Abstract

The reflection coefficient of a system of perfect quarter-wave layers is obtained in terms of the Fresnel coefficients of various boundaries by a simple summation process. This result is used to calculate the phase change on reflection from a system of approximate quarter-wave layers and this is shown to depend linearly on the phase errors of the individual layers. From the resulting formulas the important features of the transmission characteristics of all-dielectric interference filters are evaluated.

© 1959 Optical Society of America

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References

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  1. C. Dufour and A. Herpin, Optica Acta (Paris) 1, 1 (1954).
    [Crossref]
  2. P. Giacomo, Rev. opt. 35, 317, 442 (1956).
  3. P. H. Lissberger and W. L. Wilcock, J. Opt. Soc. Am. 49, 126 (1959), following paper.
    [Crossref]
  4. D. L. Caballero, J. Opt. Soc. Am. 37, 176 (1947).
    [Crossref] [PubMed]
  5. B. S. Blaisse and J. J. Van der Sande, Physica 13, 415 (1947).
    [Crossref]

1959 (1)

1956 (1)

P. Giacomo, Rev. opt. 35, 317, 442 (1956).

1954 (1)

C. Dufour and A. Herpin, Optica Acta (Paris) 1, 1 (1954).
[Crossref]

1947 (2)

B. S. Blaisse and J. J. Van der Sande, Physica 13, 415 (1947).
[Crossref]

D. L. Caballero, J. Opt. Soc. Am. 37, 176 (1947).
[Crossref] [PubMed]

Blaisse, B. S.

B. S. Blaisse and J. J. Van der Sande, Physica 13, 415 (1947).
[Crossref]

Caballero, D. L.

Dufour, C.

C. Dufour and A. Herpin, Optica Acta (Paris) 1, 1 (1954).
[Crossref]

Giacomo, P.

P. Giacomo, Rev. opt. 35, 317, 442 (1956).

Herpin, A.

C. Dufour and A. Herpin, Optica Acta (Paris) 1, 1 (1954).
[Crossref]

Lissberger, P. H.

Van der Sande, J. J.

B. S. Blaisse and J. J. Van der Sande, Physica 13, 415 (1947).
[Crossref]

Wilcock, W. L.

J. Opt. Soc. Am. (2)

Optica Acta (Paris) (1)

C. Dufour and A. Herpin, Optica Acta (Paris) 1, 1 (1954).
[Crossref]

Physica (1)

B. S. Blaisse and J. J. Van der Sande, Physica 13, 415 (1947).
[Crossref]

Rev. opt. (1)

P. Giacomo, Rev. opt. 35, 317, 442 (1956).

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Figures (4)

Fig. 1
Fig. 1

Single boundary.

Fig. 2
Fig. 2

Single layer.

Fig. 3
Fig. 3

Double layer.

Fig. 4
Fig. 4

Interference filter.

Tables (1)

Tables Icon

Table I Values of R0s+1,s+2, Cs+1,s+2, and Ls+1 for various values of s.

Equations (35)

