Abstract

Based on the three-dimensional transient heat conduction equation and the elastic stress-strain equation, the temperature rise, distortion, and equivalent stress distributions of a high-reflectivity silicon reflector and a white bijou window irradiated by a high-power sloped annularly distributed laser beam are simulated using a three-dimensional finite element model (FEM). The effects of laser intensity, output duration, beam obscure ratio, and laser intensity spatial gradient on the results are especially investigated. The effects of mirror and window thermal distortion on laser beam phase aberrations are also evaluated. This noncylindrosymmetric three-dimensional FEM can be used to evaluate high-power, high-energy, laser beam-induced thermal effects on optical components.

© 2005 Optical Society of America

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References

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  1. C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt Eng. 29, 343–350 (1990).
    [CrossRef]
  2. C. A. Klein, “Materials for high-power laser optics figures of merit for thermally induced beam distortions,” Opt. Eng. 36, 1586–1595 (1997).
    [CrossRef]
  3. J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejie, R. L. Byer, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wavefront sensor,” Appl Opt 40, 366–374 (2001).
    [CrossRef]
  4. Y. F. Peng, Z. H. Cheng, Y. N. Zhang, J. L. Qiu, “Temperature distributions and thermal deformations of mirror substrates in laser resonators,” Appl. Opt. 40, 4824–4830 (2001).
    [CrossRef]
  5. J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its affect on far-field optical spot,” High Power Laser and Particle Beam 6, 243–249 (1994).
  6. Y. Y. Ma, Z. H. Cheng, Y. N. Zhang, “Finite-element method in thermal deformation analysis of high power laser windows,” High Power Laser and Particle Beam 11, 6–10 (1999).
  7. Chen Faliang, Li Youkuan, “Thermal deformation of optical windows induced by annularly distributed laser beam,” High Power Laser and Particle Beam 15, 736–740 (2003).
  8. D. C. Harris, Infrared Window and Dome Materials, Tutorial Text in Optical Engineering (SPIE Press, 1999), pp. 126–129.
    [CrossRef]
  9. Li Jingzhen, Handbook of Optics (Shannxi Science and Technology Press, 1985), pp. 1328–1337.
  10. Sadik Kakac, Yaman Yener, Heat Conduction, 2nd ed. (Hemisphere, 1985), pp. 29–34.
  11. Y. Takeuti, translated by Tingwei Guo, Anding Li, Thermal Stress. (Science, 1977), pp. 47–50.
  12. Su Yilin, Material Mechanics (Tianjing University Press, 2001), p. 152.

2003 (1)

Chen Faliang, Li Youkuan, “Thermal deformation of optical windows induced by annularly distributed laser beam,” High Power Laser and Particle Beam 15, 736–740 (2003).

2001 (2)

Y. F. Peng, Z. H. Cheng, Y. N. Zhang, J. L. Qiu, “Temperature distributions and thermal deformations of mirror substrates in laser resonators,” Appl. Opt. 40, 4824–4830 (2001).
[CrossRef]

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejie, R. L. Byer, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wavefront sensor,” Appl Opt 40, 366–374 (2001).
[CrossRef]

1999 (1)

Y. Y. Ma, Z. H. Cheng, Y. N. Zhang, “Finite-element method in thermal deformation analysis of high power laser windows,” High Power Laser and Particle Beam 11, 6–10 (1999).

1997 (1)

C. A. Klein, “Materials for high-power laser optics figures of merit for thermally induced beam distortions,” Opt. Eng. 36, 1586–1595 (1997).
[CrossRef]

1994 (1)

J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its affect on far-field optical spot,” High Power Laser and Particle Beam 6, 243–249 (1994).

1990 (1)

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt Eng. 29, 343–350 (1990).
[CrossRef]

Byer, R. L.

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejie, R. L. Byer, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wavefront sensor,” Appl Opt 40, 366–374 (2001).
[CrossRef]

Chen, J. B.

J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its affect on far-field optical spot,” High Power Laser and Particle Beam 6, 243–249 (1994).

Cheng, Z. H.

Y. F. Peng, Z. H. Cheng, Y. N. Zhang, J. L. Qiu, “Temperature distributions and thermal deformations of mirror substrates in laser resonators,” Appl. Opt. 40, 4824–4830 (2001).
[CrossRef]

Y. Y. Ma, Z. H. Cheng, Y. N. Zhang, “Finite-element method in thermal deformation analysis of high power laser windows,” High Power Laser and Particle Beam 11, 6–10 (1999).

Faliang, Chen

Chen Faliang, Li Youkuan, “Thermal deformation of optical windows induced by annularly distributed laser beam,” High Power Laser and Particle Beam 15, 736–740 (2003).

Fejie, M. M.

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejie, R. L. Byer, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wavefront sensor,” Appl Opt 40, 366–374 (2001).
[CrossRef]

Gustafson, E. K.

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejie, R. L. Byer, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wavefront sensor,” Appl Opt 40, 366–374 (2001).
[CrossRef]

Harris, D. C.

