This paper reviews the current level of theoretical understanding and experimental confirmation of the properties of unstable optical resonators for laser oscillators. A primary conclusion is that for any laser system characterized by a moderate gain per pass and a Fresnel number greater than a few times unity the unstable resonator provides the best practical resonator design for obtaining a high-quality near-diffraction-limited output laser beam combined with efficient energy extraction from the laser medium.

Unstable optical 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.

A canonical formulation for analyzing multielement unstable resonators

Publisher Summary This chapter presents an in-depth overview of the fundamental mode structure theory of unstable optical cavities for laser applications. Much of this analysis has focused on the passive cavity mode structure properties, as it is in this purely optical arena that the more fundamental mathematical properties reside, several of which are still unresolved. In the chapter Siegman postulated an approximate geometrical analysis of the dominant, lowest loss transverse mode that is remarkably accurate in predicting the out coupling losses associated with an unstable cavity. This geometrical mode analysis is extended to the higher-order, higher loss transverse modes of such resonators. At this point it became clear that unstable resonators possess several significant advantages over their stable counterparts for many laser applications, particularly in the high energy regime.

IV Unstable Resonator Modes

We present a rigorous derivation of the reciprocity and orthogonality relations appropriate to ring resonators. In particular, we show that linear ring resonators are always reciprocal, contrary to recent assertions.

Reciprocity and orthogonality relations for ring resonators

Analysis of a confocal ring laser including thermal lensing effects in the active medium is presented. The relations for the stability criterion, beam magnification, and the positional shift in the beam minimum are derived in terms of the parameters of the resonator. It is suggested that by appropriately adjusting the separation between the confocal lens pair, the effects due to thermal lensing can be compensated.

Analysis of an unstable confocal ring laser with a thermally induced active medium

An unstable ring resonator has been studied, both theoretically and experimentally; quasi-traveling-wave operation is obtained with a high ratio of the powers of the two opposite rotating waves and with diffraction-limited output beams.

P. Pozzo

Papers

An unstable ring resonator