Abstract

A rigorous coupled-wave analysis for metallic surface-relief gratings is presented. This approach allows an arbitrary complex permittivity to be used for the material and thus avoids the infinite conductivity (perfect-conductor) approximation. Both TE and TM polarizations and arbitrary angles of incidence are treated. Diffraction characteristics for rectangular-groove gold gratings with equal groove and ridge widths are presented for free-space wavelengths of 0.5, 1.0 and 10.0 μm for all diffracted orders as a function of period, groove depth, polarization, and angle of incidence. Results include the following: (1) TM-polarization diffraction characteristics vary more rapidly than do those for TE polarization, (2) 95% first-order diffraction efficiency occurs for TM polarization at 10.0 μm, (3) <0.1% zero-order specular reflectivity occurs for both TE and TM polarizations, (4) >50% absorption of incident power occurs at 0.5 μm, and (5) the perfect-conductor approximation is not valid for TM polarization at any of the wavelengths and is not valid for TE polarization at 0.5 μm.

© 1986 Optical Society of America

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References

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  1. E.g., M. C. Hutley, Diffraction Gratings (Academic, London, 1982).
  2. K. Knop, “Reflection grating polarizer for the infrared,” Opt. Commun. 26, 281–283 (1978).
    [CrossRef]
  3. W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–356 (1982).
    [CrossRef] [PubMed]
  4. W. B. Veldkamp, “Developments in laser-beam control with holographic diffraction gratings,” in Practical Electro-Optical Instruments and Techniques, R. L. Kurtz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.255, 136–144 (1980).
    [CrossRef]
  5. V.G. Gerbig, “Holographic scanning: a review,” in Advances in Laser Scanning and Recording, L. Beiser, ed. Proc. Soc. Photo-Opt. Instrum. Eng.396, 28–35 (1983).
    [CrossRef]
  6. G. J. Swanson, W. B. Veldkamp, “Binary lenses for use at 10.6 micrometers,” Opt. Eng. 24, 791–795 (1985).
    [CrossRef]
  7. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  8. D. Maystre, R. Petit, “Diffraction par reseau lamellaire infiniment conducteur (Diffraction by an infinite conductivity lamellar grating),” Opt. Commun. 5, 90–93 (1972).
    [CrossRef]
  9. A. Hessel, J. Schmoys, D. Y. Tseng, “Bragg-angle blazing of diffraction gratings,” J. Opt. Soc. Am. 65, 380–384 (1975).
    [CrossRef]
  10. J. L. Roumiguieres, D. Maystre, R. Petit, “On the efficiencies of rectangular-groove gratings,” J. Opt. Soc. Am. 66, 772–775 (1976).
    [CrossRef]
  11. J. W. Heath, E. V. Jull, “Perfectly blazed reflection gratings with rectangular grooves,” J. Opt. Soc. Am. 68, 1211–1217 (1978).
    [CrossRef]
  12. J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
    [CrossRef]
  13. G. M. Whitman, D. M. Leskiw, F. Schwering, “Rigorous theory of scattering by perfectly conducting periodic surfaces with trapezoidal height profile, TE and TM polarization,” J. Opt. Soc. Am. 70, 1495–1503 (1980).
    [CrossRef]
  14. J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
    [CrossRef]
  15. R. Petit, M. Cadilhac, “Form of the electromagnetic field in the groove region of a perfectly conducting echelette grating,” J. Opt. Soc. Am. 73, 963–965 (1983).
    [CrossRef]
  16. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  17. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  18. W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
    [CrossRef]
  19. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  20. D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
    [CrossRef]
  21. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction—E mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  22. E.g., C. L. Liu, J. W. S. Liu, Linear Systems Analysis (McGraw-Hill, New York, 1975).
  23. Program eigrf from the International Mathematics and Statistics Library (IMSL), Houston, Texas.
  24. G. Hass, L. Hadley, “Optical properties of metals,” in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1972).
  25. P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
    [CrossRef]
  26. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  27. E.g. L. C. Shen, J. A. Kong, Applied Electromagnetism (Brooks/Cole, Monterey, Calif., 1983), p. 74.
  28. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]

1985

G. J. Swanson, W. B. Veldkamp, “Binary lenses for use at 10.6 micrometers,” Opt. Eng. 24, 791–795 (1985).
[CrossRef]

1983

1982

1981

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
[CrossRef]

1980

1979

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

1978

1976

J. L. Roumiguieres, D. Maystre, R. Petit, “On the efficiencies of rectangular-groove gratings,” J. Opt. Soc. Am. 66, 772–775 (1976).
[CrossRef]

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
[CrossRef]

1975

1972

D. Maystre, R. Petit, “Diffraction par reseau lamellaire infiniment conducteur (Diffraction by an infinite conductivity lamellar grating),” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Andrewartha, J. R.

