Generalized Sylvester theorems for periodic applications in matrix optics

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.

[1]  L. Casperson.  Beam propagation in periodic quadratic-index waveguides. , 1985, Applied optics.

[2]  Herwig Kogelnik,et al.  Imaging of optical modes — resonators with internal lenses , 1965 .

[3]  H. Kogelnik,et al.  Laser beams and resonators. , 1966, Applied optics.

[4]  P. Yeh,et al.  Electromagnetic propagation in periodic stratified media. I. General theory , 1977 .

[5]  Arthur Cayley.  III. A memoir on the theory of matrices , 1859, Proceedings of the Royal Society of London.

[6]  Dewey H. Hodges,et al.  Theory and problems , 1996 .

[7]  Wei-Ping Huang,et al.  On the transfer matrix method for distributed-feedback waveguide devices , 1992 .

[8]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[9]  J R Pierce,et al.  MODES IN SEQUENCES OF LENSES. , 1961, Proceedings of the National Academy of Sciences of the United States of America.

[10]  P. Yeh.  Extended Jones matrix method , 1982 .

[11]  Oscar E. Martínez,et al.  Matrix formalism for pulse compressors , 1988 .

[12]  A. Zinober.  Matrices: Methods and Applications , 1992 .

[13]  Pochi Yeh,et al.  Extended Jones matrix method. II , 1993 .

[14]  Miguel A. Muriel,et al.  A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing , 1990 .

[15]  Lee W. Casperson,et al.  Generalized reverse theorems for multipass applications in matrix optics , 1994 .

[16]  P. I. Richards,et al.  Manual of Mathematical Physics , 1959 .

[17]  R. Jones.  A New Calculus for the Treatment of Optical SystemsI. Description and Discussion of the Calculus , 1941 .

[18]  L. Casperson,et al.  Multipass resonators for annular gain lasers , 1981 .

[19]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[20]  L W Casperson,et al.  Gaussian light beams in inhomogeneous media. , 1973, Applied optics.

[21]  J A Arnaud.  Degenerate Optical Cavities. II: Effect of Misalignments. , 1969, Applied optics.

[22]  R. Jones.  A New Calculus for the Treatment of Optical Systems. IV. , 1942 .

[23]  John U. White.  Long Optical Paths of Large Aperture , 1942 .

[24]  D. Edwards,et al.  An Introduction to the Physics of High Energy Accelerators , 1992 .

[25]  M. Adams,et al.  Optical waves in crystals , 1984, IEEE Journal of Quantum Electronics.

[26]  L W Casperson,et al.  Gaussian modes in high loss laser resonators. , 1975, Applied optics.