Abstract

The numerical performance of a finite-difference modal method for the analysis of one-dimensional lamellar gratings in a classical mounting is studied. The method is simple and relies on first-order finite difference in the grating to solve the Maxwell differential equations. The finite-difference scheme incorporates three features that accelerate the convergence performance of the method: (1) The discrete permittivity is interpolated at the lamellar boundaries, (2) mesh points are located on the permittivity discontinuities, and (3) a nonuniform sampling with increased resolution is performed near the discontinuities. Although the performance achieved with the present method remains inferior to that achieved with up-to-date grating theories such as rigorous coupled-wave analysis with adaptive spatial resolution, it is found that the present method offers rather good performance for metallic gratings operating in the visible and near-infrared regions of the spectrum, especially for TM polarization.

© 2000 Optical Society of America

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  1. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications a` l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
    [CrossRef]
  2. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permit-tivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  3. D. Maystre, “Integral method,” in Electromagnetic Theory of Gratings, R. Petit ed. (Springer-Verlag, Berlin, 1980), Chap. 3.
  4. E. Popov, B. Bozhkov, D. Maystre, J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47–55 (1999).
    [CrossRef]
  5. T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–936 (1985).
    [CrossRef]
  6. 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]
  7. Ph. 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]
  8. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  9. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  10. L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
    [CrossRef]
  11. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods (Oxford U. Press, Oxford, UK, 1985).
  12. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).
  13. S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
    [CrossRef]
  14. T. Delort, D. Maystre, “Finite-element method for gratings,” J. Opt. Soc. Am. A 10, 2592–2601 (1993).
    [CrossRef]
  15. H. Ichikawa, “Electromagnetic analysis of diffraction gratings by the finite-difference time-domain method,” J. Opt. Soc. Am. A 15, 152–157 (1998).
    [CrossRef]
  16. M. K. Moaveni, “Plane-wave diffraction by dielectric gratings, finite difference formulation,” IEEE Trans. Antennas Propag. 37, 1026–1031 (1989).
    [CrossRef]
  17. B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. (Bellingham) 33, 3518–3526 (1994).
    [CrossRef]
  18. K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
    [CrossRef]
  19. D. W. Prather, S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
    [CrossRef]
  20. R. Pregla, W. Pasher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.
  21. Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
    [CrossRef]
  22. J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett. 28, 1013–1015 (1992).
    [CrossRef]
  23. C. M. Kim, R. V. Ramaswamy, “Modeling of graded-index channel waveguides using nonuniform finite difference method,” J. Lightwave Technol. 7, 1581–1589 (1989).
    [CrossRef]
  24. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
    [CrossRef]
  25. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  26. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  27. Ph. Lalanne, M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for TM polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
    [CrossRef]
  28. See Chap. 10 in Ref. 12, for instance.
  29. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  30. C. Vassallo, “Improvement of finite difference methods for step-index optical waveguides,” IEE Proc. J: Optoelectron. 139, 137–142 (1992).
  31. H. J. W. M. Hoekstra, G. J. M. Krijnen, P. V. Lambeck, “Efficient interface conditions for the finite difference beam propagation method,” J. Lightwave Technol. 10, 1352–1355 (1992).
    [CrossRef]
  32. S. F. Helfert, R. Pregla, “Finite difference expressions for arbitrary positioned dielectrics steps in waveguide structures,” J. Lightwave Technol. 14, 2414–2421 (1996).
    [CrossRef]
  33. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993);R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alherhand, “Erratum: Accurate theoretical analysis of photonic band-gap materials [Phys. Rev. B 48, 8434 (1993)],” Phys. Rev. B 55, 15942 (1997).
    [CrossRef]

1999 (3)

1998 (2)

Ph. Lalanne, M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for TM polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[CrossRef]

H. Ichikawa, “Electromagnetic analysis of diffraction gratings by the finite-difference time-domain method,” J. Opt. Soc. Am. A 15, 152–157 (1998).
[CrossRef]

