Abstract

Starting from the electromagnetic theory, we derive an asymptotic formalism to investigate the behavior of perfectly conducting gratings used at small wavelengths to groove pitch ratios and near normal incidence. The theory is applied to study three classical types of profiles: sinusoidal, lamellar, and blazed gratings. Results are given for both −1 and −2 Littrow (or near Littrow) mounts. The accuracy of the theory and the limits of the domain where it applies are studied by the use of rigorous electromagnetic computations. The role of a finite conductivity of the surface is also investigated.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Maystre, “Sur la Diffraction d’une onde plane par un réseau métallique de conductivite finie,” Opt. Commun. 6, 50 (1972).
    [CrossRef]
  2. M. Nevière, P. Vincent, and R. Petit, “Sur la theorie du réseaux conducteur et ses applications a l’optique,” Nouv. Rev. Opt. 5, 65 (1974).
    [CrossRef]
  3. E. Loewen, M. Nevière, and D. Maystre, “Grating Efficiency Theory as it Applies to Blazed and Holographic Gratings,” Appl. Opt. 16, 2711 (1977).
    [CrossRef] [PubMed]
  4. B. A. Lippman, “Note on the Theory of Gratings,” J. Opt. Soc. Am. 43, 408 (1953).
    [CrossRef]
  5. R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. Ser. B 262, 468 (1966).
  6. R. F. Millar, “On the Rayleigh Assumption in Scattering by a Periodic Surface,” Proc. Camb. Philos. Soc. 65, 773 (1969).
    [CrossRef]
  7. R. F. Millar, “Singularities of Two-Dimensional Exterior Solutions of the Helmholtz Equation,” Proc. Camb. Philos. Soc. 69, 175 (1971).
    [CrossRef]
  8. R. G. Barentsev, Vestn. Leningr. Univ. 1, 66 (1965).
  9. J. Pavageau, “Equation intégrale pour la diffraction electromagnétique par des conducteurs parfaits dans les problèmes à deux dimensions–Application aux réseaux,” C. R. Acad. Sci. Ser. B 264, 424 (1967).
  10. J. L. Uretsky, “The Scattering of Plane Waves from Periodic Surfaces,” Ann. Phys. (Leipzig) 33, 400 (1965).
    [CrossRef]
  11. A. Wirgin, “Considérations théoriques sur la diffraction par réflexion sur des surfaces quasiment planes; application a la diffraction par des réseaux,” C. R. Acad. Sci. (Paris),  259, 1486 (1964).
  12. M. Nevière and M. Cadilhac, “Sur la Validité du Developement de Rayleigh,” Opt. Commun. 2, 235 (1970).
    [CrossRef]
  13. R. Petit and D. Maystre, “Application des lois de l’électromagnetique a l’étude des réseaux,” Rev. Phys. Appl. 7, 427 (1972).
    [CrossRef]
  14. M. Nevière, M. Cadilhac, and R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
    [CrossRef]
  15. R. F. Millar, “On the Rayleigh Assumption in Scattering by a Periodic Surface,” Proc. Camb. Philos. Soc. 69, 217 (1971).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 64.
  17. E. G. Loewen, M. Nevière, and D. Maystre, “Efficiency Optimization of Rectangular Groove Gratings in Visible and IR Regions,” (unpublished).
  18. D. Maystre and R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur,” Opt. Commun. 5, 90 (1972).
    [CrossRef]
  19. E. G. Loewen and M. Nevière, “Simple Selection Rules for VUV and XUV Diffraction Gratings,” Appl. Opt. (to be published).

1977 (1)

1974 (1)

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

1973 (1)

M. Nevière, M. Cadilhac, and R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
[CrossRef]

1972 (3)

D. Maystre and R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur,” Opt. Commun. 5, 90 (1972).
[CrossRef]

D. Maystre, “Sur la Diffraction d’une onde plane par un réseau métallique de conductivite finie,” Opt. Commun. 6, 50 (1972).
[CrossRef]

R. Petit and D. Maystre, “Application des lois de l’électromagnetique a l’étude des réseaux,” Rev. Phys. Appl. 7, 427 (1972).
[CrossRef]

1971 (2)

R. F. Millar, “Singularities of Two-Dimensional Exterior Solutions of the Helmholtz Equation,” Proc. Camb. Philos. Soc. 69, 175 (1971).
[CrossRef]

