Abstract

We present a new rigorous integral formalism for the theoretical study of dielectric coated gratings and grating couplers. It applies in the resonance domain, where the wavelength of the incident field and the groove spacing are of the same order of magnitude. The computed program issued from this theory extends the domain of application of the previous differential or integral theories. It can be used to investigate, with a very good accuracy, the properties of bare or dielectric coated gratings, for any groove shape and any polarization, in the entire visible, ultraviolet, and infrared regions. Various classical criteria are used to control the validity of the numerical results and comparisons are made with the numerical results obtained using the previous integral and differential formalisms. Two examples of applications are given. First, we show that the new possibilities of our program lead to a better agreement between theoretical results and experimental data. Second, a theoretical study of a certain type of grating coupler is given.

© 1978 Optical Society of America

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  1. R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 3, 129–135 (1975).
    [Crossref]
  2. A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar periodic structures,” Alta Freq.,  38, special issue on URSI Symposium, 282–285 (1968).
  3. P. M. Van den Berg, “Rigorous diffraction theory of optical reflexion and transmission gratings,” thesis (, Delft, Netherlands, 1971).
  4. D. Maystre, “Sur la diffraction d’une onde plane par un réseau métallique de conductivité finie,” Opt. Commun. 1, 50–54 (1972).
    [Crossref]
  5. D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 3, 216–219 (1973).
    [Crossref]
  6. R. Petit, D. Maystre, and M. Nevière, “Practical applications of the electromagnetic theory of gratings,” Space Optics, Proceedings of the Ninth International Congress of the I.C.O., 667–681 (1972).
  7. D. Maystre, R. Petit, M. Duban, and J. Gilewicz, “Theoretical determination of the efficiencies for a conducting grating used in the near ultraviolet,” Nouv. Rev. Opt. 2, 79–85 (1974).
    [Crossref]
  8. D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” thesis (University Aix-Marseille III, CNRS A.O. 9545, 1974) (unpublished).
  9. D. Maystre and R. Petit, “Some recent theoretical results for gratings; Application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
    [Crossref]
  10. E. G. Loewen, M. Nevière, and D. Maystre, “Review of grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. (to be published).
  11. R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 5, 413–421 (1974).
    [Crossref]
  12. R. C. McPhedran and D. Maystre, “Theoretical study of the diffraction anomalies of holographic gratings,” Nouv. Rev. Opt. 4, 241–248 (1974).
    [Crossref]
  13. E. G. Loewen, D. Maystre, R. C. McPhedran, and I. Wilson, “Correlation between efficiency of diffraction gratings and theoretical calculations over a wide range,” Jpn. J. Appl. Phys. 14 (suppl.), 143–155 (1975).
  14. G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).
  15. M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 2, 65–77 (1974).
    [Crossref]
  16. M. Nevière, “Sur un formalisme différentiel pour les problèmes de diffraction dans le domaine de résonance,” thesis (University Aix-Marseille III, CNRS A.O. 11556, 1975) (unpublished).
  17. M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 2, 87–95 (1975).
    [Crossref]
  18. M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 3, 240–245 (1973).
    [Crossref]
  19. C. Müller, Foundations of the mathematical theory of electromagnetic waves (Springer Verlag, Berlin, 1969), p. 328.
  20. The linear vector space ℛ0′ of distributions, used in this paper, is the set of the linear continuous functionals on the space ℛ0 of the functions ξ(x,y) defined by the following: ξ(x,y) is differentiable everywhere any number of times; ξ(x,y) is periodic with respect to x, with period d; ξ(x;y) is equal to zero outside a bounded segment of Oy axis. Using the Dirac notation, the δ function of ℛ0′ is defined by 〈δ, ξ, (x,y)〉, = ξ(0,0).
  21. D. Maystre and R. Petit, “Application des propriétés des réseaux échelettes au filtrage des longueurs d’onde,” Opt. Commun. 5, 380–382 (1972).
    [Crossref]
  22. D. Maystre and M. Nevière, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
    [Crossref]
  23. J. Meixner, “On the behaviour of the electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 422–447 (1972).
    [Crossref]
  24. D. Maystre and R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 2, 164–167 (1974).
    [Crossref]
  25. M. Nevière, M. Cadilhac, and R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. 21, 37–46 (1973).
    [Crossref]
  26. D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 2, 196–200 (1976).
    [Crossref]
  27. M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 3, 431–436 (1976).
    [Crossref]
  28. A. Otto and W. Sohler, “Modification of the total reflectance modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
    [Crossref]

