Analytical solution for the solid angle subtended at any point by an ellipse via a point source radiation vector potential

Abstract

An axially symmetric radiation vector potential is derived for a spherically symmetric point source. This vector potential is used to derive a line integral for the solid angle subtended at a point source by a detector of arbitrary shape and location. An equivalent line integral given previously by Asvestas for optical applications is derived using this formulation. The line integral can be evaluated in closed form for important cases, and the analytical solution for the solid angle subtended by an ellipse at a general point is presented. The solution for the ellipse was obtained by considering sections of a right elliptic cone. The general solution for the ellipse requires the solution of an auxiliary cubic equation, but simpler formulas are possible for special cases. Numerical results both from the analytical formulas and direct numerical integration are given. The solution for the ellipse also provides exact solutions for the magnetic field of an elliptical current loop and the velocity field of a thin elliptical vortex. The well known solid angle formulas for a circular disk are also derived using this formulation.