Abstract

The origin of the instabilities of the Waterman method was studied previously and an improvement in the method was developed for one-dimensional gratings and s polarization [J. Opt. Soc. Am.15, 1566 (1998)]. Later, the same kind of regularization was used to improve Rayleigh’s expansion method. We show that the same well-adapted regularization process can be generalized to two-dimensional (2D) gratings. Numerical implementations show that the convergence domain of the Waterman method is extended by a factor of ∼40% in the range of groove depth. In the same way, the convergence domain of the Rayleigh expansion method is extended by a factor of ∼35% for 2D sinusoidal gratings. As a consequence, the new versions of Waterman and Rayleigh methods become simple and efficient tools for use in investigating the properties of 2D gratings that have ratios of groove depth to period up to unity.

© 1999 Optical Society of America

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  1. P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
    [CrossRef]
  2. N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
    [CrossRef]
  3. G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
    [CrossRef]
  4. J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
    [CrossRef]
  5. Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–415 (1907).
    [CrossRef]
  6. Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.
  7. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. XXI, pp. 1–67.
  8. M. Bagieu, D. Maystre, “Waterman and Rayleigh methods for diffraction problems: extension of the convergence domain,” J. Opt. Soc. Am. A 15, 1566–1576 (1998).
    [CrossRef]
  9. J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
    [CrossRef]
  10. R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
    [CrossRef]
  11. D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
    [CrossRef]
  12. M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 53–62.
  13. R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
  14. R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
    [CrossRef]
  15. M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).

1998 (1)

1994 (1)

J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
[CrossRef]

1982 (1)

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

1979 (1)

G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
[CrossRef]

1978 (1)

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

1975 (1)

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

1974 (1)

D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
[CrossRef]

1967 (1)

R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
[CrossRef]

1965 (1)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

1907 (1)

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–415 (1907).
[CrossRef]

Abramowitz, M.

M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).

Armand, G.

G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
[CrossRef]

Bagieu, M.

Botten, L. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

Cabrera, N.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Cadilhac, M.

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 53–62.

Celli, V.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Chesneaux, J. M.

J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
[CrossRef]

Derrick, G. H.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

Garcia, N.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Hill, N. R.

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

Manson, J. R.

G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
[CrossRef]

Maystre, D.

M. Bagieu, D. Maystre, “Waterman and Rayleigh methods for diffraction problems: extension of the convergence domain,” J. Opt. Soc. Am. A 15, 1566–1576 (1998).
[CrossRef]

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. XXI, pp. 1–67.

McPhedran, R. C.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
[CrossRef]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

Nevière, M.

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

Petit, R.

R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–415 (1907).
[CrossRef]

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

Stegun, A.

M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).

Uretsky, J. L.

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

Waterman, P. C.

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

Wirgin, A.

J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
[CrossRef]

Ann. Phys. (1)

J. L. Uretsky, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[CrossRef]

J. Acoust. Soc. Am. (2)

J. M. Chesneaux, A. Wirgin, “Reflection from a corrugated surface revisited,” J. Acoust. Soc. Am. 96, 1116–1129 (1994).
[CrossRef]

P. C. Waterman, “Scattering by periodic surfaces,” J. Acoust. Soc. Am. 57, 791–802 (1975).
[CrossRef]

J. Opt. (Paris) (1)

R. C. McPhedran, G. H. Derrick, M. Nevière, D. Maystre, “Metallic crossed gratings,” J. Opt. (Paris) 13, 209–218 (1982).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
[CrossRef]

Opt. Commun. (1)

D. Maystre, R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 12, 164–167 (1974).
[CrossRef]

Phys. Rev. B (2)

N. Garcia, V. Celli, N. R. Hill, N. Cabrera, “Ill-conditioned matrices in the scattering of waves from hard corrugated surfaces,” Phys. Rev. B 18, 5184–5189 (1978).
[CrossRef]

G. Armand, J. R. Manson, “Scattering from a corrugated hard wall: comparison of boundary conditions,” Phys. Rev. B 19, 4091–4099 (1979).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–415 (1907).
[CrossRef]

Other (5)

Lord Rayleigh, The Theory of Sound (Dover, New York, 1945), Vol. 2.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1984), Vol. XXI, pp. 1–67.

M. Cadilhac, “Some mathematical aspects of the grating theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 53–62.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

M. Abramowitz, A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1968).

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