D. C. Harris, Infrared Window and Dome Materials, Tutorial Text in Optical Engineering (SPIE Press, 1999), pp. 126–129.
[CrossRef]

Hennawi, J.

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejie, R. L. Byer, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wavefront sensor,” Appl Opt 40, 366–374 (2001).
[CrossRef]

Jiang, Z. P.

J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its affect on far-field optical spot,” High Power Laser and Particle Beam 6, 243–249 (1994).

Jingzhen, Li

Li Jingzhen, Handbook of Optics (Shannxi Science and Technology Press, 1985), pp. 1328–1337.

Kakac, Sadik

Sadik Kakac, Yaman Yener, Heat Conduction, 2nd ed. (Hemisphere, 1985), pp. 29–34.

Klein, C. A.

C. A. Klein, “Materials for high-power laser optics figures of merit for thermally induced beam distortions,” Opt. Eng. 36, 1586–1595 (1997).
[CrossRef]

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt Eng. 29, 343–350 (1990).
[CrossRef]

Liu, Z. J.

J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its affect on far-field optical spot,” High Power Laser and Particle Beam 6, 243–249 (1994).

Lu, Q. S.

J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its affect on far-field optical spot,” High Power Laser and Particle Beam 6, 243–249 (1994).

Ma, Y. Y.

Y. Y. Ma, Z. H. Cheng, Y. N. Zhang, “Finite-element method in thermal deformation analysis of high power laser windows,” High Power Laser and Particle Beam 11, 6–10 (1999).

Mansell, J. D.

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejie, R. L. Byer, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wavefront sensor,” Appl Opt 40, 366–374 (2001).
[CrossRef]

Peng, Y. F.

Qiu, J. L.

Takeuti, Y.

Y. Takeuti, translated by Tingwei Guo, Anding Li, Thermal Stress. (Science, 1977), pp. 47–50.

Yener, Yaman

Sadik Kakac, Yaman Yener, Heat Conduction, 2nd ed. (Hemisphere, 1985), pp. 29–34.

Yilin, Su

Su Yilin, Material Mechanics (Tianjing University Press, 2001), p. 152.

Youkuan, Li

Chen Faliang, Li Youkuan, “Thermal deformation of optical windows induced by annularly distributed laser beam,” High Power Laser and Particle Beam 15, 736–740 (2003).

Zhang, Y. N.

Y. F. Peng, Z. H. Cheng, Y. N. Zhang, J. L. Qiu, “Temperature distributions and thermal deformations of mirror substrates in laser resonators,” Appl. Opt. 40, 4824–4830 (2001).
[CrossRef]

Y. Y. Ma, Z. H. Cheng, Y. N. Zhang, “Finite-element method in thermal deformation analysis of high power laser windows,” High Power Laser and Particle Beam 11, 6–10 (1999).

Zhang, Z. W.

J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its affect on far-field optical spot,” High Power Laser and Particle Beam 6, 243–249 (1994).

Zhao, Y. J.

J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its affect on far-field optical spot,” High Power Laser and Particle Beam 6, 243–249 (1994).

Appl Opt (1)

J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejie, R. L. Byer, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wavefront sensor,” Appl Opt 40, 366–374 (2001).
[CrossRef]

Appl. Opt. (1)

High Power Laser and Particle Beam (3)

J. B. Chen, Z. J. Liu, Z. P. Jiang, Q. S. Lu, Z. W. Zhang, Y. J. Zhao, “Heating effect of DF laser unstable cavity window and its affect on far-field optical spot,” High Power Laser and Particle Beam 6, 243–249 (1994).

Y. Y. Ma, Z. H. Cheng, Y. N. Zhang, “Finite-element method in thermal deformation analysis of high power laser windows,” High Power Laser and Particle Beam 11, 6–10 (1999).

Chen Faliang, Li Youkuan, “Thermal deformation of optical windows induced by annularly distributed laser beam,” High Power Laser and Particle Beam 15, 736–740 (2003).

Opt Eng. (1)

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt Eng. 29, 343–350 (1990).
[CrossRef]

Opt. Eng. (1)

C. A. Klein, “Materials for high-power laser optics figures of merit for thermally induced beam distortions,” Opt. Eng. 36, 1586–1595 (1997).
[CrossRef]

Other (5)

D. C. Harris, Infrared Window and Dome Materials, Tutorial Text in Optical Engineering (SPIE Press, 1999), pp. 126–129.
[CrossRef]

Li Jingzhen, Handbook of Optics (Shannxi Science and Technology Press, 1985), pp. 1328–1337.

Sadik Kakac, Yaman Yener, Heat Conduction, 2nd ed. (Hemisphere, 1985), pp. 29–34.

Y. Takeuti, translated by Tingwei Guo, Anding Li, Thermal Stress. (Science, 1977), pp. 47–50.

Su Yilin, Material Mechanics (Tianjing University Press, 2001), p. 152.

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Figures (7)

Fig. 1
Fig. 1

Laser intensity distribution for simulation: (a) diagram of a laser beam section and (b) intensity distribution along the x axis.