1996 (6)

1995 (2)

1994 (2)

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. (Bellingham) 33, 3518–3526 (1994).
[CrossRef]

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permit-tivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

1993 (2)

T. Delort, D. Maystre, “Finite-element method for gratings,” J. Opt. Soc. Am. A 10, 2592–2601 (1993).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993);R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alherhand, “Erratum: Accurate theoretical analysis of photonic band-gap materials [Phys. Rev. B 48, 8434 (1993)],” Phys. Rev. B 55, 15942 (1997).
[CrossRef]

1992 (4)

C. Vassallo, “Improvement of finite difference methods for step-index optical waveguides,” IEE Proc. J: Optoelectron. 139, 137–142 (1992).

H. J. W. M. Hoekstra, G. J. M. Krijnen, P. V. Lambeck, “Efficient interface conditions for the finite difference beam propagation method,” J. Lightwave Technol. 10, 1352–1355 (1992).
[CrossRef]

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett. 28, 1013–1015 (1992).
[CrossRef]

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

1991 (1)

Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

1989 (2)

M. K. Moaveni, “Plane-wave diffraction by dielectric gratings, finite difference formulation,” IEEE Trans. Antennas Propag. 37, 1026–1031 (1989).
[CrossRef]

C. M. Kim, R. V. Ramaswamy, “Modeling of graded-index channel waveguides using nonuniform finite difference method,” J. Lightwave Technol. 7, 1581–1589 (1989).
[CrossRef]

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–936 (1985).
[CrossRef]

1982 (1)

1974 (1)

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

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993);R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alherhand, “Erratum: Accurate theoretical analysis of photonic band-gap materials [Phys. Rev. B 48, 8434 (1993)],” Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Benish, D.

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett. 28, 1013–1015 (1992).
[CrossRef]

Bozhkov, B.

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993);R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alherhand, “Erratum: Accurate theoretical analysis of photonic band-gap materials [Phys. Rev. B 48, 8434 (1993)],” Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Chandezon, J.

Chew, W. C.

Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

Cornet, G.

Delort, T.

Dupuis, M. T.

Gallagher, N. C.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. (Bellingham) 33, 3518–3526 (1994).
[CrossRef]

Gaylord, T. K.

Gedney, S. D.

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

Gerdes, J.

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett. 28, 1013–1015 (1992).
[CrossRef]

Glytsis, E. N.

Granet, G.

Grann, E. B.

Guizal, B.

Helfert, S. F.

S. F. Helfert, R. Pregla, “Finite difference expressions for arbitrary positioned dielectrics steps in waveguide structures,” J. Lightwave Technol. 14, 2414–2421 (1996).
[CrossRef]

Hirayama, K.

Hoekstra, H. J. W. M.

H. J. W. M. Hoekstra, G. J. M. Krijnen, P. V. Lambeck, “Efficient interface conditions for the finite difference beam propagation method,” J. Lightwave Technol. 10, 1352–1355 (1992).
[CrossRef]

Hoose, J.

Ichikawa, H.

Joannopoulos, J. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993);R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alherhand, “Erratum: Accurate theoretical analysis of photonic band-gap materials [Phys. Rev. B 48, 8434 (1993)],” Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Jurek, M. P.

Ph. Lalanne, M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for TM polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[CrossRef]

Kim, C. M.

C. M. Kim, R. V. Ramaswamy, “Modeling of graded-index channel waveguides using nonuniform finite difference method,” J. Lightwave Technol. 7, 1581–1589 (1989).
[CrossRef]

Krijnen, G. J. M.

H. J. W. M. Hoekstra, G. J. M. Krijnen, P. V. Lambeck, “Efficient interface conditions for the finite difference beam propagation method,” J. Lightwave Technol. 10, 1352–1355 (1992).
[CrossRef]

Lalanne, Ph.

Ph. Lalanne, M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for TM polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[CrossRef]

Ph. 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]

Lambeck, P. V.

