Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 335 / 1

DefineIf $A$ and $B$ commute, the Helmholtz equation operator factorizes asWhen $n$ varies with $x$, the exact product also contains the commutator $[A,B]$; neglecting it assumes longitudinal changes are slow. The forward-propagating factor isbecause it admits $e^{i{k}_{0}x}$ in a uniform medium.

Put $\psi =E{e}^{i{k}_{0}x}$. The one-way equation becomesForthe Taylor expansion $(1+Q{)}^{1/2}=1+Q/2+O(Q^2)$ givesThus the parabolic wave equation is

The expansion is accurate under the paraxial approximation: transverse wavenumbers satisfy $|k_z|/k_0\ll 1$, the envelope varies slowly on the carrier scale, the refractive-index contrast is weak enough for $Q^2$ to be negligible, and $n$ varies slowly in $x$ so $[A,B]$ is small. The one-way factor discards backward propagation and reflection; the square-root expansion additionally discards large-angle and higher-order diffraction, and it does not accurately represent strongly evanescent components or abrupt longitudinal interfaces.

Withthe parabolic wave equation is $E_x=(D+S)E$. Freeze $S$ at $x_0$, or preferably at the step midpoint. Over a short distance $\Delta x$, Lie-Trotter splitting givesThe reversed ordering has the same first-order accuracy, while symmetric half-steps in $S$ give the more accurate Strang form.

The commutator can be displayed explicitly. If $q=n^2-1$, thenThe splitting assumption requires $\Delta x^2[D,S]E$ to be small relative to $E$. It is favored by a short range step, a transversely smooth refractive index, and a field without unresolved large transverse wavenumbers. Freezing $S$ also requires $\Delta x{S}_x$ to be small. These conditions supplement the one-way and paraxial approximation already used in part i.

The phase-screen substep is pointwise:DefineThen solve the free-diffraction initial-value problemto $x_0+\Delta x$. In transverse Fourier transform variables, this substep is simplyThis is the split-step Fourier method.

Partition the range into planes $x_j=x_0+j\Delta x$. At each step, evaluate the refractive-index phase screen at $x_j$ or the midpoint, multiply the current envelope by $e^{\Delta x{S}_j}$, transform in $z$, multiply each transverse Fourier component by the diffraction phase $e^{-i{k}_z^2\Delta x/(2k_0)}$, and apply the inverse Fourier transform. Repeating this split-step Fourier method $m$ times produces $E(x_m,z)$ from $E(x_0,z)$. The step size must continue to resolve both longitudinal medium variation and the Lie-Trotter splitting error.