 Phase-to-intensity conversion near a deterministic phase screen

For $E(0,z)=e^{i\varphi (z)}$ and sufficiently smooth real phase, expand free paraxial propagation as $E=E_0+ix{E}_0^{\prime \prime }/(2k)-x^2{E}_0^{\prime \prime \prime \prime }/(8k^2)+O(x^3)$. Taking the squared modulus gives the displayed intensity. Negative phase curvature initially focuses the field. With $\varphi =\cos z$, $I=1+(x/k)\cos z+(x/k{)}^2\cos 2z+O(x^3)$, with transverse length scaled so the modulation wavenumber is one. Period-averaged intensity stays one.