Abstract

Multilayer coatings for high-power laser mirrors and lenses have been optimized by a particular constrained gradient method to reach together with a reduction of damage threshold (i.e., suppression of the peak electric field), good values of optical parameters (reflectance or transmittance). The problem has been solved for normal and oblique incident radiation of λ = 10.6 μm. Reflectance values of more than 0.99 have been achieved with only two layers deposited on a metal mirror.

© 1984 Optical Society of America

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References

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  1. J. R. Bettis, R. A. House, A. H. Guenther, R. Austin, in “Laser Induced Damage in Optical Materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 435 (1975), p. 289.
  2. J. H. Apfel, Appl. Opt. 16, 1880 (1977).
    [CrossRef] [PubMed]
  3. O. Arnon, Appl. Opt. 16, 2147 (1977).
    [CrossRef] [PubMed]
  4. P. H. Lissberger, Opt. Acta, 25, 291 (1978).
    [CrossRef]
  5. P. H. Lissbergber, Appl. Opt. 20, 95 (1981).
    [CrossRef]
  6. P. H. Lissberger, Opt. Acta 28, 187 (1981).
    [CrossRef]

1981 (2)

P. H. Lissbergber, Appl. Opt. 20, 95 (1981).
[CrossRef]

P. H. Lissberger, Opt. Acta 28, 187 (1981).
[CrossRef]

1978 (1)

P. H. Lissberger, Opt. Acta, 25, 291 (1978).
[CrossRef]

1977 (2)

1975 (1)

J. R. Bettis, R. A. House, A. H. Guenther, R. Austin, in “Laser Induced Damage in Optical Materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 435 (1975), p. 289.

Apfel, J. H.

Arnon, O.

Austin, R.

J. R. Bettis, R. A. House, A. H. Guenther, R. Austin, in “Laser Induced Damage in Optical Materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 435 (1975), p. 289.

Bettis, J. R.

J. R. Bettis, R. A. House, A. H. Guenther, R. Austin, in “Laser Induced Damage in Optical Materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 435 (1975), p. 289.

Guenther, A. H.

J. R. Bettis, R. A. House, A. H. Guenther, R. Austin, in “Laser Induced Damage in Optical Materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 435 (1975), p. 289.

House, R. A.

J. R. Bettis, R. A. House, A. H. Guenther, R. Austin, in “Laser Induced Damage in Optical Materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 435 (1975), p. 289.

Lissbergber, P. H.

Lissberger, P. H.

P. H. Lissberger, Opt. Acta 28, 187 (1981).
[CrossRef]

P. H. Lissberger, Opt. Acta, 25, 291 (1978).
[CrossRef]

Appl. Opt. (3)

Natl. Bur. Stand. (U.S.) Spec. Publ. 435 (1)

J. R. Bettis, R. A. House, A. H. Guenther, R. Austin, in “Laser Induced Damage in Optical Materials,” Natl. Bur. Stand. (U.S.) Spec. Publ. 435 (1975), p. 289.

Opt. Acta (2)

P. H. Lissberger, Opt. Acta 28, 187 (1981).
[CrossRef]

P. H. Lissberger, Opt. Acta, 25, 291 (1978).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Electric field amplitude profile of a fourteen-layer reflector (Table I) optimized at normal incidence.

Fig. 2
Fig. 2

Electric field amplitude profile of the same fourteen-layer reflector of Fig. 1 (Table I) when its layer thicknesses are optimized at an angle of incidence θ0 = 45°.

Fig. 3
Fig. 3

Electric field amplitude profile of a three-layer coating on a Cu mirror (L1 = 3.0435 μm, L2 = 0.5368 μm, L3 = 3.0709 μm, nH = 2.90, nL = 1.52, n = 10.3, K = 63.5) optimized at normal incidence.

Fig. 4
Fig. 4

Electric field amplitude profile of the same reflector as Fig. 3 when the angle of incidence is θ0 = 45°.

Fig. 5
Fig. 5

Electric field amplitude profile of the same reflector as Fig. 3 optimized at an angle of incidence θ0 = 45° (L1 = 3.4735 μm, L2 = 0.5992 μm, L3 = 3.2828 μm).

