Abstract

A closed-form solution for the reflectivity of an optical coating with a sine-wave index profile (rugate) is derived by using coupled-wave theory. Reflectance is calculated for such coatings on arbitrary substrates and incident media. A vector formalism is used that produces results for both s- and p- polarization components.

© 1988 Optical Society of America

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References

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  1. H. Sankur, W. H. Southwell, “Broadband gradient-index antireflection coating for ZnSe,” Appl. Opt. 23, 2770–2773 (1984).
    [CrossRef] [PubMed]
  2. J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  3. W. H. Southwell, “Use of gradient index for spectral filters,” in Solid State Optical Control Devices, P. Yeh, ed., Proc. Soc. Photo-Opt. Instrum. Eng.464, 110–114 (1984).
    [CrossRef]
  4. A. Thelen, “Design of multilayer interference filters,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1964), Vol. 5, pp. 47–85.
  5. D. L. Jaggard, C. Elachi, “Floquet and coupled-wave analysis of higher-order Bragg coupling in a periodic medium,” J. Opt. Soc. Am. 66, 674–682 (1976).
    [CrossRef]
  6. Rayleigh, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
    [CrossRef]
  7. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell. Syst. Tech. J. 48, 2909–2947 (1969).
  8. The word rugate is really an adjective meaning wrinkled; hence it is used to describe the filters with sine-wave index profiles. In thin-film applications, however, the word is also used sometimes as a noun, meaning a rugate filter.

1984 (1)

1978 (1)

1976 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell. Syst. Tech. J. 48, 2909–2947 (1969).

1917 (1)

Rayleigh, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
[CrossRef]

Dobrowolski, J. A.

Elachi, C.

Jaggard, D. L.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell. Syst. Tech. J. 48, 2909–2947 (1969).

Lowe, D.

Rayleigh,

Rayleigh, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
[CrossRef]

Sankur, H.

Southwell, W. H.

H. Sankur, W. H. Southwell, “Broadband gradient-index antireflection coating for ZnSe,” Appl. Opt. 23, 2770–2773 (1984).
[CrossRef] [PubMed]

W. H. Southwell, “Use of gradient index for spectral filters,” in Solid State Optical Control Devices, P. Yeh, ed., Proc. Soc. Photo-Opt. Instrum. Eng.464, 110–114 (1984).
[CrossRef]

Thelen, A.

A. Thelen, “Design of multilayer interference filters,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1964), Vol. 5, pp. 47–85.

Appl. Opt. (2)

Bell. Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell. Syst. Tech. J. 48, 2909–2947 (1969).

J. Opt. Soc. Am. (1)

Proc. R. Soc. London Ser. A (1)

Rayleigh, “On the reflection of light from a regularly stratified medium,” Proc. R. Soc. London Ser. A 93, 565–577 (1917).
[CrossRef]

Other (3)

The word rugate is really an adjective meaning wrinkled; hence it is used to describe the filters with sine-wave index profiles. In thin-film applications, however, the word is also used sometimes as a noun, meaning a rugate filter.

W. H. Southwell, “Use of gradient index for spectral filters,” in Solid State Optical Control Devices, P. Yeh, ed., Proc. Soc. Photo-Opt. Instrum. Eng.464, 110–114 (1984).
[CrossRef]

A. Thelen, “Design of multilayer interference filters,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1964), Vol. 5, pp. 47–85.

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Figures (3)

Fig. 1
Fig. 1

Rugate index profile, showing 10.5 cycles. Each cycle is one half-wave optical thickness at the stop-band wavelength. In this diagram, the substrate is on the right and the light is incident from the left.

Fig. 2
Fig. 2

Propagation vectors k± inside the rugate. Also shown are ê±, the unit vectors of the transverse E fields for the p component. The s component E fields are in the j ˆ direction, which is out of the page.

Fig. 3
Fig. 3

Reflectance of a 100-cycle rugate with na = 2, np = 0.1, ns = 1.52, n0 = 1, and λ1 = 0.55. Evaluation was performed at normal incidence in two ways: the coupled-wave theory described in this paper and the exact characteristic matrix approach, for comparison. The curves are superimposed.

