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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 19, Iss. 1 — Jan. 1, 2002
  • pp: 33–42

Staircase approximation validity for arbitrary-shaped gratings

Evgeny Popov, Michel Nevière, Boris Gralak, and Gérard Tayeb  »View Author Affiliations

JOSA A, Vol. 19, Issue 1, pp. 33-42 (2002)

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An electromagnetic study of the staircase approximation of arbitrary shaped gratings is conducted with three different grating theories. Numerical results on a deep aluminum sinusoidal grating show that the staircase approximation introduces sharp maxima in the local field map close to the edges of the profile. These maxima are especially pronounced in TM polarization and do not exist with the original sinusoidal profile. Their existence is not an algorithmic artifact, since they are found with different grating theories and numerical implementations. Since the number of the maxima increases with the number of the slices, a greater number of Fourier components is required to correctly represent the electromagnetic field, and thus a worsening of the convergence rate is observed. The study of the local field map provides an understanding of why methods that do not use the staircase approximation (e.g., the differential theory) converge faster than methods that use it. As a consequence, a 1% accuracy in the efficiencies of a deep sinusoidal metallic grating is obtained 30 times faster when the differential theory is used in comparison with the use of the rigorous coupled-wave theory. A theoretical analysis is proposed in the limit when the number of slices tends to infinity, which shows that even in that case the staircase approximation is not well suited to describe the real profile.

© 2002 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(050.1950) Diffraction and gratings : Diffraction gratings
(050.1960) Diffraction and gratings : Diffraction theory
(260.2110) Physical optics : Electromagnetic optics

Original Manuscript: March 23, 2001
Revised Manuscript: May 31, 2001
Manuscript Accepted: May 31, 2001
Published: January 1, 2002

Evgeny Popov, Michel Nevière, Boris Gralak, and Gérard Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002)

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  1. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981). [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982). [CrossRef]
  3. D. M. Pai, K. A. Awada, “Analysis of dielectric gratingsof arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991). [CrossRef]
  4. M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994). [CrossRef]
  5. D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976). [CrossRef]
  6. J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom-molecular reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976). [CrossRef]
  7. 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]
  8. F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998). [CrossRef]
  9. P. Lalanne, G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–783 (1996). [CrossRef]
  10. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]
  11. E. Popov, M. Nevière, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000). [CrossRef]
  12. E. Popov, M. Nevière, “Grating theory: New equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000). [CrossRef]
  13. J. M. Elson, P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995). [CrossRef]
  14. G. Tayeb, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique: comment,” J. Opt. Soc. Am. A 13, 1766–1767 (1996). [CrossRef]
  15. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981). [CrossRef]
  16. G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984). [CrossRef]
  17. S. E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: An exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993). [CrossRef]
  18. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995). [CrossRef]
  19. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., (Springer-Verlag, Berlin, 1980), Chap. 3.
  20. G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” special issue on “Generalized Multipole Techniques (GMT)” of Appl. Comput. Electromagn. Soc. J. 9, 90–100 (1994).
  21. E. Popov, M. Nevière, “Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001). [CrossRef]
  22. P. Lalanne, “Convergence performance of the coupled-wave and the differential method for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1997). [CrossRef]

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