Unstable optical resonators

A technique, firmly based on a development from ray optics, is presented for calculating the loss due to the finite sizes of curved mirrors when these form an unstable optical resonator. If paraxial rays launched within such a resonator are confined near the resonator axis, the resonator is termed stable; otherwise it is termed unstable, and is known to have high losses. Siegman has recently presented a geometrical method, brilliantly constructed ad hoc, for calculating these losses in unstable resonators, and indicated where these might be advantageous in laser application. The ray optical theory presented here, which employs the concept of ray modes in an equivalent beam waveguide, is shown to yield results equivalent to those of Siegman for all cases considered by him. However, being derived from conventional ray optics, the validity of the formulas is independently established, and these formulas are immediately applicable to re-entrant resonators and resonators containing inhomogeneous media. The fractional loss per resonator pass is equal to 1-|lambda(2)|, where |lambda(2)| < this 1 is an eigenvalue of the transfer matrix T, representing the corresponding ray transformation.