## Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the *R*-matrix propagation algorithm

JOSA A, Vol. 11, Issue 12, pp. 3241-3250 (1994)

http://dx.doi.org/10.1364/JOSAA.11.003241

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### Abstract

The analysis of gratings of arbitrary depth, profile, and permittivity is conducted by cutting the modulated region into different slices for which the differential theory of gratings is able to compute the diffracted field for both TE and TM polarization without numerical instabilities. The use of a suitable transition matrix (*R* matrix) then allows one to analyze the entire stack without encountering the numerical instabilities that generally occur with use of the *T*-transmission matrix, which is well known in stratified media theory. The use of the *R*-matrix propagation algorithm provides a breakthrough for grating theoreticians in the sense that it not only permits the study of grating of arbitrary depth but also eliminates the numerical instabilities that have plagued the differential theory in TM polarization during the past 20 years.

© 1994 Optical Society of America

**History**

Original Manuscript: February 23, 1994

Revised Manuscript: July 27, 1994

Manuscript Accepted: July 28, 1994

Published: December 1, 1994

**Citation**

F. Montiel and M. Neviere, "Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm," J. Opt. Soc. Am. A **11**, 3241-3250 (1994)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-11-12-3241

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### References

- M. T. Gale, K. Knop, R. Morf, “Zero order diffractive microstructure for security applications,” presented at the Optical Security and Anticounterfeiting Systems Conference, Los Angeles, Calif., January 15–16, 1990.
- E. G. Loewen, M. Nevière, D. Maystre, “Grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977). [CrossRef] [PubMed]
- R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980). [CrossRef]
- B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics, in Applications of Thin Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985). [CrossRef]
- M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994). [CrossRef]
- F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 and 706–782 (1950).
- D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991). [CrossRef]
- D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976). [CrossRef]
- J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976). [CrossRef]
- L. F. DeSandre, J. M. Elson, “Extinction theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991). [CrossRef]
- L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2593.
- L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction gratings,” Opt. Acta 28, 413–428 (1981). [CrossRef]
- L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981). [CrossRef]
- L. Li, “A modal analysis of lamellar diffraction gratings in conical mounting,” J. Mod. Opt. 40, 553–573 (1993). [CrossRef]
- M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nou. Rev. Opt. 5, 65–77 (1974). [CrossRef]
- M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. B 5, 1522–1524 (1988). [CrossRef]
- P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121. [CrossRef]
- P. Henrici, Discrete Variable Methods in Ordinary Differential Equations (Wiley, New York, 1962).
- H. Iwaoka, K. Akiyama, “A high-resolution laser scale interferometer,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moiré Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 135–139 (1984). [CrossRef]
- A. Teimel, “Technology and application of grating interferometers in high-precision measurements,” in Progress in Precision Engineering, P. Seysried, H. Kunzmann, T. McKeown, eds. (Springer-Verlag, Berlin, 1991), pp. 131–147.

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