1977 (1)

D. Maystre and M. Nevière, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
[Crossref]

1976 (3)

D. Maystre and R. Petit, “Some recent theoretical results for gratings; Application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
[Crossref]

D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 2, 196–200 (1976).
[Crossref]

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 3, 431–436 (1976).
[Crossref]

1975 (3)

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 3, 129–135 (1975).
[Crossref]

E. G. Loewen, D. Maystre, R. C. McPhedran, and I. Wilson, “Correlation between efficiency of diffraction gratings and theoretical calculations over a wide range,” Jpn. J. Appl. Phys. 14 (suppl.), 143–155 (1975).

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 2, 87–95 (1975).
[Crossref]

1974 (5)

D. Maystre and R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 2, 164–167 (1974).
[Crossref]

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 5, 413–421 (1974).
[Crossref]

R. C. McPhedran and D. Maystre, “Theoretical study of the diffraction anomalies of holographic gratings,” Nouv. Rev. Opt. 4, 241–248 (1974).
[Crossref]

D. Maystre, R. Petit, M. Duban, and J. Gilewicz, “Theoretical determination of the efficiencies for a conducting grating used in the near ultraviolet,” Nouv. Rev. Opt. 2, 79–85 (1974).
[Crossref]

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 2, 65–77 (1974).
[Crossref]

1973 (3)

D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 3, 216–219 (1973).
[Crossref]

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

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 3, 240–245 (1973).
[Crossref]

1972 (4)

D. Maystre and R. Petit, “Application des propriétés des réseaux échelettes au filtrage des longueurs d’onde,” Opt. Commun. 5, 380–382 (1972).
[Crossref]

J. Meixner, “On the behaviour of the electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 422–447 (1972).
[Crossref]

R. Petit, D. Maystre, and M. Nevière, “Practical applications of the electromagnetic theory of gratings,” Space Optics, Proceedings of the Ninth International Congress of the I.C.O., 667–681 (1972).

D. Maystre, “Sur la diffraction d’une onde plane par un réseau métallique de conductivité finie,” Opt. Commun. 1, 50–54 (1972).
[Crossref]

1971 (1)

A. Otto and W. Sohler, “Modification of the total reflectance modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
[Crossref]

1969 (1)

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

1968 (1)

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar periodic structures,” Alta Freq.,  38, special issue on URSI Symposium, 282–285 (1968).

Cadilhac, M.

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 3, 240–245 (1973).
[Crossref]

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

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

Cerutti-Maori, G.

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

Duban, M.

D. Maystre, R. Petit, M. Duban, and J. Gilewicz, “Theoretical determination of the efficiencies for a conducting grating used in the near ultraviolet,” Nouv. Rev. Opt. 2, 79–85 (1974).
[Crossref]

Gilewicz, J.

D. Maystre, R. Petit, M. Duban, and J. Gilewicz, “Theoretical determination of the efficiencies for a conducting grating used in the near ultraviolet,” Nouv. Rev. Opt. 2, 79–85 (1974).
[Crossref]

Hutley, M. C.

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 3, 431–436 (1976).
[Crossref]

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 2, 87–95 (1975).
[Crossref]

Loewen, E. G.

