Computing the laser beam path in optical cavities: a geometric Newton's method based approach

TL;DR

This paper introduces a geometric Newton's method-based approach to compute steady-state laser beam paths in optical cavities, achieving high accuracy with reduced computational effort by exploiting Fermat's principle and matrix manifold optimization.

Contribution

It presents a novel, efficient method for beam path computation in optical cavities that avoids overparametrization and second derivatives, improving precision and computational speed.

Findings

Method converges with second order rate

Effective for non-planar polygonal cavities

Achieves high accuracy with negligible computational effort

Abstract

In the last decade, increasing attention has been drawn to high precision optical experiments, which push resolution and accuracy of the measured quantities beyond their current limits. This challenge requires to place optical elements (e.g. mirrors, lenses, etc.) and to steer light beams with sub-nanometer precision. Existing methods for beam direction computing in resonators, e.g. iterative ray tracing or generalized ray transfer matrices, are either computationally expensive or rely on overparametrized models of optical elements. By exploiting Fermat's principle, we develop a novel method to compute the steady-state beam configurations in resonant optical cavities formed by spherical mirrors, as a function of mirror positions and curvature radii. The proposed procedure is based on the geometric Newton method on matrix manifold, a tool with second order convergence rate that relies on a second order model of the cavity optical length. As we avoid coordinates to parametrize the beam position on mirror surfaces, the computation of the second order model does not involve the second derivatives of the parametrization. With the help of numerical tests, we show that the convergence properties of our procedure hold for non-planar polygonal cavities, and we assess the effectiveness of the geometric Newton method in determining their configurations with high degree of accuracy and negligible computational effort.