Tables (1)

Tables Icon

Table 1 Calculation of Reflectance and Optical Density at the Peak and at the Edge of the Stop Banda

Equations (81)

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× H = J + D t ,
× E = B t ,
· D = ρ ,
· B = 0.
J = σ E ,
D = E ,
B = μ H .
× ( × E ) = μ ( × H ) t .
× ( × E ) = μ σ E t μ 2 E t .
E = E 0 e i ω t ,
× ( × E ) = ( μ ω 2 i μ σ ω ) E .
× ( × E ) = ( · E ) 2 E ,
ω 2 c 2 N 2 = μ ω 2 i μ σ ω ,
2 E ( · E ) + k 2 E = 0 ,
k = 2 π λ N .
n = n a + n p 2 sin ( K 1 z + ϕ ) ,
K 1 = 4 π λ 1 n a ;
E = F exp ( i k + · r ) e ˆ + + B exp ( i k · r ) e ˆ ,
r = x i ˆ + z k ˆ .
k ± = k 0 ( sin θ i ˆ ± cos θ k ˆ ) ,
k 0 = 2 π λ n a .
e ˆ ± = { j s component ± cos θ i ˆ sin θ k ˆ p component .
2 = 2 x 2 + 2 z 2
2 x 2 E = k x 2 E .
2 E = ( k x 2 + 2 z 2 ) F exp ( i k + · r ) e ˆ + + ( k x 2 + 2 z 2 ) B × exp ( i k · r ) e ˆ .
2 E = ( F 2 i k z F k 0 2 F ) exp ( i k + · r ) e ˆ + + ( B + 2 i k z B k 0 2 B ) exp ( i k · r ) e ˆ .
· E = E x x + E z Z .
E x = E · i ˆ = [ F exp ( i k + · r ) B exp ( i k · r ) ] cos θ ,
E z = E · k ˆ = [ F exp ( i k + · r ) B exp ( i k · r ) ] sin θ .
· E = [ F exp ( i k + · r ) B exp ( i k · r ) sin θ .
( · E ) = x ( · E ) i ˆ + y ( · E ) k ˆ .
( · E ) = i sin θ [ F exp ( i k + · r ) k + + B exp ( i k · r ) k ] [ F exp ( i k + · r ) + B × exp ( i k · r ) sin θ k ˆ ] .
k 2 E = ( k 0 2 + k 0 2 n p i 2 n a { exp [ i ( k 1 z + ϕ ) ] exp [ i ( k 1 z + ϕ ) ] } ) × [ F exp ( i k + · r ) e + + B exp ( i k · r ) e ˆ ] .
| F | | 2 k z F | and | B | | 2 k z B | .
2 i k z F exp ( i k z z ) e ˆ + + 2 i k z B exp ( i k z z ) e ˆ i sin θ [ exp ( i k z z ) k + + exp ( i k z z ) k ] + k 0 2 n p i 2 n a × { exp [ i ( k 1 z + ϕ ) ] exp [ j ( k 1 z + ϕ ) ] } × [ F exp ( i k z z ) e ˆ + + B exp ( i k z z ) e ˆ ] = 0.
2 i k z F exp ( i k z z ) + 2 i k z B g exp ( i k z z ) i sin θ exp ( i k z z ) h + k 0 2 n p i 2 n a { exp [ i ( K 1 z + ϕ ) ] exp [ i ( K 1 z + ϕ ) ] } × [ F exp ( i k z z ) + B g exp ( i k z z ) ] = 0 ,
e ˆ + · e ˆ = g = { 1 s component cos 2 θ p component ,
e ˆ + · k + = 0 ,
e ˆ + · k = h .
( 2 i k z F k 0 2 n p g 2 i n a B exp { i [ ( 2 k z K 1 ) z ϕ ] } ) d z = 0.
F = κ B exp { i [ ( 2 k z K 1 ) z ϕ ] } ,
κ = k 0 2 n p g k z 4 n a .
( 2 i k z B + k 0 2 n p g 2 i n a F exp { i [ ( 2 k z K 1 ) z ϕ ] } ) d z = 0.
