= Phase-to-intensity conversion near a deterministic phase screen
{title2=$I=1-(x/k)\phi''+(x^2/k^2)[(\phi'')^2+\phi'\phi''']+O(x^3)$}
For $E(0,z)=e^{i\phi(z)}$ and sufficiently smooth real phase, expand free <paraxial propagation> as $E=E_0+ixE_0''/(2k)-x^2E_0''''/(8k^2)+O(x^3)$. Taking the squared modulus gives the displayed intensity. Negative phase curvature initially focuses the field. With $\phi=\cos z$, $I=1+(x/k)\cos z+(x/k)^2\cos2z+O(x^3)$, with transverse length scaled so the modulation wavenumber is one. Period-averaged intensity stays one.
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