In stable resonators any given initially paraxial rays remain close to the axis of the structure and are, in fact, confined within well-defined contours—the envelope of the ray system. Previously, envelopes of rays in empty resonators had been found and their form identified with the variation of the spot size. This geometric optical approach is extended to general resonators, comprising arbitrary arrangements of lenses and convergent or divergent inhomogeneous focusing media. An invariant quadratic form involving parameters descriptive of any of the ray segments that result from a given initial ray segment leads to a differential equation satisfied by the ray segments and their envelope in portions of the resonator. A maximum-minimum problem for the envelope is formulated and solved. In convergent media the envelope function is found to be periodically modulated. The period of the modulation depends only on the properties of the convergent medium; the location of relative maxima and minima, as well as their ratio, depends on both the medium and associated optics. In special cases, results are compared with available solutions of the corresponding electromagnetic problem. A particularly simple resonator is analyzed, and envelope characteristics correlated with the stability limits.
Noritaka Kurauchi and Walter K. Kahn, "Rays and Ray Envelopes within Stable Optical Resonators Containing Focusing Media," Appl. Opt. 5, 1023-1029 (1966)