Abstract

The required thickness and complex refractive index of single homogeneous layers on lossy substrates to produce zero reflectivity are calculated by a rigorous impedance matching approach. The analysis is applicable to both TE and TM polarization and to any angle of incidence. The filling factor and groove depth of a rectangular-groove grating, equivalent to a single homogeneous lossy layer in the long-wavelength limit, are calculated. The method reduces to that previously found for dielectric surface-relief gratings in the limit of no losses. The antireflection behavior of the gratings is verified using the rigorous (without approximations) coupled-wave analysis of metallic surface-relief grating diffraction. It is shown that multiple zero-reflectivity solutions exist for both TE and TM polarizations and for any angle of incidence for an arbitrary complex-refractive-index substrate. Example zero-reflectivity gold gratings for incident free space wavelengths from 0.44 to 12.0 μm are presented.

© 1987 Optical Society of America

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References

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  1. P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-Selective Absorption Enhancement in Thin-Film Solar Cells,” Appl. Phys. Lett. 43, 579 (1983).
    [CrossRef]
  2. R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 48, 928 (1935).
    [CrossRef]
  3. M. C. Hutley, V. M. Bird, “A Detailed Experimental Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 20, 771 (1973).
    [CrossRef]
  4. J. Hägglund, F. Sellberg, “Reflection, Absorption, and Emission of Light by Opaque Optical Gratings,” J. Opt. Soc. Am. 56, 1031 (1966).
    [CrossRef]
  5. R. C. McPhedran, D. Maystre, “A Detailed Theoretical Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 21, 413 (1974).
    [CrossRef]
  6. D. Maystre, R. Petit, “Brewster Incidence for Metallic Gratings,” Opt. Commun. 17, 196 (1976).
    [CrossRef]
  7. G. W. Ford, W. H. Weber, “Electromagnetic Interactions of Molecules with Metal Surfaces,” Phys. Rep. 113, 195 (1984).
    [CrossRef]
  8. D. Maystre, M. Nevière, “Sur Une Méthode D'Étude Théorique Quantitative des Anomalies de Wood des Réseaux de Diffraction: Application aux Anomalies de Plasmons,” J. Opt. Paris 8, 165 (1977).
    [CrossRef]
  9. M. Nevière, D. Maystre, P. Vincent, “Application du Calcul des Modes de Propagation a L'Étude Théorique des Anomalies des Réseaux Recouverts de Diélectrique,” J. Opt. Paris 8, 231 (1977).
  10. E. G. Loewen, M. Nevière, “Dielectric Coated Gratings: A Curious Property,” Appl. Opt. 16, 3009 (1977).
    [CrossRef] [PubMed]
  11. V. Shah, T. Tamir, “Brewster Phenomena in Lossy Structures,” Opt. Commun. 23, 113 (1977).
    [CrossRef]
  12. V. Shah, T. Tamir, “Anomalous Absorption by Multi-Layered Media,” Opt. Commun. 37, 383 (1981).
    [CrossRef]
  13. M. C. Hutley, D. Maystre, “The Total Absorption of Light by a Diffraction Grating,” Opt. Commun. 19, 431 (1976).
    [CrossRef]
  14. D. Maystre, “Scattering of p-Polarized Light from Au and Ag Gratings and the Effect of Surface Polaritons: A Comment,” Surf. Sci. 164, L823 (1985).
    [CrossRef]
  15. M. G. Moharam, T. K. Gaylord, “Diffraction Analysis of Dielectric Surface-Relief Gratings,” J. Opt. Soc. Am. 72, 1385 (1982).
    [CrossRef]
  16. T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-Reflectivity High Spatial-Frequency Rectangular-Groove Dielectric Surface-Relief Gratings,” Appl. Opt. 25, 4562 (1986).
    [CrossRef] [PubMed]
  17. R. C. Enger, S. K. Case, “High Frequency Holographic Transmission Gratings in Photoresist,” J. Opt. Soc. Am. 73, 1113 (1983).
    [CrossRef]
  18. R. C. Enger, S. K. Case, “Optical Elements with Ultrahigh Spatial-Frequency Surface Corrugations,” Appl. Opt. 22, 3220 (1983).
    [CrossRef] [PubMed]
  19. S. J. Wilson, M. C. Hutley, “The Optical Properties of'Moth Eye' Antireflection Surfaces,” Opt. Acta 29, 993 (1982).
    [CrossRef]
  20. M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Metallic Surface-Relief Gratings,” J. Opt. Soc. Am. A 3, 1780 (1986).
    [CrossRef]
  21. O. Wiener, “Die Theorie des Mischkörpers für das Feld der Stationären Strömung,” Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509 (1912).
  22. Z. Hashin, S. Shtrikman, “A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials,” J. Appl. Phys. 33, 3125 (1962).
    [CrossRef]
  23. R. B. Stephens, P. Sheng, “Acoustic Reflections from Complex Strata,” Geophysics 50, 1100 (1985).
    [CrossRef]
  24. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).
  25. G. Hass, L. Hadley, “Optical Properties of Metals,” in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1972), p. 6–119.
  26. T. N. Shkliarevskii, V. G. Padalka, “Measurements of the Optical Constants of Copper, Gold, and Nickel in the Infrared Region of the Spectrum,” Opt. Spectrosc. USSR 6, 45 (1959).
  27. M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Planar-Grating Diffraction,” J. Opt. Soc. Am. 71, 811 (1981).
    [CrossRef]
  28. W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction Characteristics of Planar Absorption Gratings,” Appl. Phys. B 32, 1520 (1983).
    [CrossRef]
  29. R. E. Burge, “Zone Plates for X-Ray Microscopy at 300-Å Resolution,” J. Opt. Soc. Am. A 3 (13), P21 (1986).

