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

Measure the phase relative to propagation through the same geometrical thickness of air. If the lens's central thickness is $t_0$, its spherical surface givesThe optical-path excess is $(n-1)t(x)$, so the phase imparted by the lens isThe constant part is a spatially uniform phase. The physically important lens transmission is therefore $T_f(x)=\exp [-ik{x}^2/(2f)]$ withOne way to identify the focus is to append propagation to the on-axis point a distance $f$ away. Its path phase has the paraxial contribution $+k{x}^2/(2f)$, which cancels the lens phase. Thus all incident portions of a plane wave add with the same phase at that point. The diagram below shows the thickness change and both Fourier-transform arrangements.