A canonical formulation for analyzing multielement unstable resonators

A simple but general formulation for the unified analysis of many different varieties of unstable optical resonators is obtained by expressing the round-trip Huygens' integral for the resonator in terms of the round-trip paraxial ray matrix. Complicated multielement resonators, ring resonators, and unstable resonators with variable-reflectivity output mirrors are thereby all reduced to a single equivalent collimated-beam diffraction problem. Essential assumptions in the analysis are that all elements in the resonator obey paraxial ray theory and that there is only one significant output coupler or limiting aperture per round trip inside the resonator. The only parameters needed to describe an arbitrary unstable resonator in this formulation are the round-trip magnification
M
and either the equivalent Fresnel number N
eq
or else an equivalent collimated Fresnel number
N_{c} = [2M^{2}/(M^{2} - 1)] N_{eq}
, plus for variable-reflectivity mirrors a specification of the mirror reflectivity profile.