Fig. 2
Fig. 2

CW laser-induced (a) temperature rise, (b) distortion, (c) equivalent stress distribution on the mirror surface.

Fig. 3
Fig. 3

Influence of laser intensity spatial gradient on the temperature field of a mirror surface: (a) γ = 0, (b) γ = 0.05 cm−1, (c) γ = 0.15 cm−1.

Fig. 4
Fig. 4

Influence of laser intensity spatial gradient on the distortion distribution of the mirror surface: (a) γ = 0, (b) γ = 0.05 cm−1, (c) γ = 0.15 cm−1.

Fig. 5
Fig. 5

Influence of laser intensity spatial gradient on the equivalent stress distribution of a mirror surface: (a) γ = 0, (b) γ = 0.05 cm−1, (c) γ = 0.15 cm−1.

Fig. 6
Fig. 6

Front surface temperature, distortion, and equivalent stress distribution of a white bijou window after penetration by a high-power annular beam: (a) t = 2 s, isothermal line; (b) t = 4 s, isothermal line; (c) t = 4 s, distortion isoline; (d) t = 4 s, stress isoline.

Fig. 7
Fig. 7

Rear surface temperature and distortion distribution of the white bijou window after penetration by a high-power annular beam: (a) t = 4 s, isothermal line; (b) t = 4 s, distortion isoline.

Tables (4)

Tables Icon

Table 1 Thermodynamic Parameters of Reflector and Window Materials

Tables Icon

Table 2 Cylindrosymmetric Computational Distortion of Silicon Mirrora Irradiated by a High-Power CW Laser

Tables Icon

Table 3 Three-dimensional Computational Distortion of Silicon Mirrora Irradiated by a High-Power CW Laser

Tables Icon

Table 4 Three-dimensional Computational Distortion of White Bijoua Irradiated by a High-Power CW Laser

Equations (29)

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ρ c T t = Δ ( K T ) + Q ,
ρ c T ( x , y , z , t ) t = x ( K T x ) + y ( K T y ) + z ( K T z ) + Q ( x , y , z , t ) .
q l = ( 1 R ) I ( x , y , z ) ;
q r = ɛ σ ( T 4 T 0 4 ) ;
q c = h f ( T T 0 ) ;
T ( x , y , z , t ) t = 0 = T = 0 ;
I ( x , y , z , t ) z = 1 cm = { I 0 ( 1 + γ x ) a x 2 + y 2 b and 0 < t 4 s 0 x 2 + y 2 < a or x 2 + y 2 > b } ,
P = I ( x , y , z ) | z = 1 cm d x d y = 2 b b I 0 ( 1 + γ x ) b 2 x 2 d x 2 a a I 0 ( 1 + γ x ) a 2 x 2 d x = I 0 π ( b 2 a 2 ) ,
I 0 = P π ( b 2 a 2 ) .
Q ( x , y , z , t ) = A I ( x , y , z , t ) | z = 1 cm e A ( h z ) ,
ɛ x x = u x x = 1 E { σ x x ν ( σ y y + σ z z ) } + α τ ,
ɛ y y = u y y = 1 E { σ y y ν ( σ x x + σ z z ) } + α τ ,
ɛ z z = u z z = 1 E { σ z z ν ( σ x x + σ y y ) } + α τ ,
ɛ x y = 1 2 ( u y x + u x y ) = 1 + ν E σ x y ,
ɛ y z = 1 2 ( u z y + u y z ) = 1 + ν E σ y z ,
ɛ z x = 1 2 ( u x z + u z x ) = 1 + ν E σ z x ,
σ x x x + σ y x y + σ z x z = 0 ,
σ x y x + σ y y y + σ z y z = 0 ,
σ x z x + σ y z y + σ z z z = 0 .
x 2 + y 2 = 6 cm .
σ e = [ ( σ 1 σ 2 ) 2 + ( σ 2 σ 3 ) 2 + ( σ 3 σ 1 ) 2 ] / 2 ,
ϕ ( x , y , t ) = 2 π λ 2 u z ( x , y , t ) ,
Δ ϕ = 2 π λ Δ l ( x , y , t ) ,
l ( x , y , t ) = n L + [ Δ n thermal ( x , y , t ) ] L + [ Δ n stress ( x , y , t ) ] L + n [ Δ n thermal ( x , y , t ) ] ,
Δ l ( x , y , t ) = [ Δ n thermal ( x , y , t ) ] L + [ Δ n stress ( x , y , t ) ] L + n [ Δ n thermal ( x , y , t ) ] .
Δ n thermal ( x , y , t ) = d n d T Δ T ( x , y , z ) ,
Δ n thermal ( x , y , t ) n 3 2 ρ 12 α Δ T ( x , y , z ) ,
[ Δ l ( x , y , t ) ] rms = [ Δ n thermal ( x , y , t ) ] rms L + n [ Δ n thermal ( x , y , t ) ] rms ,
[ Δ n thermal ( x , y , t ) ] rms = d n d T [ Δ T ( x , y , t ) ] rms .

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