Abstract

We are concerned with the similarity and scaling law of the thermal effects of windows subjected to laser propagation and influences on laser beam quality. Using characteristic physical quantities and dimensionless equations, appropriate similarity relations are derived, independent of the specific properties of the materials and beams. As an example, a full-scale and a half-scale window are numerically analyzed to verify the relations. It is concluded that the phase aberration resulting from thermal deformation and thermo-optic effects comply completely with similarity-based scaling, while the phase aberration resulting from elasto-optic effects scales approximately. From the results of a model window, the performances of a full-scale window can be obtained using the similarity relations.

© 2008 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, 5029-5046 (1971).
    [CrossRef]
  2. C. A. Klein, "Optical distortion coefficients of high-power laser windows," Opt. Eng. 29, 343-350 (1997).
    [CrossRef]
  3. W. Wang, F. Tan, B. Lü, and C. Liu, "Three-dimensional calculation of high-power, annularly distributed, laser-beam-induced thermal effects on reflectors and window," Appl. Opt. 44, 7442-7450 (2005).
    [CrossRef] [PubMed]
  4. K. W. Billman, D. C. Tran, L. F. Johnson, and M. B. Moran, S. M. F. Nee, J. A. Detrio, S. M. Daigneault, and A. P. Bukley, "Development of low-OPD windows for airborne laser," Proc. SPIE 4376, 45-56 (2001).
    [CrossRef]
  5. C. A. Klein, "Materials for high-energy laser windows: oxyfluoride glass vs. fusion-cast CaF2," Proc. SPIE 5786, 280-295 (2005).
    [CrossRef]
  6. J. An, Y. Li, and X. Du, "Thermal effects of a laser beam tube consisting of a window and nonflowing gas," Opt. Lett. 29, 2899-2901 (2004).
    [CrossRef]
  7. Z. S. Zhang and G. X. Cui, Fluid Dynamics (Tsinghua, 1999).
  8. F. L. Chen and T. X. Yu, "Scaling laws for structural thermal-dynamic response and failure," Acta Mech. Solid Sinica 18, 25-37 (1997).
  9. D. Sheng, W. Jian-Guo, W. Yu-Heng, and F. Liu, "Similarity of thermo-mechanical effect induced by high energy laser beam," High Power Laser and Particle Beams 17, 1331-1334 (2005).
  10. M. J. Weber, Handbook of Optical Materials (CRC, 2002).
    [CrossRef]
  11. C. W. Sun, Q. S. Lu, Z. X. Fan, Y. Z. Zhen, C. F. Li, J. L. Guan, and C. W. Guan, Effects of Laser Radiation (National Defense Industry, 2002).

2005

W. Wang, F. Tan, B. Lü, and C. Liu, "Three-dimensional calculation of high-power, annularly distributed, laser-beam-induced thermal effects on reflectors and window," Appl. Opt. 44, 7442-7450 (2005).
[CrossRef] [PubMed]

C. A. Klein, "Materials for high-energy laser windows: oxyfluoride glass vs. fusion-cast CaF2," Proc. SPIE 5786, 280-295 (2005).
[CrossRef]

D. Sheng, W. Jian-Guo, W. Yu-Heng, and F. Liu, "Similarity of thermo-mechanical effect induced by high energy laser beam," High Power Laser and Particle Beams 17, 1331-1334 (2005).

2004

2001

K. W. Billman, D. C. Tran, L. F. Johnson, and M. B. Moran, S. M. F. Nee, J. A. Detrio, S. M. Daigneault, and A. P. Bukley, "Development of low-OPD windows for airborne laser," Proc. SPIE 4376, 45-56 (2001).
[CrossRef]

1997

C. A. Klein, "Optical distortion coefficients of high-power laser windows," Opt. Eng. 29, 343-350 (1997).
[CrossRef]

F. L. Chen and T. X. Yu, "Scaling laws for structural thermal-dynamic response and failure," Acta Mech. Solid Sinica 18, 25-37 (1997).

1971

M. Sparks, "Optical distortion by heated windows in high-power laser systems," J. Appl. Phys. 42, 5029-5046 (1971).
[CrossRef]

Acta Mech. Solid Sinica

F. L. Chen and T. X. Yu, "Scaling laws for structural thermal-dynamic response and failure," Acta Mech. Solid Sinica 18, 25-37 (1997).

Appl. Opt.

High Power Laser and Particle Beams

D. Sheng, W. Jian-Guo, W. Yu-Heng, and F. Liu, "Similarity of thermo-mechanical effect induced by high energy laser beam," High Power Laser and Particle Beams 17, 1331-1334 (2005).

J. Appl. Phys.

M. Sparks, "Optical distortion by heated windows in high-power laser systems," J. Appl. Phys. 42, 5029-5046 (1971).
[CrossRef]

Opt. Eng.

C. A. Klein, "Optical distortion coefficients of high-power laser windows," Opt. Eng. 29, 343-350 (1997).
[CrossRef]

Opt. Lett.

