  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.