H. J. W. M. Hoekstra, G. J. M. Krijnen, P. V. Lambeck, “Efficient interface conditions for the finite difference beam propagation method,” J. Lightwave Technol. 10, 1352–1355 (1992).
[CrossRef]

Lee, J. F.

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

Li, L.

Lichtenberg, B.

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. (Bellingham) 33, 3518–3526 (1994).
[CrossRef]

Liu, Q. H.

Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

Lunitz, B.

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett. 28, 1013–1015 (1992).
[CrossRef]

Maystre, D.

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993);R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alherhand, “Erratum: Accurate theoretical analysis of photonic band-gap materials [Phys. Rev. B 48, 8434 (1993)],” Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Mittra, R.

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

Moaveni, M. K.

M. K. Moaveni, “Plane-wave diffraction by dielectric gratings, finite difference formulation,” IEEE Trans. Antennas Propag. 37, 1026–1031 (1989).
[CrossRef]

Moharam, M. G.

Montiel, F.

Morris, G. M.

Nevière, M.

Pasher, W.

R. Pregla, W. Pasher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

Petit, R.

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

Pommet, D. A.

Popov, E.

Prather, D. W.

Pregla, R.

S. F. Helfert, R. Pregla, “Finite difference expressions for arbitrary positioned dielectrics steps in waveguide structures,” J. Lightwave Technol. 14, 2414–2421 (1996).
[CrossRef]

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett. 28, 1013–1015 (1992).
[CrossRef]

R. Pregla, W. Pasher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

Ramaswamy, R. V.

C. M. Kim, R. V. Ramaswamy, “Modeling of graded-index channel waveguides using nonuniform finite difference method,” J. Lightwave Technol. 7, 1581–1589 (1989).
[CrossRef]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993);R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alherhand, “Erratum: Accurate theoretical analysis of photonic band-gap materials [Phys. Rev. B 48, 8434 (1993)],” Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Shi, S.

Smith, G. D.

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods (Oxford U. Press, Oxford, UK, 1985).

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).

Vassallo, C.

C. Vassallo, “Improvement of finite difference methods for step-index optical waveguides,” IEE Proc. J: Optoelectron. 139, 137–142 (1992).

Vincent, P.

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

Wilson, D. W.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Appl. Opt. (1)

Electron. Lett. (1)

J. Gerdes, B. Lunitz, D. Benish, R. Pregla, “Analysis of slab waveguide discontinuities including radiation and absorption effects,” Electron. Lett. 28, 1013–1015 (1992).
[CrossRef]

IEE Proc. J: Optoelectron. (1)

C. Vassallo, “Improvement of finite difference methods for step-index optical waveguides,” IEE Proc. J: Optoelectron. 139, 137–142 (1992).

IEEE Trans. Antennas Propag. (2)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

M. K. Moaveni, “Plane-wave diffraction by dielectric gratings, finite difference formulation,” IEEE Trans. Antennas Propag. 37, 1026–1031 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[CrossRef]

Q. H. Liu, W. C. Chew, “Analysis of discontinuities in planar dielectric waveguides: an eigenmode propagation method,” IEEE Trans. Microwave Theory Tech. 39, 422–430 (1991).
[CrossRef]

J. Lightwave Technol. (3)

C. M. Kim, R. V. Ramaswamy, “Modeling of graded-index channel waveguides using nonuniform finite difference method,” J. Lightwave Technol. 7, 1581–1589 (1989).
[CrossRef]

H. J. W. M. Hoekstra, G. J. M. Krijnen, P. V. Lambeck, “Efficient interface conditions for the finite difference beam propagation method,” J. Lightwave Technol. 10, 1352–1355 (1992).
[CrossRef]

S. F. Helfert, R. Pregla, “Finite difference expressions for arbitrary positioned dielectrics steps in waveguide structures,” J. Lightwave Technol. 14, 2414–2421 (1996).
[CrossRef]