Fig. 6
Fig. 6

Electric field amplitude profile of a two-layer coating on a Cu mirror (L1 = 0.4000 μm, L2 = 3.0000 μm, nH = 3.413, nL = 1.52) optimized at normal incidence.

Fig. 7
Fig. 7

Electric field amplitude profile of the same reflector as Fig. 6 when the angle of incidence is θ0 = 45°.

Fig. 8
Fig. 8

Electric field amplitude profile of a reflector optimized an angle of incidence θ0 = 45° (L1 = 0.4771 μm, L2 = 3.7884 μm, nH = 3.417, nL = 1.394).

Tables (1)

Tables Icon

Table I Design of a Fourteen-Layer Dielectric Stack

Equations (25)

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E ( z , t ) = E ( z ) exp ( i ω t ) = A j exp [ i ( ω t - 2 π λ n ^ j z + α j ) ] + B j × exp [ i ( ω t + 2 π λ n ^ j z + β j ) ] , H ( z , t ) = H ( z ) exp ( i ω t ) = n ^ j { A j exp [ i ( ω t - 2 π λ n ^ j z + α j ) ] - B j × exp [ i ( ω t + 2 π λ n ^ j z + β j ) ] } .
β j = arctan [ n j + K j w w n j - K j ] - 2 π λ n j z j , B j = - 1 2 A j + 1 ( n j + 1 - n j ) cos α ¯ j + 1 * - B j + 1 φ ¯ j + 1 2 ( n j + 1 + n j ) cos β ¯ j + 1 * + A j + 1 ( K j + 1 - K j ) sin α ¯ j + 1 * - B j + 1 φ ¯ j + 1 * 2 ( K j + 1 + K j ) sin β ¯ j + 1 * φ j φ ¯ j + 1 * [ n j cos β ¯ j + K j sin β ¯ j ] , α j = arctan [ ( φ j / φ ¯ j + 1 * ) [ A j + 1 sin α ¯ j + 1 * + B j + 1 φ ¯ j + 1 * 2 sin β ¯ j + 1 * ] - B j ϕ j 2 sin β ¯ j ( φ j / φ ¯ j + 1 * ) [ A j + 1 cos α ¯ j + 1 * + B j + 1 φ ¯ j + 1 * 2 cos β ¯ j + 1 * ] - B j φ j 2 cos β ¯ j ] + 2 π λ n j z j , A j = ( φ j / φ ¯ j + 1 * ) [ A j + 1 cos α ¯ j + 1 * + B j + 1 φ ¯ j + 1 * 2 cos β ¯ j + 1 * ] - B j φ j 2 cos β ¯ j cos α ¯ j ,
α ¯ j = a j - 2 π λ n j z j ,             β ¯ j = β j + 2 π λ n j z j , α ¯ j + 1 * = α j + 1 - 2 π λ n j + 1 z j ,             β ¯ j + 1 * = β j + 1 + 2 π λ n j + 1 z j , φ j = exp ( 2 π λ K j z j ) ,             φ ¯ j + 1 * = exp ( 2 π λ K j + 1 z j ) , w = A j + 1 ( n j + 1 - n j ) cos α ¯ j + 1 * - B j + 1 φ ¯ j + 1 * 2 ( n j + n j + 1 ) cos β ¯ j + 1 * + A j + 1 ( K j + 1 - K j ) sin α ¯ j + 1 * - B j + 1 φ ¯ j + 1 * 2 ( K j + 1 + K j ) sin β ¯ j + 1 * A j + 1 ( K j - K j + 1 ) cos α ¯ j + 1 * + B j + 1 φ ¯ j + 1 2 cos β ¯ j + 1 * ( K j + K j + 1 ) + A j + 1 ( n j + 1 - n j ) sin α ¯ j + 1 * - B j + 1 φ ¯ j + 1 2 ( n j + n j + 1 ) sin β ¯ j + 1 * .
R = B 0 2 A 0 2 ,             T = n N + 1 A N + 1 2 exp ( - 4 π λ K N + 1 z N ) A 0 2 n 0 , A = 1 - T - R .
