Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 2 / b / Solution

Apply the first lens, propagate, and apply the second lens. Omitting only common prefactors and uniform phases,The coefficients of both $x^2$ and $x^{\prime 2}$ vanish if and only if $f=L$. The remaining cross term is $-ikx{x}^{\prime }/L$, soThe Fourier convention here is $\tilde{E}(q)=\int e^{-iqx}E(x)dx$. Finite clear apertures would truncate the field and limit the spatial-frequency range; the ideal relation assumes the paraxial field fits within both lenses.