Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 80 / 3 / c

Take transverse length units in which the wave phase modulation has wavenumber one. Set $E_0(z)=e^{i\cos z}$. A Taylor expansion of the free parabolic propagator givesFor a general real wave phase $\varphi$,Multiplying the envelope expansion by its complex conjugate and collecting powers of $x$ gives the phase-to-intensity conversion near a deterministic phase screen,For $\varphi =\cos z$, ${\varphi }^{\prime }=-\sin z$, ${\varphi }^{\prime \prime }=-\cos z$ and ${\varphi }^{\prime \prime \prime }=\sin z$, henceThe first-order brightening occurs where the wave phase curvature is negative. Both correction terms average to zero over a transverse period, in agreement with power conservation. If a dimensional modulation $\cos (qz)$ is used instead, the two terms are $(q^{2}x/k)\cos (qz)$ and $(q^4{x}^2/k^2)\cos (2qz)$.