Proc. SPIE

K. W. Billman, D. C. Tran, L. F. Johnson, and M. B. Moran, S. M. F. Nee, J. A. Detrio, S. M. Daigneault, and A. P. Bukley, "Development of low-OPD windows for airborne laser," Proc. SPIE 4376, 45-56 (2001).
[CrossRef]

C. A. Klein, "Materials for high-energy laser windows: oxyfluoride glass vs. fusion-cast CaF2," Proc. SPIE 5786, 280-295 (2005).
[CrossRef]

Other

M. J. Weber, Handbook of Optical Materials (CRC, 2002).
[CrossRef]

C. W. Sun, Q. S. Lu, Z. X. Fan, Y. Z. Zhen, C. F. Li, J. L. Guan, and C. W. Guan, Effects of Laser Radiation (National Defense Industry, 2002).

Z. S. Zhang and G. X. Cui, Fluid Dynamics (Tsinghua, 1999).

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Figures (4)

Fig. 1
Fig. 1

Phase aberration distribution along radius direction caused by thermal deformation and thermo-optic effects, for both a full-scale window and a half-scale model.

Fig. 2
Fig. 2

Phase aberration distribution along radius direction, of radial and azimuthal polarized plane waves, caused by elasto-optic effects, for a full-scale window and a half-scale model.

Fig. 3
Fig. 3

Total phase aberration distribution along radius direction, of radial and azimuthal polarized plane waves, for a full-scale window and a half-scale model.

Fig. 4
Fig. 4

(Color online) RMS of total phase aberration of radial and azimuthal polarized plane waves evolves during laser radiation, for a full-scale window and a half-scale model.

Tables (2)

Tables Icon

Table 1 Similarity Relations of Laser Induced Window Thermal Effects

Tables Icon

Table 2 Calculation Parameters for a Full-Scale Window and a Half-Scale Model

Equations (45)

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ϕ = k ( n 1 ) h ,
Δ ϕ = k [ ( n 1 ) Δ h + h Δ n ] .
Δ ϕ = k [ ( n 1 ) 0 h δ h + 0 h δ n d l ] .
0 h δ h = 0 h ε z ( r , θ , z ) d l = u h u 0 ,
δ n = ( δ n ) temp + ( δ n ) stress .
( δ n ) temp = ( n T ) σ = 0 δ T = d n d T δ T ,
( δ n ) stress = n 3 2 [ q σ ρ + q ( σ θ + σ z ) ] ,
( δ n ) stress = n 3 2 [ q σ θ + q ( σ ρ + σ z ) ] ,
q = 1 2 ( q 11 + q 12 + q 44 ) ,
q = 1 6 ( q 11 + 5 q 12 q 44 ) ,
Δ ϕ rms = Δ ϕ 2 Δ ϕ 2 .
ρ c p T t κ 2 T = β V I ,
T z | z = 0 = 1 κ β s I ,
T z | z = h = 1 κ β s I ,
x = x / C x , t = t / C t , I = I / C I , T = T / C T .
C T C t C I ρ C p T t C T C x 2 C I κ 2 T = β v I ,
C T C x C I T z | z = 0 = 1 κ β s I ,
C T C x C I T z | z = h / C x = 1 κ β s I .
C T / C x 2 C I ,     C T / C x C I .
ρ c p T t κ ( 2 T x 2 + 2 T y 2 ) = ( β V + 2 β s h ) I ,
T r | r = R = 0 ,
T | t = 0 = 0 .
ε i j = 1 2 ( u i , j + u j , i ) ,
σ i j = λ θ δ i j + 2 G ε i j ( 3 λ + 2 G ) α l T δ i j ,
σ i j , j = 0 ,
θ = ε x + ε y + ε z ,
λ = E ν / [ ( 1 + ν ) ( 1 2 ν ) ] , G = E / [ 2 ( 1 + ν ) ] ,
C T = 1 κ h 2 Q p ,     C ε = 1 κ h 2 α l Q p ,
C σ = E κ h 2 α l Q p , C u = 1 κ h 3 α l Q p .
x = x / h , t = t / t 0 , Q = Q / Q p ,
T = T / C T , ε i j = ε i j / C ε , σ i j = σ i j / C σ , u i = u i / C u .
ρ c p κ T t t 0 h 2 2 T x i 2 = Q t 0 h 2 ,
T r | r = R / h = 0 ,
T | t = 0 = 0 ,
ε i j = 1 2 ( u i , j + u j , i ) ,
σ i j = 1 E ( λ θ δ i j + 2 G ε i j ) 1 E ( 3 λ + 2 G ) T δ i j ,
σ i j , j = 0 ,
θ = ε x + ε y + ε z .
t 0 h 2 .
T f ( x , y , z , t ) = T h ( x / γ , y / γ , z / γ , t / γ 2 ) γ 2 η ,
ε f ( x , y , z , t ) = ε h ( x / γ , y / γ , z / γ , t / γ 2 ) γ 2 η ,
σ f ( x , y , z , t ) = σ h ( x / γ , y / γ , z / γ , t / γ 2 ) γ 2 η ,
u f ( x , y , z , t ) = u h ( x / γ , y / γ , z / γ , t / γ 2 ) γ 3 η ,
Δ ϕ f ( x , y , z , t ) = Δ ϕ h ( x / γ , y / γ , z / γ , t / γ 2 ) γ 3 η ,
η = 1 / γ 2 ,

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