understanding of the Maxwell-Lorentz electrodynamics[4,5,6,7,8,9,10,11] and a working knowledge of vector calculus, differential equations, Fourier transformation, and complex-plane integration techniques.[12]. 4. The Huygens-Fresnel-Kirchhoff theory of diffraction.[1,2,3,9] Consider a scalar function ߰(࢘) that satisfies the homogeneous Helmholtz equation (ߘଶ + ݇బଶ)߰(࢘) = 0 everywhere within a volume ܸ of free space enclosed by a surface ܵ. Applying the Kirchhoff formula in Eq(22), where the integral is over the closed surface ܵ = ܵభ + ܵమ, and the function ߰(࢘) is any scalar field that satisfies the Helmholtz equation, to a Cartesian component of the ‫ܧ‬-field, say, ‫ܧ‬ೣ, we write. A similar argument can be advanced for the ‫ܤ‬-field and, the following vector diffraction equations are generally valid for a planar surface ܵభ: ࡱ(࢘బ) = (2ߨ)ିଵࢺబ × ‫׬‬ೄభ[࢔ෝ × ࡱ(࢘)]‫࢘(ܩ‬, ࢘బ)d‫ݏ‬.

Normalized intensity

Recalling that