Paraxial wave equation

Remarks

Exactly, $\Delta U + k^2 U = e^{ikz} \bigl(\partial_x^2 A + \partial_y^2 A + \partial_z^2 A + 2ik \, \partial_z A\bigr)$, so the equation is the paraxial approximation of the Helmholtz equation. Here the time dependence is $e^{-i\omega t}$; with the convention $e^{-ikz}$ the sign of the last term changes. In the form $i \, \partial_z A = -\frac{1}{2k} (\partial_x^2 + \partial_y^2) A$ it is the free Schrödinger equation in the plane, with $z$ in place of time.

Example: let $z_R > 0$, $q(z) = z - i z_R$ and $A = q^{-1} \exp\bigl(ik(x^2 + y^2)/(2q)\bigr)$. With $r^2 = x^2 + y^2$, one computes $2ik \, \partial_z A = \bigl(-2ik/q^2 + k^2 r^2/q^3\bigr) q A$. Also $(\partial_x^2 + \partial_y^2) A = \bigl(2ik/q^2 - k^2 r^2/q^3\bigr) q A$, so the sum vanishes. This solution is the Gaussian beam.

Non-example: $A = e^{i k_x x}$ with $k_x \ne 0$ does not solve the equation, since $\partial_x^2 A = -k_x^2 A \ne 0$ while $\partial_z A = 0$. The tilted beam $A = e^{i k_x x - i k_x^2 z/(2k)}$ does solve it.