Abstract

Using theorems of Fourier factorization, a recent paper [J. Opt. Soc. Am. A 13, 1870 (1996)] has shown that the truncated Fourier series of products of discontinuous functions that were used in the differential theory of gratings during the past 30 years are not converging everywhere in TM polarization. They turn out to be converging everywhere only at the limit of infinitely low modulated gratings. We derive new truncated equations and implement them numerically. The computed efficiencies turn out to converge about as fast as in the TE-polarization case with respect to the number of Fourier harmonics used to represent the field. The fast convergence is observed on both metallic and dielectric gratings with sinusoidal, triangular, and lamellar profiles as well as with cylindrical and rectangular rods, and examples are shown on gratings with 100% modulation. The new formulation opens a new wide range of applications of the method, concerning not only gratings used in TM polarization but also conical diffraction, crossed gratings, three-dimensional problems, nonperiodic objects, rough surfaces, photonic band gaps, nonlinear optics, etc. The formulation also concerns the TE polarization case for a grating ruled on a magnetic material as well as gratings ruled on anisotropic materials. The method developed is applicable to any theory that requires the Fourier analysis of continuous products of discontinuous periodic functions; we propose to call it the fast Fourier factorization method.

© 2000 Optical Society of America

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  1. M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  2. M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
    [CrossRef]
  3. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
  4. P. Vincent, “New improvement of the differential formalism for high-modulated gratings,” in Periodic Structures, Gratings, Moiré Patterns and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 147–154 (1980).
    [CrossRef]
  5. M. Nevière, J. Flamand, “Electromagnetic theory as it applies to X-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
    [CrossRef]
  6. M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft X-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
    [CrossRef]
  7. A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
    [CrossRef]
  8. R. A. Depine, J. M. Simon, “Comparison between the differential and integral methods used to solve the grating problem in the H ∥ case,” J. Opt. Soc. Am. A 4, 834–838 (1987).
    [CrossRef]
  9. M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
    [CrossRef]
  10. 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]
  11. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2593 (1993).
    [CrossRef]
  12. M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
    [CrossRef]
  13. F. Montiel, M. Nevière, “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).
    [CrossRef]
  14. N. Chateau, J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
    [CrossRef]
  15. 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]
  16. 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 transmission matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  17. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]
  18. F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
    [CrossRef]
  19. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  20. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  21. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  22. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
    [CrossRef]
  23. P. Lalanne, “Convergence performance of the coupled-wave and the differential method for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1977).
    [CrossRef]
  24. M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
    [CrossRef]
  25. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit ed. (Springer-Verlag, Berlin, 1980), pp. 63–100.
    [CrossRef]
  26. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [CrossRef]
  27. E. Popov, M. Nevière, “Differential theory for diffraction gratings: new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000).
    [CrossRef]
  28. P. Vincent, M. Nevière, D. Maystre, “Computation of the efficiencies and polarization effects of XUV gratings used in classical and conical mountings,” Nucl. Instrum. Methods 152, 123–126 (1978).
    [CrossRef]
  29. D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
    [CrossRef]
  30. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–2961978.
    [CrossRef]
  31. M. Nevière, E. Popov, R. Reinisch, G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, London, UK) (to be published).
  32. P. L. Knight, ed., Special issue on Photonic Band Structures, J. Mod. Opt.41(2) (1994).
  33. M. Nevière, E. Popov, “Grating electromagnetic theory user guide,” J. Imaging Sci. Technol. 41, 315–323 (1997).

2000 (1)

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

1997 (1)

M. Nevière, E. Popov, “Grating electromagnetic theory user guide,” J. Imaging Sci. Technol. 41, 315–323 (1997).

1996 (4)

1995 (2)

1994 (3)

1993 (1)

1991 (1)

1988 (2)

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

1987 (1)

1982 (1)

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft X-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

1980 (1)

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to X-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

1978 (3)

P. Vincent, M. Nevière, D. Maystre, “Computation of the efficiencies and polarization effects of XUV gratings used in classical and conical mountings,” Nucl. Instrum. Methods 152, 123–126 (1978).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–2961978.
[CrossRef]

1977 (1)

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

1973 (3)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

Awada, K. A.

Brinkman, A. C.

A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
[CrossRef]

Cadilhac, M.

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Chateau, N.

Craig, W. W.

A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
[CrossRef]

de Korte, P. A. J.

A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
[CrossRef]

den Boggende, A. J. F.

