Abstract

Three-dimensional vectorial diffraction analysis of gratings is presented based on Legendre polynomial expansion of electromagnetic fields. In contrast to conventional rigorous coupled wave analysis (RCWA) in which the solution is obtained using state variables representation of the coupled wave amplitudes, here the solution of first-order coupled Maxwell’s equations is expanded in terms of Legendre polynomials, where Maxwell’s equations are analytically projected in the Hilbert space spanned by Legendre polynomials. This approach yields well-behaved algebraic equations for deriving diffraction efficiencies and electromagnetic field profiles without facing the problem of numerical instability. The proposed approach can be applied in the analysis of two cases: first, arbitrarily oriented planar gratings with slanted yet homogeneous fringes; second, nonslanted but longitudinally inhomogeneous gratings. The method is then applied to various test cases within the above-mentioned two categories, comparison to other methods already reported in the literature is made, and the presented approach is justified. Different aspects of the proposed method such as numerical stability and convergence rate are also investigated. Special attention is given to how the resonant frequency of frequency selective structures varies with introducing tilt angles and/or longitudinal inhomogenity in the permittivity profile.

© 2007 Optical Society of America

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  1. T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
    [CrossRef]
  2. R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).
    [CrossRef]
  3. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Dekker, 1997).
  4. S. Peng and G. M. Morris, "Efficient implementation of rigorous coupled wave analysis for surface relief gratings," J. Opt. Soc. Am. A 12, 1087-1096 (1995).
    [CrossRef]
  5. M. G. Moharam, D. A. Pommet, E. B. Grann, and 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]
  6. L. Li and C. W. Haggans, "Convergence of the coupled wave method for metallic lamellar diffraction gratings," J. Opt. Soc. Am. A 10, 1184-1189 (1993).
    [CrossRef]
  7. M. Neviere and E. Popov, Light Propagation in Periodic Media (Dekker, 2002).
  8. M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 71, 811-818 (1981).
    [CrossRef]
  9. M. G. Moharam and T. K. Gaylord, "Three-dimensional vector coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 73, 1105-1112 (1983).
    [CrossRef]
  10. E. N. Glytsis and T. K. Gaylord, "Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction," J. Opt. Soc. Am. A 7, 1399-1420 (1990).
    [CrossRef]
  11. M. G. Moharam, E. B. Grann, D. A. Pommet, and 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]
  12. E. N. Glytsis and T. K. Gaylord, "Rigorous 3-D coupled wave diffraction analysis of multiple superposed gratings in anisotropic media," Appl. Opt. 28, 2401-2421 (1989).
    [CrossRef] [PubMed]
  13. E. J. Restall and A. C. Walker, "Rigorous coupled-wave method applied to fan-out gratings," IEE Proc.: Optoelectron. 145, 165-169 (1998).
    [CrossRef]
  14. 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]
  15. R. C. Hall, R. Mittra, and K. M. Mitzner, "Analysis of multilayered periodic structures using generalized scattering matrix theory," IEEE Trans. Antennas Propag. 36, 511-517 (1988).
    [CrossRef]
  16. O. Forslund and S. He, "Electromagnetic scattering from an inhomogeneous grating using a wave-splitting approach," Prog. Electromagn. Res. 19, 147-171 (1998).
    [CrossRef]
  17. 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]
  18. L. Li, "Exponentially convergent and numerically efficient solution of Maxwell's equations for lamellar gratings: comment," J. Opt. Soc. Am. A 13, 541-542 (1996).
    [CrossRef]
  19. M. Chamanzar, K. Mehrany, and B. Rashidian, "Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields," IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
    [CrossRef]
  20. Ph. Lalanne and G. M. Morris, "Highly improved convergence of the coupled wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  21. G. Garnet and 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]
  22. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  23. D. Maystre, "Integral methods" in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980).
    [CrossRef]
  24. S. L. Chaung and J. A. Kong, "Wave scattering from a periodic dielectric surface for a general angle of incidence," Radio Sci. 17, 545-557 (1982).
    [CrossRef]
  25. P. Cornet, J. Chandezon, and C. Faure, "Conical diffraction of a plane wave by an inclined parallel-plate grating," J. Opt. Soc. Am. A 14, 437-449 (1997).
    [CrossRef]
  26. L. Mashev and E. Popov, "Reflection gratings in conical diffraction mounting," J. Opt. (Paris) 18, 3-8 (1987).
    [CrossRef]
  27. E. Popov and L. Mashev, "Conical diffraction mounting generalization of a rigorous differential method," J. Opt. (Paris) 17, 175-180 (1986).
    [CrossRef]
  28. D. Lacour, G. Granet, J. Plumey, and A. Mure-Ravaud, "Polarization independence of a one-dimensional grating in conical mounting," J. Opt. Soc. Am. A 20, 1546-1552 (2003).
    [CrossRef]
  29. K. Mehrany and B. Rashidian, "Polynomial expansion for extraction of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).
    [CrossRef]
  30. A. F. Nikiforov, Special Functions of Mathematical Physics: a Unified Introduction with Applications (Birkhauser, 1988).
  31. M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis, " in Proceedings of the 2004 International Symposium on Antennas and Propagation (IEEE, 2004), 161-164.
  32. B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
    [CrossRef]
  33. E. Popov and M. Neviere, "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]
  34. M. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B 23, 969-977 (2006).
    [CrossRef]
  35. H. L. Bertoni, L. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
    [CrossRef]
  36. A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria, "Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method," IEEE Trans. Antennas Propag. 52, 2091-2099 (2004).
    [CrossRef]

