Abstract

It is shown that the performance of metal–dielectric interference filters of the Fabry–Perot and Spacerless Induced Transmission types in parallel radiation incident obliquely can be described in terms of that of a single equivalent dielectric layer of refractive index μE. Expressions for μE are deduced in terms of the optical constants of the layer materials, and their validity is established by the satisfactory agreement that is shown to exist between experimental and calculated values of μE.

© 1967 Optical Society of America

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References

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  1. P. H. Lissberger, W. L. Wilcock, J. Opt. Soc. Am. 49, 126 (1959).
    [CrossRef]
  2. G. Koppelmann, K. Krebs, Z. Physik 157, 172 (1960).
    [CrossRef]
  3. C. R. Pidgeon, S. D. Smith, J. Opt. Soc. Am. 54, 1459 (1964).
    [CrossRef]
  4. I. H. Blifford, Appl. Opt. 5, 105 (1966).
    [CrossRef] [PubMed]
  5. P. H. Berning, A. F. Turner, J. Opt. Soc. Am. 47, 230 (1957).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon Press Ltd., London, 1959).
  7. P. H. Berning, in Physics of Thin Films—Advances in Research and Development, Georg Hass, Ed. (Academic Press, London, 1963), vol. 1.
  8. J. F. Hall, W. F. G. Ferguson, J. Opt. Soc. Am. 45, 714 (1955).
    [CrossRef]
  9. L. G. Schulz, J. Opt. Soc. Am. 44, 357 (1954).
    [CrossRef]

1966 (1)

1964 (1)

1960 (1)

G. Koppelmann, K. Krebs, Z. Physik 157, 172 (1960).
[CrossRef]

1959 (1)

1957 (1)

1955 (1)

1954 (1)

Berning, P. H.

P. H. Berning, A. F. Turner, J. Opt. Soc. Am. 47, 230 (1957).
[CrossRef]

P. H. Berning, in Physics of Thin Films—Advances in Research and Development, Georg Hass, Ed. (Academic Press, London, 1963), vol. 1.

Blifford, I. H.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press Ltd., London, 1959).

Ferguson, W. F. G.

Hall, J. F.

Koppelmann, G.

G. Koppelmann, K. Krebs, Z. Physik 157, 172 (1960).
[CrossRef]

Krebs, K.

G. Koppelmann, K. Krebs, Z. Physik 157, 172 (1960).
[CrossRef]

Lissberger, P. H.

Pidgeon, C. R.

Schulz, L. G.

Smith, S. D.

Turner, A. F.

Wilcock, W. L.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press Ltd., London, 1959).

Appl. Opt. (1)

J. Opt. Soc. Am. (5)

Z. Physik (1)

G. Koppelmann, K. Krebs, Z. Physik 157, 172 (1960).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon Press Ltd., London, 1959).

P. H. Berning, in Physics of Thin Films—Advances in Research and Development, Georg Hass, Ed. (Academic Press, London, 1963), vol. 1.

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Figures (4)

Fig. 1
Fig. 1

Fabry-Perot filter.

Fig. 2
Fig. 2

Transmittance, reflectance, and absorptance of half-filter.

Fig. 3
Fig. 3

Complex amplitude reflectances of general filter structure.

Fig. 4
Fig. 4

Complex amplitude reflectances at boundaries of matching layer.

Tables (3)

Tables Icon

Table I Comparison of Optical Thickness of Matching Layer Calculated from General and Approximate Expressions

Tables Icon

Table II Expressions a for μE in Terms of Optical Constants of Layer Materials: μE = [(1 − Q)/P]½ = {[1 − (VQ/V)]/(VP/V)}½

Tables Icon

Table III Comparison of Experimental and Theoretical Values of μE

Equations (42)

