Abstract

The thermal strain in a laser rod with a longitudinal temperature increase is modeled and analytically derived through the method of thermoelastic displacement potential and the method of Love displacement function. The analytical results show that in the absence of external forces, the longitudinal rise of fluid temperature has an unnoticeable effect on the thermal stress profile in the laser rod. However, the thermal strain field caused by the temperature distribution under the traction free boundary condition has an evident variation in the longitudinal direction, which will considerably affect the laser transmission characteristics and the beam quality.

© 2009 Optical Society of America

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  1. P. Hello, E. Durand, P. K. Fritschelt, and C. N. Man, J. Mod. Opt. 41, 1371 (1994).
    [Crossref]
  2. B. A. Usievich, V. A. Sychugov, F. Pigeon, and A. Tishchenko, IEEE J. Quantum Electron. 37, 1210 (2001).
    [Crossref]
  3. S. Chénais, F. Balembois, F. Druon, G. L. Leclin, and P. Georges, IEEE J. Quantum Electron. 40, 1217 (2004).
    [Crossref]
  4. W. Xie, Y. Kwon, W. Hu, and F. Zhou, Opt. Eng. (Bellingham) 42, 1787 (2003).
    [Crossref]
  5. X. Peng, L. Xu, and A. Asundi, Opt. Eng. (Bellingham) 43, 2454 (2004).
    [Crossref]
  6. M. Ostermeyer, D. Mudge, P. J. Veitch, and J. Munch, Appl. Opt. 45, 5368 (2006).
    [Crossref] [PubMed]
  7. Z. Li, X. Huai, and Y. Tao, Appl. Phys. B 87, 301 (2007).
    [Crossref]
  8. Y. Takeuchi, Thermal Stress (Science, 1977).

2007 (1)

Z. Li, X. Huai, and Y. Tao, Appl. Phys. B 87, 301 (2007).
[Crossref]

2006 (1)

2004 (2)

X. Peng, L. Xu, and A. Asundi, Opt. Eng. (Bellingham) 43, 2454 (2004).
[Crossref]

S. Chénais, F. Balembois, F. Druon, G. L. Leclin, and P. Georges, IEEE J. Quantum Electron. 40, 1217 (2004).
[Crossref]

2003 (1)

W. Xie, Y. Kwon, W. Hu, and F. Zhou, Opt. Eng. (Bellingham) 42, 1787 (2003).
[Crossref]

2001 (1)

B. A. Usievich, V. A. Sychugov, F. Pigeon, and A. Tishchenko, IEEE J. Quantum Electron. 37, 1210 (2001).
[Crossref]

1994 (1)

P. Hello, E. Durand, P. K. Fritschelt, and C. N. Man, J. Mod. Opt. 41, 1371 (1994).
[Crossref]

Asundi, A.

X. Peng, L. Xu, and A. Asundi, Opt. Eng. (Bellingham) 43, 2454 (2004).
[Crossref]

Balembois, F.

S. Chénais, F. Balembois, F. Druon, G. L. Leclin, and P. Georges, IEEE J. Quantum Electron. 40, 1217 (2004).
[Crossref]

Chénais, S.

S. Chénais, F. Balembois, F. Druon, G. L. Leclin, and P. Georges, IEEE J. Quantum Electron. 40, 1217 (2004).
[Crossref]

Druon, F.

S. Chénais, F. Balembois, F. Druon, G. L. Leclin, and P. Georges, IEEE J. Quantum Electron. 40, 1217 (2004).
[Crossref]

Durand, E.

P. Hello, E. Durand, P. K. Fritschelt, and C. N. Man, J. Mod. Opt. 41, 1371 (1994).
[Crossref]

Fritschelt, P. K.

P. Hello, E. Durand, P. K. Fritschelt, and C. N. Man, J. Mod. Opt. 41, 1371 (1994).
[Crossref]

Georges, P.

S. Chénais, F. Balembois, F. Druon, G. L. Leclin, and P. Georges, IEEE J. Quantum Electron. 40, 1217 (2004).
[Crossref]

Hello, P.

P. Hello, E. Durand, P. K. Fritschelt, and C. N. Man, J. Mod. Opt. 41, 1371 (1994).
[Crossref]

Hu, W.

W. Xie, Y. Kwon, W. Hu, and F. Zhou, Opt. Eng. (Bellingham) 42, 1787 (2003).
[Crossref]

Huai, X.

Z. Li, X. Huai, and Y. Tao, Appl. Phys. B 87, 301 (2007).
[Crossref]

Kwon, Y.