1986 (3)

1985 (2)

D. Maystre, “Scattering of p-Polarized Light from Au and Ag Gratings and the Effect of Surface Polaritons: A Comment,” Surf. Sci. 164, L823 (1985).
[CrossRef]

R. B. Stephens, P. Sheng, “Acoustic Reflections from Complex Strata,” Geophysics 50, 1100 (1985).
[CrossRef]

1984 (1)

G. W. Ford, W. H. Weber, “Electromagnetic Interactions of Molecules with Metal Surfaces,” Phys. Rep. 113, 195 (1984).
[CrossRef]

1983 (4)

P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-Selective Absorption Enhancement in Thin-Film Solar Cells,” Appl. Phys. Lett. 43, 579 (1983).
[CrossRef]

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction Characteristics of Planar Absorption Gratings,” Appl. Phys. B 32, 1520 (1983).
[CrossRef]

R. C. Enger, S. K. Case, “Optical Elements with Ultrahigh Spatial-Frequency Surface Corrugations,” Appl. Opt. 22, 3220 (1983).
[CrossRef] [PubMed]

R. C. Enger, S. K. Case, “High Frequency Holographic Transmission Gratings in Photoresist,” J. Opt. Soc. Am. 73, 1113 (1983).
[CrossRef]

1982 (2)

M. G. Moharam, T. K. Gaylord, “Diffraction Analysis of Dielectric Surface-Relief Gratings,” J. Opt. Soc. Am. 72, 1385 (1982).
[CrossRef]

S. J. Wilson, M. C. Hutley, “The Optical Properties of'Moth Eye' Antireflection Surfaces,” Opt. Acta 29, 993 (1982).
[CrossRef]

1981 (2)

1977 (4)

E. G. Loewen, M. Nevière, “Dielectric Coated Gratings: A Curious Property,” Appl. Opt. 16, 3009 (1977).
[CrossRef] [PubMed]

D. Maystre, M. Nevière, “Sur Une Méthode D'Étude Théorique Quantitative des Anomalies de Wood des Réseaux de Diffraction: Application aux Anomalies de Plasmons,” J. Opt. Paris 8, 165 (1977).
[CrossRef]

M. Nevière, D. Maystre, P. Vincent, “Application du Calcul des Modes de Propagation a L'Étude Théorique des Anomalies des Réseaux Recouverts de Diélectrique,” J. Opt. Paris 8, 231 (1977).