R. F. Millar, “On the Rayleigh Assumption in Scattering by a Periodic Surface,” Proc. Camb. Philos. Soc. 69, 217 (1971).
[CrossRef]

1970 (1)

M. Nevière and M. Cadilhac, “Sur la Validité du Developement de Rayleigh,” Opt. Commun. 2, 235 (1970).
[CrossRef]

1969 (1)

R. F. Millar, “On the Rayleigh Assumption in Scattering by a Periodic Surface,” Proc. Camb. Philos. Soc. 65, 773 (1969).
[CrossRef]

1967 (1)

J. Pavageau, “Equation intégrale pour la diffraction electromagnétique par des conducteurs parfaits dans les problèmes à deux dimensions–Application aux réseaux,” C. R. Acad. Sci. Ser. B 264, 424 (1967).

1966 (1)

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. Ser. B 262, 468 (1966).

1965 (2)

R. G. Barentsev, Vestn. Leningr. Univ. 1, 66 (1965).

J. L. Uretsky, “The Scattering of Plane Waves from Periodic Surfaces,” Ann. Phys. (Leipzig) 33, 400 (1965).
[CrossRef]

1964 (1)

A. Wirgin, “Considérations théoriques sur la diffraction par réflexion sur des surfaces quasiment planes; application a la diffraction par des réseaux,” C. R. Acad. Sci. (Paris),  259, 1486 (1964).

1953 (1)

Barentsev, R. G.

R. G. Barentsev, Vestn. Leningr. Univ. 1, 66 (1965).

Cadilhac, M.

M. Nevière, M. Cadilhac, and R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
[CrossRef]

M. Nevière and M. Cadilhac, “Sur la Validité du Developement de Rayleigh,” Opt. Commun. 2, 235 (1970).
[CrossRef]

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. Ser. B 262, 468 (1966).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 64.

Lippman, B. A.

Loewen, E.

Loewen, E. G.

E. G. Loewen, M. Nevière, and D. Maystre, “Efficiency Optimization of Rectangular Groove Gratings in Visible and IR Regions,” (unpublished).

E. G. Loewen and M. Nevière, “Simple Selection Rules for VUV and XUV Diffraction Gratings,” Appl. Opt. (to be published).

Maystre, D.

E. Loewen, M. Nevière, and D. Maystre, “Grating Efficiency Theory as it Applies to Blazed and Holographic Gratings,” Appl. Opt. 16, 2711 (1977).
[CrossRef] [PubMed]

D. Maystre, “Sur la Diffraction d’une onde plane par un réseau métallique de conductivite finie,” Opt. Commun. 6, 50 (1972).
[CrossRef]

D. Maystre and R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur,” Opt. Commun. 5, 90 (1972).
[CrossRef]

R. Petit and D. Maystre, “Application des lois de l’électromagnetique a l’étude des réseaux,” Rev. Phys. Appl. 7, 427 (1972).
[CrossRef]

E. G. Loewen, M. Nevière, and D. Maystre, “Efficiency Optimization of Rectangular Groove Gratings in Visible and IR Regions,” (unpublished).

Millar, R. F.

R. F. Millar, “On the Rayleigh Assumption in Scattering by a Periodic Surface,” Proc. Camb. Philos. Soc. 69, 217 (1971).
[CrossRef]

R. F. Millar, “Singularities of Two-Dimensional Exterior Solutions of the Helmholtz Equation,” Proc. Camb. Philos. Soc. 69, 175 (1971).
[CrossRef]

R. F. Millar, “On the Rayleigh Assumption in Scattering by a Periodic Surface,” Proc. Camb. Philos. Soc. 65, 773 (1969).
[CrossRef]

Nevière, M.

E. Loewen, M. Nevière, and D. Maystre, “Grating Efficiency Theory as it Applies to Blazed and Holographic Gratings,” Appl. Opt. 16, 2711 (1977).
[CrossRef] [PubMed]

M. Nevière, P. Vincent, and R. Petit, “Sur la theorie du réseaux conducteur et ses applications a l’optique,” Nouv. Rev. Opt. 5, 65 (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. AP-21, 37 (1973).
[CrossRef]

M. Nevière and M. Cadilhac, “Sur la Validité du Developement de Rayleigh,” Opt. Commun. 2, 235 (1970).
[CrossRef]

E. G. Loewen and M. Nevière, “Simple Selection Rules for VUV and XUV Diffraction Gratings,” Appl. Opt. (to be published).

