Generalized Sylvester theorems for periodic applications in matrix
Sylvester's theorem is often applied to problems involving light propagation
through periodic optical systems represented by unimodular 2 x 2 transfer
matrices. We extend this theorem to apply to broader classes of optics-related
matrices. These matrices by be 2 x 2 or take on an important augmented
3 x 3 form. The results, which are summarized in tabular form, are used
for the analysis and the synthesis of a variety of optical systems, such
as those that contain periodic distributed-feedback lasers, lossy birefringent
filters, periodic pulse compressors, and misaligned lenses and mirrors.
The results are also applicable to other types of systems such as periodic
electric circuits with intracavity independent sources, high-energy particle
accelerators, and periodic computer graphics manipulations that may include
object translation. As an example, we use the 3 x 3 form of Sylvester's
theorem to examine Gaussian beam propagation in a misaligned resonator.
Date Modified January, 1998
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