Numerical calculation of mode-fields of the spherical plano-convex unstable resonator with Gaussian reflectivity mirror by means of the finite element method

In order to study the mode and the phase characteristic of laser in an unstable resonator with Gaussian reflectivity mirrors,a finite element method is presented to numerically calculate mode-fields of the spherical plano-convex unstable resonator with Gaussian reflectivity mirrors,which is based on the canonical formulation of SIEGMAN's theory for analyzing multi-element unstable resonators.The round-trip cylindrical Huygen's integral is expressed for the resonator in terms of spherical resonator analysis and ABCDM matrix.The Huygen's integral is transformed to a finite sum format by dividing the output mirror into several width-equal circles.The eigenmode and phase result of self-reproduce field of perfect empty resonator and misalignment empty resonator are obtained via numerical calculation.The calculation result indicates that when the radius and Gaussian reflectivity of output mirror are appropriately selected,the fundamental mode distribution of Gaussian beam which has a spot radius of 0.3cm can be obtained.The calculation result accords with the experiment result.Simultaneously the influence of the output mirror on the beam quality is also discussed.