2006 (2)

M. Chamanzar, K. Mehrany, and B. Rashidian, "Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields," IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B 23, 969-977 (2006).
[CrossRef]

2004 (1)

A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria, "Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method," IEEE Trans. Antennas Propag. 52, 2091-2099 (2004).
[CrossRef]

2003 (2)

2001 (2)

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

E. Popov and M. Neviere, "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]

1998 (2)

E. J. Restall and A. C. Walker, "Rigorous coupled-wave method applied to fan-out gratings," IEE Proc.: Optoelectron. 145, 165-169 (1998).
[CrossRef]

O. Forslund and S. He, "Electromagnetic scattering from an inhomogeneous grating using a wave-splitting approach," Prog. Electromagn. Res. 19, 147-171 (1998).
[CrossRef]

1997 (1)

1996 (5)

1995 (4)

1993 (1)

1990 (1)

1989 (2)

H. L. Bertoni, L. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

E. N. Glytsis and T. K. Gaylord, "Rigorous 3-D coupled wave diffraction analysis of multiple superposed gratings in anisotropic media," Appl. Opt. 28, 2401-2421 (1989).
[CrossRef] [PubMed]

1988 (1)

R. C. Hall, R. Mittra, and K. M. Mitzner, "Analysis of multilayered periodic structures using generalized scattering matrix theory," IEEE Trans. Antennas Propag. 36, 511-517 (1988).
[CrossRef]

1987 (1)

L. Mashev and E. Popov, "Reflection gratings in conical diffraction mounting," J. Opt. (Paris) 18, 3-8 (1987).
[CrossRef]

1986 (1)

E. Popov and L. Mashev, "Conical diffraction mounting generalization of a rigorous differential method," J. Opt. (Paris) 17, 175-180 (1986).
[CrossRef]

1985 (1)

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

1983 (1)

1982 (1)

S. L. Chaung and J. A. Kong, "Wave scattering from a periodic dielectric surface for a general angle of incidence," Radio Sci. 17, 545-557 (1982).
[CrossRef]

1981 (1)

Andrés, M. V.

A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria, "Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method," IEEE Trans. Antennas Propag. 52, 2091-2099 (2004).
[CrossRef]

Bertoni, H. L.

H. L. Bertoni, L. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

Boria, V. E.

A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria, "Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method," IEEE Trans. Antennas Propag. 52, 2091-2099 (2004).
[CrossRef]

Chamanzar, M.

M. Chamanzar, K. Mehrany, and B. Rashidian, "Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields," IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B 23, 969-977 (2006).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis, " in Proceedings of the 2004 International Symposium on Antennas and Propagation (IEEE, 2004), 161-164.

