Abstract

Using rigorous coupled-wave analysis, high spatial-frequency rectangular-groove surface-relief phase gratings are shown to be capable of exhibiting zero reflectivity. Thus these corrugated surfaces may act as antireflection coatings in a variety of applications. The diffraction characteristics of rectangular-groove surface-relief gratings are presented for several ratios of incident wavelength to grating period as a function of filling factor, groove depth, angle of incidence, and polarization. The conditions for zero reflectivity are identified. Results are compared with single-homogeneous-layer approximate theory results. In the limit of long wavelengths for an electromagnetic wave in a dielectric of refractive index n1 normally incident on a dielectric of index n2, it is determined that for antireflection behavior, the grating groove depth should be λ/4(n1n2)1/2 and the filling factor should be n1/(n1 + n2) or n2/(n1 + n2) for the electric field perpendicular or parallel to the grating vector, respectively. The spectral and angular responses of these gratings are like those of single-homogeneous-layer antireflection coatings. These gratings also exhibit birefringent retardation.

© 1986 Optical Society of America

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References

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  1. M. G. Moharam, T. K. Gaylord, “Diffraction Analysis of Dielectric Surface-Relief Gratings,” J. Opt. Soc. Am. 72, 1385 (1982).
    [CrossRef]
  2. R. C. Enger, S. K. Case, “High-Frequency Holographic Transmission Gratings in Photoresist,” J. Opt. Soc. Am. 73, 1113 (1983).
    [CrossRef]
  3. R. C. Enger, S. K. Case, “Optical Elements with Ultrahigh Spatial-Frequency Surface Corrugations,” Appl. Opt. 22, 3220 (1983).
    [CrossRef] [PubMed]
  4. Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection Effect in Ultrahigh Spatial-Frequency Holographic Relief Gratings,” in Technical Digest, Topical Meeting on Holography (Optical Society of America, Washington, DC, 1986), paper TuB2.
  5. P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-Selective Absorption Enhancement in Thin-Film Solar Cells,” Appl. Phys. Lett. 43, 579 (1983).
    [CrossRef]
  6. H. Dammann, “Color Separation Gratings,” Appl. Opt. 17, 2273 (1978).
    [CrossRef] [PubMed]
  7. K. Knop, “Diffraction Gratings for Color Filtering in the Zero Diffraction Order,” Appl. Opt. 17, 3598 (1978).
    [CrossRef] [PubMed]
  8. T. K. Gaylord, M. G. Moharam, “Analysis and Applications of Optical Diffraction by Gratings,” Proc. IEEE 73, 894 (1985).
    [CrossRef]
  9. K. Knop, “Rigorous Diffraction Theory for Transmission Phase Gratings with Deep Rectangular Grooves,” J. Opt. Soc. Am. 68, 1206 (1978).
    [CrossRef]
  10. 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 (1982).
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, G. T. Sincerbox, H. Werlich, B. Yung, “Diffraction Characteristics of Photoresist Surface-Relief Gratings,” Appl. Opt. 23, 3214 (1984).
    [CrossRef] [PubMed]
  12. T. K. Gaylord, F. K. Tittel, “Angular Selectivity of Lithium Niobate Volume Holograms,” J. Appl. Phys. 44, 4771 (1973).
    [CrossRef]
  13. M. G. Moharam, T. K. Gaylord, “Rigorous Coupled-Wave Analysis of Planar-Grating Diffraction,” J. Opt. Soc. Am. 71, 811 (1981).
    [CrossRef]
  14. O. Wiener, Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509 (1912).
  15. 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]
  16. R. B. Stephens, P. Sheng, “Acoustic Reflections from Complex Strata,” Geophysics 50, 1100 (1985).
    [CrossRef]
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980).

1985 (2)

T. K. Gaylord, M. G. Moharam, “Analysis and Applications of Optical Diffraction by Gratings,” Proc. IEEE 73, 894 (1985).
[CrossRef]

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

1984 (1)

1983 (3)

1982 (2)

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

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 (1982).
[CrossRef]

1981 (1)

1978 (3)

1973 (1)

T. K. Gaylord, F. K. Tittel, “Angular Selectivity of Lithium Niobate Volume Holograms,” J. Appl. Phys. 44, 4771 (1973).
[CrossRef]

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]

1912 (1)

O. Wiener, Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509 (1912).

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).

Case, S. K.

Dammann, H.

Enger, R. C.

Gaylord, T. K.

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]

Kimura, Y.

Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection Effect in Ultrahigh Spatial-Frequency Holographic Relief Gratings,” in Technical Digest, Topical Meeting on Holography (Optical Society of America, Washington, DC, 1986), paper TuB2.

Knop, K.

Moharam, M. G.

Nishida, N.

Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection Effect in Ultrahigh Spatial-Frequency Holographic Relief Gratings,” in Technical Digest, Topical Meeting on Holography (Optical Society of America, Washington, DC, 1986), paper TuB2.

