Abstract

High-spatial-frequency, surface-relief binary gratings have been shown to have diffraction properties that are similar to homogeneous layers of equivalent refractive indices, which depend on the grating characteristics, angle of incidence, and polarization. Thus these gratings in the long-wavelength limit could be used as equivalent thin-film coatings. Because of their polarization discrimination these gratings can function as polarization-selective mirrors. A procedure for designing these gratings to be antireflective for one polarization (TE or TM) and to maximize their reflectivity for the orthogonal polarization (TM or TE) is presented. Multilevel stairstep gratings can similarly exhibit characteristics that resemble those of multilayer antireflection coatings (quarter-wave impedance transformers), thus permitting a broader wavelength bandpass. A systematic procedure for designing multilevel stairstep gratings to operate as multilayer thin-film antireflection surfaces is presented. These design methods are valid for both TE and TM polarizations and for any angle of incidence. Example designs are presented, and the rigorous coupled-wave diffraction analysis is used to evaluate the performance of these gratings as functions of the ratio of their period to the incident wavelength. Comparisons are included with homogeneous layers that are equivalent to the gratings in the long-wavelength limit.

© 1992 Optical Society of America

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References

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    [CrossRef] [PubMed]
  7. N. F. Hartman, T. K. Gaylord, “Antireflection gold surface-relief gratings: experimental characteristics,” Appl. Opt. 27, 3738–3743 (1988).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  17. E. D. Palik, Ed., Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).
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    [CrossRef]
  19. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  20. E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. 4, 2061–2080 (1987).
    [CrossRef]
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    [CrossRef]

1990 (1)

1988 (2)

1987 (3)

1986 (1)

1985 (2)

R. B. Stephens, P. Sheng, “Acoustic reflections from complex strata,” Geophysics 50, 1100–1107 (1985).

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1983 (4)

1982 (1)

1978 (2)

1962 (1)

Z. Hashin, S. Shtrikman, “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” J. Appl. Phys. 33, 3125–3131 (1962).
[CrossRef]

1912 (1)

O. Wiener, “Die Theorie des Mischkorpers fur das Feld der Stationären Stromung,” Abh. Math. Phys. K1. Sechs. Akad. Wiss. Leipzig 32, 509–604 (1912).

Baird, W. E.

Bloch, A. N.

P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43, 579–581 (1983).
[CrossRef]

Born, M.

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

Case, S. K.

Collin, R. E.

R. E. Collin, Foundations of Microwave Engineering (McGraw-Hill, New York, 1966).

Dammann, H.

Enger, R. C.

Gaylord, T. K.

E. N. Glytsis, T. K. Gaylord, “Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction,” J. Opt. Soc. Am. A 7, 1399–1420 (1990).
[CrossRef]

N. F. Hartman, T. K. Gaylord, “Antireflection gold surface-relief gratings: experimental characteristics,” Appl. Opt. 27, 3738–3743 (1988).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Antireflection surface structure: dielectric layer(s) over high spatial-frequency surface-relief grating on a lossy substrate,” Appl. Opt. 27, 4288–4304 (1988).
[CrossRef] [PubMed]

T. K. Gaylord, E. N. Glytsis, M. G. Moharam, “Zero-reflectivity homogeneous layers and high spatial-frequency surface-relief gratings on lossy substrates,” Appl. Opt. 26, 3123–3135 (1987).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. 4, 2061–2080 (1987).
[CrossRef]

T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
[CrossRef] [PubMed]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

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]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

Glytsis, E. N.

Hartman, N. F.

Hashin, Z.

Z. Hashin, S. Shtrikman, “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” J. Appl. Phys. 33, 3125–3131 (1962).
[CrossRef]

Kimura, Y.

Knop, K.

Moharam, M. G.

Nishida, N.

Ohta, Y.

Ono, Y.

Sheng, P.

R. B. Stephens, P. Sheng, “Acoustic reflections from complex strata,” Geophysics 50, 1100–1107 (1985).

P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43, 579–581 (1983).
[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, 3125–3131 (1962).
[CrossRef]

Stephens, R. B.

R. B. Stephens, P. Sheng, “Acoustic reflections from complex strata,” Geophysics 50, 1100–1107 (1985).

Stepleman, R. S.

P. Sheng, A. N. Bloch, R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43, 579–581 (1983).
[CrossRef]

Wiener, O.

