Abstract

We describe the behavior of the equivalent refractive index (or Herpin index) and equivalent phase thickness in relation to the phase thicknesses of the layers in a dielectric stack. This relation is visualized by a diagram that provides insight into the existing solutions for given combinations of the Herpin index and the equivalent phase thickness. Furthermore, it can be used for the explanation of the occurrence of stop bands and of the dispersion of equivalent layer stacks.

© 1995 Optical Society of America

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References

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  1. L. I. Epstein, “The design of optical filters,” J. Opt. Soc. Am. 42, 806–810 (1952).
    [CrossRef]
  2. P. H. Berning, “Use of equivalent films in the design of infrared multilayer antireflection coatings,” J. Opt. Soc. Am. 52, 431–436 (1962).
    [CrossRef]
  3. A. Thelen, “Equivalent layers in multilayer filters,” J. Opt. Soc. Am. 56, 1533–1538 (1966).
    [CrossRef]
  4. M. C. Ohmer, “Design of three-layer equivalent films,” J. Opt. Soc. Am. 68, 137–139 (1978).
    [CrossRef]
  5. H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Hilger, London, 1986), Chap. 6, p. 188.
  6. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988), Chap. 3, p. 41.
  7. Z. Knittl, Optics of Thin Films (Wiley, London, 1976), Chap. 6, p. 252.
  8. H. M. Liddel, Computer-aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, England, 1981), Chap. 2, p. 50.
  9. J. H. Apfel, “Phase retardance of periodic multilayer mirrors,” Appl. Opt. 21, 733–738 (1982).
    [CrossRef] [PubMed]

1982 (1)

1978 (1)

1966 (1)

1962 (1)

1952 (1)

Apfel, J. H.

Berning, P. H.

Epstein, L. I.

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), Chap. 6, p. 252.

Liddel, H. M.

H. M. Liddel, Computer-aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, England, 1981), Chap. 2, p. 50.

Macleod, H. A.

H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Hilger, London, 1986), Chap. 6, p. 188.

Ohmer, M. C.

Thelen, A.

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

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988), Chap. 3, p. 41.

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Other (4)

H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Hilger, London, 1986), Chap. 6, p. 188.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, New York, 1988), Chap. 3, p. 41.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), Chap. 6, p. 252.

H. M. Liddel, Computer-aided Techniques for the Design of Multilayer Filters (Hilger, Bristol, England, 1981), Chap. 2, p. 50.

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Figures (5)

Fig. 1
Fig. 1

Equivalent refractive index N as a function of relative wave number g for δq/2δp = 1.00 [curves (a)] and δq/2δp = 1.43 [curves (b)]. Refractive indices np = 1.38 and nq = 2.30.

Fig. 2
Fig. 2

Lines of constant equivalent phase thickness Γ in the 2δp, δq plane (0 < δq < π and 0 < 2δq < π), calculated for ρ = 2.300/1.380 = 1.667. Variables 2δp and δq as well as parameter Γ are indicated in units of π.

Fig. 3
Fig. 3

Lines of constant Γ (dotted curves) together with lines of constant H = N/np (solid curves) in the 2δp, δq plane (0 < δq < π and 0 < 2δp < π), calculated for ρ = 2.300/1.380 = 1.667. Variables 2δp and δq are indicated in units of π, whereas the constant H curves are indicated by their α values.

Fig. 4
Fig. 4

Lines of constant Γ (dotted curves) together with lines of constant H = N/np (solid curves) in the 2δp, δq plane (0 < δq < 2π and 0 < 2δp < 2π), calculated for ρ = 1.667. Variables 2δp and δq are indicated in units of π, whereas the constant H curves are indicated by their α values.

Fig. 5
Fig. 5

Approximation of a layer with refractive index 2.077 and phase thickness 0.6π by periods of equivalent layers for which np = 1.38 and nq = 2.30. In contrast with Figs. 4 and 5, the equivalent refractive index is used to indicate the curves. Phase thicknesses 2δp and δq, as well as Γ, are indicated in radians.

Tables (2)

Tables Icon

Table 1 Approximation of a Single Layera by Several Periods of the Equivalent Layer Combination for which np = 1.38 and nq = 2.300

Tables Icon

Table 2 Bandwidth of an Equivalent Layer with np = 1.38 and nq = 2.30

Equations (36)

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M k = [ cos δ k i n k 1 sin δ k i n k sin δ k cos δ k ]
M p q p = M p M q M p = [ m 11 m 12 m 21 m 22 ] ,
N = m 21 / m 12 ,
cos Γ = m 11 .
m 11 = m 22 = cos 2 δ p cos δ q A sin 2 δ p sin δ q ,
m 12 = ( i / n p ) ( sin 2 δ p cos δ q + A cos 2 δ p sin δ q B sin δ q ) ,
m 21 = i n p ( sin 2 δ p cos δ q + A cos 2 δ p sin δ q + B sin δ q ) ,
A = ( ρ + ρ 1 ) / 2 ,
B = ( ρ ρ 1 ) / 2 ,
ρ = n q / n p .
N = n p ( sin 2 δ p cos δ q + A cos 2 δ p sin δ q + B sin δ q sin 2 δ p cos δ q + A cos 2 δ p sin δ q B sin δ q ) 1 / 2 ,
cos Γ = cos 2 δ p cos δ q A sin 2 δ p sin δ q .
g = λ 1 / λ .
2 δ p = 2 δ p 1 g ,
δ q = δ q 1 g .
2 δ p 1 + δ q 1 = π .
H = N / n p .
cos δ q = cos 2 δ p cos Γ ± A sin 2 δ p ( B 2 sin 2 2 δ p + sin 2 Γ ) 1 / 2 1 + B 2 sin 2 2 δ p .
cos 2 δ p = cos δ q cos Γ ± A sin δ q ( B 2 sin 2 δ q + sin 2 Γ ) 1 / 2 1 + B 2 sin 2 δ q .
sin δ q = sin 2 δ p cos δ q A cos 2 δ p B C ,
C = H + H 1 H H 1 .
cos δ q = ± B C A cos 2 δ q ( B 2 cos 2 2 δ p 2 A B C cos 2 δ p + B 2 C 2 + 1 ) 1 / 2 .
H = ρ α ,
( 1 ) γ = cos 2 δ p cos δ q A sin 2 δ p sin δ q ,
g = γ + Δ g .
0 = 2 ( A + 1 ) cos ( Δ g s π ) + ( 1 ) γ ( A 1 ) × cos { [ ( t 1 / t + 1 ) ] ( γ + Δ g s ) π } ,
t = δ q / 2 δ p
Δ g = ± 2 π arcsin | n q n p n q + n p | .
H ( ρ , δ q , 2 δ p ) = H 1 ( ρ , π δ q , π 2 δ p ) ,
H ( ρ , δ q , 2 δ p ) = H 1 ( ρ 1 , δ q , 2 δ p ) ,
Γ ( ρ , 2 δ p , δ q ) = Γ ( ρ , δ q , 2 δ p ) ,
Γ ( ρ , 2 δ p , δ q ) = Γ ( ρ 1 , 2 δ p , δ q ) .
sin δ q = H H 1 ρ ρ 1 sin Γ ,
sin Γ extr = ρ ρ 1 H H 1 ,
Δ N = N ( g max ) N ( g min ) .
μ = Γ ( g max ) Γ ( g min ) .

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