Chandezon, J.

Chaung, S. L.

S. L. Chaung and J. A. Kong, "Wave scattering from a periodic dielectric surface for a general angle of incidence," Radio Sci. 17, 545-557 (1982).
[CrossRef]

Cheo, L. S.

H. L. Bertoni, L. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

Chernov, B.

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

Cornet, P.

Coves, A.

A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria, "Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method," IEEE Trans. Antennas Propag. 52, 2091-2099 (2004).
[CrossRef]

Faure, C.

Forslund, O.

O. Forslund and S. He, "Electromagnetic scattering from an inhomogeneous grating using a wave-splitting approach," Prog. Electromagn. Res. 19, 147-171 (1998).
[CrossRef]

Garnet, G.

Gaylord, T. K.

Gil, J.

A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria, "Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method," IEEE Trans. Antennas Propag. 52, 2091-2099 (2004).
[CrossRef]

Gimeno, B.

A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria, "Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method," IEEE Trans. Antennas Propag. 52, 2091-2099 (2004).
[CrossRef]

Glytsis, E. N.

Granet, G.

Grann, E. B.

Guizal, B.

Haggans, C. W.

Hall, R. C.

R. C. Hall, R. Mittra, and K. M. Mitzner, "Analysis of multilayered periodic structures using generalized scattering matrix theory," IEEE Trans. Antennas Propag. 36, 511-517 (1988).
[CrossRef]

He, S.

O. Forslund and S. He, "Electromagnetic scattering from an inhomogeneous grating using a wave-splitting approach," Prog. Electromagn. Res. 19, 147-171 (1998).
[CrossRef]

Kong, J. A.

S. L. Chaung and J. A. Kong, "Wave scattering from a periodic dielectric surface for a general angle of incidence," Radio Sci. 17, 545-557 (1982).
[CrossRef]

Lacour, D.

Lalanne, Ph.

Li, L.

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Dekker, 1997).

Mashev, L.

L. Mashev and E. Popov, "Reflection gratings in conical diffraction mounting," J. Opt. (Paris) 18, 3-8 (1987).
[CrossRef]

E. Popov and L. Mashev, "Conical diffraction mounting generalization of a rigorous differential method," J. Opt. (Paris) 17, 175-180 (1986).
[CrossRef]

Maystre, D.

D. Maystre, "Integral methods" in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980).
[CrossRef]

Mehrany, K.

M. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B 23, 969-977 (2006).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields," IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

K. Mehrany and B. Rashidian, "Polynomial expansion for extraction of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis, " in Proceedings of the 2004 International Symposium on Antennas and Propagation (IEEE, 2004), 161-164.

Mittra, R.

R. C. Hall, R. Mittra, and K. M. Mitzner, "Analysis of multilayered periodic structures using generalized scattering matrix theory," IEEE Trans. Antennas Propag. 36, 511-517 (1988).
[CrossRef]

Mitzner, K. M.

R. C. Hall, R. Mittra, and K. M. Mitzner, "Analysis of multilayered periodic structures using generalized scattering matrix theory," IEEE Trans. Antennas Propag. 36, 511-517 (1988).
[CrossRef]

Moharam, M. G.

Morf, R. H.

Morris, G. M.

Mure-Ravaud, A.

Neviere, M.

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

E. Popov and M. Neviere, "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]

M. Neviere and E. Popov, Light Propagation in Periodic Media (Dekker, 2002).

Nikiforov, A. F.

A. F. Nikiforov, Special Functions of Mathematical Physics: a Unified Introduction with Applications (Birkhauser, 1988).

Peng, S.

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).
[CrossRef]

Plumey, J.

Pommet, D. A.

Popov, E.

E. Popov and M. Neviere, "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]

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

L. Mashev and E. Popov, "Reflection gratings in conical diffraction mounting," J. Opt. (Paris) 18, 3-8 (1987).
[CrossRef]

E. Popov and L. Mashev, "Conical diffraction mounting generalization of a rigorous differential method," J. Opt. (Paris) 17, 175-180 (1986).
[CrossRef]

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Dekker, 1997).