O. Wiener, “Die Theorie des Mischkorpers fur das Feld der Stationären Stromung,” Abh. Math. Phys. K1. Sechs. Akad. Wiss. Leipzig 32, 509–604 (1912).

Wolf, E.

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

Abh. Math. Phys. K1. Sechs. Akad. Wiss. Leipzig (1)

O. Wiener, “Die Theorie des Mischkorpers fur das Feld der Stationären Stromung,” Abh. Math. Phys. K1. Sechs. Akad. Wiss. Leipzig 32, 509–604 (1912).

Appl. Opt. (8)

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, 579–581 (1983).
[CrossRef]

Geophysics (1)

R. B. Stephens, P. Sheng, “Acoustic reflections from complex strata,” Geophysics 50, 1100–1107 (1985).

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, 3125–3131 (1962).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (1)

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other (3)

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

R. E. Collin, Foundations of Microwave Engineering (McGraw-Hill, New York, 1966).

E. D. Palik, Ed., Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

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Figures (11)

Fig. 1
Fig. 1

(a) Diffraction geometry of a single-level binary surface-relief grating. (b) The equivalent layer model for the single-level grating in the long-wavelength limit.

Fig. 2
Fig. 2

(a) Diffraction geometry of a multilevel (N-level) stairstep surface-relief grating. (b) The equivalent multilayer (N-layer) model for the multilevel stairstep grating in the long-wavelength limit.

Fig. 3
Fig. 3

(a) Transmission line model of the single-layer model. b) Transmission line model of the multilayer model.

Fig. 4
Fig. 4

Reflectivity R of a silicon single-level binary surface-relief grating polarization-selective mirror as a function of the grating spatial frequency (expressed as λ0/Λ) for λ0 = 1.5 μm for TE (solid curve) and TM (dotted–dashed curve) polarizations. The grating characteristics are given in Table 1. (a) AR for TE – max R for TM design and (b) the AR for TM – max R for TE design.

Fig. 5
Fig. 5

Reflectivity R of a silicon single-level binary surface-relief grating polarization-selective mirror as a function of the filling factor F (around the designed value) for λ0 = 1.5 μm and for TE (solid curve) and TM (dotted–dashed curve) polarizations. The grating characteristics are given in Table 1. (a) The AR for TE – max R for TM design and (b) the AR for TM – max R for TE design.

Fig. 6
Fig. 6

Reflectivity R of a silicon single-level binary surface-relief grating polarization-selective mirror as a function of the normalized groove depth (expressed as d0) for λ0 = 1.5 μm, for TE and TM polarization, and for normal (ϕ1 = 0°) incidence. The solid curves represent the RCWA results, while the dashed curves represent the equivalent homogeneous layer results. The grating characteristics are given in Table 1. (a) The AR for TE – max R for TM design and (b) the AR for TM – max R for TE design.

Fig. 7
Fig. 7

Reflectivity R of a silicon single-level binary surface-relief grating polarization-selective mirror as a function of the normalized groove depth (expressed as d0) for λ0 = 1.5 μm, for TE and TM polarization, and for oblique (ϕ1 = 30°) incidence. The solid curves represent the RCWA results, while the dashed curves represent the equivalent homogeneous layer results. The grating characteristics are given in Table 1. (a) The AR for TE – max R for TM design and (b) the AR for TM – max R for TE design.

Fig. 8
Fig. 8

Reflectivity R of a silicon multilevel stairstep surface-relief grating quarter-wave maximally flat impedance transformer (for two and three sections) as a function of the inverse free-space wavelength (1/λ0) for normal (ϕ1 = 0°) incidence. The solid curves represent the RCWA results, while the dashed curves represent the equivalent homogeneous layer results. The grating characteristics are given in Table 2. (a) TE polarization and (b) TM polarization.

Fig. 9
Fig. 9

Reflectivity R of a glass multilevel stairstep surface-relief grating quarter-wave maximally flat impedance transformer (for two and three sections) as a function of the inverse free-space wavelength (1/λ0) for oblique (ϕ1 = 30°) incidence. The solid curves represent the RCWA results, while the dashed curves represent the equivalent homogeneous layer results. The grating characteristics are given in Table 3. (a) TE polarization and (b) TM polarization.

Fig. 10
Fig. 10

Reflectivity R of a silicon multilevel stairstep surface-relief grating quarter-wave equal-ripple impedance transformer (for two and three sections) as a function of the inverse free-space wavelength (1/λ0) for normal (ϕ1 = 0°) incidence. The solid curves represent the RCWA results, while the dashed curves represent the equivalent homogeneous layer results. The grating characteristics are given in Table 2. (a) TE polarization and (b) TM polarization.