Ohta, Y.

Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection Effect in Ultrahigh Spatial-Frequency Holographic Relief Gratings,” in Technical Digest, Topical Meeting on Holography (Optical Society of America, Washington, DC, 1986), paper TuB2.

Ono, Y.

Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection Effect in Ultrahigh Spatial-Frequency Holographic Relief Gratings,” in Technical Digest, Topical Meeting on Holography (Optical Society of America, Washington, DC, 1986), paper TuB2.

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 (1982).
[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]

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 (1982).
[CrossRef]

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]

Sincerbox, G. T.

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]

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 (1982).
[CrossRef]

Tittel, F. K.

T. K. Gaylord, F. K. Tittel, “Angular Selectivity of Lithium Niobate Volume Holograms,” J. Appl. Phys. 44, 4771 (1973).
[CrossRef]

Werlich, H.

Wiener, O.

O. Wiener, Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509 (1912).

Wolf, E.

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

Yung, B.

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

O. Wiener, Abh. Math. Phys. Kl. Saechs. Akad. Wiss. Leipzig 32, 509 (1912).

Appl. Opt. (4)

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. (2)

T. K. Gaylord, F. K. Tittel, “Angular Selectivity of Lithium Niobate Volume Holograms,” J. Appl. Phys. 44, 4771 (1973).
[CrossRef]

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. Soc. Am. (4)

Phys. Rev. B (1)

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 (1982).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and Applications of Optical Diffraction by Gratings,” Proc. IEEE 73, 894 (1985).
[CrossRef]

Other (2)

Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection Effect in Ultrahigh Spatial-Frequency Holographic Relief Gratings,” in Technical Digest, Topical Meeting on Holography (Optical Society of America, Washington, DC, 1986), paper TuB2.

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

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Figures (7)

Fig. 1
Fig. 1

Diffraction geometry of surface-relief grating.

Fig. 2
Fig. 2

Minimum reflected power at normal incidence as a function of filling factor with λ/Λ = 1 for (a) EK polarization, (b) EK polarization, and (c) random polarization.

Fig. 3
Fig. 3

Minimum reflected power at normal incidence as a function of filling factor with λ/Λ = 1.5 for (a) EK polarization, (b) EK polarization, and (c) random polarization.

Fig. 4
Fig. 4

Minimum reflected power at normal incidence as a function of filling factor with λ/Λ = 2.5 for (a) EK polarization, (b) EK polarization, and (c) random polarization.

Fig. 5
Fig. 5

Minimum reflected power at normal incidence as a function of filling factor with λ/Λ = 10 for (a) EK polarization, (b) EK polarization, and (c) random polarization.

Fig. 6
Fig. 6

Reflected power at normal incidence as a function of ratio of groove depth (or layer thickness) to incident wavelength for (a) rectangular-groove surface-relief phase grating using rigorous coupled-wave theory (RCWT) for EK polarization, (b) single-homogeneous-layer approximate theory with polarization-dependent refractive index ( n ¯ E K), and (c) single-homogeneous-layer approximate theory with polarization-independent refractive index ( n ¯). Zero reflectivity occurs for the grating at d/λ = 0.205 (or d/λ = 0.5125).

Fig. 7
Fig. 7

Reflected power as a function of angle of incidence for (a) rectangular-groove surface-relief phase grating using rigorous coupled-wave theory (RCWT) for EK (TE polarization), (b) single-homogeneous-layer approximate theory with polarization-dependent refractive index ( n ¯ E K), and (c) single-homogeneous-layer approximate theory with polarization-independent refractive index ( n ¯).

Tables (1)

Tables Icon

Table I Parameters of Rectangular-Profile Grating of n2 = 1.5 to Produce Zero Reflectivity (<5 × 10−5 %) at Normal Incidence in Air (n1 = 1.0)

Equations (10)

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i λ = Λ ( n 1 sin θ + n 2 sin θ i ) ,
i λ / n 1 = Λ ( sin θ + sin θ i ) .
m λ / n 1 = 2 Λ sin θ ,
n ¯ = n 1 + F ( n 2 - n 1 ) .
n ¯ E K = [ n 1 2 ( 1 - F ) + n 2 2 F ] 1 / 2 .
n ¯ E K = [ ( 1 - F ) / n 1 2 + F / n 2 2 ] - 1 / 2 .
d = λ / 4 ( n 1 n 2 ) 1 / 2 ,
F E K = n 1 / ( n 1 + n 2 )
F E K = n 2 / ( n 1 + n 2 )
Γ = - π [ ( n 2 / n 1 ) - 1 ] 2 / 2 3 / 2 [ ( n 2 / n 1 ) 2 + 1 ] 1 / 2 ( n 2 / n 1 ) 1 / 2 .

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