M. Neviere and E. Popov, Light Propagation in Periodic Media (Dekker, 2002).

Rashidian, B.

M. Chamanzar, K. Mehrany, and B. Rashidian, "Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields," IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B 23, 969-977 (2006).
[CrossRef]

K. Mehrany and B. Rashidian, "Polynomial expansion for extraction of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis, " in Proceedings of the 2004 International Symposium on Antennas and Propagation (IEEE, 2004), 161-164.

Restall, E. J.

E. J. Restall and A. C. Walker, "Rigorous coupled-wave method applied to fan-out gratings," IEE Proc.: Optoelectron. 145, 165-169 (1998).
[CrossRef]

San Blas, A. A.

A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria, "Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method," IEEE Trans. Antennas Propag. 52, 2091-2099 (2004).
[CrossRef]

Tamir, T.

H. L. Bertoni, L. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

Walker, A. C.

E. J. Restall and A. C. Walker, "Rigorous coupled-wave method applied to fan-out gratings," IEE Proc.: Optoelectron. 145, 165-169 (1998).
[CrossRef]

Appl. Opt. (1)

IEE Proc.: Optoelectron. (1)

E. J. Restall and A. C. Walker, "Rigorous coupled-wave method applied to fan-out gratings," IEE Proc.: Optoelectron. 145, 165-169 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

R. C. Hall, R. Mittra, and K. M. Mitzner, "Analysis of multilayered periodic structures using generalized scattering matrix theory," IEEE Trans. Antennas Propag. 36, 511-517 (1988).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields," IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

H. L. Bertoni, L. S. Cheo, and T. Tamir, "Frequency-selective reflection and transmission by a periodic dielectric layer," IEEE Trans. Antennas Propag. 37, 78-83 (1989).
[CrossRef]

A. Coves, B. Gimeno, J. Gil, M. V. Andrés, A. A. San Blas, and V. E. Boria, "Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method," IEEE Trans. Antennas Propag. 52, 2091-2099 (2004).
[CrossRef]

J. Opt. (Paris) (2)

L. Mashev and E. Popov, "Reflection gratings in conical diffraction mounting," J. Opt. (Paris) 18, 3-8 (1987).
[CrossRef]

E. Popov and L. Mashev, "Conical diffraction mounting generalization of a rigorous differential method," J. Opt. (Paris) 17, 175-180 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (14)

E. Popov and M. Neviere, "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]

D. Lacour, G. Granet, J. Plumey, and A. Mure-Ravaud, "Polarization independence of a one-dimensional grating in conical mounting," J. Opt. Soc. Am. A 20, 1546-1552 (2003).
[CrossRef]

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]

M. G. Moharam, E. B. Grann, D. A. Pommet, and 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]

M. G. Moharam, D. A. Pommet, E. B. Grann, and 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]

S. Peng and G. M. Morris, "Efficient implementation of rigorous coupled wave analysis for surface relief gratings," J. Opt. Soc. Am. A 12, 1087-1096 (1995).
[CrossRef]

P. Cornet, J. Chandezon, and C. Faure, "Conical diffraction of a plane wave by an inclined parallel-plate grating," J. Opt. Soc. Am. A 14, 437-449 (1997).
[CrossRef]

E. N. Glytsis and T. K. Gaylord, "Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction," J. Opt. Soc. Am. A 7, 1399-1420 (1990).
[CrossRef]

L. Li and C. W. Haggans, "Convergence of the coupled wave method for metallic lamellar diffraction gratings," J. Opt. Soc. Am. A 10, 1184-1189 (1993).
[CrossRef]

L. Li, "Exponentially convergent and numerically efficient solution of Maxwell's equations for lamellar gratings: comment," J. Opt. Soc. Am. A 13, 541-542 (1996).
[CrossRef]

Ph. Lalanne and 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. Garnet and 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]

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]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

B. Chernov, M. Neviere, and E. Popov, "Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings," Opt. Commun. 194, 289-297 (2001).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

Prog. Electromagn. Res. (1)

O. Forslund and S. He, "Electromagnetic scattering from an inhomogeneous grating using a wave-splitting approach," Prog. Electromagn. Res. 19, 147-171 (1998).
[CrossRef]

Radio Sci. (1)

S. L. Chaung and J. A. Kong, "Wave scattering from a periodic dielectric surface for a general angle of incidence," Radio Sci. 17, 545-557 (1982).
[CrossRef]

Other (6)

D. Maystre, "Integral methods" in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980).
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings (Springer, 1980).
[CrossRef]

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Dekker, 1997).