Fig. 11
Fig. 11

Reflectivity R of a glass multilevel stairstep surface-relief grating quarter-wave equal-ripple impedance transformer (for two and three sections) as a function of the inverse free-space wavelength (1/λ0) for oblique (ϕ1 = 30°) incidence. The solid curves represent the RCWA results, while the dashed curves represent the equivalent homogeneous layer results. The grating characteristics are given in Table 3. (a) TE polarization and (b) TM polarization.

Tables (3)

Tables Icon

Table 1 Polarization-Selective Grating Mirrorsa

Tables Icon

Table 2 Grating Broadband Antireflection Surfaces on Silicona

Tables Icon

Table 3 Grating Broadband Antireflection Surfaces on Glassa

Equations (19)

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N E K = N 2 , e = n 2 cos ϕ 2 = [ ( 1 - F ) n 1 2 cos 2 ϕ 1 + F n 3 2 cos 2 ϕ 3 ] 1 / 2 = [ ( 1 - F ) N 1 , e 2 + F N 3 , e 2 ] 1 / 2 ,
N H K = N 2 , h = n 2 / cos ϕ 2 = [ ( 1 - F ) cos 2 ϕ 1 / n 1 2 + F cos 2 ϕ 3 / n 3 2 ] - 1 / 2 = [ ( 1 - F ) / N 1 , h 2 + F / N 3 , h 2 ] - 1 / 2 ,
Z in , e = Z 2 , e Z 3 , e + j Z 2 , e tan ( k 0 N 2 , e d ) Z 2 , e + j Z 3 , e tan ( k 0 N 2 , e d ) = Z 1 , e ,
F e = N 1 , e N 1 , e + N 3 , e = n 1 cos ϕ 1 n 1 cos ϕ 1 + n 3 cos ϕ 3 ,
d = ( 2 m + 1 ) λ 0 ( N 1 , e N 3 , e ) 1 / 2             m = 0 , 1 , 2 , .
k 0 N 2 , h d = l π ,             l = 0 , 1 , 2 , ,
N 2 , h N 2 , e = 2 l 2 m + 1 l = 2 m + 1 2 N 2 , h N 2 , e n 2 , h 2 - n 1 2 sin 2 ϕ 1 n 2 , h 2 ,
F h = N 3 , h N 1 , h + N 3 , h = ( n 3 / cos ϕ 3 ) ( n 1 / cos ϕ 1 ) + ( n 3 / cos ϕ 3 ) ,
d = ( 2 m + 1 ) λ 0 ( N 1 , h N 3 , h ) 1 / 2 n 2 , h 2 - n 1 2 sin 2 ϕ 1 n 2 , h 2 m = 0 , 1 , 2 , ,
k 0 N 2 , e d = l π ,             l = 0 , 1 , 2 , ,
N 2 , e N 2 , h = 2 l 2 m + 1 l = 2 m + 1 2 N 2 , h N 2 , e n 2 , h 2 n 2 , h 2 - n 1 2 sin 2 ϕ 1 .
Γ = b N ( - θ ) + ρ N a N ( - θ ) α N ( θ ) + ρ N b N ( θ ) ,
Γ 2 exp ( - j N θ ) [ ρ 0 cos N θ + ρ 1 cos ( N - 2 ) θ + + ρ i cos ( N - 2 i ) θ + ] ,
F i , e = N 2 i , e 2 - N 1 , e 2 N 3 , e 2 - N 1 , e 2             for TE polarization , F i , h = ( 1 / N 2 i , h 2 ) - ( 1 / N 1 , h 2 ) ( 1 / N 3 , h 2 ) - ( 1 / N 1 , h 2 )             for TM polarization .
d i = { λ 0 4 N 2 i , e for TE polarization , λ 0 4 N 2 i , h for TM polarization .
ρ i = 2 - N N 1 - N 3 N 1 + N 3 N ! ( N - i ) ! i ! ,
ln [ N 2 i N 2 ( i + 1 ) ] = 2 - N N ! ( N - i ) ! i ! ln ( N 1 N 3 ) ,
Γ = - r m exp ( - j N θ ) T N ( sec θ m cos θ ) ,
T N ( sec θ m ) = 1 r m N 3 - N 1 N 3 + N 1 .

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