Fresnel propagator ($U_x=\exp (ix{\Delta }_{\perp }/(2k_0))$)

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.