E. G. Loewen, D. Maystre, R. C. McPhedran, and I. Wilson, “Correlation between efficiency of diffraction gratings and theoretical calculations over a wide range,” Jpn. J. Appl. Phys. 14 (suppl.), 143–155 (1975).

E. G. Loewen, M. Nevière, and D. Maystre, “Review of grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. (to be published).

Maystre, D.

D. Maystre and M. Nevière, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
[Crossref]

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 3, 431–436 (1976).
[Crossref]

D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 2, 196–200 (1976).
[Crossref]

D. Maystre and R. Petit, “Some recent theoretical results for gratings; Application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
[Crossref]

E. G. Loewen, D. Maystre, R. C. McPhedran, and I. Wilson, “Correlation between efficiency of diffraction gratings and theoretical calculations over a wide range,” Jpn. J. Appl. Phys. 14 (suppl.), 143–155 (1975).

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 5, 413–421 (1974).
[Crossref]

R. C. McPhedran and D. Maystre, “Theoretical study of the diffraction anomalies of holographic gratings,” Nouv. Rev. Opt. 4, 241–248 (1974).
[Crossref]

D. Maystre, R. Petit, M. Duban, and J. Gilewicz, “Theoretical determination of the efficiencies for a conducting grating used in the near ultraviolet,” Nouv. Rev. Opt. 2, 79–85 (1974).
[Crossref]

D. Maystre and R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 2, 164–167 (1974).
[Crossref]

D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 3, 216–219 (1973).
[Crossref]

R. Petit, D. Maystre, and M. Nevière, “Practical applications of the electromagnetic theory of gratings,” Space Optics, Proceedings of the Ninth International Congress of the I.C.O., 667–681 (1972).

D. Maystre, “Sur la diffraction d’une onde plane par un réseau métallique de conductivité finie,” Opt. Commun. 1, 50–54 (1972).
[Crossref]

D. Maystre and R. Petit, “Application des propriétés des réseaux échelettes au filtrage des longueurs d’onde,” Opt. Commun. 5, 380–382 (1972).
[Crossref]

E. G. Loewen, M. Nevière, and D. Maystre, “Review of grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. (to be published).

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” thesis (University Aix-Marseille III, CNRS A.O. 9545, 1974) (unpublished).

McPhedran, R. C.

E. G. Loewen, D. Maystre, R. C. McPhedran, and I. Wilson, “Correlation between efficiency of diffraction gratings and theoretical calculations over a wide range,” Jpn. J. Appl. Phys. 14 (suppl.), 143–155 (1975).

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 2, 87–95 (1975).
[Crossref]

D. Maystre and R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 2, 164–167 (1974).
[Crossref]

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 5, 413–421 (1974).
[Crossref]

R. C. McPhedran and D. Maystre, “Theoretical study of the diffraction anomalies of holographic gratings,” Nouv. Rev. Opt. 4, 241–248 (1974).
[Crossref]

Meixner, J.

J. Meixner, “On the behaviour of the electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 422–447 (1972).
[Crossref]

Müller, C.

C. Müller, Foundations of the mathematical theory of electromagnetic waves (Springer Verlag, Berlin, 1969), p. 328.

Neureuther, A. R.

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar periodic structures,” Alta Freq.,  38, special issue on URSI Symposium, 282–285 (1968).

Nevière, M.

D. Maystre and M. Nevière, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
[Crossref]

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 2, 87–95 (1975).
[Crossref]

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 2, 65–77 (1974).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 3, 240–245 (1973).
[Crossref]

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

R. Petit, D. Maystre, and M. Nevière, “Practical applications of the electromagnetic theory of gratings,” Space Optics, Proceedings of the Ninth International Congress of the I.C.O., 667–681 (1972).

E. G. Loewen, M. Nevière, and D. Maystre, “Review of grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. (to be published).

M. Nevière, “Sur un formalisme différentiel pour les problèmes de diffraction dans le domaine de résonance,” thesis (University Aix-Marseille III, CNRS A.O. 11556, 1975) (unpublished).

