Stability Criteria for Resonators and Periodic Optical Structures by Ray Optic Analysis:
Consider a prototypical periodic guiding lens system or an equivalent resonator.
Using the appropriate ABCD matrix with the indicated reference planes, we may write
[ IV1a ]

and
[ IV1b ]

From the first equation we write
and substitute into Equation [ IV1b ] to obtain
[ IV2a ]

The determinant of the coefficients so that
[ IV2a' ]

We see that
[ IV3 ]

Thus, stable ray propagation may characterized by bound solutions of the form which are possible if and only if
[ IV4 ]

Therefore propagation is stable  i.e. the rays are confined  when so that
[ IV5 ]

Ray stability of rays in a periodic system may be usefully characterized in terms of the variables and as follows:
Stability (Confinement) Diagram for Periodic Systems
Stability of a Spherical Mirror Resonators  Using Solutions of the Paraxial Equation
Consider a HermiteGaussian mode confined in an asymmetrical spherical cavity:
In order to sustain a resonant mode in such a cavity, the beam's radius of curvature must match each mirror's radius of curvature at the mirror's surface and, thus, the following conditions must hold (see Equation [ III20 ] ):
[ IV6a ]

where . Hence, we see that
[ IV6b ]

with a lot of algebra we can show that
[ IV7 ]

where now and .
For a symmetric resonator and
[ IV8a ]

and
[ IV8b ]

For an asymmetric resonator, it can be shown with a bit more algebra that
[ IV9a ]

[ IV9b ]

As a measure of the effect of resonator length and mirror radius on diffraction loss consider the ratio:
[ IV10 ]

where is beam width at the mirror for the confocal configuration  i.e., when both mirrors have their focal points at the midpoint of the cavity
Resonance Frequencies of the Optical Resonator:
[ IV11a ]

[ IV11b ]

After much algebra, it can be shown that:
[ IV10b']

[1] Where for reference , we see that
,