V. Shah, T. Tamir, “Brewster Phenomena in Lossy Structures,” Opt. Commun. 23, 113 (1977).
[CrossRef]

1976 (2)

M. C. Hutley, D. Maystre, “The Total Absorption of Light by a Diffraction Grating,” Opt. Commun. 19, 431 (1976).
[CrossRef]

D. Maystre, R. Petit, “Brewster Incidence for Metallic Gratings,” Opt. Commun. 17, 196 (1976).
[CrossRef]

1974 (1)

R. C. McPhedran, D. Maystre, “A Detailed Theoretical Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 21, 413 (1974).
[CrossRef]

1973 (1)

M. C. Hutley, V. M. Bird, “A Detailed Experimental Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 20, 771 (1973).
[CrossRef]

1966 (1)

1962 (1)

Z. Hashin, S. Shtrikman, “A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials,” J. Appl. Phys. 33, 3125 (1962).
[CrossRef]

1959 (1)

T. N. Shkliarevskii, V. G. Padalka, “Measurements of the Optical Constants of Copper, Gold, and Nickel in the Infrared Region of the Spectrum,” Opt. Spectrosc. USSR 6, 45 (1959).

1935 (1)

R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 48, 928 (1935).
[CrossRef]

1912 (1)

O. Wiener, “Die Theorie des Mischkörpers für das Feld der Stationären Strömung,” Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509 (1912).

Baird, W. E.

T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-Reflectivity High Spatial-Frequency Rectangular-Groove Dielectric Surface-Relief Gratings,” Appl. Opt. 25, 4562 (1986).
[CrossRef] [PubMed]

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction Characteristics of Planar Absorption Gratings,” Appl. Phys. B 32, 1520 (1983).
[CrossRef]

Bird, V. M.

M. C. Hutley, V. M. Bird, “A Detailed Experimental Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 20, 771 (1973).
[CrossRef]

Bloch, A. N.

P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-Selective Absorption Enhancement in Thin-Film Solar Cells,” Appl. Phys. Lett. 43, 579 (1983).
[CrossRef]

Born, M.

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

Burge, R. E.

R. E. Burge, “Zone Plates for X-Ray Microscopy at 300-Å Resolution,” J. Opt. Soc. Am. A 3 (13), P21 (1986).

Case, S. K.

Enger, R. C.

Ford, G. W.

G. W. Ford, W. H. Weber, “Electromagnetic Interactions of Molecules with Metal Surfaces,” Phys. Rep. 113, 195 (1984).
[CrossRef]

Gaylord, T. K.

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), p. 6–119.

Hägglund, J.

Hashin, Z.

Z. Hashin, S. Shtrikman, “A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials,” J. Appl. Phys. 33, 3125 (1962).
[CrossRef]

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), p. 6–119.

Hutley, M. C.

S. J. Wilson, M. C. Hutley, “The Optical Properties of'Moth Eye' Antireflection Surfaces,” Opt. Acta 29, 993 (1982).
[CrossRef]

M. C. Hutley, D. Maystre, “The Total Absorption of Light by a Diffraction Grating,” Opt. Commun. 19, 431 (1976).
[CrossRef]

M. C. Hutley, V. M. Bird, “A Detailed Experimental Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 20, 771 (1973).
[CrossRef]

Loewen, E. G.

Maystre, D.

D. Maystre, “Scattering of p-Polarized Light from Au and Ag Gratings and the Effect of Surface Polaritons: A Comment,” Surf. Sci. 164, L823 (1985).
[CrossRef]

D. Maystre, M. Nevière, “Sur Une Méthode D'Étude Théorique Quantitative des Anomalies de Wood des Réseaux de Diffraction: Application aux Anomalies de Plasmons,” J. Opt. Paris 8, 165 (1977).
[CrossRef]

M. Nevière, D. Maystre, P. Vincent, “Application du Calcul des Modes de Propagation a L'Étude Théorique des Anomalies des Réseaux Recouverts de Diélectrique,” J. Opt. Paris 8, 231 (1977).

D. Maystre, R. Petit, “Brewster Incidence for Metallic Gratings,” Opt. Commun. 17, 196 (1976).
[CrossRef]

M. C. Hutley, D. Maystre, “The Total Absorption of Light by a Diffraction Grating,” Opt. Commun. 19, 431 (1976).
[CrossRef]

R. C. McPhedran, D. Maystre, “A Detailed Theoretical Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 21, 413 (1974).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, D. Maystre, “A Detailed Theoretical Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 21, 413 (1974).
[CrossRef]

Moharam, M. G.

Nevière, M.

