# Complex Source-Point Theory of the Electromagnetic Open Resonator

A. L. Cullen , P. K. Yu

## Abstract

Using the complex source-point method, we deduce simple approximate formulae for the six rectangular components of the electromagnetic field of a Gaussian beam. It is shown that in the standing-wave pattern formed by two oppositely directed Gaussian beams, nodal surfaces exist on which the tangential component of the electric field vector is zero. These surfaces can be regarded as defining mirror shapes for an open resonator supporting this Gaussian standing-wave pattern. These mirror shapes are nearly, but not exactly, spherical. The resonant frequencies for the fundamental transverse mode of such a resonator have been found as a function of the geometry and the axial mode number. Using a simple perturbation method the resonant frequency of an open resonator with spherical mirrors has been found. The result, though approximate is accurate to within $(kw_{0})^{-6}$, in contrast to the commonly quoted result, which is only accurate to within $(kw_{0})^{-4}$, where k is the phase coefficient for TEM waves in free space and $w_{0}$ is the scale radius of the Gaussian beam at the beam waist.