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Δ λ / λ 0 = - θ 2 / 2 μ E 2 ,
τ = τ max / ( 1 + F sin 2 ρ 0 ) ,
r 0 e i ρ 0 = r 1 e i ϕ 1 + r e i ρ e 2 i D ( n + i k ) 1 + r 1 r e i ϕ 1 e i ρ e 2 i D ( n + i k ) ,
r 0 e i ρ 0 = r 1 e i ϕ 1 + r e i ϕ 1 + r 1 r e i ( ϕ 1 + ϕ ) ,
tan ρ 0 = r 1 ( 1 - r 2 ) sin ϕ 1 + r ( 1 - r 1 2 ) sin ϕ r 1 ( 1 + r 2 ) cos ϕ 1 + r ( 1 + r 1 2 ) cos ϕ
tan ρ 0 = tan ϕ 1 cosh 2 ξ + ( cos ϕ cos ϕ 1 ) sinh 2 ξ ,
ρ 0 = ρ 0 + ( 4 π / λ ) μ 0 d 0 .
r e - 2 q = r 1.
ρ 0 = ϕ 1 + ( 4 π / λ ) μ 0 d 0 ,
K π = ϕ 1 + ( 4 π / λ 0 ) μ 0 d 0 ,
K π = ( ϕ 1 + ϕ 1 λ Δ λ ) + 4 π λ 0 + Δ λ ( μ 0 + μ 0 λ Δ λ ) d 0 ( 1 - θ 2 2 μ 0 2 )
μ E 2 = ( 1 - Q ) / P
P = 1 / μ 0 2 , Q = λ 0 μ 0 μ 0 λ + λ 0 S η 1 λ , η 1 = ϕ 1 / π ,
S = K - η 1 .
Δ μ = μ E - μ 0 .
r = r e - 2 q = 1 ,
r = e 2 q 1.
r e i ρ = r 2 e i ϕ 2 + r D e i ρ D e i δ M 1 + r 2 r D e i ϕ 2 e i ρ D e i δ M
r 2 = r 2 2 + r D 2 + 2 r 2 r D cos ( ρ D + δ M - ϕ 2 ) 1 + r 2 2 r D 2 + 2 r 2 r D cos ( ρ D + δ M + ϕ 2 ) .
r = | cos ( α / 2 ) cos ( β / 2 ) | ,
β = ρ D + δ M + ϕ 2 = ( 2 K + 1 ) π
μ M d M = M λ 0 / 4 ,
ϕ = tan - 1 [ 2 μ k / ( k 2 - μ 2 ) ] .
M = ( 2 K + 1 ) - ( 1 / π ) tan - 1 [ 2 μ H k / ( k 2 - μ H 2 ) ] ,
M = 2 K - ( 1 / π ) tan - 1 [ 2 μ L k / ( k 2 - μ L 2 ) ] .
ρ D ( λ 0 + Δ λ , θ ) + [ 4 π / ( λ 0 + Δ λ ) ] [ μ M ( λ 0 ) + ( μ M / λ ) Δ λ ] d M ( 1 - θ 2 / 2 μ M 2 ) + [ ϕ 2 ( λ 0 ) + ( ϕ 2 / λ ) Δ λ ] = ( 2 K + 1 ) π .
ρ D ( λ 0 + Δ λ , θ ) = 4 π λ 0 + Δ λ j = 1 l + 1 L j [ μ j ( λ 0 ) + μ j λ Δ λ ] ( 1 - θ 2 2 μ j 2 ) d j - π j = 2 l + 1 L j ,
4 μ M d M λ 0 ( 1 - A M Δ λ λ 0 - θ 2 2 μ M 2 ) + 4 L j μ j d j λ 0 ( 1 - A j Δ λ λ 0 - θ 2 2 μ j 2 ) - L j + η 2 + λ 0 n 2 λ Δ λ λ 0 = 2 K + 1 ,
Δ λ / λ 0 = - θ 2 / 2 μ E 2 ,
μ E 2 = M A M + L j A j - λ 0 η / λ ( M / μ M 2 ) + ( L j / μ j 2 ) .
μ E 2 = ( 1 - Q ) / P ,
P = 1 V ( M μ M 2 + L j μ j 2 ) , Q = 1 V [ M λ 0 μ M μ M λ + L j λ 0 μ j μ j λ + λ 0 η 2 λ ] , V = M + L j .
η = ϕ / π = ( 1 / π ) tan - 1 [ 2 μ k / ( k 2 - μ 2 ) ]
η λ = 1 a μ λ - 1 b k λ ,
μ H [ 1 - ( λ O / μ H ) μ H / λ ]
( 1 - x 1 ) ,
ϕ 1 = tan - 1 [ 2 U 1 k / ( k 2 - U 1 2 ) ] ,
U 1 = μ 0 + Δ U 1
U 1 = μ 0 + Δ U 1 ,
Δ η 1 = Δ ϕ 1 / π = ( 2 / π ) [ k / ( μ 0 2 + k 2 ) ] Δ U 1 .
Δ η 1 = - Δ η 1 = ( 2 / π ) [ μ 0 k / ( k 2 + μ 0 2 ) ] ( θ 2 / μ 0 2 ) .
x 1 = - x 1 = ( 2 / π S ) μ 0 k / ( k 2 + μ 0 2 ) .

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