Abstract

Two-dimensional symmetric and asymmetric subwavelength binary gratings are investigated. A method for determining the three effective indices of a two-dimensional (2-D) subwavelength grating is presented, as well as a theoretical formalization for the effective index parallel with the normal to the surface. It is shown that a 2-D asymmetric binary grating on the surface of a dielectric substrate is analogous to a biaxial thin film. If the grating is symmetric, then the two effective indices perpendicular to the normal are equal, and the grating is analogous to a uniaxial thin film. Using these effective indices and the quarter-wave Tschebyscheff synthesis technique, we designed two- and three-level binary gratings to suppress reflections over a broad band. It is shown that for a substrate index of ns = 3.0 a three-level 2-D binary grating reduced reflections below 0.1% from 8 μm to 12 μm.

© 1994 Optical Society of America

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References

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  2. D. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
    [CrossRef] [PubMed]
  3. D. Raguin, G. M. Morris, “Analysis of antireflection-structured surfaces with continuous one-dimensional surface profiles,” Appl. Opt. 32, 2582–2598 (1993).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. 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]
  7. E. N. Glytsis, T. K. Gaylord, “High-spatial-frequency binary and multilevel stairstep gratings: polarization-selective mirrors and broadband antireflection surfaces,” Appl. Opt. 31, 4459–4470 (1992).
    [CrossRef] [PubMed]
  8. D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
    [CrossRef]
  9. P. van der Werf, J. Haisma, “Broadband antireflection coatings for fiber-communication optics,” Appl. Opt. 23, 499–503 (1984).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1980), Chap. 1, pp. 100–117.
  16. 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]
  17. P. Vincent, “A finite difference method for dielectric and conducting cross gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  18. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  19. Soon Ting Han, Yuh-Luen Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2343–2352 (1992).
    [CrossRef] [PubMed]
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    [CrossRef]
  21. L. Young, “Synthesis of multiple antireflection films over a prescribed band,” J. Opt. Soc. Am. 51, 967–974 (1961).
    [CrossRef]

1993 (3)

1992 (3)

1991 (1)

1987 (1)

1986 (1)

1984 (1)

1983 (2)

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[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]

1982 (1)

1979 (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

1978 (1)

P. Vincent, “A finite difference method for dielectric and conducting cross gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

1961 (1)

1957 (1)

H. J. Riblet, “General synthesis of quarter-wave impedance transformers,” IRE Trans. Microwave Theory Tech. MTT-5, 36–43 (1957).
[CrossRef]

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–474 (1956).

Baird, W. E.

Becker, M. F.

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1980), Chap. 1, pp. 100–117.

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Dobrowolski, J. A.

Flanders, D. C.

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Gunning, W. J.

Haggans, C. W.

Hainer, H.

Haisma, J.

Han, Soon Ting

Ho, F.

Kimura, Y.

Kipfer, P.

Kostuk, R. K.

Li, L.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, New York, 1989).

Maystre, D.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Motamedi, M. E.

Nevière, M.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Nishida, N.

Ohta, Y.

Ono, Y.

Raguin, D.

Riblet, H. J.

H. J. Riblet, “General synthesis of quarter-wave impedance transformers,” IRE Trans. Microwave Theory Tech. MTT-5, 36–43 (1957).
[CrossRef]

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–474 (1956).

Southwell, W. H.

Stork, W.

Striebl, N.

Tsao, Yuh-Luen

van der Werf, P.

Vincent, P.

P. Vincent, “A finite difference method for dielectric and conducting cross gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Walser, R. M.

Young, L.

Appl. Opt. (9)

Y. Ono, Y. Kimura, Y. Ohta, N. Nishida, “Antireflection effect in ultrahigh spatial-frequency holographic relief gratings,” Appl. Opt. 26, 1142–1146 (1987).
[CrossRef] [PubMed]

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]

E. N. Glytsis, T. K. Gaylord, “High-spatial-frequency binary and multilevel stairstep gratings: polarization-selective mirrors and broadband antireflection surfaces,” Appl. Opt. 31, 4459–4470 (1992).
[CrossRef] [PubMed]

