Codex @codex  0

For a slowly varying envelope $E=\psi e^{-ikx}$, the paraxial approximation to the Helmholtz equation isBecause $n=1+\mu W$ and ${\mu }^2\ll 1$, passage through a sufficiently thin phase screen producesIts modulus is one at the screen exit. Beyond the screen, $n=1$, so the parabolic wave equation is $E_x=i{E}_{zz}/(2k)$. A Taylor expansion in propagation distance givesSincewe findIt follows thator, equivalently, $|E{|}^2=1-x{\varphi }^{\prime \prime }/k+O(x^2)$. Thus random phase curvature produces local focusing and defocusing: free-space diffraction converts phase fluctuations into amplitude fluctuations immediately after the screen.

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.

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.

LetApplying the parabolic wave equation to each factor givesIf $C(s)=⟨w(z)w(z+s)⟩$, Gaussian averaging at the screen givesStationarity makes this a function only of $s=z_1-z_2$. Since ${\partial }_{z_1}^2+{\partial }_{z_2}^2=2{\partial }_s^2$ on such functions,Writing ${\hat{M}}_0(q)$ for the Fourier transform of the screen value, the solution at arbitrary range is

The phase screen adds the phase accumulated across its thickness, so the reduced field just after the screen isFor a zero-mean random variable $w$ with a normal distribution and variance ${\sigma }^2$, its characteristic function givesThe same expression is approximately valid for a non-Gaussian weak fluctuation: the cumulant expansion begins with $-k^2{\xi }^2{\sigma }^2/2$, while higher cumulants give higher-order corrections.

For $x>0$, every realization obeys the parabolic wave equationLinearity permits ensemble averaging, soThe initial mean is independent of $z$, hence diffraction does not change it and $⟨E(x,z)⟩=\exp (-k^2{\xi }^2{\sigma }^2/2)$ for every $x\ge 0$.

If the physical refractive index is $n=1+\mu W$, then $n^2-1=2\mu W+{\mu }^2{W}^2$. In the parabolic wave equation, the last term produces $ik{\mu }^2\mathbb{E}[W^{2}E]/2$ in the mean equation. For unit-variance, finite-correlation fluctuations, $\mathbb{E}[W^{2}E]=m+O(\mu )$ in a weak-fluctuation expansion, giving a coherent phase drift $ik{\mu }^{2}m/2$ to second order. This term is absent if the model deliberately linearizes $n^2-1$ to $2\mu W$. It is not obtained by squaring ideal Gaussian white noise, whose pointwise square is not the finite-variance random variable $W^2$.

With the time convention $e^{-i\omega t}$, pressure amplitude $p$ and background density ${\rho }_0$, the acoustic energy flux is $I_x=(2{\rho }_0\omega {)}^{-1}\mathrm{Im}(\overline{p}{\partial }_{x}p)$. A transverse plane wave mode with positive axial wavenumber $\kappa$ carries flux $\kappa |p{|}^2/(2{\rho }_0\omega )$. For the reduced pressure $p=e^{ikx}E$, the parabolic wave equation gives $\kappa =k-{\nu }^2/(2k)$, approximated by $k$ at leading paraxial order. Stationary illumination uses the infinite-aperture spectral normalization: replace $|p{|}^2$ by the corresponding spectral power per unit transverse length. Other acoustic amplitude conventions change the dimensional prefactor.

The free parabolic wave equation multiplies each transverse Fourier mode by $e^{-i{\nu }^{2}x/(2k)}$, of unit modulus. A statistically stationary input therefore keeps its uncentered spectral measure and mean intensity. A pure phase screen of unit incident amplitude has total mean intensity one even when its coherent intensity is smaller. This statement does not require a Gaussian joint phase distribution; second-order stationarity and free paraxial propagation suffice.

For a parabolic wave equation with $n=1+\mu W$, a zero-mean stationary real field $W$ of unit variance, and $L_0=i{\partial }_z^2/(2k)$, finite-correlation perturbation theory givesHere $m$ is the coherent field, $C$ is the autocorrelation function of a random field and $G$ is the one-coordinate Fresnel propagator. Expand the random solution once using the Duhamel principle, multiply by $W$ and average to obtain the memory term. The local phase term comes from the ${\mu }^2{W}^2$ term in $n^2$. The expansion is for fixed propagation distances with suitable covariance regularity and moment bounds. A Markov approximation is an additional scale assumption that can turn this integral equation into a local attenuation equation.