Abstract

A three-step design procedure is developed for dielectric stacks which are required to be nonpolarizing for a given wavelength λr and angle of incidence θ0,r, at which the reflectance Rr is prescribed. The method leads to solutions in which only three layer materials occur and can be applied for a wide range of values of θ0,r and Rr. The media can be chosen from the available coating materials. Furthermore, the procedure offers the possibility of optimizing with respect to the behavior of the reflectance in the neighborhood of λr and θ0,r. An example is elaborated, and its results are compared with an actually produced coating.

© 1983 Optical Society of America

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References

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  1. P. Baumeister, Opt. Acta 8, 105 (1961).
    [CrossRef]
  2. H. F. Mahlein, Opt. Acta 21, 577 (1974).
    [CrossRef]
  3. V. R. Costich, Appl. Opt. 9, 866 (1970).
    [CrossRef] [PubMed]
  4. D. M. Cordray, T. A. Wiggins, Appl. Opt. 12, 2242 (1973).
    [CrossRef] [PubMed]
  5. A. J. Thelen, Appl. Opt. 15, 2983 (1976).
    [CrossRef] [PubMed]
  6. Z. Knittl, Appl. Opt. 20, 105 (1981).
    [CrossRef] [PubMed]
  7. Z. Knittl, H. Houserková, Appl. Opt. 21, 2055 (1982).
    [CrossRef] [PubMed]
  8. H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1969).
  9. C. M. de Sterke, “Niet-polariserende deelspiegels en -kubussen,” Master’s thesis, Delft (1982); in Dutch.

1982 (1)

1981 (1)

1976 (1)

1974 (1)

H. F. Mahlein, Opt. Acta 21, 577 (1974).
[CrossRef]

1973 (1)

1970 (1)

1961 (1)

P. Baumeister, Opt. Acta 8, 105 (1961).
[CrossRef]

Baumeister, P.

P. Baumeister, Opt. Acta 8, 105 (1961).
[CrossRef]

Cordray, D. M.

Costich, V. R.

de Sterke, C. M.

C. M. de Sterke, “Niet-polariserende deelspiegels en -kubussen,” Master’s thesis, Delft (1982); in Dutch.

Houserková, H.

Knittl, Z.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1969).

Mahlein, H. F.

H. F. Mahlein, Opt. Acta 21, 577 (1974).
[CrossRef]

Thelen, A. J.

Wiggins, T. A.

Appl. Opt. (5)

Opt. Acta (2)

P. Baumeister, Opt. Acta 8, 105 (1961).
[CrossRef]

H. F. Mahlein, Opt. Acta 21, 577 (1974).
[CrossRef]

Other (2)

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1969).

C. M. de Sterke, “Niet-polariserende deelspiegels en -kubussen,” Master’s thesis, Delft (1982); in Dutch.

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Figures (7)

Fig. 1
Fig. 1

Basic, preliminary, and final stacks for a fourteen-layer coating. In the basic and preliminary stacks, the order of the λ/4 layers with refractive indices nL or nH is undetermined yet. The given order is one of the possibilities. The order in the final stack yields an optimal behavior of the reflectance R as a function of λ in the neighborhood of λr.

Fig. 2
Fig. 2

Numbering of layers and related quantities.

Fig. 3
Fig. 3

(Symmetrical) relation between the reflectance R and 10log(A).

Fig. 4
Fig. 4

n M / ( n L β n H β 1 ) as a function of β for three values of S. (nL = 1.299 and nH = 2.325).

Fig. 5
Fig. 5

Reflectances Rs (solid curves) and Rp (dotted curves) as a function of g = λr/λ for four layer orders (parameters as in Sec. IV).

Fig. 6
Fig. 6

Reflectances Rs and Rp as functions of λ for the optimal design (solid curve) and for the actually produced mirror (dotted curve). The order of the refractive indices is n0, nL, nM, nH, nM, nH, nM, nH, nM, nH, nM, nL, nM, nH, nM, nS (n0 = 1.000, nL = 1.299, nM = 1.753, nH = 2.325, nS = 1.515, while θ0,r = ¼π). The phase thicknesses are δ1 = 0.086, δj = π/2 (j = 2,3,…,13), and δ14 = 0.479.

Fig. 7
Fig. 7

Curves for the reflectance Rs and Rp as functions of λ (a) and θ0 (b) for a BK7 beam-splitting prism design. (λr = 632.8 nm and θ0,r = π/4.) The order of refractive indices is n0, nL, nM, nH, nM, nH, nM, nH, nM, nL, nM, nH, nM, nH, nS (n0 = nS = 1.515, nL = 1.299, nM = 1.753, and nH = 2.325). The phase thicknesses are δ1 = 1.347, δj = π/2 (j = 2,3,…,12) and δ13 = 0.943.