Otto, A.

A. Otto and W. Sohler, “Modification of the total reflectance modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
[Crossref]

Petit, R.

D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 2, 196–200 (1976).
[Crossref]

D. Maystre and R. Petit, “Some recent theoretical results for gratings; Application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
[Crossref]

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 3, 129–135 (1975).
[Crossref]

D. Maystre, R. Petit, M. Duban, and J. Gilewicz, “Theoretical determination of the efficiencies for a conducting grating used in the near ultraviolet,” Nouv. Rev. Opt. 2, 79–85 (1974).
[Crossref]

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 2, 65–77 (1974).
[Crossref]

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

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 3, 240–245 (1973).
[Crossref]

D. Maystre and R. Petit, “Application des propriétés des réseaux échelettes au filtrage des longueurs d’onde,” Opt. Commun. 5, 380–382 (1972).
[Crossref]

R. Petit, D. Maystre, and M. Nevière, “Practical applications of the electromagnetic theory of gratings,” Space Optics, Proceedings of the Ninth International Congress of the I.C.O., 667–681 (1972).

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

Sohler, W.

A. Otto and W. Sohler, “Modification of the total reflectance modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
[Crossref]

Van den Berg, P. M.

P. M. Van den Berg, “Rigorous diffraction theory of optical reflexion and transmission gratings,” thesis (, Delft, Netherlands, 1971).

Verrill, J. P.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 2, 87–95 (1975).
[Crossref]

Vincent, P.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 2, 87–95 (1975).
[Crossref]

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 2, 65–77 (1974).
[Crossref]

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 3, 240–245 (1973).
[Crossref]

Wilson, I.

E. G. Loewen, D. Maystre, R. C. McPhedran, and I. Wilson, “Correlation between efficiency of diffraction gratings and theoretical calculations over a wide range,” Jpn. J. Appl. Phys. 14 (suppl.), 143–155 (1975).

Zaki, K.

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar periodic structures,” Alta Freq.,  38, special issue on URSI Symposium, 282–285 (1968).

Alta Freq. (1)

A. R. Neureuther and K. Zaki, “Numerical methods for the analysis of scattering from nonplanar periodic structures,” Alta Freq.,  38, special issue on URSI Symposium, 282–285 (1968).

C. R. Acad. Sci. Paris (1)

G. Cerutti-Maori, R. Petit, and M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. Paris 268, 1060–1063 (1969).

IEEE Trans. Antennas Propag. (2)

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

J. Meixner, “On the behaviour of the electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 422–447 (1972).
[Crossref]

J. Opt. (1)

D. Maystre and M. Nevière, “Quantitative theoretical study on the plasmon anomalies of diffraction gratings,” J. Opt. 8, 165–174 (1977).
[Crossref]

Jpn. J. Appl. Phys. (1)

E. G. Loewen, D. Maystre, R. C. McPhedran, and I. Wilson, “Correlation between efficiency of diffraction gratings and theoretical calculations over a wide range,” Jpn. J. Appl. Phys. 14 (suppl.), 143–155 (1975).

Nouv. Rev. Opt. (6)

R. C. McPhedran and D. Maystre, “Theoretical study of the diffraction anomalies of holographic gratings,” Nouv. Rev. Opt. 4, 241–248 (1974).
[Crossref]

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 3, 129–135 (1975).
[Crossref]

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, and P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 2, 87–95 (1975).
[Crossref]

D. Maystre, R. Petit, M. Duban, and J. Gilewicz, “Theoretical determination of the efficiencies for a conducting grating used in the near ultraviolet,” Nouv. Rev. Opt. 2, 79–85 (1974).
[Crossref]

D. Maystre and R. Petit, “Some recent theoretical results for gratings; Application to their use in the very far ultraviolet,” Nouv. Rev. Opt. 3, 165–180 (1976).
[Crossref]

