Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 78 / 3 / b / i / Solution

Factor the forward carrier as $E=e^{ikx}U$. Substitution into the scalar Helmholtz equation gives the exact envelope equationIf $|U_{xx}|\ll 2k|U_x|$, drop the second longitudinal derivative to obtain the parabolic wave equationTypical sufficient conditions are a transverse width $w$ with $kw\gg 1$, longitudinal envelope and material variation on scales much longer than $k^{-1}$, and weak refractive contrast so that factoring out the free-space carrier leaves a slowly varying envelope. A paraxial ordering has transverse propagation angles $O(\eta )$ and ${ϵ}_{\mathrm{rel}}-1=O({\eta }^2)$, with $U_x=O(k{\eta }^{2}U)$ and $U_{xx}$ higher order. A large refractive contrast may require a different, medium-adapted carrier even when the ray direction is nearly axial.

The paraxial approximation retains transverse diffraction and slowly accumulated refraction, but neglects backward propagation, strong reflection from rapid material changes, wide-angle propagation and, in this scalar model, polarization coupling. The supplied vector Helmholtz equation is itself a model assumption: in a source-free inhomogeneous dielectric, Maxwell equations additionally require $\nabla \cdot (ϵ\mathbf{E})=0$, and generally supply a longitudinal-gradient term when the vector wave equation is expanded. An independent scalar component is justified only with a compatible polarization and the relevant small-gradient approximation.