J. Mod. Opt. (1)

Ph. Lalanne, M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for TM polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (12)

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
[CrossRef]

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permit-tivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

T. Delort, D. Maystre, “Finite-element method for gratings,” J. Opt. Soc. Am. A 10, 2592–2601 (1993).
[CrossRef]

H. Ichikawa, “Electromagnetic analysis of diffraction gratings by the finite-difference time-domain method,” J. Opt. Soc. Am. A 15, 152–157 (1998).
[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 transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

Ph. 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 lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

K. Hirayama, E. N. Glytsis, T. K. Gaylord, D. W. Wilson, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

D. W. Prather, S. Shi, “Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffractive optical elements,” J. Opt. Soc. Am. A 16, 1131–1142 (1999).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
[CrossRef]

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

Nouv. Rev. Opt. (1)

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

Opt. Eng. (Bellingham) (1)

B. Lichtenberg, N. C. Gallagher, “Numerical modeling of diffractive devices using the finite element method,” Opt. Eng. (Bellingham) 33, 3518–3526 (1994).
[CrossRef]

Phys. Rev. B (1)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993);R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alherhand, “Erratum: Accurate theoretical analysis of photonic band-gap materials [Phys. Rev. B 48, 8434 (1993)],” Phys. Rev. B 55, 15942 (1997).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–936 (1985).
[CrossRef]

Other (5)

D. Maystre, “Integral method,” in Electromagnetic Theory of Gratings, R. Petit ed. (Springer-Verlag, Berlin, 1980), Chap. 3.

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods (Oxford U. Press, Oxford, UK, 1985).

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995).

R. Pregla, W. Pasher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

See Chap. 10 in Ref. 12, for instance.

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Figures (8)

Fig. 1
Fig. 1

Parameter definition for the classical grating diffraction problems considered in this paper.

Fig. 2
Fig. 2

Grating period discretization.

Fig. 3
Fig. 3

Uniform discretization: computational error for the reflected zeroth order of a metallic grating: (a) TM, (b) TE. The values of the parameters are: θ=30°, f=0.5, n1=n=1, n3=n=0.22+6.71i, and λ=Λ=h=1.0 μm. The exact value of the reflected zeroth order is 84.848% for TM and 73.428% for TE. Pluses, present method; circles, RCWA.

Fig. 4
Fig. 4

Step-by-step procedure for the generation of the discrete sampling points. As in Fig. 2, crosses and circles correspond to xi and xi point locations, respectively.

Fig. 5
Fig. 5

Nonuniform discretization (r=1). Minus-first reflected order for the grating considered in Ref. 27. The values of the parameters are θ=arcsin(λ/2/Λ), f=0.57, n1=n=1, n3=n=1+40i, λ=1 μm, Λ=1.2361 μm, and h=0.4Λ. (a) TM polarization, (b) TE polarization. Pulses, present method; circles, RCWA.

Fig. 6
Fig. 6

TM polarization and nonuniform discretization (r=0.5). Computational error for the transmitted minus-first order of a dielectric grating. The dashed curve represents the deviation to the energy conservation obtained with the present method. The values of the parameters are θ=30°, f=0.234, n1=n=1, n3=1.5, n=2.3, λ=1 μm, Λ=2 μm, and h=1 μm. The exact value for the diffraction efficiency is 51.062%. Pluses, present method; circles, RCWA.

Fig. 7
Fig. 7

Same as in Fig. 3(a) with an ideal lossless metal (n=n3=6.71i) and a nonuniform discretization (r=0.5).

Fig. 8
Fig. 8

Notation for the discretization at an interface between two media with relative permittivities 1 and 2. The interface between the media is perpendicular to the x-grid coordinate.

