From Maxwell to paraxial wave optics

TLDR

The paper studies light beam propagation in inhomogeneous, isotropic media with possibly nonlinear refractive indices. The authors derive a systematic expansion of Maxwell’s equations in powers of \(w_{0}/l\), yielding equations for successive field orders. They demonstrate that standard paraxial approximations are inconsistent with Maxwell’s equations, derive explicit longitudinal first‑order corrections, and clarify the limits of paraxial wave optics.

Abstract

In this paper we are concerned with the propagation of a light beam through an inhomogeneous, isotropic medium with a possibly nonlinear index of refraction. The customary paraxial approximations of neglecting grad $\mathrm{div}\mathcal{E}$ and seeking a plane-polarized solution are shown to be incompatible with the exact Maxwell equations. By starting from Maxwell's equations, and scaling transverse and longitudinal distances by the beam waist ${w}_{0}$ and diffraction length $l$, respectively, an expansion procedure in powers of $\frac{{w}_{0}}{l}$ is developed. The exact equations obeyed by the zeroth-order fields are not Maxwell's equations but the customary paraxial approximation to Maxwell's equations. Equations for the first-, second-, and third-order fields are developed. The first-order field is found to be a longitudinal field. It is solved for explicitly in terms of the zeroth-order field which is transverse. Thus a precise knowledge of the meaning and accuracy of paraxial wave optics is obtained.