A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
[CrossRef]

Depine, R. A.

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Flamand, J.

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft X-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to X-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

Gaylord, T. K.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Granet, G.

Grann, E. B.

Guizal, B.

Hailey, C. J.

A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
[CrossRef]

Hugonin, J. P.

Kahn, S. M.

A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
[CrossRef]

Lalanne, P.

Lerner, J. M.

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft X-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Li, L.

Maystre, D.

P. Vincent, M. Nevière, D. Maystre, “Computation of the efficiencies and polarization effects of XUV gratings used in classical and conical mountings,” Nucl. Instrum. Methods 152, 123–126 (1978).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit ed. (Springer-Verlag, Berlin, 1980), pp. 63–100.
[CrossRef]

Moharam, M. G.

Montiel, F.

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

F. Montiel, M. Nevière, “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).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

Morf, R. H.

Morris, G. M.

Nevière, M.

E. Popov, M. Nevière, “Differential theory for diffraction gratings: new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000).
[CrossRef]

M. Nevière, E. Popov, “Grating electromagnetic theory user guide,” J. Imaging Sci. Technol. 41, 315–323 (1997).

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

F. Montiel, M. Nevière, “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).
[CrossRef]

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft X-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to X-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

P. Vincent, M. Nevière, D. Maystre, “Computation of the efficiencies and polarization effects of XUV gratings used in classical and conical mountings,” Nucl. Instrum. Methods 152, 123–126 (1978).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, E. Popov, R. Reinisch, G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, London, UK) (to be published).

A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
[CrossRef]

Pai, D. M.

Petit, R.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Peyrot, P.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

Pommet, D. A.

Popov, E.

E. Popov, M. Nevière, “Differential theory for diffraction gratings: new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598–600 (2000).
[CrossRef]

M. Nevière, E. Popov, “Grating electromagnetic theory user guide,” J. Imaging Sci. Technol. 41, 315–323 (1997).

M. Nevière, E. Popov, R. Reinisch, G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, London, UK) (to be published).

Reinisch, R.

M. Nevière, E. Popov, R. Reinisch, G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, London, UK) (to be published).

Simon, J. M.

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Videler, P. H.

A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
[CrossRef]

Vincent, P.

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
[CrossRef]

P. Vincent, M. Nevière, D. Maystre, “Computation of the efficiencies and polarization effects of XUV gratings used in classical and conical mountings,” Nucl. Instrum. Methods 152, 123–126 (1978).
[CrossRef]

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–2961978.
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

P. Vincent, “New improvement of the differential formalism for high-modulated gratings,” in Periodic Structures, Gratings, Moiré Patterns and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 147–154 (1980).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
[CrossRef]

Vitrant, G.

M. Nevière, E. Popov, R. Reinisch, G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, London, UK) (to be published).

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

J. Imaging Sci. Technol. (1)

M. Nevière, E. Popov, “Grating electromagnetic theory user guide,” J. Imaging Sci. Technol. 41, 315–323 (1997).

J. Mod. Opt. (1)

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1988).
[CrossRef]

J. Opt. (Paris) (1)

D. Maystre, M. Nevière, “Electromagnetic theory of crossed gratings,” J. Opt. (Paris) 9, 301–306 (1978).
[CrossRef]

J. Opt. Soc. Am. A (13)

N. Chateau, J. P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
[CrossRef]

F. Montiel, M. Nevière, “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).
[CrossRef]

P. Lalanne, “Convergence performance of the coupled-wave and the differential method for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1977).
[CrossRef]

R. A. Depine, J. M. Simon, “Comparison between the differential and integral methods used to solve the grating problem in the H ∥ case,” J. Opt. Soc. Am. A 4, 834–838 (1987).
[CrossRef]

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]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2593 (1993).
[CrossRef]

P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

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]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

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 transmission matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
[CrossRef]

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

Nucl. Instrum. Methods (3)

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to X-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft X-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

P. Vincent, M. Nevière, D. Maystre, “Computation of the efficiencies and polarization effects of XUV gratings used in classical and conical mountings,” Nucl. Instrum. Methods 152, 123–126 (1978).
[CrossRef]

Opt. Commun. (4)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–2961978.
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

Opt. Lett. (1)

Other (6)

