Sylvester’s theorem is often applied to problems involving light propagation through periodic optical systems represented by unimodular 2 × 2 transfer matrices. We extend this theorem to apply to broader classes of optics-related matrices. These matrices may be 2 × 2 or take on an important augmented 3 × 3 form. The results, which are summarized in tabular form, are useful 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 system 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 × 3 form of Sylvester’s theorem to examine Gaussian beam propagation in a misaligned resonator.
© 1995 Optical Society of America
Original Manuscript: June 9, 1994
Revised Manuscript: September 19, 1994
Manuscript Accepted: September 20, 1994
Published: March 1, 1995
Anthony A. Tovar and Lee W. Casperson, "Generalized Sylvester theorems for periodic applications in matrix optics," J. Opt. Soc. Am. A 12, 578-590 (1995)