Spectral Solution of the Helmholtz and Paraxial Wave Equations and Classical Diffraction Formulae

Abstract

Abstract : This report examines the general nonlinear vector wave equation implied by the Maxwell equations in a nonmagnetic, isotropic medium and discusses the various approximations under which this general result reduces to the familiar scalar Helmholtz equation and the paraxial wave equation. We see why the Helmholtz equation may be regarded as a singular perturbation of the paraxial wave equation and bow some of the difficulties arising in the solution of the former partial differential equation are related to this fact. Standard integral transform methods are used to obtain general solutions of the Helmholtz equation in a linear medium and of the paraxial wave equation in a linear medium. We show that these solutions are equivalent, respectively, to the exact Rayleigh-Sommerfeld diffraction integral and the Rayleigh-Sommerfeld integral in the Fresnel approximation. We discuss the use of these linear solutions in numerical procedures for treating the corresponding nonlinear beam propagation problems. The general linear solutions are specialized to situations with axial symmetry, and the results are used to treat the example of a clipped Gaussian beam.