Tables (2)

Tables Icon

Table 1 Reflected Zero-Order Efficiencies for Various Truncation Orders N and for the Grating of Fig. 3 (TM Polarization)

Tables Icon

Table 2 Reflected Zero-Order Efficiencies for Various Truncation Orders N and for the Grating of Fig. 3 (TE Polarization)

Equations (48)

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Ψinc=exp{-jk0n1[sin(θ)x+cos(θ)z]}.
Ψ1=Ψinc+iRi exp[-j(kxix-k1,ziz)],
Ψ3=iTi exp{-j[kxix+k3,zi(z-h)]},
2Ψz2+ x 1 Ψx+k02Ψ=0.
2Ψz2+EΨ=0,
E=1(D22-1D1+k02I).
1(i, i)=11/(x)[xi-1;xi],
2(i, i)=(x)[xi;xi+1].
D1=h1-1(1)h1-1(2)h1-1(N)×-11-1-11α-1,
D2=h2-1(1)h2-1(2)h2-1(N)×1-α-111-11.
Ψ(z)=m=1NWm{cm+ exp(-λmz)+cm- exp[λm(z-h)]},
recti(x)=1ifxi<x<xi+10otherwise.
exp(-jk0n1 sin θx)+iRi exp(-jkxix)
p=1Nm=1Nwm,p rectp (x)[cm++cm- exp(-λmh)],
-jk0 cos θn1 exp(-jk0n1 sin θx)+i jk1,zi1 Ri exp(-jkxix)
p=1Nm=1Nλmvm,p rectp (x)[-cm++cm- exp(-λmh)],
iTi exp(-jkxix)
p=1Nm=1Nwm,p rectp(x)[cm+ exp(-λmh)+cm-],
i -jk3,zi3 Ti exp(-jkxix)
p=1Nm=1Nλmvm,p rectp(x)[-cm+ exp(-λmh)+cm-].
pi,m=exp(jkxmxi){exp[jkxm(xi-xi)]-exp[jkxm(xi-1-xi)]}/(jkxmΛ).
δi,0jδi,0k0 cos θ/n1 + I-jZ1R=PWPWXPV-PVX c+c-,
IjZ2T=PWXPWPVX-PV c+c-,
DEri=RiRi* Re(k1.zi/k0n1 cos θ),
DEti=TiTi* Re(k3,zin1/k0n32 cos θ).
2Ψz2+EΨ=0,
whereE=D2D1+k023.
3(i,i)=(x)[xi-1;xi].
δi,0jδi,0k0n1 cos θ + I-jY1R=PWPWXPV-PVX c+c-,
IjY2T=PWXPWPVX-PV c+c-,
DEri=RiRi* Re(k1,zi/k0n1 cos θ),
DEti=TiTi* Re(k3,zi/k0n1 cos θ).
Ψ1(x, z)=exp(-jqz)[α1 exp(-jrx)+β1 exp(jrx)].
Ψ2(x, z)=exp(-jqz)[α2 exp(-jrx)+β2 exp(jrx)],
r2+q2=1k2,r2 + q2=2k2.
α1+β1=α2+β2,r1 (α1-β1)=r2 (α2-β2).
jω0Ez=xΨ,
jω0Ex=-zΨ,
jωμ0Ψ=xEz-zEx.
Ez1=-rω01 exp(-jqz)[α1 exp(-jrx)-β1 exp(jrx)],
Ez2=-rω02 exp(-jqz)[α2 exp(-jrx)-β2 exp(jrx)],
jωizEz,i=1h1 (Ψi+1-Ψi),
iz r2 (β2-α2)h1=(β2+α2-β1-α1)+r(1-f1)(β2-α2)+rf1(β1-α1)h+O(h12),
iz=f11+(1-f1)2.
ixk2Ψi+jω0h2 (Ez,i-Ez,i-1)=q2Ψi,
1xjr1 (α1-β1)-r2 (α2-β2)+h2k2(α2+β2)-r21 (α1+β1)f2-r22 (α2+β2)(1-f2)
=q2(α2+β2)h2+O(h22).
ix=1f2/1+(1-f2)/2,

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