Abstract

A general optical model is given which predicts optical wave front distortions and birefringence due to stress and temperature variations in laser heated and pressure loaded windows for cubic lattice window materials. A computer code is described that integrates stress and thermal computations with an optical model to predict the wave front distortions. Restrictive approximations, which have been used previously to predict window temperature and stress distributions, are avoided by using stress and thermal codes to predict these distributions within the windows. Comparisons between code predictions and experimental results are given.

© 1985 Optical Society of America

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References

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  1. M. Sparks, “Optical Distortion by Heated Windows in High-Power Laser Systems,” J. Appl. Phys. 42, 5029 (1971).
    [CrossRef]
  2. C. Klein, “Stress-Induced Birefringence, Critical Window Orientation, and Thermal Lensing Experiments,” in Laser-Induced Damage of Optical Materials: 1980,” Nat. Bur. Stand. U.S. Spec. Pub. 620, Washington (1981), pp. 117–128.
    [CrossRef]
  3. E. G. Bernal, J. S. Loomis, “Interferometric Measurements of Laser Heated Windows,” in Laser-Induced Damage in Optical Materials: 1976, Natl. Bur. Stand. U.S. Spec. Publ. 462, Washington (1967), pp. 36–44.
  4. J. F. Nye, Physical Properties of Crystals (Oxford U. P., London, 1957).
  5. C. Klein, “Concept of an Effective Optical Distortion Parameter: Application to KCl Laser Windows,” Infrared Phys. 17, 343 (1977).
    [CrossRef]
  6. J. Turley, G. Sines, “The Anisotropy of Young’s Modulus, Shear Modulus and Poisson’s Ratio in Cubic Materials,” J. Phys. D 4, 264 (1971).
    [CrossRef]

1977 (1)

C. Klein, “Concept of an Effective Optical Distortion Parameter: Application to KCl Laser Windows,” Infrared Phys. 17, 343 (1977).
[CrossRef]

1971 (2)

J. Turley, G. Sines, “The Anisotropy of Young’s Modulus, Shear Modulus and Poisson’s Ratio in Cubic Materials,” J. Phys. D 4, 264 (1971).
[CrossRef]

M. Sparks, “Optical Distortion by Heated Windows in High-Power Laser Systems,” J. Appl. Phys. 42, 5029 (1971).
[CrossRef]

Bernal, E. G.

E. G. Bernal, J. S. Loomis, “Interferometric Measurements of Laser Heated Windows,” in Laser-Induced Damage in Optical Materials: 1976, Natl. Bur. Stand. U.S. Spec. Publ. 462, Washington (1967), pp. 36–44.

Klein, C.

C. Klein, “Concept of an Effective Optical Distortion Parameter: Application to KCl Laser Windows,” Infrared Phys. 17, 343 (1977).
[CrossRef]

C. Klein, “Stress-Induced Birefringence, Critical Window Orientation, and Thermal Lensing Experiments,” in Laser-Induced Damage of Optical Materials: 1980,” Nat. Bur. Stand. U.S. Spec. Pub. 620, Washington (1981), pp. 117–128.
[CrossRef]

Loomis, J. S.

E. G. Bernal, J. S. Loomis, “Interferometric Measurements of Laser Heated Windows,” in Laser-Induced Damage in Optical Materials: 1976, Natl. Bur. Stand. U.S. Spec. Publ. 462, Washington (1967), pp. 36–44.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U. P., London, 1957).

Sines, G.

J. Turley, G. Sines, “The Anisotropy of Young’s Modulus, Shear Modulus and Poisson’s Ratio in Cubic Materials,” J. Phys. D 4, 264 (1971).
[CrossRef]

Sparks, M.

M. Sparks, “Optical Distortion by Heated Windows in High-Power Laser Systems,” J. Appl. Phys. 42, 5029 (1971).
[CrossRef]

Turley, J.

