Free-space diffraction

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