E j ( z ) = { A j 2 exp ( - 4 π λ K j z ) + B j 2 exp ( 4 π λ K j z ) + 2 A j B j [ cos ( α j - β j ) + sin ( α j - β j ) sin ( 4 π λ n j z ) - 2 sin 2 ( 2 π λ n j z ) cos ( α j - β j ) ] } 1 / 2 .
n j , s = L j n j - U j K j ;             K j , s = L j K j + U j n j ,
L j = { ½ [ ( P j 2 + Q j 2 ) 1 / 2 + P j ] } 1 / 2 ;             U j = { ½ [ ( P j 2 + Q j 2 ) 1 / 2 - P j ] } 1 / 2 , P j = 1 + ( K j 2 - n j 2 ) · ( n 0 sin θ 0 n j 2 + K j 2 ) 2 , Q j = - 2 n j K j ( n 0 sin θ 0 n j 2 + K j 2 ) 2 .
n j , p = n j L j + K j U j L j 2 + U j 2 ,             K j , p = K j L j - n j U j L j 2 + U j 2 .
β j , p = arctan [ n j , p + K j , p w p n j , p w p - K j , p ] - 2 π λ n j , s z j , B j , p = - 1 2 + A j + 1 , p sin α ¯ j + 1 , p * ( K j + 1 , p - K j , p ) - B j + 1 , p · δ ¯ j + 1 * 2 sin β ¯ j + 1 , ρ * ( K j + 1 , p + K j , p ) + A j + 1 , ρ cos α ¯ j + 1 , p * ( n j + 1 , p - n j , p ) - B j + 1 , p cos β ¯ j + 1 , p * δ ¯ j + 1 * 2 ( n j + 1 , ρ + n j , p ) δ j δ ¯ j + 1 * [ n j , p cos β ¯ j , p + K j , p sin β ¯ j , p ] , α j , p = arctan [ ( δ j / δ ¯ j + 1 * ) · [ A j + 1 , p sin α ¯ j + 1 , p * + B j + 1 , p δ ¯ j + 1 * 2 sin β ¯ j + 1 , p * ] - B j , p δ j 2 sin β ¯ j , p ( δ j / δ ¯ j + 1 * ) · [ A j + 1 , p cos α ¯ j + 1 , p * + B j + 1 , p δ ¯ j + 1 * 2 cos β ¯ j + 1 , p * ] - B j , p δ j 2 sin β ¯ j , p ] + 2 π λ n j , s z j , A j , p = ( δ j / δ ¯ j + 1 * ) · [ A j + 1 , p cos α ¯ j + 1 , p * + B j + 1 , p δ ¯ j + 1 * 2 cos β ¯ j + 1 , p * ] - B j , p δ j 2 cos β ¯ j , p cos α ¯ j , p ,
α ¯ j , p = α j , p - 2 π λ n j , s z j ,             α ¯ j + 1 , p * = α j + 1 , p - 2 π λ n j + 1 , s z j , β ¯ j , p = β j , p + 2 π λ n j , s z j ,             β ¯ j + 1 , p * = β j + 1 , p + 2 π λ n j + 1 , s z j , δ j = exp ( 2 π λ K j , s z j ) ,             δ ¯ j + 1 * = exp ( 2 π λ K j + 1 , s z j ) , w p = A j + 1 , p ( n j + 1 , p - n j , p ) cos α ¯ j + 1 , p * - B j + 1 , p δ ¯ j + 1 * 2 cos β ¯ j + 1 , p * ( n j + 1 , p + n j , p ) + A j + 1 , p ( K j + 1 , p - K j , p ) sin α ¯ j + 1 , p * - B j + 1 , p δ ¯ j + 1 * 2 ( K j + 1 , p + K j , p ) sin β ¯ j + 1 , p * A j + 1 , p ( K j , p - K j + 1 , p ) cos α ¯ j + 1 , p * × B j + 1 , p δ ¯ j + 1 2 cos β ¯ j + 1 , p * ( K j , p + K j + 1 , p ) + A j + 1 , p ( n j + 1 , p - n j , p ) sin α ¯ j + 1 , p * - B j + 1 , p δ ¯ j + 1 * 2 ( n j , p + n j + 1 , p ) sin β ¯ j + 1 , p *