J. Turley, G. Sines, “The Anisotropy of Young’s Modulus, Shear Modulus and Poisson’s Ratio in Cubic Materials,” J. Phys. D 4, 264 (1971).
[CrossRef]

Infrared Phys. (1)

C. Klein, “Concept of an Effective Optical Distortion Parameter: Application to KCl Laser Windows,” Infrared Phys. 17, 343 (1977).
[CrossRef]

J. Appl. Phys. (1)

M. Sparks, “Optical Distortion by Heated Windows in High-Power Laser Systems,” J. Appl. Phys. 42, 5029 (1971).
[CrossRef]

J. Phys. D (1)

J. Turley, G. Sines, “The Anisotropy of Young’s Modulus, Shear Modulus and Poisson’s Ratio in Cubic Materials,” J. Phys. D 4, 264 (1971).
[CrossRef]

Other (3)

C. Klein, “Stress-Induced Birefringence, Critical Window Orientation, and Thermal Lensing Experiments,” in Laser-Induced Damage of Optical Materials: 1980,” Nat. Bur. Stand. U.S. Spec. Pub. 620, Washington (1981), pp. 117–128.
[CrossRef]

E. G. Bernal, J. S. Loomis, “Interferometric Measurements of Laser Heated Windows,” in Laser-Induced Damage in Optical Materials: 1976, Natl. Bur. Stand. U.S. Spec. Publ. 462, Washington (1967), pp. 36–44.

J. F. Nye, Physical Properties of Crystals (Oxford U. P., London, 1957).

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Figures (13)

Fig. 1
Fig. 1

Cartesian coordinate systems showing the index ellipse rotated due to stresses.

Fig. 2
Fig. 2

General propagation directions with respect to crystallographic planes of the crystal. [h0l] is a general direction in the x-z plane. [hkl] is a general propagation direction.

Fig. 3
Fig. 3

Window cut with α = β = γ = 0. Here the window faces are each parallel to the 〈001〉 Miller plane while the window edges are cut parallel to either the 〈100〉 or 〈010〉 Miller planes.

Fig. 4
Fig. 4

Window cut with α = β = 0 and γ ± 0. Window faces are each parallel to the 〈001〉 plane, but edges are not parallel to Miller planes.

Fig. 5
Fig. 5

Window as viewed along the [111] direction of propagation with α = 45°, β = cos 1 ( 1 / 3 ). Two values of γ are shown.

Fig. 6
Fig. 6

Data transfer process among thermal, stress, and optics codes.

Fig. 7
Fig. 7

Phase profile of the x polarization of light after a single pass through a segmented CaF2 window assembly. Input phase profile was perfectly plane x-polarized light. Window parameters are: thickness = 1.83 cm; window full aperture = 33 cm; n = 1.446; α = β = γ = 0; propagation direction is [001]; pressure differential = 4.67 atm, q11 = −0.38 × 10−12, q12 = 1.04 × 10−12, q44 = 0.71 × 10−12, s11 = 6.9 × 10−12, s12 = −1.45 × 10−12, s44 = 2.97 × 10−11 (all in m2/n), λ = 351 nm.

Fig. 8
Fig. 8

Region of the segmented window assembly over which the phase profile is shown in Fig. 7.

Fig. 9
Fig. 9

Sections of the phase profile shown in Fig. 7.

Fig. 10
Fig. 10

Sections of the phase profile for the x polarization of light for the [111] direction of propagation. All parameters are the same as in Fig. 7 except α = 45°, β = 54.74°, and γ = 0°.

Fig. 11
Fig. 11

Zygo transmission interferograms for horizontally polarized light under 0- and 100-psi pressure loading: λ = 632.8 nm. Sample 4, CaF2, α = 40.25°, β = 24°, γ = 0. Sample rested on a metallic backup structure with a 2.2-mm (0.09-in.) seat. Sample dimensions were 10.54 × 10.54 × 2.17 cm with a wedge of 20 ± 2 min of arc.

Fig. 12
Fig. 12

Single quadrant phase difference between 0- and 100-psi test cases. Expiermental error bars are shown.

Fig. 13
Fig. 13

Single quadrant phase profile predicted by the code for the experimental conditions given in the caption to Fig. 11 for a 100-psi pressure loading.

