= Solution
Factor the forward carrier as $E=e^{ikx}U$. Substitution into the scalar <Helmholtz equation> gives the exact envelope equation
$$
U_{xx}+2ikU_x+\nabla_\perp^2U+k^2(\epsilon_{\rm rel}-1)U=0.
$$
If $|U_{xx}|\ll2k|U_x|$, drop the second longitudinal derivative to obtain the <parabolic wave equation>
$$
\boxed{2ikU_x+\nabla_\perp^2U+k^2(\epsilon_{\rm rel}-1)U=0,\qquad U_x=iLU,\qquad
L=\frac1{2k}\nabla_\perp^2+\frac k2(\epsilon_{\rm rel}-1).}
$$
Typical sufficient conditions are a transverse width $w$ with $kw\gg1$, 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 $\epsilon_{\rm rel}-1=O(\eta^2)$, with $U_x=O(k\eta^2U)$ 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 $\nabla\cdot(\epsilon\mathbf E)=0$, 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.
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