M. Nevière, P. Vincent, and R. Petit, “Theory of conducting gratings and their applications to optics,” Nouv. Rev. Opt. 2, 65–77 (1974).
[Crossref]

Opt. Acta (1)

R. C. McPhedran and D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 5, 413–421 (1974).
[Crossref]

Opt. Commun. (8)

M. Nevière, P. Vincent, R. Petit, and M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 3, 240–245 (1973).
[Crossref]

D. Maystre, “Sur la diffraction d’une onde plane par un réseau métallique de conductivité finie,” Opt. Commun. 1, 50–54 (1972).
[Crossref]

D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 3, 216–219 (1973).
[Crossref]

D. Maystre and R. Petit, “Application des propriétés des réseaux échelettes au filtrage des longueurs d’onde,” Opt. Commun. 5, 380–382 (1972).
[Crossref]

D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. 2, 196–200 (1976).
[Crossref]

M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 3, 431–436 (1976).
[Crossref]

A. Otto and W. Sohler, “Modification of the total reflectance modes in a dielectric film by one metal boundary,” Opt. Commun. 3, 254–258 (1971).
[Crossref]

D. Maystre and R. C. McPhedran, “Le théorème de réciprocité pour les réseaux de conductivité finie: démonstration et applications,” Opt. Commun. 2, 164–167 (1974).
[Crossref]

Proceedings of the Ninth International Congress of the I.C.O. (1)

R. Petit, D. Maystre, and M. Nevière, “Practical applications of the electromagnetic theory of gratings,” Space Optics, Proceedings of the Ninth International Congress of the I.C.O., 667–681 (1972).

Other (6)

P. M. Van den Berg, “Rigorous diffraction theory of optical reflexion and transmission gratings,” thesis (, Delft, Netherlands, 1971).

E. G. Loewen, M. Nevière, and D. Maystre, “Review of grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. (to be published).

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” thesis (University Aix-Marseille III, CNRS A.O. 9545, 1974) (unpublished).

C. Müller, Foundations of the mathematical theory of electromagnetic waves (Springer Verlag, Berlin, 1969), p. 328.

The linear vector space ℛ0′ of distributions, used in this paper, is the set of the linear continuous functionals on the space ℛ0 of the functions ξ(x,y) defined by the following: ξ(x,y) is differentiable everywhere any number of times; ξ(x,y) is periodic with respect to x, with period d; ξ(x;y) is equal to zero outside a bounded segment of Oy axis. Using the Dirac notation, the δ function of ℛ0′ is defined by 〈δ, ξ, (x,y)〉, = ξ(0,0).

M. Nevière, “Sur un formalisme différentiel pour les problèmes de diffraction dans le domaine de résonance,” thesis (University Aix-Marseille III, CNRS A.O. 11556, 1975) (unpublished).

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Figures (6)

FIG. 1
FIG. 1

Schematic presentation of the problem and notations.

FIG. 2
FIG. 2

The unknown function ϕP is equal to the discontinuity of the normal derivative of EP+1 on P.

FIG. 3
FIG. 3

The unknown function ϕP–1 is equal to the discontinuity of the normal derivative of FP on P–1, and is linearly linked with ϕP.

FIG. 4
FIG. 4

Calculation of ϕ1 by resolution of a Fredholm equation.

FIG. 5
FIG. 5

Comparison of experimental and theoretical efficiencies for a blazed grating used in Littrow mount. The solid line gives our theoretical curve, the dashed line the experimental one, and the dotted line shows the theoretical curves obtained by Nevière using the differential formalism, and supposing the metal to be infinitly conducting for S polarization. The wavelength λ0 is expressed in microns and the position where the −2 and +1 orders pass off is indicated by the arrow. (A) noncoated grating, P polarization; (B) coated grating, P polarization; (C) noncoated grating, S polarization; (D) coated grating, S polarization.

