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

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.

Assume the relative permittivity is real and the field decays sufficiently at transverse infinity, or impose boundary conditions with zero transverse flux. Then $L(x)$ is self-adjoint on its transverse domain by integration by parts. The scalar parabolic wave equation givesThis is norm conservation for the scalar parabolic wave equation. It remains true when $L$ depends explicitly on $x$; its instantaneous self-adjointness suffices. Equivalently, the local balance isSince $|E|=|U|$, the invariant can also be writtenFor a unit transverse polarization in a fixed reference medium with impedance $Z_0=\sqrt{{\mu }_0/{ϵ}_0}$, the leading forward paraxial Poynting vector has $S_x\simeq |E{|}^2/(2Z_0)$. Thus the scalar model conserves its leading reference-normalized power $P_0=\mathcal{N}/(2Z_0)$. This is the intended consequence of the Hermitian-operator hint. Complex permittivity would add absorption or gain and invalidate this norm invariant.

Two qualifications are necessary for the printed physical interpretation. First, $(1,0,1)$ is neither a unit vector nor transverse to nearly axial propagation: its longitudinal electric component is as large as its transverse one. In a homogeneous source-free medium, the example $\mathbf{E}=(1,0,1)e^{ikx}$ satisfies the componentwise Helmholtz equation but violates $\nabla \cdot \mathbf{E}=0$. It therefore is not the stated near-axial electromagnetic plane wave. The same scalar calculation formally conserves $\int |\mathbf{U}{|}^2=2\mathcal{N}$ for that fixed vector, but does not make it a physical Maxwell field. A unit transverse polarization, for example $(0,0,1)$, is the consistent paraxial choice, with only small longitudinal corrections.

Second, for varying local refractive index $n(x)=\sqrt{{ϵ}_{\mathrm{rel}}(x)}$, the quoted local plane-wave impedance relation gives $Z(x)=Z_0/n(x)$ and hence a flux proportional to $n(x)|E{|}^2$, not merely $|E{|}^2$. The scalar norm proof alone cannot make this exactly constant for arbitrary $n(x)$. For example a transversely uniform envelope obeying the printed parabolic equation hasso its intensity is constant while the locally inferred plane-wave flux changes with $n$. This supplies a direct obstruction to the literal exact-flux claim within the supplied scalar assumptions.

The consistent physical statement follows from Poynting theorem: for a lossless Maxwell solution and no lateral energy leakage,For a slowly varying transverse plane wave, the corresponding forward WKB field has amplitude proportional to $n^{-1/2}$, making $n|E{|}^2$ constant. In one dimension this is also encoded in the exact conserved flux $(2{\mu }_0\omega {)}^{-1}\mathrm{Im}(E^{\ast }{E}_x)$. Thus the requested physical conservation holds for the corrected lossless Maxwell description, while the given scalar equation proves the reference-normalized paraxial invariant above. The supplied fixed longitudinal polarization and arbitrary local-impedance inference cannot be used to claim more.