P. van der Werf, J. Haisma, “Broadband antireflection coatings for fiber-communication optics,” Appl. Opt. 23, 499–503 (1984).
[CrossRef] [PubMed]

J. A. Dobrowolski, F. Ho, “High performance step-down AR coatings for high refractive-index IR materials,” Appl. Opt. 21, 288–292 (1982).
[CrossRef] [PubMed]

M. E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in silicon using binary optics technology,” Appl. Opt. 31, 4371–4376 (1992).
[CrossRef] [PubMed]

D. Raguin, G. M. Morris, “Antireflection structured surfaces for the infrared spectral region,” Appl. Opt. 32, 1154–1167 (1993).
[CrossRef] [PubMed]

D. Raguin, G. M. Morris, “Analysis of antireflection-structured surfaces with continuous one-dimensional surface profiles,” Appl. Opt. 32, 2582–2598 (1993).
[CrossRef] [PubMed]

Soon Ting Han, Yuh-Luen Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2343–2352 (1992).
[CrossRef] [PubMed]

Appl. Phys. (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Cross gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Appl. Phys. Lett. (1)

D. C. Flanders, “Submicrometer periodicity gratings as artificial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494 (1983).
[CrossRef]

IRE Trans. Microwave Theory Tech. (1)

H. J. Riblet, “General synthesis of quarter-wave impedance transformers,” IRE Trans. Microwave Theory Tech. MTT-5, 36–43 (1957).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Commun. (1)

P. Vincent, “A finite difference method for dielectric and conducting cross gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Opt. Lett. (1)

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–474 (1956).

Other (3)

M. G. Moharam, “Coupled-wave analysis of two-dimensional dielectric gratings,” in Holographic Optics: P Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).
[CrossRef]

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1980), Chap. 1, pp. 100–117.

H. A. Macleod, Thin-Film Optical Filters (McGraw-Hill, New York, 1989).

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Figures (19)

Fig. 1
Fig. 1

1-D subwavelength grating corresponding to a uniaxial thin film. d, depth; Λ, grating period; L, length of grating ridge; ɛeff, permittivity of each axis.

Fig. 2
Fig. 2

2-D subwavelength grating corresponding to a biaxial thin film. d, depth; Λx,y, grating period along the x and y axes; Lx,y, length of grating ridge along the x and y axes; ɛeff, permittivity of each axis.

Fig. 3
Fig. 3

Comparison of the reflected field, transmitted field, and reflected power methods, for ni = 1.0 and ns = 3.0.

Fig. 4
Fig. 4

Effective permittivity ɛz(0) for a 2-D subwavelength grating. Wave propagation is perpendicular to the normal of the surface, resulting in an effective permittivity parallel to the normal of the surface.

Fig. 5
Fig. 5

Effective index of refraction for a 2-D symmetric subwavelength grating at normal incidence with ns = 1.5, for Λ/λ = 0.001, 0.1, and 0.25.

Fig. 6
Fig. 6

Effective index of refraction for a 2-D symmetric subwavelength grating at normal incidence with ns = 3.0, for Λ/λ = 0.001, 0.05, and 0.2.

Fig. 7
Fig. 7

Comparison of symmetric 2-D EMT and RCWA as a function of angle of incidence θ. RCWA parameters: ni = 1.0, ns = 3.0, fx = fy = 75.97%, depth = 0.20λ, and Λ/λ = 0.001. Thin-film uniaxial parameters: ni = 1.0, ns = 3.0, nx = 1.732, ny = 1.732, nz = 2.374, and depth = 0.20λ.

Fig. 8
Fig. 8

Effective index ny for a 2-D asymmetric subwavelength grating. Grating parameters: ni = 1.0, ns = 1.5, and Λ/λ = 0.001.

Fig. 9
Fig. 9

Effective index ny for a 2-D asymmetric subwavelength grating. Grating parameters: ni = 1.0, ns = 3.0, and Λ/λ = 0.001.

Fig. 10
Fig. 10

Effective index nz for a 2-D asymmetric subwavelength grating. Grating parameters: ni = 1.0, ns = 1.5, and Λ/λ = 0.001.

Fig. 11
Fig. 11

Effective index nz for a 2-D asymmetric subwavelength grating. Grating parameters: ni = 1.0, ns = 3.0, and Λ/λ = 0.001.

