Mirrors with designed spherical aberration for multi-pass cavities

We present a novel multi-pass cavity design based on the use of a rotationally symmetric end mirror having a specifically designed spherical aberration so that its focal length varies inversely as the ray height from the optical axis. We provide a detailed discussion of how ray tracing can be done for this system and show with numerical simulations that a very rich set of exotic spot patterns can be formed on the end mirrors. We further show a specific q-preserving configuration where the q-parameters of the input and output beams remain the same. Finally, we derive the polar form of the mirror surface profile that gives this offset-dependent focal length.