M. Neviere and E. Popov, Light Propagation in Periodic Media (Dekker, 2002).

A. F. Nikiforov, Special Functions of Mathematical Physics: a Unified Introduction with Applications (Birkhauser, 1988).

M. Chamanzar, K. Mehrany, and B. Rashidian, "Polynomial expansion of electromagnetic fields for grating diffraction analysis, " in Proceedings of the 2004 International Symposium on Antennas and Propagation (IEEE, 2004), 161-164.

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Figures (14)

Fig. 1
Fig. 1

General form of a slanted grating in conical mounting.

Fig. 2
Fig. 2

Transmitted diffraction efficiency D E 30 versus the number of polynomial terms at different values of the normalized thicknesses d Λ G : triangles, d Λ G = 1 ; circles, d Λ G = 2 ; squares, d Λ G = 5 .

Fig. 3
Fig. 3

Computation time versus M, i.e., number of retained polynomial basis functions, solid line, standard RCWA for a homogeneous grating; squares, the proposed method for a homogeneous grating; diamonds, the proposed method for a longitudinally inhomogeneous grating.

Fig. 4
Fig. 4

Computation time versus N, i.e., number of retained space harmonics, circles: standard RCWA for a homogeneous grating; squares, the proposed method for a homogeneous grating; diamonds, the proposed method for a longitudinally inhomogeneous grating.

Fig. 5
Fig. 5

Dotted curve represents the error in computing the TM-polarized reflected diffraction efficiency D E 10 for metallic lamellar grating computed by employing our method, where N = 41 space harmonics are retained. The solid line represents the error of RCWA with the same number of space harmonics N = 41 .

Fig. 6
Fig. 6

Dotted curve represents the error in computing the TM-polarized reflected diffraction efficiency D E 11 for metallic lamellar grating computed by employing our proposed method, where N = 41 space harmonics are retained. The solid line represents the error of RCWA with the same number of space harmonics N = 41 .

Fig. 7
Fig. 7

Dotted curve represents the error in computing the TM-polarized reflected diffraction efficiency D E 10 for metallic lamellar grating computed by employing our proposed method, where M = 9 polynomial terms are retained. The solid line represents the error of RCWA with the same number of space harmonics N = 41 .

Fig. 8
Fig. 8

Dotted curve represents the error in computing the TM-polarized reflected diffraction efficiency D E 11 for metallic lamellar grating computed by employing our proposed method, where M = 9 polynomial terms are retained. The solid line represents the error of RCWA with the same number of space harmonics N = 41 .

Fig. 9
Fig. 9

Dotted and solid curves, respectively, represent the reflected diffraction efficiencies D E 10 and D E 11 , for metallic lamellar grating in conical mounting computed by employing our proposed method, where M = 9 polynomial terms are retained.

Fig. 10
Fig. 10

Binary grating in conical mounting.

Fig. 11
Fig. 11

Reflectance versus the normalized frequency for the dielectric frequency selective structure with f = 0.5 , ε I = ε I I I = 1 , ε g r o o v e = 1.44 , ε r i d g e = 2.56 , d = 1.713 Λ G , and α = 45 ° for in plane incidence.

Fig. 12
Fig. 12

Variation of resonant frequency in longitudinally homogeneous FSS by introducing tilt angle, solid curve, δ = 0 ° ; dashed curve, δ = 3 ° ; and dotted curve, δ = 5 ° .

Fig. 13
Fig. 13

Longitudinal permittivity profile of the ridges in the inhomogeneous frequency selective structure.