Equations (42)

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n L < n M < n H .
, n U , n M , n U , n M , n U , ,
[ n ˆ × E j ( μ / ) 1 / 2 H j ; tan ] = M j [ n ˆ × E j + 1 ( μ / ) 1 / 2 H j + 1 ; tan ] ,
M j = [ cos ( δ j ) i N j 1 sin ( δ j ) i N j sin ( δ j ) cos ( δ j ) ] ,
δ j = 2 π n j t j cos ( θ j ) / λ ,
M j = ( 0 i N j 1 i N j 0 ) .
n j sin ( θ j ) = n 0 sin ( θ 0 ) = S ( j = 0,1,2 , , k , s ) ,
N j s = n j cos ( θ j ) for s polarization ( electric field perpendicular to the plane of incidence ) , N j p = n j / cos ( θ j ) for p polarization ( electric field in the plane of incidence ) .
M = M 1 M 2 M j M k = ( m 11 i m 12 i m 21 m 22 ) .
( B C ) = M ( 1 N s ) ;
Y = C / B .
A = Y / N 0 .
r = ( 1 A ) / ( 1 + A ) ,
R = | r | 2 .
R = ( Q 1 ) / ( Q + 1 ) ,
Q = 1 2 ( N 0 N s 1 m 11 2 + N 0 N s m 12 2 + N 0 1 N s 1 m 21 2 + N 0 1 N s m 22 2 ) .
R s = R p ;
A s = A p
A s = 1 / A p .
A = N 2 2 N 4 2 N k 2 2 N 3 2 N 5 2 N k 1 2 N k N 1 ( k even ) .
A = N 2 2 N 4 2 N k 1 2 N 3 2 N 5 2 N k 2 2 1 N 1 N k ( k odd ) .
A = ( N M k 1 N L a N H b ) ± 1 ,
a + b = k 1.
β = a / ( k 1 ) ,
A = ( N M N L β N H 1 β ) ± ( k 1 ) .
cos ( θ M ) = cos β ( θ L ) cos 1 β ( θ H ) .
M 2 M k 1 = ( 1 ) k / 2 1 ( ( A N 1 / N k ) 1 / 2 0 0 ( A N 1 / N k ) 1 / 2 ) ( k even )
M 2 M k 1 = ( 1 ) ( k + 1 ) / 2 [ 0 i ( A N 1 N k ) 1 / 2 i ( A N 1 N k ) 1 / 2 0 ] ( k odd ) ,
M = 1 2 ( 1 ) k / 2 1 ( 1 + A ) ( A N 1 / N k ) 1 / 2 [ c + i N k 1 s i N 1 s + ( N 1 / N k ) c ] ( k even ) ] ,
M = 1 2 ( 1 ) ( k + 1 ) / 2 ( 1 + A ) ( A N 1 N k ) 1 / 2 [ N k s i c + i N 1 N k c N 1 s + ] ( k odd ) ] ,
c ± = cos ( δ 1 + δ k ) ± r b cos ( δ 1 δ k ) , s ± = sin ( δ 1 + δ k ) ± r b sin ( δ 1 δ k ) .
( 1 + R ) / ( 1 R ) = a 1 + a 2 cos ( 2 δ 1 ) + a 3 cos ( 2 δ k ) + a 4 cos ( 2 δ 1 ) cos ( 2 δ k ) + a 5 sin ( 2 δ 1 ) sin ( 2 δ k ) ,
a 1 = 1 4 ( F c + F c 1 + F s + F s 1 ) ( 1 + r b 2 ) / ( 1 r b 2 ) , a 2 = 1 2 ( F c F c 1 F s + F s 1 ) r b / ( 1 r b 2 ) , a 3 = 1 2 ( F c F c 1 + F s F s 1 ) r b / ( 1 r b 2 ) , a 4 = 1 4 ( F c + F c 1 F s F s 1 ) ( 1 + r b 2 ) / ( 1 r b 2 ) , a 5 = 1 4 ( F c + F c 1 F s F s 1 ) ,
F c = N 0 N k / N 1 N s , F s = N 1 N k / N 0 N s ( k even ) , F c = N 0 N s / N 1 N k , F s = N 1 N s / N 0 N k ( k odd ) .
A 1 ( 2 δ p ) cos ( 2 δ q ) + A 2 ( 2 δ p ) sin ( 2 δ q ) = A 3 ( 2 δ p ) .
Δ R = P δ Δ λ + ( Q N + Q δ ) Δ θ 0 .
A = ( N M 13 N L 3 N H 10 ) ± 1 ,
r s = ± 0.775 , r p = ± 0.712 ,
R s = 0.601 , R p = 0.507.
R s = R p = R r = 0.500.
4.465 1.761 cos ( 2 δ 1 ) + 0.770 cos ( 2 δ 14 ) 0.324 cos ( 2 δ 1 ) cos ( 2 δ 14 ) + 0.081 sin ( 2 δ 1 ) sin ( 2 δ 14 ) = 3.000 ( s - polarization ) , 3.089 0.264 cos ( 2 δ 1 ) + 0.325 cos ( 2 δ 14 ) 0.031 cos ( 2 δ 1 ) cos ( 2 δ 14 ) + 0.010 sin ( 2 δ 1 ) sin ( 2 δ 14 ) = 3.000 ( p - polarization ) ,
δ 1 = 0.086 , δ 14 = 0.479 , δ 1 = 3.056 , δ 14 = 2.663 , δ 1 = 3.020 , δ 14 = 0.480 , δ 1 = 0.121 , δ 14 = 2.662.

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