E. G. Loewen, M. Nevière, “Dielectric Coated Gratings: A Curious Property,” Appl. Opt. 16, 3009 (1977).
[CrossRef] [PubMed]

D. Maystre, M. Nevière, “Sur Une Méthode D'Étude Théorique Quantitative des Anomalies de Wood des Réseaux de Diffraction: Application aux Anomalies de Plasmons,” J. Opt. Paris 8, 165 (1977).
[CrossRef]

M. Nevière, D. Maystre, P. Vincent, “Application du Calcul des Modes de Propagation a L'Étude Théorique des Anomalies des Réseaux Recouverts de Diélectrique,” J. Opt. Paris 8, 231 (1977).

Padalka, V. G.

T. N. Shkliarevskii, V. G. Padalka, “Measurements of the Optical Constants of Copper, Gold, and Nickel in the Infrared Region of the Spectrum,” Opt. Spectrosc. USSR 6, 45 (1959).

Petit, R.

D. Maystre, R. Petit, “Brewster Incidence for Metallic Gratings,” Opt. Commun. 17, 196 (1976).
[CrossRef]

Sellberg, F.

Shah, V.

V. Shah, T. Tamir, “Anomalous Absorption by Multi-Layered Media,” Opt. Commun. 37, 383 (1981).
[CrossRef]

V. Shah, T. Tamir, “Brewster Phenomena in Lossy Structures,” Opt. Commun. 23, 113 (1977).
[CrossRef]

Sheng, P.

R. B. Stephens, P. Sheng, “Acoustic Reflections from Complex Strata,” Geophysics 50, 1100 (1985).
[CrossRef]

P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-Selective Absorption Enhancement in Thin-Film Solar Cells,” Appl. Phys. Lett. 43, 579 (1983).
[CrossRef]

Shkliarevskii, T. N.

T. N. Shkliarevskii, V. G. Padalka, “Measurements of the Optical Constants of Copper, Gold, and Nickel in the Infrared Region of the Spectrum,” Opt. Spectrosc. USSR 6, 45 (1959).

Shtrikman, S.

Z. Hashin, S. Shtrikman, “A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials,” J. Appl. Phys. 33, 3125 (1962).
[CrossRef]

Stephens, R. B.

R. B. Stephens, P. Sheng, “Acoustic Reflections from Complex Strata,” Geophysics 50, 1100 (1985).
[CrossRef]

Stepleman, R. S.

P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-Selective Absorption Enhancement in Thin-Film Solar Cells,” Appl. Phys. Lett. 43, 579 (1983).
[CrossRef]

Tamir, T.

V. Shah, T. Tamir, “Anomalous Absorption by Multi-Layered Media,” Opt. Commun. 37, 383 (1981).
[CrossRef]

V. Shah, T. Tamir, “Brewster Phenomena in Lossy Structures,” Opt. Commun. 23, 113 (1977).
[CrossRef]

Vincent, P.

M. Nevière, D. Maystre, P. Vincent, “Application du Calcul des Modes de Propagation a L'Étude Théorique des Anomalies des Réseaux Recouverts de Diélectrique,” J. Opt. Paris 8, 231 (1977).

Weber, W. H.

G. W. Ford, W. H. Weber, “Electromagnetic Interactions of Molecules with Metal Surfaces,” Phys. Rep. 113, 195 (1984).
[CrossRef]

Wiener, O.

O. Wiener, “Die Theorie des Mischkörpers für das Feld der Stationären Strömung,” Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509 (1912).

Wilson, S. J.

S. J. Wilson, M. C. Hutley, “The Optical Properties of'Moth Eye' Antireflection Surfaces,” Opt. Acta 29, 993 (1982).
[CrossRef]

Wolf, E.

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

Wood, R. W.

R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 48, 928 (1935).
[CrossRef]

Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig (1)

O. Wiener, “Die Theorie des Mischkörpers für das Feld der Stationären Strömung,” Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509 (1912).

Appl. Opt. (3)

Appl. Phys. B (1)

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction Characteristics of Planar Absorption Gratings,” Appl. Phys. B 32, 1520 (1983).
[CrossRef]

Appl. Phys. Lett. (1)

P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-Selective Absorption Enhancement in Thin-Film Solar Cells,” Appl. Phys. Lett. 43, 579 (1983).
[CrossRef]

Geophysics (1)

R. B. Stephens, P. Sheng, “Acoustic Reflections from Complex Strata,” Geophysics 50, 1100 (1985).
[CrossRef]

J. Appl. Phys. (1)

Z. Hashin, S. Shtrikman, “A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials,” J. Appl. Phys. 33, 3125 (1962).
[CrossRef]

J. Opt. Paris (2)

D. Maystre, M. Nevière, “Sur Une Méthode D'Étude Théorique Quantitative des Anomalies de Wood des Réseaux de Diffraction: Application aux Anomalies de Plasmons,” J. Opt. Paris 8, 165 (1977).
[CrossRef]

M. Nevière, D. Maystre, P. Vincent, “Application du Calcul des Modes de Propagation a L'Étude Théorique des Anomalies des Réseaux Recouverts de Diélectrique,” J. Opt. Paris 8, 231 (1977).

