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 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']

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^Footnotes

[1] Where for reference , we see that  ,
 

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