Fig. 12
Fig. 12

Comparison of asymmetric 2-D EMT and RCWA as a function of the angle of incidence θ at ϕ = 0°. RCWA parameters: ni = 1.0, ns = 3.0, fx = 40%, fy = 80%, depth = 0.20λ, and Λ/λ = 0.001. Thin-film biaxial parameters: ni = 1.0, ns = 3.0, nx = 1.2464, ny = 1.5311, nz = 1.8868, and depth = 0.20λ.

Fig. 13
Fig. 13

Comparison of asymmetric 2-D EMT and RCWA as a function of the angle of incidence θ at ϕ = 30°. RCWA parameters: ni = 1.0, ns = 3.0, fx = 40%, fy = 80%, depth = 0.20λ, and Λ/λ = 0.001. Thin-film biaxial parameters: ni = 1.0, ns = 3.0, nx = 1.2464, ny = 1.5311, nz = 1.8868, and depth = 0.20λ.

Fig. 14
Fig. 14

Comparison of asymmetric 2-D EMT and RCWA as a function of the angle of incidence θ at ϕ = 45°. RCWA parameters: ni = 1.0, n = 3.0, fx = 40%, fy = 80%, depth = 0.20λ, and Λ/λ = 0.001. Thin-film biaxial parameters: ni = 1.0, ns = 3.0, nx = 1.2464, ny = 1.5311, nz = 1.8868, and depth = 0.20λ.

Fig. 15
Fig. 15

Convergence of RCWA with the following grating parameters: ni = 1.0, ns = 1.5, fx = fy = 50%, and Λ/λ = 0.001.

Fig. 16
Fig. 16

Convergence of RCWA with the following grating parameters: ni = 1.0, ns = 3.0, fx = fy = 70%, and Λ/λ = 0.001.

Fig. 17
Fig. 17

Four-level 2-D symmetric subwavelength antireflection structure.

Fig. 18
Fig. 18

Comparison of single-layer antireflection surfaces for a 1-D grating (unpolarized and polarized light), a 2-D grating (unpolarized light), and a thin film: ni = 1.0, ns = 3.0, Λ/λ = 0.05, and θ = 0°.

Fig. 19
Fig. 19

Comparison of two-layer antireflection surfaces for a 1-D grating (unpolarized and polarized light), a 2-D dimensional grating (unpolarized light), and a thin film: ni = 1.0, ns = 3.0, Λ/λ = 0.05, and θ = 0°.

Equations (16)

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k 0 2 n s 2 = k x 2 + k y 2 + k z 2 ,
k 0 2 n s 2 < k x a 2 + k y b 2 ,
k x = k x a = ( 2 π / λ ) n i sin θ i cos ϕ i - 2 π a / Λ x ,
k y = k y b = ( 2 π / λ ) n i sin θ i sin ϕ i - 2 π b / Λ y ,
n s 2 < ( n i sin θ i cos ϕ i - a λ / Λ x ) 2 + ( n i sin θ i sin ϕ i - b λ / Λ y ) 2 .
Λ x / λ < 1 / { [ max ( n s 2 , n i 2 ) - ( n i sin θ i sin ϕ i ) 2 ] 1 / 2 + n i sin θ i cos ϕ i } ,
Λ y / λ < 1 / { [ max ( n s 2 , n i 2 ) - ( n i sin θ i cos ϕ i ) 2 ] 1 / 2 + n i sin θ i sin ϕ i } ,
( Λ x / λ , Λ y / λ ) < 1 / [ max ( n s , n i ) + n i ] .
f = L / Λ .
n eff = λ / 4 d λ / 4 .
n in = n eff [ n s + j n eff tan ( 2 π λ n eff d ) n eff + j n s tan ( 2 π λ n eff d ) ] ,
P reflected = ( n in - n i n in + n i ) 2 ,
ɛ E K = ɛ s f y + ɛ i ( 1 - f y ) = ɛ s .
ɛ eff = ɛ s f x + ɛ i ( 1 - f x ) .
ɛ eff = ɛ s f x f y + ɛ i ( 1 - f x f y ) = ɛ z ( 0 ) .
n f = n s n i ,

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