W. Xie, Y. Kwon, W. Hu, and F. Zhou, Opt. Eng. (Bellingham) 42, 1787 (2003).
[Crossref]

Leclin, G. L.

S. Chénais, F. Balembois, F. Druon, G. L. Leclin, and P. Georges, IEEE J. Quantum Electron. 40, 1217 (2004).
[Crossref]

Li, Z.

Z. Li, X. Huai, and Y. Tao, Appl. Phys. B 87, 301 (2007).
[Crossref]

Man, C. N.

P. Hello, E. Durand, P. K. Fritschelt, and C. N. Man, J. Mod. Opt. 41, 1371 (1994).
[Crossref]

Mudge, D.

Munch, J.

Ostermeyer, M.

Peng, X.

X. Peng, L. Xu, and A. Asundi, Opt. Eng. (Bellingham) 43, 2454 (2004).
[Crossref]

Pigeon, F.

B. A. Usievich, V. A. Sychugov, F. Pigeon, and A. Tishchenko, IEEE J. Quantum Electron. 37, 1210 (2001).
[Crossref]

Sychugov, V. A.

B. A. Usievich, V. A. Sychugov, F. Pigeon, and A. Tishchenko, IEEE J. Quantum Electron. 37, 1210 (2001).
[Crossref]

Takeuchi, Y.

Y. Takeuchi, Thermal Stress (Science, 1977).

Tao, Y.

Z. Li, X. Huai, and Y. Tao, Appl. Phys. B 87, 301 (2007).
[Crossref]

Tishchenko, A.

B. A. Usievich, V. A. Sychugov, F. Pigeon, and A. Tishchenko, IEEE J. Quantum Electron. 37, 1210 (2001).
[Crossref]

Usievich, B. A.

B. A. Usievich, V. A. Sychugov, F. Pigeon, and A. Tishchenko, IEEE J. Quantum Electron. 37, 1210 (2001).
[Crossref]

Veitch, P. J.

Xie, W.

W. Xie, Y. Kwon, W. Hu, and F. Zhou, Opt. Eng. (Bellingham) 42, 1787 (2003).
[Crossref]

Xu, L.

X. Peng, L. Xu, and A. Asundi, Opt. Eng. (Bellingham) 43, 2454 (2004).
[Crossref]

Zhou, F.

W. Xie, Y. Kwon, W. Hu, and F. Zhou, Opt. Eng. (Bellingham) 42, 1787 (2003).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

Z. Li, X. Huai, and Y. Tao, Appl. Phys. B 87, 301 (2007).
[Crossref]

IEEE J. Quantum Electron. (2)

B. A. Usievich, V. A. Sychugov, F. Pigeon, and A. Tishchenko, IEEE J. Quantum Electron. 37, 1210 (2001).
[Crossref]

S. Chénais, F. Balembois, F. Druon, G. L. Leclin, and P. Georges, IEEE J. Quantum Electron. 40, 1217 (2004).
[Crossref]

J. Mod. Opt. (1)

P. Hello, E. Durand, P. K. Fritschelt, and C. N. Man, J. Mod. Opt. 41, 1371 (1994).
[Crossref]

Opt. Eng. (Bellingham) (2)

W. Xie, Y. Kwon, W. Hu, and F. Zhou, Opt. Eng. (Bellingham) 42, 1787 (2003).
[Crossref]

X. Peng, L. Xu, and A. Asundi, Opt. Eng. (Bellingham) 43, 2454 (2004).
[Crossref]

Other (1)

Y. Takeuchi, Thermal Stress (Science, 1977).

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Figures (4)

Fig. 1
Fig. 1

Schematic of the convective cooling of a laser rod in an annular passage.

Fig. 2
Fig. 2

The temperature contours in the laser rod based on analytical solution [K].

Fig. 3
Fig. 3

Radial one-dimensional (1D) thermal stress profile in an arbitrary cross section of the laser rod [MPa]. (The longitudinal variation of the thermal stress components are within 0.07%, which is negligible.)

Fig. 4
Fig. 4

Thermal strain distribution in the laser rod [(a) ε r r , (b) ε ϕ ϕ , and (c) ε z z ] (unit: 10 4 ).

