Abstract

Zero-reflectance conditions for quarter-wave-thick, high-spatial-frequency, rectangular-groove dielectric gratings are analyzed further at oblique incidence by a combination of the effective medium theory and the anisotropic thin-film theory. Numerical examples are given for gratings on glass and silicon substrates with refractive indices of 1.5 and 3.5, respectively.

© 1996 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 705–708.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

1994 (2)

1993 (1)

1992 (1)

1991 (1)

1990 (1)

1987 (1)

1979 (1)

1972 (1)

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 705–708.

Bräuer, R.

Brundrelt, D. L.

Bryngdahl, O.

Fritsch, M.

Gaylord, T. K.

Glytsis, E. N.

Gu, C.

Gunning, W. J.

Haas, G.

Kimura, Y.

Liang, Q.-T.

Mlynski, D. A.

Motamedi, M. E.

Nishida, N.

Ohta, Y.

Ono, Y.

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Southwell, W. H.

Wöhler, H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 705–708.

Yeh, P.

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

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

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other (2)

Ref. 5, Sect. 1.6.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), pp. 705–708.

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Figures (4)

Fig. 1
Fig. 1

(a) Rectangular-groove grating in cross section.3 (b) General orientation of vector K: θ is the angle between the K and the z direction, and ϕ is the angle between the projection of K on the x–y plane and the y direction.

Fig. 2
Fig. 2

Specific orientation of the high-spatial-frequency grating (a) with vector K in the y–z incident plane and parallel to the y direction (ϕ = 0); (b) with K perpendicular to the incident plane and parallel to the x direction (ϕ = π/2). In both cases θ = π/2.

Fig. 3
Fig. 3

Variations (a) in the filling factor F and (b) in the remnant reflectance Rp when Rs = 0 against the incident angle θ1 at ϕ = 0 (solid curves) and ϕ = π/2 (dashed curves) for two substrate refractive indices, n 3 = 1.5 and 3.5.

Fig. 4
Fig. 4

Variations in the filling factor F against the incident angle θ1 at Rp = 0 when (a) n 3 = 1.5 and (b) n 3 = 3.5 at ϕ = 0 (solid curves) and ϕ = π/2 (dashed curves). The dotted curves at ϕ = 0 correspond to the approximation given by Eq. (18). θ B 3 is the Brewster angle of the substrate. Variation in the remnant reflectance Rs at Rp = 0 against θ1 when (c) n 3 = 1.5 and (d) n 3 = 3.5 at ϕ = 0 (solid curves) and ϕ = π/2 (dashed curves).

Equations (36)

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n E K = [ ( 1 F ) / n 1 2 + F / n 3 2 ] 1 / 2 ,
n E K = [ ( 1 F ) n 1 2 + F n 3 2 ] 1 / 2 .
( E rs E rp ) = ( r ss r sp r ps r pp ) ( E s E p ) .
r = ( r 12 + r 23 X ) / ( 1 + r 12 r 23 X ) ,
r ik = ( N i N k ) / ( N i + N k ) ,
r = ( r 12 r 23 ) / ( 1 r 12 r 23 ) .
N 2 2 = N 1 N 3 .
F = N 1 / ( N 1 + N 3 ) .
d / λ = 1 / ( 4 N 2 ) .
sin 2 θ B s = 1 ( 3 / 1 1 ) F 2 / ( 1 2 F ) .
F = 3 N 1 / ( 1 N 3 + 3 N 1 ) ,
sin 2 θ B s = 3 [ 3 ( 1 F ) 2 1 F 2 ] / [ 3 2 ( 1 F ) 2 1 2 F 2 ] .
ξ 2 = e ( 0 ζ 2 ) / 0 ,
n 2 2 = ξ 2 + ζ 2 .
N 2 = ( cos θ 2 + sin θ 2 tan δ ) / n 2 ,
tan δ = ( e o ) tan θ 2 / ( o + e tan 2 θ 2 ) .
N 2 = ( ξ / e ) [ o e + ( e o ) ζ 2 ] / [ o e ( e o ) ζ 2 ] .
N 2 = ξ / e .
i = 0 6 a i ( 3 1 ) i F i = 0
d / λ = 1 / ( 4 ξ ) .
F = [ 1 2 1 N 1 N 3 + ( 1 4 ζ 2 N 1 N 3 ) 1 / 2 ] / [ 2 ( 3 1 ) N 1 N 3 ] .
d / λ = 1 / ( 4 n 2 cos θ 2 ) .
sin 2 θ B p = { σ o 4 2 o μ 2 o 2 ( 3 1 ) × [ o 4 4 μ 2 F ( 1 F ) ] 1 / 2 } / [ 2 1 ( o 4 μ 2 ) ] ,
sin 2 θ B p = σ [ 1 ( 1 + 4 μ / σ 2 ) 1 / 2 ] / ( 2 1 ) .
Δ = 3 1 ,
Q = [ ( 1 ζ 2 ) ( 3 ζ 2 ) ] 1 / 2 ,
g 1 = 1 3 Δ ζ 2 ,
g 2 = Δ + ζ 2 Q ,
g 3 ± = 1 Q ± 3 ( 1 ζ 2 ) .
a 0 = 1 4 3 2 g 3 ,
a 1 = 1 2 3 [ 1 2 3 ( 4 Q g 2 ) 2 g 1 g 3 + ] ,
a 2 = 1 4 3 2 + g 1 2 g 3 2 1 2 3 [ g 1 g 2 + ζ 2 ( g 3 + 4 Q ) ] ,
a 3 = g 1 ( 2 ζ 2 g 3 g 1 g 2 ) + 2 1 2 3 [ g 1 ζ 2 ( g 2 + 4 Q ) ] + 4 1 3 Δ ζ 2 Q ,
a 4 = g 1 2 + ζ 2 [ 2 1 3 ( 1 2 Q ) 2 g 1 g 2 + ζ 2 g 3 ] ,
a 5 = ζ 2 ( 2 g 1 ζ 2 g 2 ) ,
a 6 = ζ 4 .

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