Fig. 14
Fig. 14

Variation of resonant frequency in longitudinally inhomogeneous FSS by introducing tilt angle, solid curve, δ = 0 ° ; dashed curve, δ = 3 ° ; and dotted curve, δ = 5 ° .

Tables (1)

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Table 1 Resonant Frequency as a Function of Tilt Angle for the Frequency Selective Structure a

Equations (56)

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ε ( r + Λ G ) = ε ( r ) ,
K G = 2 π Λ G [ sin ( ϕ ) x ̂ + cos ( ϕ ) z ̂ ] = K G x x ̂ + K G z z ̂ .
E 1 = u ̑ e j k 10 r + i = + R i e j k 1 i r ,
E 3 = i = + T i e j k 3 i ( r d z ̂ ) ,
k l x i = k x i = k 1 sin α cos δ i K G sin ϕ ,
k l y i = k y = k 1 sin α sin δ ,
k l z i = k l 2 k l x i 2 k y 2
u ̑ = ( cos ψ cos α cos δ sin ψ sin δ ) x ̂ + ( cos ψ cos α sin δ + sin ψ cos δ ) y ̂ cos ψ sin α z ̂ .
× E = j ω μ H ,
× H = j ω ε 0 ε ( x , z ) E .
E = i [ S x i ( z ) x ̂ + S y i ( z ) y ̂ + S z i ( z ) z ̂ ] exp [ j σ i . r ] ,
H = ε 0 μ 0 i [ U x i ( z ) x ̂ + U y i ( z ) y ̂ + U z i ( z ) z ̂ ] exp [ j σ i . r ] ,
ε ( x , z ) = h ε h e j h K G r ,
1 ε ( x , z ) = h ε h 1 e j h K G r ,
d S x i ( z ) d z = j { i K G z S x i ( z ) + k x i k 0 p [ [ ε ] ] i p 1 [ k y U x p ( z ) k x p U y p ( z ) ] + k 0 U y i ( z ) } ,
d S y i ( z ) d z = j { i K G z S y i ( z ) + k y k 0 p [ [ ε ] ] i p 1 [ k y U x p ( z ) k x p U y p ( z ) ] k 0 U x i ( z ) } ,
d U x i ( z ) d z = j { i K G z U x i ( z ) + k x i k 0 [ k y S x i ( z ) k x i S y i ( z ) ] + k 0 p [ [ ε ] ] i p S y p ( z ) } ,
d U y i ( z ) d z = j { i K G z U y i ( z ) k y k 0 [ k y S x i ( z ) k x i S y i ( z ) ] + k 0 p [ [ 1 ε ] ] i p 1 S x p ( z ) } .
S x i ( z ) = m = 0 + q m i P m ( ξ ) ,
S y i ( z ) = m = 0 + h m i P m ( ξ ) ,
U x i ( z ) = m = 0 + t m i P m ( ξ ) ,
U y i ( z ) = m = 0 + l m i P m ( ξ ) ,
m q m i d d z P m ( ξ ) + j m P m ( ξ ) { i K G z q m i + k x i k 0 p [ [ ε ] ] i p 1 [ k y t m p k x p l m p ] + k 0 l m i } = 0 ,
m h m i d d z P m ( ξ ) + j m P m ( ξ ) { i K G z h m i + k y k 0 p [ [ ε ] ] i p 1 [ k y t m p k x p l m p ] k 0 t m i } = 0 ,
m t m i d d z P m ( ξ ) + j m P m ( ξ ) { i K G z t m i k x i k 0 [ k y q m i k x i h m i ] k 0 p [ [ ε ] ] i p h m p } = 0 ,
m l m i d d z P m ( ξ ) + j m P m ( ξ ) { i K G z l m i k y k 0 [ k y q m i k x i h m i ] + k 0 p [ [ 1 ε ] ] i p 1 q m p } = 0 .
