Huygens principle: Exact and approximate wavefronts propagated through conic lenses

Exact and approximate formulae for refracted wavefronts through singlet lenses are obtained by considering an incident plane wavefront propagating along the optical axis. We provide two different approaches for the wavefronts approximated at the second order based on the Huygens principle and the Malus–Dupin theorem. We have in the first method found a way to use iterative wavelets instead of the usual evaluated integral to arrive at these formulae, showing a physical and mathematical correspondence between both methods. Finally, we introduce a parabolic wavefront into the irradiance transport equation in order to provide an analytical solution for the distribution of illumination.