Abstract

The concept of equivalent layers, previously applied only to all-dielectric multilayer systems, is extended to include symmetrical systems containing absorbing layers. To maximize the potential transmittance of such a system, its parameters must be chosen so that the refractive index of the equivalent layer is real. This choice simultaneously leads to a minimum of the imaginary part of the equivalent phase thickness. The proof involves a novel technique of representing the parameters of the layers by those of a spherical triangle. In the mathematical process, a simple expression is derived for the phase thickness of the dielectric layers of the symmetrical system appropriate to the optimization of its potential transmittance.

© 1972 Optical Society of America

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References

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  1. P. H. Berning and A. F. Turner, J. Opt. Soc. Am. 47, 230 (1957).
    [Crossref]
  2. S. D. Smith, J. Opt. Soc. Am. 48, 43 (1958).
    [Crossref]
  3. A. Herpin, Compt. Rend. 225, 182 (1947).
  4. R. J. Holloway and P. H. Lissberger, Appl. Opt. 8, 653 (1969).
    [Crossref] [PubMed]
  5. R. J. Holloway, Ph.D. thesis, University of Salford (1967).
  6. L. I. Epstein, J. Opt. Soc. Am. 42, 806 (1952).
    [Crossref]
  7. T. M. MacRobert and W. Arthur, Trigonometry, Part IV: Spherical Trigonometry (Methuen, London, 1938).
  8. A. Thelen, J. Opt. Soc. Am. 56, 1533 (1966).
    [Crossref]

1969 (1)

1966 (1)

1958 (1)

1957 (1)

1952 (1)

1947 (1)

A. Herpin, Compt. Rend. 225, 182 (1947).

Arthur, W.

T. M. MacRobert and W. Arthur, Trigonometry, Part IV: Spherical Trigonometry (Methuen, London, 1938).

Berning, P. H.

Epstein, L. I.

Herpin, A.

A. Herpin, Compt. Rend. 225, 182 (1947).

Holloway, R. J.

R. J. Holloway and P. H. Lissberger, Appl. Opt. 8, 653 (1969).
[Crossref] [PubMed]

R. J. Holloway, Ph.D. thesis, University of Salford (1967).

Lissberger, P. H.

MacRobert, T. M.

T. M. MacRobert and W. Arthur, Trigonometry, Part IV: Spherical Trigonometry (Methuen, London, 1938).

Smith, S. D.

Thelen, A.

Turner, A. F.

Appl. Opt. (1)

Compt. Rend. (1)

A. Herpin, Compt. Rend. 225, 182 (1947).

J. Opt. Soc. Am. (4)

Other (2)

R. J. Holloway, Ph.D. thesis, University of Salford (1967).

T. M. MacRobert and W. Arthur, Trigonometry, Part IV: Spherical Trigonometry (Methuen, London, 1938).

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Figures (9)

F. 1
F. 1

Structure and parameters of an induced-transmission filter.

F. 2
F. 2

Parameters of (a) an induced-transmission filter and (b) the corresponding equivalent layers.

F. 3
F. 3

Representation of equivalent-layer parameters by spherical trigonometry.

F. 4
F. 4

Plane analog of spherical triangle.

F. 5
F. 5

Normalized real part, δE′/π, of the phase thickness of the central equivalent layer as a function of that, δH, of the phase-adjusting layers. Computation constants: nH = 2.35 (ZnS), n = 0.075 + 3.41i, and δ = πn/5 (50-nm Ag) at a wavelength of 500 nm.

F. 6
F. 6

Imaginary part, δE, of the phase thickness of the central equivalent layer as a function of that, δH, of the phase-adjusting layers. Computation constants: same as for Fig. 5.

F. 7
F. 7

Normalized real part, nE′/nH, of the refractive index of the central equivalent layer as a function of the phase thickness, δH, of the phase-adjusting layers. Computation constants: same as for Fig. 5.

F. 8
F. 8

Normalized imaginary part, nE″/nH, of the refractive index of the central equivalent layer as a function of the phase thickness, δH, of the phase-adjusting layers. Computation constants: same as for Fig. 5.

F. 9
F. 9

An induced-transmission filter with three equal absorbing layers. (a) Actual structure; (b) equivalent structure.

