Electromagnetic theory of an open resonator

Particular higher-order sources give rise to electromagnetic Gaussian beams, which are linearly polarized and have their maximum in the propagation direction. For this dipolar beam the cross-sectional shape changes in the propagation direction. Nodal surfaces exist on which the tangential component of the electric field vanishes in the standing wave that is formed by the two oppositely directed dipolar, electromagnetic Gaussian beams. These surfaces are identified as the mirror shapes for an open resonator that supports this standing wave. For standing waves that have a particular cross-sectional shape at the waist the cross section of the beam near the mirror surfaces is circular. The resonant frequencies for the fundamental transverse mode of such a resonator have been determined as a function of the geometry and the axial mode number. By a perturbation technique the resonant frequency of an open resonator with spherical mirrors has been obtained. This result is valid in only the paraxial approximation. Illustrative numerical results are included.