The connection of geometrical optics with the propagation of gaussian beams and the theory of optical resonators

Abstract

The propagation of light near the axis of astigmatic optical systems may be described by the geometrical-optics approximation with the aid of ray-matrices. The application of the theory of diffraction to the propagation of light in such systems leads to integrals containing essentially the elements of the ray-matrices as parameters. The ABCD-law is derived by evaluating these integrals for gaussian beams. Integral equations applicable to astigmatic optical resonators, having nearly vanishing diffraction losses, are set up. They are only valid under certain conditions, which are comprehensively discussed. The eigensolutions and the eigenvalues of these integral equations are given. The spot-sizes at the resonator mirrors are derived from the eigensolutions, and the eigenvalues lead to the resonance condition. Spot-sizes and resonance condition appear as functions of the elements of the characteristic resonator matrices. The methods described here are applied to the propagation of gaussian beams through gas-lenses and to a resonator containing an internal gas-lens.1. Introduction H. Kogelnik [1] indicated that the geometrical optics ray-matrices (ABCD-matrices) are formally connected with the propagation of gaussian beams. By analogy he was led to the derivation of the ABCD-law joining the curvature of the phase front and the spot-size at two different cross sections of the gaussian beam. However, the elements of the ray-matrices appear likewise in the diffraction integrals describing the propagation of light of arbitrary field distribution through astigmatic lens-like systems. These integrals result from Huygens' principle in inhomogeneous, isotropic media for short wavelength [2]. The ABCD-Iaw will be shown to follow from them in a simple manner, if the field distribution is expanded in Hermite-Gauss-functions. At the same time the transformation of gaussian beams by astigmatic lens-like systems is obtained. The integral equations for the field distributions at the mirrors of astigmatic optical resonators in the limit of vanishing diffraction losses will be solved by applying this transformation.

Simple multi-lens intracavity Gaussian laser beam simulation using complex rays

“…Note that an efficient alternative to Kogelnik’s method (skips the evaluation of the principal planes, the g factors, and effective resonator length) for computing the mirror spot sizes is the following simple equation 21 , 22 ω12=(λπ)[−BDAC]12.…”

Laser resonator modeling by field tracing: a flexible approach for fully vectorial transversal eigenmode calculation

“…We classify Sziklas and Siegman's combination of the Fresnel integral with TEA [17], the BPM [18,19], the Collins integral concept [23][24][25] and the Gaussian beam propagation [26] as special cases of the field tracing approach.…”