P. Vincent, “New improvement of the differential formalism for high-modulated gratings,” in Periodic Structures, Gratings, Moiré Patterns and Diffraction Phenomena, C. H. Chi, E. G. Loewen, C. L. O’Bryan, eds., Proc. SPIE240, 147–154 (1980).
[CrossRef]

A. J. F. den Boggende, P. A. J. de Korte, P. H. Videler, A. C. Brinkman, S. M. Kahn, W. W. Craig, C. J. Hailey, M. Nevière, “Efficiencies of X-ray reflection gratings,” in X-Ray Instruments, Multilayers and Sources, J. F. Marshall, ed., Proc. SPIE982, 283–298 (1988).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
[CrossRef]

M. Nevière, E. Popov, R. Reinisch, G. Vitrant, Electromagnetic Resonances in Nonlinear Optics (Gordon & Breach, London, UK) (to be published).

P. L. Knight, ed., Special issue on Photonic Band Structures, J. Mod. Opt.41(2) (1994).

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit ed. (Springer-Verlag, Berlin, 1980), pp. 63–100.
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Grating geometry and notation.

Fig. 2
Fig. 2

Convergence of the previous and new versions of the differential theory for the case of a dielectric lamellar grating with high contrast. Squares, old version of the differential theory for TM polarization; open triangles, new version, TM polarization; solid triangles, TE polarization.

Fig. 3
Fig. 3

Convergence of the previous and new versions of the differential theory, and the Lalanne and Morris19 equations for a slanted (45°) lamellar grating. The grating material is the same as in Fig. 2. Solid circles are obtained with Eqs. (21) and (22).

Fig. 4
Fig. 4

Same as in Fig. 2, but for a sinusoidal profile.

Fig. 5
Fig. 5

Same as in Fig. 2, but for an aluminum sinusoidal profile. The solid circles are obtained with Eqs. (21) and (22).

Fig. 6
Fig. 6

Same as in Fig. 5, but for a triangular (echelette) profile with 30° blaze angle, 90° apex angle.

Fig. 7
Fig. 7

Zero-order transmitted efficiency of a silver rectangular rod grating lying on a glass substrate, as a function of the wavelength. Filling ratio 0.978.

Fig. 8
Fig. 8

Same as in Fig. 2, but for a cylindrical rod grating, r=d/2; solid circles are obtained with Eqs. (21) and (22). The continuation of c2 is made according to Eq. (B5).

Fig. 9
Fig. 9

Same as in Fig. 8, but for a metallic rod, r=d/4. Open triangles, continuation of c2 according to Eq. (B5); diamonds, continuation of c2 according to Eqs. (B7).

Fig. 10
Fig. 10

Same as in Fig. 9, but for r=d/2.

Fig. 11
Fig. 11

Convergence of the method on a highly conducting sinusoidal grating; refractive index 0.1+i20.

Fig. 12
Fig. 12

Geometrical parameters of an echelette profile.

Fig. 13
Fig. 13

Geometrical parameters of a trapezoidal profile.

Fig. 14
Fig. 14

Geometrical parameters of a cylindrical rod grating.

Equations (87)