B = κ F exp { i [ ( 2 k z K 1 ) z ϕ ] } .
F = [ i ( 2 k z K 1 ) κ B + κ B ] exp { i [ ( 2 k z K 1 ) z ϕ ] } ,
F i α F κ 2 F = 0 ,
α = 2 k z K 1 .
γ 2 i α γ κ 2 = 0.
γ ± = i α / 2 ± s ,
s = [ κ 2 ( α / 2 ) 2 ] 1 / 2 .
F = C 1 exp ( γ + z ) + C 2 exp ( γ z ) = ( C 1 e s z + C 2 e s z ) exp ( i α 2 z ) ,
B = [ C 1 ( s + i α / 2 ) e s z C 2 ( s i α / 2 ) e s z ] × exp [ i ( α 2 z ϕ ) ] / κ .
r = B F exp ( i 2 k z z ) .
r s = [ C 1 ( s + i α / 2 ) e s L C 2 ( s i α / 2 ) e s L ] ( C 1 e s L + C 2 e s L ) κ exp [ i ( α L ϕ ) ] .
C 2 = { s + i α / s r 2 κ exp [ i ( α L ϕ ) ] } e s L C 1 { s i α / 2 + r s κ exp [ i ( α L ϕ ) ] } e s L .
r = ( { κ + ( s + i α / 2 ) r s exp [ i ( K 1 L + ϕ ) ] } exp [ s ( z L ) ] { κ ( s i α / 2 ) r s exp [ i ( K 1 L + ϕ ) ] } exp [ s ( z L ) ] { s i α / 2 + r s κ exp [ i ( K 1 L + ϕ ) ] } exp [ s ( z L ) + { s + i α / 2 r s κ exp [ i ( K 1 L + ϕ ) ] } exp [ s ( z L ) ] ) exp [ i ( K 1 z + ϕ ) ] .
ρ = r F + r 1 + r F r ,
r F = { n 0 cos θ 0 n i cos θ n 0 cos θ 0 + n i cos θ s component n 0 cos θ n i cos θ 0 n 0 cos θ + n i cos θ 0 p component ,
n i = n a + n p 2 sin ( ϕ ) ,
n 0 sin θ 0 = n i sin θ .
r s = { n L cos θ L n s cos θ s n L cos θ L + n s cos θ s s component n L cos θ s n s cos θ L n L cos θ s + n s cos θ L p component ,
n L = n a + n p 2 sin ( K 1 L + ϕ ) ,
n L sin θ L = n s sin θ s = n 0 sin θ 0 .
R = ρ ρ * ,
r = κ sinh ( s L ) e i ϕ s cosh ( s L ) + i α 2 sinh ( s L ) ,
R = κ 2 sinh 2 ( s L ) s 2 cosh 2 ( s L ) + ( α / 2 ) 2 sinh 2 ( s L ) .
R max = tanh 2 ( K L ) ,
κ L = { π n p N 4 n a 1 cos θ s component π n p N 4 n a cos 2 θ cos θ p component ,
L = λ 1 2 n a N .
λ max = λ 1 cos θ ,
λ max = λ 1 ( 1 sin 2 θ 0 n a 2 ) 1 / 2 .
Δ λ λ = sin 2 θ 0 2 n a 2 .
λ ± = λ 1 ( cos ± n p g 4 n a cos θ ) .
Δ λ λ 1 = { n p 2 n a 1 cos θ s component n p 2 n a cos 2 θ cos θ p component .
R e = κ 2 L 2 1 + κ 2 L 2 .
OD = log 10 ( 1 tanh 2 κ L ) = 2 log 10 ( sech κ L ) .
OD = 2 log 10 ( 2 e κ L ) = 2 ( log 10 2 + log 10 e κ L ) .
OD = { 0.6822 n p n a N 1 cos θ 0.6021 s component 0.6822 n p n a N cos 2 θ cos θ 0.6021 p component ,
angle shift = sin 2 θ 0 2 n a 2 .
bandwidth = n p 2 n a .
OD = 0.6822 n p n a N log 10 ( 4 n 0 n s ) .

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