Abstract

Surface-relief structures having the form of an array of modified pyramids (having curved rather than flat triangular shown to produce an index-matching layer that will reduce surface reflection by several orders of sides) are Such structures are equivalent to a quintic (fifth-order polynomial) gradient-index layer, which has magnitude. been shown to be near optimum for reducing reflection at dielectric interfaces.

© 1991 Optical Society of America

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References

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  1. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986).
    [CrossRef]
  2. C. G. Bernhard, “Adaptation in a visual system,” Endeavour 26, 79–84 (1967).
  3. S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
    [CrossRef]
  4. B. S. Thornton, “Limit of the moth’s eye principle and other impedance-matching corrugations for solar-absorber design,”J. Opt. Soc. Am. 65, 267–270 (1975).
    [CrossRef]
  5. M. J. Minot, “Single-layer, gradient refractive index antireflective films effective from 0.35 to 2.5 μ,” J. Opt. Soc. Am. 66, 515–519 (1976).
    [CrossRef]
  6. W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high-power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
    [CrossRef]
  7. W. H. Southwell, “Gradient-index antireflection coatings,” Opt. Lett. 8, 584–586 (1983).
    [CrossRef] [PubMed]
  8. W. H. Southwell, “Coating design using very thin high- and low-index layers,” Appl. Opt. 24, 457–460 (1985).
    [CrossRef] [PubMed]
  9. R. Jacobsson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” Phys. Thin Films 8, 51–98 (1975).
  10. E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in Si using binary optics technology,” Appl. Opt. (to be published).

1985 (1)

1983 (1)

1982 (1)

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

1980 (1)

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high-power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[CrossRef]

1976 (1)

1975 (2)

B. S. Thornton, “Limit of the moth’s eye principle and other impedance-matching corrugations for solar-absorber design,”J. Opt. Soc. Am. 65, 267–270 (1975).
[CrossRef]

R. Jacobsson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” Phys. Thin Films 8, 51–98 (1975).

1967 (1)

C. G. Bernhard, “Adaptation in a visual system,” Endeavour 26, 79–84 (1967).

Bernhard, C. G.

C. G. Bernhard, “Adaptation in a visual system,” Endeavour 26, 79–84 (1967).

Gunning, W. J.

E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in Si using binary optics technology,” Appl. Opt. (to be published).

Hutley, M. C.

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Jacobsson, R.

R. Jacobsson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” Phys. Thin Films 8, 51–98 (1975).

Lowdermilk, W. H.

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high-power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986).
[CrossRef]

Milam, D.

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high-power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[CrossRef]

Minot, M. J.

Motamedi, E.

E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in Si using binary optics technology,” Appl. Opt. (to be published).

Southwell, W. H.

Thornton, B. S.

Wilson, S. J.

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W. H. Lowdermilk, D. Milam, “Graded-index antireflection surfaces for high-power laser applications,” Appl. Phys. Lett. 36, 891–893 (1980).
[CrossRef]

Endeavour (1)

C. G. Bernhard, “Adaptation in a visual system,” Endeavour 26, 79–84 (1967).

J. Opt. Soc. Am. (2)

Opt. Acta (1)

S. J. Wilson, M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29, 993–1009 (1982).
[CrossRef]

Opt. Lett. (1)

Phys. Thin Films (1)

R. Jacobsson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” Phys. Thin Films 8, 51–98 (1975).

Other (2)

E. Motamedi, W. H. Southwell, W. J. Gunning, “Antireflection surfaces in Si using binary optics technology,” Appl. Opt. (to be published).

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Refractive index of the optical-thickness-dependent quintic given by Eqs. (1) and (4). (b) Refractive index for a physical-thickness-dependent quintic, evaluated by replacing u in Eq. (1) by z. [Curve (b) would also be the optical-thickness-dependent quintic if the abscissa were optical thickness u instead of physical thickness z as shown.]

Fig. 2
Fig. 2

Log10 of the reflectance of the two refractive-index distributions shown in Fig. 1: (a) optical-thickness-dependent quintic shown in Fig. 1(a); (b) physical-thickness-dependent quintic. The substrates refractive index was 2.4.

Fig. 3
Fig. 3

Comparison of reflectance for quintics: (a) 0.5 μm, (b) 1.0 μm. Note that the reflectance at wavelength λ scales directly with thickness of the quintic. The substrate’s refractive index in this example is 2.4.

Fig. 4
Fig. 4

Reflectance at a surface having a 1.0-μm physical thickness quintic for substrates having refractive-index values of (a) 1.5, (b) 2.4, (c) 3.2, (d) 4.0.

Fig. 5
Fig. 5

Incremental layer at height z above the base of the unit cell. The equivalent index in that layer is determined by the volume fraction of the substrate material it contains.

Fig. 6
Fig. 6

Surface-relief pattern for an equivalent quintic on a substrate whose refractive index is 1.45.

Fig. 7
Fig. 7

Surface-relief pattern for an equivalent quintic on a substrate whose refractive index is 3.42.

Equations (24)

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n = n s - ( n s - 1 ) ( 10 u 3 - 15 u 4 + 6 u 5 ) ,
u = 0 z n d x .
d u = n d z .
z = 0 u d u n ( u ) .
Δ z = Δ u n .
Z = 0 1 d u n ( u ) .
n = n s [ 1 - q ( u ) Δ n / n s ] ,
q ( u ) = 10 u 3 - 15 u 4 + 6 u 5 .
Z = 0 0.5 d u n ( u ) + 0.5 1 d u n ( u ) .
n = 1 + Δ n q ( 1 - u ) ,
Z = 1 n s 0 0.5 d u 1 - q ( u ) Δ n / n s + 0 0.5 d u 1 + q ( u ) Δ n .
1 1 - α q = 1 + α q 1 - α q .
0 0.5 d u 1 - α q ( u ) = 1 2 + α 1 - α q ( β ) 0 0.5 q ( u ) d ( u ) ,
0 0.5 q ( u ) d u = 5 / 64 .
Z = ( n s + 1 ) 2 n s + [ 1 n s ( n s - g 1 Δ n ) - 1 ( 1 + g 2 Δ n ) ] 5 Δ n 64 .
h = Z O T ,
h > λ 0 , Refl . < 0.0001 for all λ < λ 0 , h > λ 0 / 2 , Refl . < 0.005 for all λ < λ 0 ,
f = w 2 / p 2 ,
n 2 = f n 1 2 + ( 1 - f ) n 2 2 ,
n 2 = f n s 2 + 1 - f .
w 2 p 2 = n 2 - 1 n s - 1 .
G = [ n s - ( n s - 1 ) ( 10 u 3 - 15 u 4 + 6 u 5 ) ] 2 - 1 + v 2 ( n s 2 - 1 ) = 0.
z = 0 u d u n ( u ) .
p < λ / n s .

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