FIG. 6
FIG. 6

Schematic representation of the retrocoupler; ν2 = 1.54, ν3 = 0.625 + 5.48 i, e = 0.5513 μm, d = 0.6 μm, α = 10°, λ0 = 0.488 μm.

Tables (1)

Tables Icon

TABLE I Approximate and exact values γg and γc of the propagation constant of the guided modes for retrocoupler of Fig. 6.

Equations (40)

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E i ( x , y ) = exp ( - i k 1 y ) .
E ( x , y ) = n = - + A p , n exp ( i n K x - i χ p , n y ) + n = - + B p , n exp ( i n K x + i χ p , n y ) ,
χ p , n = ( k p 2 - n 2 K 2 ) 1 / 2 ,
p , n , Re ( χ p , n ) + Im ( χ p , n ) > 0.
A 1 , n = δ n 0 ,
n , B P + 1 , n = 0.
Δ E + k p 2 E = 0.
p ( 1 , P ) ,             E p = J p ϕ p ,
D p = C p ϕ p ,
p ( 2 , P ) ,             ϕ p = p - 1 ϕ p - 1 .
N ϕ 1 = ,
under P :             F P + 1 ( x , y ) = E ( x , y ) ,
over P = Δ F P + 1 + k P + 1 2 F P + 1 = 0 ,
F P + 1 satisfies , for y + , the Sommerfeld condition ,
F P + 1 is continuous , in particular on P .
Δ F P + 1 + k P + 1 2 F P + 1 = ϕ P δ P ,
u δ p , ξ ( x , y ) = 1 period of p u ξ ( x , y ) d l p ,
Δ G P + 1 + k P + 1 2 G P + 1 = δ ,
F P + 1 = G P + 1 * ϕ P δ P .
E P = lim N P M P ( F P + 1 ( N P ) ) = J P ϕ P ,
D P = lim N P M P ( n ( M P ) · grad N P ( F P + 1 ) ) = C P ϕ P ,
lim N P M P ( F P + 1 ( N P ) ) = F P + 1 ( M P ) ,
lim N P M P ( n ( M P ) · grad N P ( F P + 1 ) ) = - ϕ P / 2 + n P ( M P ) · grad M P ( F P + 1 ) .
J P ϕ P = 1 period of P S ( M P , M P ) ϕ P ( M P ) d l P ,
under P :             F P = 0 ,
between P and P - 1 :             F P = E ( x , y ) ,
over P P + 1 :             Δ F P + k P 2 F P = 0 ,
F P satisfies , for y + , the outgoing wave condition ,
F P is continuous , in particular in P - 1 , except naturally in P .
Δ F P + k P 2 F P = ϕ P - 1 δ P - 1 + D P δ P + div ( n p E P δ P ) .
lim N P M P ( F P ( N P ) ) = - E P / 2 + F P ( M P ) = - J P ϕ P / 2 + V P ϕ P + W P - 1 ϕ P - 1 = 0 ,
P - 1 = - ( - J P 2 + V P ) - 1 W P - 1 .
E 1 d = J 1 ϕ 1 - E 1 i ,
D 1 d = C 1 ϕ 1 - D 1 i .
Δ F 1 + k 1 2 F 1 = D 1 d δ 1 + div ( n 1 E 1 d δ 1 ) .
F 1 = G 1 * [ D 1 d ( ϕ 1 ) δ 1 + div ( n 1 E 1 d ( ϕ 1 ) δ 2 ) ] .
lim N 1 M 1 ( F 1 ( N 1 ) ) = - E 1 d / 2 + F 1 ( M 1 ) = - J 1 ϕ 1 2 + E 1 i 2 + V 1 ϕ 1 + V 1 E 1 i + V 1 D 1 i = 0.
( V 1 - J 1 / 2 ) ϕ 1 = - ( E i 2 + V 1 E 1 i + V 1 D 1 i ) .
γ c = 17.128 + i 0.0886.
θ c = arcsin ( 1 k 1 Re ( γ c ) - λ 0 d )