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Baird, W. E.

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Cadilhac, M.

DeSanto, J. A.

J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
[CrossRef]

Fox, J. R.

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Gaylord, T. K.

Gerbig, V.G.

V.G. Gerbig, “Holographic scanning: a review,” in Advances in Laser Scanning and Recording, L. Beiser, ed. Proc. Soc. Photo-Opt. Instrum. Eng.396, 28–35 (1983).
[CrossRef]

Hadley, L.

G. Hass, L. Hadley, “Optical properties of metals,” in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1972).

Hass, G.

G. Hass, L. Hadley, “Optical properties of metals,” in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1972).

Heath, J. W.

Hessel, A.

Hutley, M. C.

E.g., M. C. Hutley, Diffraction Gratings (Academic, London, 1982).

Jull, E. V.

Kastner, C. J.

Knop, K.

K. Knop, “Reflection grating polarizer for the infrared,” Opt. Commun. 26, 281–283 (1978).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kong, J. A.

E.g. L. C. Shen, J. A. Kong, Applied Electromagnetism (Brooks/Cole, Monterey, Calif., 1983), p. 74.

Leskiw, D. M.

Liu, C. L.

E.g., C. L. Liu, J. W. S. Liu, Linear Systems Analysis (McGraw-Hill, New York, 1975).

Liu, J. W. S.

E.g., C. L. Liu, J. W. S. Liu, Linear Systems Analysis (McGraw-Hill, New York, 1975).

Marcuse, D.

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
[CrossRef]

Maystre, D.

J. L. Roumiguieres, D. Maystre, R. Petit, “On the efficiencies of rectangular-groove gratings,” J. Opt. Soc. Am. 66, 772–775 (1976).
[CrossRef]

D. Maystre, R. Petit, “Diffraction par reseau lamellaire infiniment conducteur (Diffraction by an infinite conductivity lamellar grating),” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

Moharam, M. G.

Petit, R.

Roumiguieres, J. L.

Sanda, P. N.

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Schmoys, J.

Schwering, F.

Shen, L. C.

E.g. L. C. Shen, J. A. Kong, Applied Electromagnetism (Brooks/Cole, Monterey, Calif., 1983), p. 74.

Sheng, P.

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Swanson, G. J.

G. J. Swanson, W. B. Veldkamp, “Binary lenses for use at 10.6 micrometers,” Opt. Eng. 24, 791–795 (1985).
[CrossRef]

Tseng, D. Y.

Veldkamp, W. B.

G. J. Swanson, W. B. Veldkamp, “Binary lenses for use at 10.6 micrometers,” Opt. Eng. 24, 791–795 (1985).
[CrossRef]

W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–356 (1982).
[CrossRef] [PubMed]

W. B. Veldkamp, “Developments in laser-beam control with holographic diffraction gratings,” in Practical Electro-Optical Instruments and Techniques, R. L. Kurtz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.255, 136–144 (1980).
[CrossRef]

Whitman, G. M.

Wilson, I. J.

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

Appl. Opt.

Appl. Phys. B

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

Bell Syst. Tech. J.

D. Marcuse, “Exact theory of TE-wave scattering from blazed dielectric gratings,” Bell Syst. Tech. J. 55, 1295–1317 (1976).
[CrossRef]

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

J. R. Andrewartha, J. R. Fox, I. J. Wilson, “Resonance anomalies in the lamellar grating,” Opt. Acta 26, 69–89 (1979).
[CrossRef]

Opt. Commun.

D. Maystre, R. Petit, “Diffraction par reseau lamellaire infiniment conducteur (Diffraction by an infinite conductivity lamellar grating),” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

K. Knop, “Reflection grating polarizer for the infrared,” Opt. Commun. 26, 281–283 (1978).
[CrossRef]

Opt. Eng.

G. J. Swanson, W. B. Veldkamp, “Binary lenses for use at 10.6 micrometers,” Opt. Eng. 24, 791–795 (1985).
[CrossRef]

Phys. Rev. B

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Radio Sci.