E. G. Loewen, M. Nevière, and D. Maystre, “Efficiency Optimization of Rectangular Groove Gratings in Visible and IR Regions,” (unpublished).

Pavageau, J.

J. Pavageau, “Equation intégrale pour la diffraction electromagnétique par des conducteurs parfaits dans les problèmes à deux dimensions–Application aux réseaux,” C. R. Acad. Sci. Ser. B 264, 424 (1967).

Petit, R.

M. Nevière, P. Vincent, and R. Petit, “Sur la theorie du réseaux conducteur et ses applications a l’optique,” Nouv. Rev. Opt. 5, 65 (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. AP-21, 37 (1973).
[CrossRef]

R. Petit and D. Maystre, “Application des lois de l’électromagnetique a l’étude des réseaux,” Rev. Phys. Appl. 7, 427 (1972).
[CrossRef]

D. Maystre and R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur,” Opt. Commun. 5, 90 (1972).
[CrossRef]

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. Ser. B 262, 468 (1966).

Uretsky, J. L.

J. L. Uretsky, “The Scattering of Plane Waves from Periodic Surfaces,” Ann. Phys. (Leipzig) 33, 400 (1965).
[CrossRef]

Vincent, P.

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

Wirgin, A.

A. Wirgin, “Considérations théoriques sur la diffraction par réflexion sur des surfaces quasiment planes; application a la diffraction par des réseaux,” C. R. Acad. Sci. (Paris),  259, 1486 (1964).

Ann. Phys. (Leipzig) (1)

J. L. Uretsky, “The Scattering of Plane Waves from Periodic Surfaces,” Ann. Phys. (Leipzig) 33, 400 (1965).
[CrossRef]

Appl. Opt. (1)

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

A. Wirgin, “Considérations théoriques sur la diffraction par réflexion sur des surfaces quasiment planes; application a la diffraction par des réseaux,” C. R. Acad. Sci. (Paris),  259, 1486 (1964).

C. R. Acad. Sci. Ser. B (2)

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infiniment conducteur,” C. R. Acad. Sci. Ser. B 262, 468 (1966).

J. Pavageau, “Equation intégrale pour la diffraction electromagnétique par des conducteurs parfaits dans les problèmes à deux dimensions–Application aux réseaux,” C. R. Acad. Sci. Ser. B 264, 424 (1967).

IEEE Trans. Antennas Propag. (1)

M. Nevière, M. Cadilhac, and R. Petit, “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Propag. AP-21, 37 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

Nouv. Rev. Opt. (1)

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

Opt. Commun. (3)

D. Maystre, “Sur la Diffraction d’une onde plane par un réseau métallique de conductivite finie,” Opt. Commun. 6, 50 (1972).
[CrossRef]

D. Maystre and R. Petit, “Diffraction par un réseau lamellaire infiniment conducteur,” Opt. Commun. 5, 90 (1972).
[CrossRef]

M. Nevière and M. Cadilhac, “Sur la Validité du Developement de Rayleigh,” Opt. Commun. 2, 235 (1970).
[CrossRef]

Proc. Camb. Philos. Soc. (3)

R. F. Millar, “On the Rayleigh Assumption in Scattering by a Periodic Surface,” Proc. Camb. Philos. Soc. 65, 773 (1969).
[CrossRef]

R. F. Millar, “Singularities of Two-Dimensional Exterior Solutions of the Helmholtz Equation,” Proc. Camb. Philos. Soc. 69, 175 (1971).
[CrossRef]

R. F. Millar, “On the Rayleigh Assumption in Scattering by a Periodic Surface,” Proc. Camb. Philos. Soc. 69, 217 (1971).
[CrossRef]

Rev. Phys. Appl. (1)

R. Petit and D. Maystre, “Application des lois de l’électromagnetique a l’étude des réseaux,” Rev. Phys. Appl. 7, 427 (1972).
[CrossRef]

Vestn. Leningr. Univ. (1)

R. G. Barentsev, Vestn. Leningr. Univ. 1, 66 (1965).

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 64.

E. G. Loewen, M. Nevière, and D. Maystre, “Efficiency Optimization of Rectangular Groove Gratings in Visible and IR Regions,” (unpublished).

E. G. Loewen and M. Nevière, “Simple Selection Rules for VUV and XUV Diffraction Gratings,” Appl. Opt. (to be published).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

FIG. 1
FIG. 1

Schematic representation of the grating and notations.