[ [ ε ] ] ( ξ ) [ a 0 ] + [ a 1 ] ξ + [ a 2 ] ξ 2 + [ a 3 ] ξ 3 + + [ a N 1 ] ξ N 1 ,
[ [ ε ] ] 1 ( ξ ) [ b 0 ] + [ b 1 ] ξ + [ b 2 ] ξ 2 + [ b 3 ] ξ 3 + + [ b N 1 ] ξ N 1 ,
[ [ 1 ε ] ] 1 ( ξ ) [ c 0 ] + [ c 1 ] ξ + [ c 2 ] ξ 2 + [ c 3 ] ξ 3 + + [ c N 1 ] ξ N 1 ,
ξ P m ( ξ ) = m + 1 2 m + 1 P m + 1 ( ξ ) + m 2 m + 1 P m 1 ( ξ ) .
m = 0 + ξ q m P m ( ξ ) = m = 0 + χ m P m ( ξ ) ,
χ m = m 2 m 1 q m 1 + m + 1 2 m + 3 q m + 1 .
[ χ ¯ m ] = [ χ ] [ q ¯ m ] .
2 d [ q ¯ m i ] + j { i K G z [ q ¯ m i ] + k x i k 0 p n [ b n ] i p [ χ ] n ( k y [ t ¯ m p ] k x p [ l ¯ m p ] ) + k 0 [ l ¯ m i ] } = 0 ,
2 d [ h ¯ m i ] + j { i K G z [ h ¯ m i ] + k y k 0 p n [ b n ] i p [ χ ] n ( k y [ t ¯ m p ] k x p [ l ¯ m p ] ) k 0 [ t ¯ m i ] } = 0 ,
2 d [ t ¯ m i ] + j { i K G z [ t ¯ m i ] k x i k 0 ( k y [ q ¯ m i ] k x i [ h ¯ m i ] ) k 0 p n [ a n ] i p [ χ ] n [ h ¯ m p ] } = 0 ,
2 d [ l ¯ m i ] + j { i K G z [ l ¯ m i ] k y k 0 ( k y [ q ¯ m i ] k x i [ h ¯ m i ] ) + k 0 p n [ c n ] i p [ χ ] n [ q ¯ m p ] } = 0 .
x m i = ( 2 m + 1 ) l = m + 1 l + m odd M i x l i ,
u x δ i 0 + R x i = S x i ( 0 ) ,
u y δ i 0 + R y i = S y i ( 0 ) ,
δ i 0 ( k y u z k 1 cos α u y ) k z 1 i R y i + k y R z i = k 0 U x i ( 0 ) ,
δ i 0 ( k 1 cos α u x k x 0 u z ) + k z 1 i R x i k x i R z i = k 0 U y i ( 0 ) ,
T x i = S x i ( d ) exp ( j i K G z d ) ,
T y i = S y i ( d ) exp ( j i K G z d ) ,
k z 3 i T y i + k y T z i = k 0 U x i ( d ) exp ( j i K G z d ) ,
k z 3 i T x i k x i T z i = k 0 U y i ( d ) exp ( j i K G z d ) .
k x i R x i + k y R y i + k z 1 i R z i = 0 ,
k x i T x i + k y T y i + k z 3 i T z i = 0 .
m ( 1 ) m [ k z 1 i 2 + k x i 2 k z 1 i q m i + k x i k y k z 1 i h m i k 0 l m i ] = δ i 0 [ k x 0 u z k 1 z u x + k z 1 i 2 + k x i 2 k z 1 i u x + k x i k y k z 1 i u y ] ,
m ( 1 ) m [ k z 1 i 2 + k y 2 k z 1 i h m i + k x i k y k z 1 i q m i + k 0 t m i ] = δ i 0 [ k y u z k 1 z u y + k y 2 + k z 1 i 2 k z 1 i u y + k x i k y k z 1 i u x ] ,
m [ k z 3 i 2 + k y 2 k z 3 i h m i + k x i k y k z 3 i q m i + k 0 t m i ] = 0 ,
m [ k z 3 i 2 + k x i 2 k z 3 i q m i + k x i k y k z 3 i h m i k 0 l m i ] = 0 .
D E 1 i = Re ( k 1 i z k 10 z ) R i R i * ,
D E 3 i = Re ( k 3 i z k 10 z ) T i T i * .
i D E 1 i + D E 3 i = 1 .
ε = n 2 κ 2 j 2 n κ ,

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