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r x y = ( U y - U x ) / ( U y + U x ) ,
U y = μ y cos θ y ,             U x = μ x cos θ x ,
U y = μ y cos θ y ,             U x = μ x cos θ x
r x y = ( μ y - μ x ) / ( μ y + μ x )
R 23 = r 23 + r 12 e - i α 2 1 + r 23 r 12 e - i α 2 ,
R 34 = r 34 + R 23 e - i α 3 1 + r 34 R 23 e - i α 3 ,             α 3 = 2 π λ · 2 μ 3 t 3 .
R k + 1 , k + 2 = r k + 1 , k + 2 + R k , k + 1 e - i α k + 1 1 + r k + 1 , k + 2 R k , k + 1 e - i α k + 1 , α k + 1 = ( 2 π / λ ) · 2 μ k + 1 t k + 1 , r k + 1 , k + 2 = μ k + 2 - μ k + 1 μ k + 2 + μ k + 1 .
R 23 = tanh ( θ 23 - θ 12 ) .
R k + 1 , k + 2 = tanh ϕ k + 1 , k + 2 ,
ϕ k + 1 , k + 2 = θ k + 1 , k + 2 - θ k , k + 1 ( - 1 ) k + 1 - s θ s , s + 1 ( - 1 ) k θ 12 = s = 1 k + 1 ( - 1 ) k + 1 - s θ s , s + 1 .
ϕ k + 1 , k + 2 = θ k + 1 , k + 2 + ( - 1 ) k [ θ 12 - ( k - 1 ) θ ] .
α s = ( 2 π / λ ) · 2 μ s t s = π + δ s ,
R 23 = r 23 - r 12 ( 1 - i δ 2 ) 1 - r 23 r 12 ( 1 - i δ 2 ) = r 23 - r 12 1 - r 23 r 12 { 1 + r 12 r 23 - r 12 · i δ 2 1 + r 23 r 12 1 - r 23 r 12 · i δ 2 } .
R 23 = R 0 23 { 1 - ( r 23 r 12 1 - r 23 r 12 - r 12 r 23 - r 12 ) · i δ 2 } ,
R 0 23 = r 23 - r 12 1 - r 23 r 12 = tanh ϕ 23 ;
R 23 = R 0 23 ( 1 - i δ 23 ) ,
δ 23 = C 23 δ 2 ,
C 23 = r 23 r 12 1 - r 23 r 12 - r 12 r 23 - r 12 .
R 34 = r 34 - R 0 23 ( 1 - i δ 23 ) ( 1 - i δ 3 ) 1 - r 34 R 0 23 ( 1 - i δ 23 ) ( 1 - i δ 3 ) = r 34 - R 0 23 [ 1 - i ( δ 23 + δ 3 ) ] 1 - r 34 R 0 23 [ 1 - i ( δ 23 + δ 3 ) ] ( second-order terms in δ being neglected ) = r 34 - R 0 23 ( 1 - i δ 3 ) 1 - r 34 R 0 23 ( 1 - i δ 3 ) ,
R 34 = R 0 34 ( 1 - i δ 34 ) ,
R 0 34 = r 34 - R 0 23 1 - r 34 R 0 23 = tanh ϕ 34 , δ 34 = C 34 δ 3 ,
C 34 = r 34 R 0 23 1 - r 34 R 0 23 - R 0 23 r 34 - R 0 23 .
R k + 1 , k + 2 = R 0 k + 1 , k + 2 ( 1 - i δ k + 1 , k + 2 ) ,
R 0 k + 1 , k + 2 = r k + 1 , k + 2 - R 0 k , k + 1 1 - r k + 1 , k + 2 R 0 k , k + 1 = tanh ϕ k + 1 , k + 2 , δ k + 1 , k + 2 = C k + 1 , k + 2 δ k + 1 ,             δ k + 1 = δ k + 1 + δ k , k + 1 ,
C k + 1 , k + 2 = r k + 1 , k + 2 R 0 k , k + 1 1 - r k + 1 , k + 2 R 0 k , k + 1 - R 0 k , k + 1 r k + 1 , k + 2 - R 0 k , k + 1 .
C k + 1 , k + 2 = 2 μ k + 1 μ k + 2 ( μ k + 2 2 + μ k + 1 2 ) - ( μ k + 2 2 - μ k + 1 2 ) coth 2 ϕ k , k + 1 .
δ k + 1 , k + 2 = C k + 1 , k + 2 δ k + 1 + C k + 1 , k + 2 C k , k + 1 δ k + ( s = s k + 1 C s , s + 1 ) δ s + ( s = 2 k + 1 C s , s + 1 ) δ 2 = L k + 1 δ k + 1 + L k δ k + L s δ s + L 2 δ 2 ,
δ k + 1 , k + 2 = s = 2 k + 1 L s δ s ,
L s = s = s k + 1 C s , s + 1 .
τ λ = τ max 1 + F sin 2 χ / 2 ,
χ = δ + δ G k + 1 , k + 2 + δ A k + 1 , k + 2 ,             δ = ( 2 π / λ ) · 2 μ t , δ G k + 1 , k + 2 = s = 2 k + 1 L G s δ G s ,             δ A k + 1 , k + 2 = s = 2 k + 1 L A s δ A s .
τ max = [ 1 - ( R 0 G k + 1 , k + 2 ) 2 ] [ 1 - ( R 0 A k + 1 , k + 2 ) 2 ] ( 1 - R 0 G k + 1. k + 2 R 0 A k + 1 , k + 2 ) 2
F = 4 R 0 G k + 1 , k + 2 R 0 A k + 1 , k + 2 ( 1 - R 0 G k + 1 , k + 2 R 0 A k + 1 , k + 2 ) 2 .
χ = ( 4 π / λ ) [ μ t + s = 2 k + 1 ( L A s μ A s t A s + L G s μ G s t G s ) ] - π s = 2 k + 1 ( L A s + L G s ) .
μ t + s = 2 k + 1 ( L A s μ A s t A s + L G s μ G s t G s ) = [ 2 n + s = 2 k + 1 ( L A s + L G s ) ] λ 0 / 4 ,