J. Opt. Soc. Am. (4)

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

R. E. Burge, “Zone Plates for X-Ray Microscopy at 300-Å Resolution,” J. Opt. Soc. Am. A 3 (13), P21 (1986).

M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Metallic Surface-Relief Gratings,” J. Opt. Soc. Am. A 3, 1780 (1986).
[CrossRef]

Opt. Acta (3)

M. C. Hutley, V. M. Bird, “A Detailed Experimental Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 20, 771 (1973).
[CrossRef]

R. C. McPhedran, D. Maystre, “A Detailed Theoretical Study of the Anomalies of a Sinusoidal Diffraction Grating,” Opt. Acta 21, 413 (1974).
[CrossRef]

S. J. Wilson, M. C. Hutley, “The Optical Properties of'Moth Eye' Antireflection Surfaces,” Opt. Acta 29, 993 (1982).
[CrossRef]

Opt. Commun. (4)

D. Maystre, R. Petit, “Brewster Incidence for Metallic Gratings,” Opt. Commun. 17, 196 (1976).
[CrossRef]

V. Shah, T. Tamir, “Brewster Phenomena in Lossy Structures,” Opt. Commun. 23, 113 (1977).
[CrossRef]

V. Shah, T. Tamir, “Anomalous Absorption by Multi-Layered Media,” Opt. Commun. 37, 383 (1981).
[CrossRef]

M. C. Hutley, D. Maystre, “The Total Absorption of Light by a Diffraction Grating,” Opt. Commun. 19, 431 (1976).
[CrossRef]

Opt. Spectrosc. USSR (1)

T. N. Shkliarevskii, V. G. Padalka, “Measurements of the Optical Constants of Copper, Gold, and Nickel in the Infrared Region of the Spectrum,” Opt. Spectrosc. USSR 6, 45 (1959).

Phys. Rep. (1)

G. W. Ford, W. H. Weber, “Electromagnetic Interactions of Molecules with Metal Surfaces,” Phys. Rep. 113, 195 (1984).
[CrossRef]

Phys. Rev. (1)

R. W. Wood, “Anomalous Diffraction Gratings,” Phys. Rev. 48, 928 (1935).
[CrossRef]

Surf. Sci. (1)

D. Maystre, “Scattering of p-Polarized Light from Au and Ag Gratings and the Effect of Surface Polaritons: A Comment,” Surf. Sci. 164, L823 (1985).
[CrossRef]

Other (2)

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

G. Hass, L. Hadley, “Optical Properties of Metals,” in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1972), p. 6–119.

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Figures (13)

Fig. 1
Fig. 1

Geometry of an electromagnetic wave in a lossless medium (region 1) incident on a lossy layer (region 2) on top of a lossy substrate (region 3).

Fig. 2
Fig. 2

Diffraction geometry of a lossy rectangular-groove surface-relief grating.

Fig. 3
Fig. 3

Schematic of a transmission line of characteristic impedance Z2 and length d terminated with a characteristic impedance of Z3.

Fig. 4
Fig. 4

(a) Real part n2 of the complex refractive index, (b) the extinction coefficient κ2, (c) the filling factor F, and (d) the normalized groove depth d0 to produce zero reflectivity for EK polarization in the range of free-space wavelengths from λ0 = 0.44 to 12.0 μm at normal incidence on gold.

Fig. 5
Fig. 5

(a) Real part n2 of the complex refractive index, (b) the extinction coefficient κ2, (c) the filling factor F, and (d) the normalized groove depth d0 to produce zero reflectivity for HK polarization in the range of free-space wavelengths from λ0 = 0.44 to 12.0 μm. at normal incidence on gold.

Fig. 6
Fig. 6

(a) Real part n2 of the complex refractive index, (b) the extinction coefficient κ2, (c) the filling factor F, and (d) the normalized groove depth d0 to produce zero reflectivity for TE polarization of free-space wavelength λ0 = 0.5 μm for the entire range of incident angles on gold.