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Hzi=A exp[i(α0x-β0y)],
Hz(x, y)=n=-+Hn(y)exp(iαnx),
div1k2(x, y)grad Hz+Hz=0.
E˜x(x, y)=1k2(x, y)Hzy,
E˜y(x, y)=-1k2(x, y)Hzx.
Hzy=k2(x, y)E˜x. 
y E˜x=-Hz+E˜yx=-Hz-x1k2Hzx.
dHndy=m=-+(k2)n-mE˜m,
dE˜ndy=αnm=-+αm1k2n-mHm-Hn.
hn=m=-+fn-mgm.
[h]=f[g].
hn(N)=m=-NNfn-mgm. 
hn(N)=m=-NN1f(N)n,m-1gm.
tx=cos θ,ty=sin θ,
Et=cos θEx+sin θEy,
En=-sin θEx+cos θEy.
c=cos θ,s=sin θ.
E˜x=cE˜t-sE˜n,
Ey=sEt+cEn,
k2E˜x=ck2E˜t-sk2E˜n.
[k2E˜x]=ck2[E˜t]-s1/k2-1[E˜n].
[E˜t]=c[E˜x]+s[E˜y].
[E˜n]=-s[E˜x]+c[E˜y].
[k2E˜x]=(ck2c+s1/k2-1s)[E˜x]+ck2s-s1/k2-1c[E˜y]. 
[k2E˜x]=k2[E˜x],
k2E˜x=k2cE˜t-k2sE˜n,
[k2E˜x]=k2[cE˜t]-1/k2-1[sE˜n].
[k2E˜x]=k2[c2E˜x+csE˜y]+1/k2-1[s2E˜x-csE˜y],
[k2E˜x]=(k2c2+1/k2-1s2)[E˜x]+(k2-1/k2-1)cs[E˜y].
Δ=(k2-1/k2-1),A=Δc2,B=Δcs.
[k2E˜x]=(A+1/k2-1)[E˜x]+B[E˜y].
[k2E˜x]=A+1k2-1[E˜x]+B-1k2Hzx. 
k2Ey=k2sEt+k2cEn,
[k2Ey]=k2[sEt]+1/k2-1[cEn].
[k2Ey]=k2[csEx+s2Ey]+1/k2-1[-csEx+c2Ey].
[k2Ey]=k2(cs[Ex]+s2[Ey])+1/k2-1×(-cs[Ex]+c2[Ey]),
[k2Ey]=B[Ex]+(k2-A)[Ey].
[Ey]=(k2-A)-1([k2Ey]-B[Ex]),
1k2Hzx=(k2-A)-1(iα[Hz]+B[E˜x]), 
Hzy=(A+1/k2-1-B(k2-A)-1B)[E˜x]-B(k2-A)-1iα[Hz],
E˜xy=α(k2-A)-1α[Hz]-[Hz]-iα(k2-A)-1B[E˜x].
Hzy=k2[E˜x], 
E˜xy=α1k2α[Hz]-[Hz]. 
Hzy=1k2-1[E˜x],
E˜xy=αk2-1α[Hz]-[Hz].
1Npn=1Npηn-η˜nηn2,
e=ck2,
en,m=p=-+cn-p(k2)p-m.
en,m=p=-+cn-m-(p-m)(k2)p-m.
en,m=p=-+cn-m-p(k2)p.
gn=r=-+kn-r2cr=s=-+cn-s(k2)s.
en,m=r=-+(k2)n-m-rcr.
en,m=t=-+(k2)n-m-t+mct-m=t=-+(k2)n-tct-m.
[k2c](N)=k2[c]=m=-N+Nkn-m2cm,
[k2c](N)=[ck2](N)=c[k2]=m=-N+Ncn-mkm2.
atpoint(x1, y):c=cos θB,s=-sin θB,
atpoint(x2, y):c=cos θ,s=sin θ.
c(x)=cos θB,x  [0, b],
=cos θ,x ]b, d],
s(x)=-sin θB,x [0, b],
=sin θ,x  ]b, d].
c(x)=cos θBx  [0, l],
=cos θ,x[l, d],
s(x)=-sin θB,x  ]0, l],
=sin θ,x  ]l, d].
tan θ=dfdx=-πadsin Kx.
c(x1)=1(1+tan2 θ)1/2=1[1+(π2a2)/d2sin2(Kx1)]1/2.
cos(Kx1)=2ya-1,
sin2(Kx1)=1-2ya-12=4ya1-ya.
c(x1) 1[1+K2y(a-y)]1/2c(x2).
s(x1)=tan θ(1+tan2θ)1/2=K[y(a-y)]1/2[1+K2y(a-y)]1/2=-s(x2).
c2(x1)=c2(x2)=11+K2y(a-y),
cs(x1)=-cs(x2)=K[y(a-y)]1/21+K2y(a-y),
1/(1+tan2 θ)=1/(1+f2)=1/1+π2a2d2sin2 Kx,
-πadsin(Kx)/1+π2a2d2sin2 Kx.
x=a2cos ϕ,
y=a2 (1+sin ϕ).
c(x1)=c(x2) a2sin ϕ  y-a2,
s(x1)=-s(x2)  -[y(a-y)]1/2.
c(x1)=c(x2)=(y-a/2)/y-a22+y(a-y)1/2=y-rr,
s(x1)=-s(x2)=-[y(a-y)]1/2/y-a22+y(a-y)1/2=x1r.
c2(x)=1-x2/r2,
cs(x)=±{c2(x)[1-c2(x)]}1/2.
c2(x)=(y-r)2/r2,
cs(x)=-sign(x)(y-r)[y(2r-y)]1/2/r2.
c2(x1)=c2(x2)=(y-r)2/r2,
c2(0)=c2(d/2)=c2(d)=1,|y-r|>2r/30,|y-r|<2r/3.

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