Equations (51)

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n E 1 = ( n E n S ) 1 2
δ E 1 = ( 2 n + 1 ) π / 2 .
ψ = 1 | r E | 2 + 2 Γ E | r E | sin ρ E exp ( 2 δ E ) | r E | 2 exp ( 2 δ E ) + 2 Γ E | r E | sin ( 2 δ E + ρ E ) ,
ψ = 1 | r E | 2 exp ( 2 δ E ) | r E | 2 exp ( 2 δ E ) .
ψ = ψ max = exp ( 2 δ E )
cos δ E = cos 2 δ H cos δ Y + sin 2 δ H sin δ
( n E n H ) 2 = sin 2 δ H cos δ + Y + cos 2 δ H sin δ + Y sin δ sin 2 δ H cos δ + Y + cos 2 δ H sin δ Y sin δ ,
Y + = 1 2 ( n n H + n H n ) and Y = 1 2 ( n n H n H n ) .
n / n H = e i Δ ,
Δ = Δ + i Δ ,
Δ = tan 1 ( n / n )
Δ = log c ( n H / | n | ) ,
Y + = cos Δ ,
Y = i sin Δ .
cos δ E = cos 2 δ H cos δ sin 2 δ H sin δ cos Δ
( n E n H ) 2 = sin 2 δ H cos δ + cos 2 δ H sin δ cos Δ + i sin δ sin Δ sin 2 δ H cos δ + cos 2 δ H sin δ cos Δ i sin δ sin Δ .
cos a = cos b cos c + sin b sin c cos A ,
cos b = cos c cos a + sin c sin a cos B ,
cos c = cos a cos b + sin a sin b cos C ,
sin a sin A = sin b sin B = sin c sin C .
a = δ E ( comlex in general ) , b = 2 δ H ( real ) , c = δ ( complex in general ) ,
A = π Δ ( complex in general ) .
( n E n H ) 2 = sin b cos c cos b sin c cos A + i sin c sin A sin b cos c cos b sin c cos A i sin c sin A .
( n E n H ) 2 = cos c cos a cos b + i sin a sin b sin C cos c cos a cos b i sin a sin b sin C .
( n E n H ) 2 = cos C + i sin C cos C i sin C = e 2 i C
cos C = 1 2 ( n E n H + n H n E ) .
sin a a b = sin b cos c + cos b sin c cos A .
cos d = sin b cos c + cos b sin c cos A .
cos d = sin a ( cos C ) .
a b = cos C ,
δ E δ H = δ E δ H + i δ E δ H = ( n E n H + n H n E ) .
sin a 2 a b 2 cos a ( a b ) 2 = cos b cos c sin b sin c cos A = cos a [ from Eq . ( 15 a ) ] .
2 a b 2 = [ 1 ( a b ) 2 ] cot a
2 δ E δ H 2 = 2 δ E δ H 2 + i 2 δ E δ H 2 = [ 4 ( δ E δ H ) 2 ] cot δ E .
tan C = sin δ sin Δ / ( sin 2 δ H cos δ + cos 2 δ H sin δ cos Δ ) .
Re [ sin δ sin Δ ( sin 2 δ H cos δ * + cos 2 δ H sin δ * cos Δ * ) ] = 0 .
tan 2 δ H = sin Δ cos Δ [ cosh 2 δ cos 2 δ ] cos Δ sinh Δ sinh 2 δ sin Δ cosh Δ sin 2 δ
δ H = ( s π ξ ) / 2 ,
ξ = tan 1 [ 2 n H n n ( cosh 2 δ cos 2 δ ) n ( | n | 2 n H 2 ) sinh 2 δ + n ( | n | 2 + n H 2 ) sin 2 δ ] .
cos a = cos b cos c + sin b sin c cos A ,
cos a * = cos b cos c * + sin b sin c * cos A * ,
sin a d a d b = cos b sin c cos A sin b cos c ,
sin a * d a d b = cos b sin c * cos A * sin b cos c * ,
| cos c sin c cos A 0 cos a cos c * sin c * cos A * 0 cos a * sin c cos A cos c sin a 0 sin c * cos A * cos c * sin a * 0 | = 0 .
sin ( a + a * ) [ cos ( a a * ) | cos c | 2 | sin c | 2 ( | cos A | 2 | sin A | 2 ) ] = 0 .
cos ( a a * ) = | cos c | 2 + | sin c | 2 ( | cos A | 2 | sin A | 2 ) , cosh 2 a = cos 2 A cosh 2 c + sin 2 A cos 2 c ,
cosh 2 δ E = cosh 2 δ + Γ 2 cos 2 δ 1 + Γ 2 = σ ,
Γ = tan Δ = n / n .
ψ max = σ ( σ 2 1 ) 1 2 ,
δ E E = N δ E
n E E = n E .