Equations (25)

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B 1 x 2 + B 2 y 2 + B 3 z 2 + 2 B 4 y z + 2 B 5 x z + 2 B 6 x y = 1 .
B 1 x 2 + B 2 y 2 + 2 B 6 x y = 1 .
x = x cos θ y sin θ , y = x sin θ + y cos θ .
θ = 1 2 tan 1 ( 2 B 6 B 1 B 2 ) ,
1 n x 2 = B 1 cos 2 θ + B 2 sin 2 θ + 2 B 6 sin θ cos θ , 1 n y 2 = B 1 sin 2 θ + B 2 cos 2 θ 2 B 6 sin θ cos θ .
1 n x , y 2 = B 1 + B 2 2 ± ( B 1 B 2 ) 2 2 ( B 1 B 2 ) 2 + 4 B 6 2 ± 2 B 6 2 ( B 1 B 2 ) 2 + 4 B 6 2 ,
B 1 = B ° + q 1 j σ j , B 2 = B ° + q 2 j σ j , B 6 = q 6 j σ j ,
σ ¯ = ( σ x x σ y y σ z z σ y z σ x z σ x y ) ,
B ° = 1 / n 2 .
1 n x , y 2 1 n 2 = f 1 + f 2 2 ± ( f 1 f 2 ) 2 2 ( f 1 f 2 ) 2 + 4 f 6 2 ± 2 f 6 2 ( f 1 f 2 ) 2 + 4 f 6 2 ,
θ = 1 2 tan 1 ( 2 f 6 f 1 f 2 ) ,
f 1 = q 1 j σ j , f 2 = q 2 j σ j , f 6 = q 6 j σ j .
Δ n x = n x n Δ n y = n y n
Δ n x , y = n 3 4 [ f 1 + f 2 ± ( f 1 f 2 ) 2 ( f 1 f 2 ) 2 + 4 f 6 2 ± 4 f 6 2 ( f 1 f 2 ) 2 + 4 f 6 2 ] .
Δ n x , y = n 3 4 [ f 1 + f 2 ± ( f 1 f 2 ) 2 ( f 1 f 2 ) 2 + 4 f 6 2 ± 4 f 6 2 ( f 1 f 2 ) 2 + 4 f 6 2 ] + d n d T T ,
q ˜ = [ q 11 q 12 q 13 0 0 0 q 21 q 11 q 23 0 0 0 q 31 q 32 q 11 0 0 0 0 0 0 q 44 0 0 0 0 0 0 q 44 0 0 0 0 0 0 q 44 ] .
q i j k l = q 1122 δ i j δ k l + q 1212 ( δ i k δ j l + δ i l δ j k ) + ( q 1111 q 1122 2 q 1212 ) A i u A j u A k u A l u .
A ˜ = [ cos α cos β sin α cos β sin β ( cos α sin β sin γ + sin α cos γ ) ( sin α sin β sin γ cos α cos γ ) cos β sin γ ( cos α sin β cos γ sin α sin γ ) ( sin α sin β cos γ + cos α sin γ ) cos β cos γ ] .
q i j k l = q m n when m and n = 1 , 2 , or 3 , 2 q i j k l = q m n when m or n = 4 , 5 , or 6 .
E ¯ i = E x exp ( i ϕ x ) x ˆ + E y exp ( i ϕ y ) y ˆ .
E ¯ 0 = exp [ i ( Δ n x + Δ n y ) 2 k l ] × ( { [ E x exp ( i ϕ x ) cos 2 θ + E y exp ( i ϕ y ) sin θ cos θ ] × exp [ i ( Δ n x Δ n y ) 2 k l ] + [ E y exp ( i ϕ y ) cos θ sin θ + E x exp ( i ϕ x ) sin 2 θ ] × exp [ i ( Δ n y Δ n x ) 2 k l ] } x ˆ + { [ E x exp ( i ϕ x ) cos θ sin θ + E y exp ( i ϕ y ) sin 2 θ ) × exp [ i ( Δ n x Δ n y ) 2 k l ] + [ E y exp ( i ϕ y ) cos 2 θ E x exp ( i ϕ x ) sin θ cos θ ] × exp [ i ( Δ n y Δ n x ) 2 k l ] } y ˆ ) .
ϕ x n = ( Δ n x + Δ n y ) 2 k l + tan 1 [ Im ( A ) Re ( A ) ] , ϕ y n = ( Δ n x + Δ n y ) 2 k l + tan 1 [ Im ( B ) Re ( B ) ] , E x n = | A | , E y n = | B | ,
OPL g = ( n 1 ) k 0 L 3 d z ,
OPL g = ( n 1 ) k 0 L ( s 3 j σ j + α T ) d z ,
s i j k l = s m n when m and n = 1 , 2 , or 3 , 2 s i j k l = s m n when m or n = 4 , 5 , or 6 , 4 s i j k l = s m n when m and n = 4 , 5 , or 6 .

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