Abstract

Plastic optical parts require antireflective as well as hard coatings. A novel design concept for coating plastics combines both functions. Symmetrical three-layer periods with a phase thickness of 3/2π are arranged in a multilayer to achieve a step-down refractive-index profile. It is shown mathematically that the equivalent index of symmetrical periods can be lower than the lowest refractive index of a material used in the design, if the phase thickness of the symmetrical period is set equal to 3/2π instead of the usual π/2. The straightforward application of the concept to the design of antireflection coatings in general is demonstrated by example.

© 2003 Optical Society of America

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References

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  1. F. Samson, “Ophthalmic lens coatings,” Surf. Coat. Technol. 81, 79–86 (1996).
    [CrossRef]
  2. A. Musset, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 203–237.
    [CrossRef]
  3. U. Schulz, U. Schallenberg, N. Kaiser, “Antireflective coating,” PCT/DE 01/02501 (2000).
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    [CrossRef] [PubMed]
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    [CrossRef]
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2002 (1)

1996 (2)

1982 (1)

1978 (1)

1966 (2)

1962 (1)

1952 (1)

Berning, P. H.

DeBell, G. W.

Dobrowolski, J. A.

Epstein, L. I.

Ho, F.

Jacobsson, R.

Kaiser, N.

Macleod, A.

A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, London, 2001).
[CrossRef]

Martensson, J. O.

Musset, A.

A. Musset, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 203–237.
[CrossRef]

Ohmer, M. C.

Samson, F.

F. Samson, “Ophthalmic lens coatings,” Surf. Coat. Technol. 81, 79–86 (1996).
[CrossRef]

Schallenberg, U.

Schulz, U.

Thelen, A.

A. Thelen, “Equivalent layers in multilayer filters,” J. Opt. Soc. Am. 56, 1533–1538 (1966).
[CrossRef]

A. Musset, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 203–237.
[CrossRef]

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989) S.59.

Tikhonravov, A. V.

Trubetskov, M. K.

Appl. Opt. (4)

J. Opt. Soc. Am. (4)

Surf. Coat. Technol. (1)

F. Samson, “Ophthalmic lens coatings,” Surf. Coat. Technol. 81, 79–86 (1996).
[CrossRef]

Other (5)

A. Musset, A. Thelen, “Multilayer antireflection coatings,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1970), Vol. 8, pp. 203–237.
[CrossRef]

U. Schulz, U. Schallenberg, N. Kaiser, “Antireflective coating,” PCT/DE 01/02501 (2000).

A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, London, 2001).
[CrossRef]

Essential Macleod, Version 8.2 © 2000 (Thin Film Center, Inc., 2745 East Via Rotunda, Tucson, Ariz. 85716).

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1989) S.59.

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Figures (6)

Fig. 1
Fig. 1

Index profiles and performance of AR-hard coatings consisting of 7 (AR-hard-7), 13 A(R-hard-13), and 25 (AR-hard-25) layers.

Fig. 2
Fig. 2

Dispersion of the equivalent layers A′, B′, C′, and D′ belonging to the symmetric periods of the AR-hard-9 design for a design wavelength of 516 nm.

Fig. 3
Fig. 3

Reflectance versus wavelength of the AR-hard-9 design: sub/2.433L 0.144H 2.83L 0.226H 2.704L 0.366H 2.55L 0.534H 1.233L/air and for the corresponding step-down design: sub/1L 3A′3B′3C′3D′/air (the dispersion of equivalent layers shown in Fig. 2 is considered). Design AR-hard-9a: sub/2.443L 0.106H 2.823L 0.226H 2.687L 0.366H 2.529L 0.534H 1.222L/air was achieved after Herpin replacement of equivalent layers A′, B′, C′, and D′.

Fig. 4
Fig. 4

Equivalent index N of a symmetric period ϕ1L ϕ2H ϕ1L depending on period thickness at wavelength 516 nm (example: equivalent layer D′ with ϕ2 = 0.534 = const.).

Fig. 5
Fig. 5

Index profiles of step-down AR coatings: layer sequence L ABCD′ corresponding to the equivalent layers that build up design AR-hard-9 and layer sequence L FGKJ′ with refractive indices calculated by use of a so-called maximally flat formula.12

Fig. 6
Fig. 6

Reflectance versus wavelength of step-down design sub/1L 3F′ 3G′ 3J′ 3K′/air and of the corresponding design AR-hard-9b after Herpin replacement: sub/2.450L 0.093H 2.780L 0.321H 2.509L 0.622H 2.179L 1H 1L/air.

Tables (3)

Tables Icon

Table 1 AR-hard-9 Designa

Tables Icon

Table 2 Rearranged AR-hard-9 Designa

Tables Icon

Table 3 Step-Down Design sub/L 3F′ 3G′ 3J′ 3K′/aira

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

sub/2.433L 0.144H 2.83L 0.226H 2.704L 0.366H 2.55L 0.534H 1.233L/air,
sub/215.8L 7H 250L 13.9H 239L 22.5H 225.3L 32.8H 108.9L/air.
N2=n12sin 2ϕ1 cos ϕ2+12n1n2+n2n1cos 2ϕ1 sin ϕ2-12n1n2-n2n1sin ϕ2sin 2ϕ1 cos ϕ2+12n1n2+n2n1cos 2ϕ1 sin ϕ2+12n1n2-n2n1sin ϕ2.
cos Γ=cos 2ϕ1 cos ϕ2-12n1n2+n2n1sin 2ϕ1 sin ϕ2,
ϕ1=2πλ n1d1,
ϕ2=2πλ n2d2,
sub/1L1.433L 0.144H 1.433L×1.387L0.226H 1.387L1.317L 0.366H 1.317L×1.233L 0.534H 1.233L/air.
sin ϕ2=n1/N-N/n1n1/n2+n2/n1sin Γ,
cot 2ϕ1=12n1n2+n2n1tan ϕ2
2ϕ1+ϕ2=π/2.
ϕ1=ϕ1+π/2.
2ϕ1+ϕ2=3π/2 or uneven numbers of 3π/2.
N2=n12-sin 2ϕ1 cos ϕ2-12n1n2+n2n1cos 2ϕ1 sin ϕ2-12n1n2-n2n1sin ϕ2-sin 2ϕ1 cos ϕ2-12n1n2+n2n1cos 2ϕ1 sin ϕ2+12n1n2-n2n1sin ϕ2.
N=n12/N.
N min=n12/n2 if n1<n2.
sub/2L 1A 2L 1B 2L 1C 2L 1D 1L/air,

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