Fig. 7
Fig. 7

(a) Real part n2 of the complex refractive index, (b) the extinction coefficient κ2, (c) the filling factor F, and (d) the normalized groove depth d0 to produce zero reflectivity for TM polarization of free-space wavelength λ0 = 0.5 μm for the entire range of incident angles on gold.

Fig. 8
Fig. 8

(a) Fraction of power reflected as a function of thickness (solid curve) for a free-space wavelength λ0 = 0.5 μm normally incident on a lossy layer of n2 = 0.78590 and κ2 = 0.21655 on gold; zero reflectivity occurs for d0 = 0.47779. Fraction of power reflected as a function of groove depth (dotted curve) for a EK polarized wave with λ0 = 0.5 μm normally incident on a gold rectangular-groove surface-relief grating with F = 0.11689 and λ/″ = 10. (b) Fraction of power reflected as a function of thickness (solid curve) for a free-space wavelength λ0 = 0.5 μm normally incident on a lossy layer of n2 = 1.84736 and k2 = 0.41372 on gold; zero reflectivity occurs for d0 = 0.083629. Fraction of power reflected as a function of groove depth (dotted curve) for a HK polarized wave with λ0 = 0.5 μm normally incident on a gold rectangular-groove surface-relief grating with F = 0.63841 and λ/″ = 20.

Fig. 9
Fig. 9

(a) Fraction of power reflected as a function of thickness for a free-space wavelength λ0 = 1.0 μm normally incident on a lossy layer of n2 = 0.42484 and κ2 = 0.062228 on gold. These data correspond to a EK polarized wave with λ0 = 1.0 μm normally incident on a rectangular-groove surface-relief grating with F = 0.017909 in the long-wavelength limit. Zero reflectivity occurs for a thickness or a groove depth of d0 = 1.12499. (b) Fraction of power reflected as a function of thickness for a free-space wavelength λ0 = 1.0 μm normally incident on a lossy layer of n2 = 10.4651 and κ2 = 0.81736 on gold. These data correspond to a HK polarized wave with λ0 = 1.0 μm normally incident on a rectangular-groove surface-relief grating with F = 0.96957 in the long-wavelength limit. Zero reflectivity occurs for a thickness or groove depth of d0 = 0.0087496.

Fig. 10
Fig. 10

(a) Fraction of power reflected as a function of thickness for a free-space wavelength λ0 = 10.0 μm normally incident on a lossy layer of n2 = 0.64895 and κ2 = 0.16372 on gold. These data correspond to a EK polarized wave with λ0 = 10.0 μm normally incident on a rectangular-groove surface-relief grating with F = 0.00013687 in the long-wavelength limit. Zero reflectivity occurs for a thickness or groove depth of d/λ0 = 0.70263. (b) Fraction of power reflected as a function of thickness for a free-space wavelength λ0 = 10.0 μm normally incident on a lossy layer of n2 = 17.4963 and κ2 = 0.18852 on gold. These data correspond to a HK polarized wave with λ0 = 10.0 μm normally incident on a rectangular-groove surface-relief grating with F = 0.99653 in the long-wavelength limit. Zero reflectivity occurs for a thickness or groove depth of d0 = 0.012049.

Fig. 11
Fig. 11

Fraction of power reflected from a gold rectangular-groove surface-relief grating as a function of grating spatial frequency (expressed as λ/Λ) at λ0 = 0.5 μm for EK and HK polarizations as calculated by rigorous coupled-wave grating diffraction analysis. The filling factors and groove depths are those obtained from analysis of the present work for (F = 0.11689 and d0 = 0.47779 for EK polarization and F = 0.63841 and d0 = 0.083629 for HK polarization).

Fig. 12
Fig. 12

Schematic illustration of possible equivalence between stairstep gratings and multilayer coatings.

Fig. 13
Fig. 13

Flow chart showing calculational procedure for obtaining the roots of the normalized equations as given in the Appendix.