FIG. 2
FIG. 2

Efficiency curves for a sinusoidal grating used in a −2 order Littrow mounting. Theoretical results given by rigorous electromagnetic theory.

FIG. 3
FIG. 3

Schematic representation of the profile of a lamellar grating, with groove width c and height h.

FIG. 4
FIG. 4

Representation of three distorted profiles given by formula (21). Curve 1: = −0.033; curve 2: = 0; curve 3: = +0.033.

FIG. 5
FIG. 5

Representation of three distorted profiles given by formula (22). Curve 1: = −0.033; curve 2: = 0; curve 3: = +0.033.

FIG. 6
FIG. 6

Representation of three distorted profiles given by formula (23). Curve 1: = −0.033; curve 2: = 0; curve 3: = +0.033.

Tables (3)

Tables Icon

TABLE I Comparison of the efficiency for the two classical cases of polarization. ξ designates the sum of the efficiencies in all the spectral orders. Lamellar gold grating 1200 g/mm, groove to pitch ratio 0.5, 150 Å groove depth, calculated for λ = 1216 Å.

Tables Icon

TABLE II −1 order efficiency of perfectly conduting gratings same geometry as Table I. Also reflectances of gold for both polarizations as functions of angle of incidence.

Tables Icon

TABLE III −1 order efficiency in P and S planes of polarization, of gold gratings of the same lamellar geometry as Tables I and II. Left columns derived from Table II, right columns by formal computation.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

E i ( x , y ) = exp i k [ ( x sin θ - y cos θ ) ]
E d ( x , y ) = n = - + B n exp ( i χ n y ) exp ( i γ n x ) ,
γ n = n + k sin θ ,
χ n = ( k 2 - γ n 2 ) 1 / 2             or             i ( γ n 2 - k 2 ) 1 / 2 .
γ n = n - ( m / 2 ) .
n = - + B n exp [ i n x + i χ n h g ( x ) ] = - exp [ - i χ 0 h g ( x ) ] .
n B n e i n x = - exp [ - 2 i w g ( χ ) ] .
B n = - 1 2 π 0 2 π exp { - i [ n x + 2 w g ( x ) ] } d x .
B - 1 = - 1 2 π 0 2 π e i ( x - w sin x ) d x .
J n ( z ) = 1 2 π 0 2 π e - i ( z sin θ - n θ ) d θ ,
ξ - 1 = B - 1 2 = J 1 ( w ) 2 .
λ = 3.412 h .
ξ - 1 = ξ - 1 max = J 1 ( 1.8412 ) 2 = ( 0.581 ) 2 = 0.338.
B - 2 = - 1 2 π 0 2 π exp [ i ( 2 x - w sin x ) ] d x = - J 2 ( w ) .
ξ - n = J n ( w ) 2 .
ξ - 2 = J 2 ( w ) 2 .
λ = 2.057 h .
ξ - 2 max = J 2 ( 3.054 ) 2 0.23.
ξ - n = B - n 2 = ( 4 / n 2 π 2 ) sin 2 [ n ( c / 2 ) ] sin 2 w             if             n 0
ξ 0 = B 0 2 = ( 1 / 4 π 2 ) c + e - 2 i w ( 2 π - c ) 2 .
ξ - 1 = ( 4 / π 2 ) sin 2 ( c / 2 ) sin 2 w .
w = π / 2             or             λ = 4 h .
ξ - 1 max = 4 / π 2 0.4053 ,
ξ 0 = cos w 2 = 0.
ξ - 2 = ( 1 / π 2 ) sin 2 c sin 2 w .
y 1 = ( h / 2 ) sin x + ( / 2 ) sin 2 x ,
y 2 = ( h / 2 ) sin x + ( / 2 ) cos 2 x ,
y 3 = ( h / 2 ) sin x + ( / 2 ) sin 3 x .
B - 1 = - { J 1 ( k h ) - ( k / 2 ) [ J 1 ( k h ) + J 3 ( k h ) ] } .
B - 1 = - { J 1 ( k h ) - ( i k / 2 ) [ J 3 ( k h ) - J 1 ( k h ) ] } .
B - 1 = - { J 1 ( k h ) - ( k / 2 ) [ J 4 ( k h ) - J 2 ( k h ) ] } .
h g ( x ) = λ m x / 2 ( 2 π ) = x tan α x sin α ,
λ = 2 ( 2 π / m ) sin α ,