# On Classical Electromagnetic Fields (cont.)

## IV. Optical Reonators (pdf copy)

### 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
 [ IV-1a ]
and
 [ IV-1b ]
From the first equation we write
.
and substitute into Equation [ IV-1b ] to obtain
 [ IV-2a ]
The determinant of the coefficients  so that
 [ IV-2a' ]
We see that
 [ IV-3 ]
Thus, stable ray propagation may characterized by bound solutions of the form  which are possible if and only if
 [ IV-4 ]
Therefore propagation is stable -- i.e. the rays are confined -- when  so that
 [ IV-5 ]
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 Hermite-Gaussian 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 [ III-20 ] ):
 [ IV-6a ]
where .  Hence, we see that
 [ IV-6b ]
with a lot of algebra we can show that
 [ IV-7 ]
where now  and  .
For a symmetric resonator  and
 [ IV-8a ]
and
 [ IV-8b ]
For an asymmetric resonator, it can be shown with a bit more algebra that
 [ IV-9a ]
 [ IV-9b ]

As a measure of the effect of resonator length and mirror radius on diffraction loss consider the ratio:
 [ IV-10 ]
where  is beam width at the mirror for the confocal configuration -- i.e., when both mirrors have their focal points at the mid-point of the cavity

Resonance Frequencies of the Optical Resonator:
 [ IV-11a ]

 [ IV-11b ]
After much algebra, it can be shown that:
 [ IV-10b']

### Footnotes

[1] Where for reference , we see that  ,

This page was prepared and is maintained by R. Victor Jones, jones@deas.harvard.edu
Last updated February 10, 2000