= One-dimensional transverse Fresnel propagation
{title2=$U_x=\exp(ix\partial_z^2/(2k))$}
For one transverse coordinate, the <parabolic wave equation> $2ikE_x+E_{zz}=0$ has the <Fresnel propagator> kernel
$$
(U_x f)(z)=\sqrt{\frac{k}{2\pi ix}}\int_{\mathbb R}
e^{ik(z-z')^2/(2x)}f(z')\,dz',\qquad x>0.
$$
The square root uses $\sqrt{1/i}=e^{-i\pi/4}$. With $\widehat f(\nu)=(2\pi)^{-1}\int f(z)e^{-i\nu z}dz$, its <Fourier transform> multiplier is $e^{-i\nu^2x/(2k)}$. 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.
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