R = 1 2 B 0 , s 2 A 0 , s 2 + 1 2 B 0 , p 2 A 0 , p 2 , T = 1 2 A N + 1 , s 2 ( L N + 1 n N + 1 - U N + 1 K N + 1 ) exp [ - 4 π λ z N ( K N + 1 L N + 1 + n N + 1 U N + 1 ) ] A 0 , s 2 cos ϑ 0 + 1 2 { A N + 1 , p 2 ( L N + 1 n N + 1 + U N + 1 K N + 1 ) exp [ - 4 π λ z N ( K N + 1 L N + 1 - n N + 1 U N + 1 ) / ( L N + 2 + U N + 1 2 ) ] } / ( L N + 1 2 + U N + 1 2 ) A 0 , p 2 · cos ϑ 0 .
E j ( z ) = [ E j , s 2 + ( Real p 2 + Real p 2 ) + ( Im p 2 + Im p 2 ) ] 1 / 2 ,
Real p = 1 δ ¯ j [ A j , p cos α ¯ ¯ j , p + B j , p δ ¯ j 2 cos β ¯ ¯ j , p ] , Real p = 1 δ ¯ j [ A j , v cos α ¯ ¯ j , p + B j , v δ ¯ j 2 sin β ¯ ¯ j , p ] , Im p = 1 δ ¯ j [ A j , p sin α ¯ ¯ j , p + B j , p δ ¯ j 2 sin β ¯ ¯ j , p ] , Im p = 1 δ ¯ j [ A j , v sin α ¯ ¯ j , p + B j , v δ ¯ j 2 sin β ¯ ¯ j , p ] , α ¯ ¯ j , p = α j , p - 2 π λ n j , s z , β ¯ ¯ j , p = β j , p + 2 π λ n j , s z , δ ¯ j = exp ( 2 π λ K j , s z ) , A j , v = D · δ j sin β ¯ ¯ j , p - C δ j cos β ¯ ¯ j , p cos α ¯ ¯ j , p sin β ¯ ¯ j , p - cos β ¯ j , p sin α ¯ ¯ j , p , B j , v = δ j - 2 [ C cos α ¯ ¯ j , p - D sin α ¯ ¯ j , p ] cos α ¯ ¯ j , p sin β ¯ ¯ j , p - cos β ¯ ¯ j , p sin α ¯ ¯ j , p , C = ( δ ¯ j + 1 * ) - 1 [ A j + 1 , v sin α ¯ ¯ j + 1 , p * + B j + 1 , v δ ¯ j + 1 * 2 sin β ¯ ¯ j + 1 , p * ] n j + 1 2 n j 2 , D = 1 δ ¯ j + 1 * [ A j + 1 , v cos α ¯ ¯ j + 1 , p * + B j + 1 , v δ ¯ j + 1 * 2 cos p ¯ j + 1 , p * ] n j + 1 2 n j 2 .
R = R ( L 1 , L 2 L N ) ;             T = T ( L 1 , L 2 L N ) ; E max = E max ( L 1 , L 2 L N ) .
P 0 ( L i ) = | L 01 L 02 L 0 N |
gradR 0 ( L i ) = | R / l 1 R / L 2 R / L N | .
Δ P 0 = | Δ L 1 Δ L 2 Δ L n | ,             gradR 0
Δ L i = κ R / L i P = P 0 ,
P ( L 11 , L 12 L 1 N ) = | L 01 + Δ L 1 L 02 + Δ L 2 L 0 N + Δ L N | .
Δ L i = - κ ( E max / L i )
V = gradR - gradE max · ( gradR × gradE max ) · 1 gradE max 2 ,
V = | V 1 V 2 V N | = | R / L 1 R / L 2 R / L N | - | E max / L 1 E max / L 2 E max / L N | · i = 1 N R L i E max L i i = 1 N ( E max / L i ) 2 .
Δ L i = κ V i .
Δ P = | Δ L 1 Δ L 2 Δ L N |
L 1 ( BaF 2 , n L = 1.39 ) - L 2 ( ZnS , n H = 2.16 ) - lens ( ZnSe , n = 2.42 ) - L 3 ( ZnS ) - L 4 ( BaF 2 ) ,

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