Lenses that provide the transformation between two given wavefronts

•Lenses for transformation between two given wavefronts.•Four types of lenses are given for solving the problem.•The lenses profiles are envelopes of Cartesian ovals of revolution.

We give an original method to design four types of lenses solving the following problems: focusing a given wavefront in a given point, and performing the transformation between two arbitrary incoming and outgoing wavefronts. The method to design the lenses profiles is based on the optical properties of the envelopes of certain families of Cartesian ovals of revolution.