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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 20, Iss. 4 — Apr. 1, 2003
  • pp: 655–660

Note on the S-matrix propagation algorithm

Lifeng Li  »View Author Affiliations

JOSA A, Vol. 20, Issue 4, pp. 655-660 (2003)

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A set of full-matrix recursion formulas for the W S variant of the S-matrix algorithm is derived, which includes the recent results of some other authors as a subset. In addition, a special type of symmetry that is often found in the structure of coefficient matrices (W matrices) that appear in boundary-matching conditions is identified and fully exploited for the purpose of increasing computation efficiency. Two tables of floating-point operation (flop) counts for both the new W S variant and the old W t S variant of the S-matrix algorithm are given. Comparisons of flop counts show that in performing S-matrix recursions in the absence of the symmetry, it is more efficient to go directly from W matrices to S matrices. In the presence of the symmetry, however, using t matrices is equally and sometimes more advantageous, provided that the symmetry is utilized.

© 2003 Optical Society of America

OCIS Codes
(000.3870) General : Mathematics
(050.1950) Diffraction and gratings : Diffraction gratings
(050.2770) Diffraction and gratings : Gratings
(050.7330) Diffraction and gratings : Volume gratings

Original Manuscript: July 19, 2002
Revised Manuscript: November 4, 2002
Manuscript Accepted: November 4, 2002
Published: April 1, 2003

Lifeng Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003)

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  1. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996). [CrossRef]
  2. K. Fu, Z. Wang, D. Zhang, J. Wen, J. Tang, “A vector analytical method of phase diffraction grating,” Acta Opt. Sin. 17, 1652–1659 (1997) (in Chinese).
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  5. X. Tang, K. Fu, Z. Wang, X. Liu, “Analysis of rigorous modal theory for arbitrary dielectric gratings made withanisotropic materials,” Acta Opt. Sin. 22, 774–779 (2002) (in Chinese).
  6. E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157–1161 (2002). [CrossRef]
  7. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995). [CrossRef]
  8. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited ,” J. Opt. Soc. Am. A JOAOD6 11, 2816–2828 (1994); errata: J. Opt. Soc. Am. A JOAOD6 13, 543 (1996). [CrossRef]
  9. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997). [CrossRef]
  10. In this paper the flop counts of matrix operations are based on information provided in G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins University Press, Baltimore, Md., 1983, 1989, and 1996). To be consistent with Table 1 of Ref. 1, the meaning of a flop follows the original definition given by the authors in the first edition of their book. See the footnote on page 18 of the third edition.
  11. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993). [CrossRef]
  12. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]

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