Tables (1)

Tables Icon

Table I Complex Refractive Index, n22, for Bulk Gold for Various Free Space Wavelengths λ0 (Ref. 25)

Equations (45)

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n 1 sin ϕ 1 = ( n 2 j κ 2 ) sin ϕ 2 = ( n 3 j κ 3 ) sin ϕ 3 .
[ 1 E r ] = 1 τ 12 τ 23 [ 1 ± ρ 12 ± ρ 12 1 ] [ exp ( γ d ) 0 0 exp ( γ d ) ] × [ 1 ± ρ 23 ± ρ 23 1 ] [ E t 0 ] ,
E r = ± ρ 12 exp ( γ d ) + ρ 23 exp ( γ d ) exp ( γ d ) + ρ 12 ρ 23 exp ( γ d ) ,
τ 12 = 2 Z 2 / ( Z 2 + Z 1 ) , τ 23 = 2 Z 3 / ( Z 3 + Z 2 ) , ρ 12 = ( Z 2 Z 1 ) / ( Z 2 + Z 1 ) , ρ 23 = ( Z 3 Z 2 ) / ( Z 3 + Z 2 ) ,
Z 1 = Z 0 / n 1 cos ϕ 1 , Z 2 = Z 0 / ( n 2 j κ 2 ) cos ϕ 2 , Z 3 = Z 0 / ( n 3 j κ 3 ) cos ϕ 3 ,
τ 12 = 2 Z 2 cos ϕ 1 / ( Z 2 + Z 1 ) cos ϕ 2 , τ 23 = 2 Z 3 cos ϕ 2 / ( Z 3 + Z 2 ) cos ϕ 3 , ρ 12 = ( Z 2 Z 1 ) / ( Z 2 + Z 1 ) , ρ 23 = ( Z 3 Z 2 ) / ( Z 3 + Z 2 ) ,
Z 1 = Z 0 cos ϕ 1 / n 1 , Z 2 = Z 0 cos ϕ 2 / ( n 2 j κ 2 ) , Z 3 = Z 0 cos ϕ 3 / ( n 3 j κ 3 ) .
n E K = [ ( 1 F ) n 1 2 cos 2 ϕ 1 + F ( n 3 j κ 3 ) 2 cos 2 ϕ 3 ) ] 1 / 2 .
n H K = [ ( 1 F ) cos 2 ϕ 1 / n 1 2 + F cos 2 ϕ 3 / ( n 3 j κ 3 ) 2 ] 1 / 2 .
Z in = Z 2 Z 3 [ 1 + j tanh ( α d ) ] tan ( β d ) + Z 2 [ tanh ( α d ) + j tan ( β d ) ] Z 2 [ 1 + j tanh ( α d ) ] tan ( β d ) + Z 3 [ tanh ( α d ) + j tan ( β d ) ] .
N 1 = n 1 cos ϕ 1 , N 2 = Re [ ( n 2 j κ 2 ) cos ϕ 2 ] , N 3 = Re [ ( n 3 j κ 3 ) cos ϕ 3 ] , K 1 = 0 , K 2 = Im [ ( n 2 j κ 2 ) cos ϕ 2 ] , K 3 = Im [ ( n 3 j κ 3 ) cos ϕ 3 ] ,
N 1 = n 1 cos ϕ 1 , N 2 = Re [ ( n 2 j κ 2 ) cos ϕ 2 ] , N 3 = Re [ ( n 3 j κ 3 ) cos ϕ 3 ] , K 1 = 0 , K 2 = Im [ ( n 2 j κ 2 ) cos ϕ 2 ] , K 3 = Im [ ( n 3 j κ 3 ) cos ϕ 3 ] ,
Z in = Z 1 .
N 1 N 3 tanh ( k 0 K 2 d ) + N 1 K 3 tan ( k 0 K 2 d ) + ( N 1 N 2 N 2 N 3 + K 2 K 3 ) + ( N 1 K 2 N 2 K 3 K 2 N 3 ) tanh ( k 0 K 2 d ) tan ( k 0 K 2 d ) ( N 2 2 K 2 2 ) tanh ( k 0 K 2 d ) 2 N 2 K 2 tan ( k 0 K 2 d ) = 0 .
