Analysis of Optical Resonators Involving Focusing Elements

The electromagnetic fields in optical resonators involving focusing elements are considered. The work is restricted to cases where the electromagnetic fields are small at the edges of any optical elements so that there is no significant aperturing. The paper presents the analytical solution for the fields across two reference surfaces (often the end mirrors) within the resonator, and describes graphically the fields inside and outside the resonator. The analytical solution uses Huygens’ principle, but maintains generality in that the expressions relating the geometry of the particular optical resonator are left in terms of constants, which can be simply evaluated for the particular resonator. The expressions for the fields (over the reference surfaces), resonant frequencies, and stability conditions are given. The graphical description of the fields inside and outside the resonator is in terms of two experimentally observable variables: the radius of curvature of phase fronts and the spot size. A chart is utilized which permits a simple numerical presentation of those variables as a function of position along the axis, and enables a simple calculation of the higher-order transverse mode frequencies.