Precise multipass Herriott cell design: derivation of controlling design equations

Multipass Herriott cells are often designed using the thin lens approximation, which results in approximate dimensions or imperfect patterns due to both spherical aberration and the finite width of the optic. We derive the design equations for exact solutions to the Herriott cell problem. We also show that Herriott cells using spherical mirrors cannot be designed such that multiple concentric beam patterns all meet their reentrant condition. We derive a solution for elliptical mirrors that allows this condition to be met simultaneously for many beams.