Equations (33)

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t s ( r , z ) = q V 4 k s r 2 + C 1 I 0 ( β 1 r ) cos ( β 1 z ) + m = 2 C m I 0 ( β m r ) cos ( β m z ) ,
C 1 = t ave , b + ( 1 + 2 B i r ) q l 4 π k s ,
C m = 2 Δ t b H i β m 2 L 2 ( 1 ) m 1 1 β m I 1 ( β m r i ) + H i I 0 ( β m r i ) ,
m = 2 , 3 , ,
θ s ( r , z ) = t s t ave , b = θ s ( 1 ) + m = 2 θ s ( m ) ,
σ ( r , z ) = σ ( 1 ) + m = 2 σ ( m ) .
σ r r ( 1 ) = { E α T q V [ 16 ( 1 ν ) k s ] } ( r 2 r i 2 ) ,
σ θ θ ( 1 ) = { E α T q V [ 16 ( 1 ν ) k s ] } ( 3 r 2 r i 2 ) ,
σ z z ( 1 ) = { E α T q V [ 8 ( 1 ν ) k s ] } ( 2 r 2 r i 2 ) .
2 Φ = [ ( 1 + ν ) ( 1 ν ) ] α T θ s .
2 2 L = [ 2 r 2 + ( 1 r ) r + 2 z 2 ] 2 L = 0 .
σ r r = E 1 + ν [ 2 Φ r 2 2 Φ + 2 r 2 ( L z ) ν 2 ( L z ) ] ,
σ r z = E 1 + ν [ 2 Φ r z + 2 r z ( L z ) ( 1 ν ) r ( 2 L ) ] ,
σ ϕ ϕ = E 1 + ν [ 1 r Φ r 2 Φ + 1 r r ( L z ) ν 2 ( L z ) ] ,
σ z z = E 1 + ν [ 2 Φ z 2 2 Φ + 2 z 2 ( L z ) ( 2 ν ) 2 ( L z ) ] .
Φ m ( r , z ) = 1 + ν 1 ν α T C m 2 β m r I 1 ( β m r ) cos ( β m z ) .
L m ( r , z ) = 1 + ν 1 ν α T 2 β m 3 [ B 1 m I 0 ( β m r ) + B 2 m β m r I 1 ( β m r ) ] sin ( β m z ) .
σ r r ( m ) = E α T 2 ( 1 ν ) { [ C m + B 1 m + ( 1 2 ν ) B 2 m ] I 0 ( β m r ) , + [ C m β m r B 1 m ( β m r ) + B 2 m β m r ] I 1 ( β m r ) } cos ( β m z ) ,
σ r z ( m ) = E α T 2 ( 1 ν ) { ( C m + B 2 m ) β m r I 0 ( β m r ) + [ B 1 m + 2 ( 1 ν ) B 2 m ] I 1 ( β m r ) } sin ( β m z ) ,
σ ϕ ϕ ( m ) = E α T 2 ( 1 ν ) { [ C m ( 1 2 ν ) B 2 m ] I 0 ( β m r ) [ B 1 m ( β m r ) ] I 1 ( β m r ) } cos ( β m z ) ,
σ z z ( m ) = E α T 2 ( 1 ν ) { [ 2 C m + B 1 m + 2 ( 2 ν ) B 2 m ] I 0 ( β m r ) + ( C m + B 2 m ) β m r I 1 ( β m r ) } cos ( β m z ) .
[ σ r r ( m ) ] r = r i = [ σ r z ( m ) ] r = r i = 0 .
B 1 m = [ D m P m + G m C m β m r i I 0 ( β m r i ) ] [ R m P m G m I 1 ( β m r i ) ] ,
B 2 m = [ R m C m β m r i I 0 ( β m r i ) + D m I 1 ( β m r i ) ] [ G m I 1 ( β m r i ) R m P m ] ,
D m = C m [ I 0 ( β m r i ) β m r i I 1 ( β m r i ) ] ,
G m = ( 1 2 ν ) I 0 ( β m r i ) + β m r i I 1 ( β m r i ) ,
P m = β m r i I 0 ( β m r i ) + 2 ( 1 ν ) I 1 ( β m r i ) ,
R m = I 0 ( β m r i ) I 1 ( β m r i ) β m r i .
Δ t s = t s t i n , b ,
ε r r = [ σ r r ν ( σ ϕ ϕ + σ z z ) ] E + α T Δ t s ,
ε ϕ ϕ = [ σ ϕ ϕ ν ( σ r r + σ z z ) ] E + α T Δ t s ,
ε z z = [ σ z z ν ( σ r r + σ ϕ ϕ ) ] E + α T Δ t s ,
Δ ε r r = Δ ε ϕ ϕ = Δ ε z z = α T Δ t s .

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