Parabolic wave equation

The parabolic wave equation is a one-way, slowly varying envelope approximation to the Helmholtz equation. For carrier wavenumber $k$ and transverse coordinate $z$, a common form is $2ik{E}_x+E_{zz}=0$.

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.

A parabolic wave equation driven by longitudinal Gaussian white noise with transverse covariance kernel $C$ has the displayed closed mutual coherence equation. The noise is the derivative of a longitudinal Brownian field with transverse covariance; the physical smooth-noise limit uses the Stratonovich integral. Conversion to the Itô integral supplies the attenuation drift and the cross variation supplies the $C({\mathbf{s}}_1-{\mathbf{s}}_2)$ term. For unit plane-wave incidence and transverse statistical homogeneity, $\Gamma =\exp [-k^2{\mu }^{2}x(C(0)-C({\mathbf{s}}_1-{\mathbf{s}}_2))]$. General Gaussian colored media do not have this local closure.

For $U_x=iL(x)U$ with a self-adjoint transverse differential operator $L(x)$, differentiating the squared L2 norm gives $i⟨U,LU⟩-i⟨LU,U⟩=0$. Reality of the refractive coefficient and vanishing transverse boundary flux are required. This is exact for the scalar envelope model. Its physical interpretation as leading paraxial power uses a fixed reference impedance; a varying local impedance or a longitudinal electromagnetic polarization requires a consistent Maxwell calculation and cannot be inferred from this invariant alone.

For $E_x=i{\Delta }_{\perp }E/(2k)+ik\mu WE$, a jointly Gaussian colored medium has covariance $C(s,\mathbf{r})$. Averaging the equation gives a term $⟨WE⟩$; the Furutsu–Novikov formula expresses it through the random propagator and the functional derivative of $E$. It does not generally close in the mean field alone. A phase-screen product followed by Gaussian averaging gives a formal exact mean propagator, with factor $\exp [-k^2{\mu }^2\Delta x^2{\sum }_{j,l}C(x_j-x_l,{\mathbf{r}}_j-{\mathbf{r}}_l)/2]$ for each transverse path. In the longitudinal white-noise limit the covariance collapses to equal slices, and the mean closes as $m_x=i{\Delta }_{\perp }m/(2k)-k^2{\mu }^{2}A(0)m/2$.

The paraxial approximation describes waves whose propagation directions make small angles with a preferred axis. Factoring out a carrier $e^{ikx}$ and neglecting the envelope's second longitudinal derivative turns the Helmholtz equation into a parabolic wave equation and suppresses backward propagation.

The split-step Fourier method alternates local phase accumulation with free diffraction. For $E_x=(D+S)E$ over a step $\Delta x$, Lie-Trotter splitting givesand a transverse Fourier transform applies the constant-coefficient diffraction exponential efficiently.

In a homogeneous medium, a paraxial envelope satisfies $E_x=i{\Delta }_{\perp }E/(2k)$. This free-space diffraction spreads transverse Fourier modes by the phase factor $\exp (-ix|\mathbf{p}{|}^2/(2k))$.

For a Neumann screen on the negative horizontal axis, the Helmholtz equation has an exact solution in Fresnel integrals. With time dependence $e^{i\omega t}$, a wave incident from angle $0<{\theta }_0<\pi$ has total field $e^{ikr\cos (\theta -{\theta }_0)}\mathcal{F}(s_{-})+e^{ikr\cos (\theta +{\theta }_0)}\mathcal{F}(-s_{+})$, where $s_{\pm }=\sqrt{2kr}\cos [(\theta \pm {\theta }_0)/2]$ and $\mathcal{F}(s)=\frac{1}{2}\mathrm{erfc}(-e^{i\pi /4}s)$. The angular derivatives cancel on both faces of the screen. Away from the two shadow boundaries, the Fresnel tail expansion splits this into incident/reflected geometrical optics fields and an outgoing edge wave proportional to $r^{-1/2}{e}^{-ikr}$.

For a rigid screen $x<0$, use time dependence $e^{i\omega t}$ and inverse Fourier transform $e^{-i\kappa x}$. The scattered field is odd in $y$. Its upper-face trace is supported on $x<0$, while its normal derivative on $x>0$ is unknown. With the outgoing factor $\gamma =\sqrt{{\kappa }^2-k_0^2}$, these traces satisfy $\gamma {\hat{\varphi }}^{-}+G^{+}=b/(\kappa +a)$, where $a=k_0\cos {\theta }_0$ and $b=k_0\sin {\theta }_0$. Factor $\gamma =\sqrt{\kappa -k_0}\sqrt{\kappa +k_0}$ into upper/lower analytic factors. Pole subtraction in a Wiener-Hopf equation and the finite-energy edge condition give the displayed trace. The forcing-pole residue yields geometrical optics, while stationary phase yields the outgoing edge waveThe angular formula excludes the shadow-boundary transition regions; the exact integral supplies their uniform continuation.

In two transverse dimensions, the Fresnel propagator has kernel $k_0(2\pi ix{)}^{-1}\exp (i{k}_0|\mathbf{z}-{\mathbf{z}}^{\prime }{|}^2/(2x))$ for $x>0$. It evolves the homogeneous parabolic wave equation.

For one transverse coordinate, the parabolic wave equation $2ik{E}_x+E_{zz}=0$ has the Fresnel propagator kernelThe square root uses $\sqrt{1/i}=e^{-i\pi /4}$. With $\hat{f}(\nu )=(2\pi {)}^{-1}\int f(z)e^{-i\nu z}dz$, its Fourier transform multiplier is $e^{-i{\nu }^{2}x/(2k)}$. This distinguishes the one-transverse-coordinate prefactor from the two-coordinate kernel. The operator is unitary on $L^2(\mathbb{R})$, although its spatial integral is interpreted as an oscillatory integral.

For $m_x=(i/k)m_{r_1{r}_2}$, the free-space fourth-moment propagator multiplies the two-dimensional Fourier transform by $e^{-ix{p}_1{p}_2/k}$. In physical coordinates its oscillatory kernel is