N 1 K 3 tanh ( k 0 K 2 d ) + N 1 N 3 tan ( k 0 K 2 d ) ( N 1 K 2 N 2 K 3 K 2 N 3 ) + ( N 1 N 2 N 2 N 3 + K 2 K 3 ) tanh ( k 0 K 2 d ) tan ( k 0 K 2 d ) ( N 2 2 K 2 2 ) tan ( k 0 K 2 d ) + 2 N 2 K 2 tanh ( k 0 K 2 d ) = 0 .
N 2 j K 2 = n E K ,
N 2 2 K 2 2 = ( 1 F ) N 1 2 + F ( N 3 2 K 3 2 ) ,
N 2 K 2 = F N 3 K 3 .
N 2 j K 2 = n H K ,
N 2 2 K 2 2 = N 1 2 [ ( 1 F ) ( N 3 2 + K 3 2 ) 2 + F N 1 2 ( N 3 2 + K 3 2 ) ] [ ( 1 F ) ( N 3 2 K 3 2 ) + F N 1 2 ] 2 + [ 2 ( 1 F ) N 3 K 3 ] 2 ,
N 2 K 2 = F N 1 4 N 3 K 3 [ ( 1 F ) ( N 3 2 K 3 2 ) + F N 1 2 ] 2 + [ 2 ( 1 F ) N 3 K 3 ] 2 .
tan ( k 0 K 2 d ) = E + G tanh ( k 0 N 2 d ) ,
tanh 2 ( k 0 K 2 d ) + [ ( N 3 + K 3 G + D E A B G ) / D G ] tanh ( k 0 K 2 d ) + 1 = 0 ,
N 2 = N 2 = [ ( N 1 / 2 ) 1 / 2 B ] / [ ( A 2 + B 2 ) 1 / 2 A ] 1 / 2 ,
K 2 = K 2 = { ( N 1 / 2 ) [ ( A 2 + B 2 ) 1 / 2 A ] } 1 / 2 ,
A = [ ( 1 F ) N 1 2 + F ( N 3 2 K 3 2 ) ] / N 1 ,
B = 2 F N 3 K 3 / N 1 ,
C = N 2 ( N 2 N 3 K 2 K 3 ) / N 1 ,
D = K 2 ( N 2 K 3 + K 2 N 3 ) / N 1 ,
E = ( C 2 + D 2 ) / ( C K 3 D N 3 B C + A D ) ,
G = ( C N 3 + D K 3 A C B D ) / ( C K 3 D N 3 B C + A D ) .
n 2 = [ ( n 1 / 2 ) 1 / 2 B 0 ] / [ ( A 0 2 + B 0 2 ) 1 / 2 A 0 ] 1 / 2 ,
κ 2 = { ( n 1 / 2 ) [ ( A 0 2 + B 0 2 ) 1 / 2 A 0 ] } 1 / 2 ,
A 0 = [ ( 1 F ) n 1 2 + F ( n 3 2 κ 3 2 ) ] / n 1 ,
B 0 = 2 F n 3 κ 3 / n 1 .
N 2 = ( 1 / 2 ) [ N 2 ( 1 ± r 0 ) ± s 0 K 2 ] ,
K 2 = ( 1 / 2 ) [ K 2 ( 1 ± r 0 ) s 0 N 2 ] ,
r 0 = Φ 1 / 2 cos ( ϑ / 2 ) ,
s 0 = Φ 1 / 2 sin ( ϑ / 2 ) ,
Φ = [ ( N 2 2 n 1 sin ϕ 1 ) 2 + K 2 2 ] 1 / 2 [ ( N 2 + 2 n 1 sin ϕ 1 ) 2 + K 2 2 ] 1 / 2 N 2 2 + K 2 2 ,
ϑ = tan 1 [ K 2 / ( N 2 + 2 n 1 sin ϕ 1 ) ] + tan 1 [ K 2 / ( N 2 2 n 1 sin ϕ 1 ) ] 2 tan 1 ( K 2 / N 2 ) .
A = N 1 [ ( 1 F ) ( N 3 2 + K 3 2 ) 2 + F N 1 2 ( N 3 2 K 3 2 ) ] [ ( 1 F ) ( N 3 2 K 3 2 ) F N 1 2 ] 2 + [ 2 ( 1 F ) N 3 K 3 ] 2 ,
B = 2 F N 1 3 N 3 K 3 [ ( 1 F ) ( N 3 2 K 3 2 ) F N 1 2 ] 2 + [ 2 ( 1 F ) N 3 K 3 ] 2 ,
A 0 = [ 2 n 1 sin 2 ϕ 1 ( 1 ± r 0 ) ] / [ ( 1 ± r 0 ) 2 + s 0 2 ] ,
B 0 = ± ( 2 n 1 sin 2 ϕ 1 s 0 ) / [ ( 1 ± r 0 ) 2 + s 0 2 ] ,

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