Theory of light scattering on a paraboloid of revolution

Abstract

A theory for light scattering on a paraboloid of revolution is developed by the use of Hertz vectors. The expansion of a scalar plane wave traveling on any direction was found in terms of separable functions which are solutions of the scalar Helmholtz equation. This result was used to find the expansion of plane waves and a focused field. The solution of the parabolic Helmholtz equation is expressed in terms of Laguerre functions similar to the solution of the hydrogen atom in parabolic coordinates. Many identities related to those functions are demonstrated including a multiplication theorem. A method for solving the boundary conditions is also presented in which the corresponding coefficients of the scattered and incident fields are calculated.