J. A. DeSanto, “Scattering from a perfectly reflecting arbitrary periodic surface: an exact theory,” Radio Sci. 16, 1315–1326 (1981).
[CrossRef]

Other

E.g., M. C. Hutley, Diffraction Gratings (Academic, London, 1982).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

W. B. Veldkamp, “Developments in laser-beam control with holographic diffraction gratings,” in Practical Electro-Optical Instruments and Techniques, R. L. Kurtz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.255, 136–144 (1980).
[CrossRef]

V.G. Gerbig, “Holographic scanning: a review,” in Advances in Laser Scanning and Recording, L. Beiser, ed. Proc. Soc. Photo-Opt. Instrum. Eng.396, 28–35 (1983).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

E.g. L. C. Shen, J. A. Kong, Applied Electromagnetism (Brooks/Cole, Monterey, Calif., 1983), p. 74.

E.g., C. L. Liu, J. W. S. Liu, Linear Systems Analysis (McGraw-Hill, New York, 1975).

Program eigrf from the International Mathematics and Statistics Library (IMSL), Houston, Texas.

G. Hass, L. Hadley, “Optical properties of metals,” in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1972).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the metallic rectangular-groove surface-relief grating diffraction problem analyzed.

Fig. 2
Fig. 2

Diffraction efficiencies, DEi, and losses for gold rectangular-groove surface-relief gratings as a function of groove depth, d, for an incident wave of λ = 0.5 μm at the first Bragg angle (θ = 30.0°). The gratings have a period of Λ = 0.5 μm and equal groove and ridge widths (i.e., groove width equal to half of the grating period).

Fig. 3
Fig. 3

Same as Fig. 2 with λ = Λ = 1.0 μm.

Fig. 4
Fig. 4

Same as Fig. 2 with λ = Λ = 10.0 μm.

Fig. 5
Fig. 5

Allowed propagating backward-diffracted orders for normal incidence, first Bragg angle incidence, and second Bragg angle incidence for a grating with a period equal to twice the incident wavelength.

Fig. 6
Fig. 6

Diffraction efficiencies, DEi, and losses for gold rectangular-groove gratings as a function of groove depth, d, with an optical wave normally incident and with polarization parallel to the grating grooves (TE). The gratings have periods equal to twice the incident wavelength.

Fig. 7
Fig. 7

Diffraction efficiencies, DEi, and losses for gold rectangular-groove gratings as a function of angle of incidence, θ, for an incident wave of λ = 0.5 μm. The gratings have a period of Λ = 1.0 μm and equal groove and ridge widths (i.e., groove width equal to half of the grating period). The groove depth is d = 0.3λ = 0.15 μm. The angles include normal incidence (0°), the first Bragg angle (14.47°), and the second Bragg angle (30.0°).

Fig. 8
Fig. 8

Same as Fig. 7 with λ = Λ = 1.0 μm.

Fig. 9
Fig. 9

Same as Fig. 7 with λ = Λ = 10.0 μm.

Tables (1)

Tables Icon

Table 1 Complex Refractive Index, n = n, and Intensity Reflectivity, R, for Bulk Gold at the Visible and Infrared Free-Space Wavelengths, λ, Treated and Angles of Incidence of 0° and 30° (Ref. 24)

Equations (15)

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

( x , z ) = ( x + Λ , z ) = p p ( z ) exp ( j p K x ) ,
E I = exp [ j ( k x 0 x + k z 0 z ) ] + i R i exp [ j ( k x i x k z i I z ) ] ,
E III = i T i exp { j [ k x i x + k z i III ( z d ) ] } ,
E II = i S i ( z ) exp [ j ( k x i x + k z 0 z ) ] ,
2 E II + k 2 ( x , z ) E II = 0.
d 2 S i ( z ) d z 2 j 2 k z 0 d S i ( z ) d z = ( k x i 2 + k x 0 2 ) S i ( z ) k 2 p p ( z ) S i p ( z ) .
S i ( z ) = m C m w i m exp ( λ m z ) ,
δ i 0 + R i = m C m w i m ,
( δ i 0 R i ) k z i = m C m w i m ( k z 0 + j λ m ) ,
T i = m C m w i m exp ( λ m d ) ,
T i k z i = C m w i m ( k z 0 + j λ m ) exp ( λ m d ) .
DE i = | R i | 2 Re ( k z i / k z 0 ) .
= n 2 κ 2 j 2 n κ .
m λ / ( I ) 1 / 2 = 2 Λ sin θ ,
i λ / ( I ) 1 / 2 = Λ ( sin θ sin θ i ) ,

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