Abstract

In this paper, we introduce analytical derivations for the thermal lensing of an end-pumped laser disk. These derivations are done for pump beam shapes from Gaussian to top hat, assuming that the thermal conductivity is either constant with the temperature or not. We give examples in two temperature regions, where the thermal conductivity exhibits T1 or T2 dependence. Numerical applications are given for a Q-CW pumped Yb:YAG disk laser.

© 2010 Optical Society of America

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  1. G. L. Bourdet, “Numerical simulation of a high average power diode pumped ytterbium doped YAG laser with an unstable cavity and a super Gaussian mirror,” Appl. Opt. 44, 1018–1027 (2005).
    [CrossRef] [PubMed]
  2. G. L. Bourdet, J.-C Chanteloup, A. Fülöp, Y. Julien, and A. Migus, “The LUCIA project: a high average power ytterbium diode pumped solid state laser chain,” Proc. SPIE 5478, 4–7 (2004).
    [CrossRef]
  3. A. Bayramian, “A high average power femto-petawatt laser for high intensity applications,” presented at the International Conference on Ultrahigh Intensity Lasers (ICUIL), Shanghai-Tongli, China, 27–31 October 2008.
  4. C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
    [CrossRef]
  5. J. J. Zayhowski, “Thermal guiding in microchip lasers,” in Advanced Solid State Lasers, G.Dube, ed., Vol. 6 of OSA Proceedings Series (Optical Society of America, 1990), pp. 9–13.
  6. G. L. Bourdet and G. Lescroart, “Theoretical modeling of mode formation in Tm3+:YVO4 microchip lasers,” Opt. Commun. 150136–140 (1998).
    [CrossRef]
  7. M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematic Functions (Dover, 1972), Eq. (5.1.1), p. 228.
  8. M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematic Functions, (Dover, 1972), Eq. (5.1.11), p. 229.
  9. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  10. C. Palma and V. Bagnini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
    [CrossRef]
  11. R. Borghi and M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
    [CrossRef]
  12. Y. Li, “New expansion for a flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
    [CrossRef]
  13. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155-4), p. 5.
  14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155-3), p. 5.
  15. T. Numazawa, O. Arai, Q. Hu, and T. Noda, “Thermal conductivity measurements for evaluation of crystal perfection at low temperatures,” Meas. Sci. Technol. 12, 2089–2094(2001).
    [CrossRef]
  16. G. L. Bourdet and H. Yu, “Longitudinal temperature distribution in a end-pumped solid state amplifier medium: application to a high average power diode pumped Yb:YAG thin disk amplifier,” Appl. Opt. 46, 6033–6041(2007).
    [CrossRef] [PubMed]

2007 (1)

2005 (1)

2004 (1)

G. L. Bourdet, J.-C Chanteloup, A. Fülöp, Y. Julien, and A. Migus, “The LUCIA project: a high average power ytterbium diode pumped solid state laser chain,” Proc. SPIE 5478, 4–7 (2004).
[CrossRef]

2002 (1)

Y. Li, “New expansion for a flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

2001 (1)

T. Numazawa, O. Arai, Q. Hu, and T. Noda, “Thermal conductivity measurements for evaluation of crystal perfection at low temperatures,” Meas. Sci. Technol. 12, 2089–2094(2001).
[CrossRef]

2000 (1)

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

1998 (2)

G. L. Bourdet and G. Lescroart, “Theoretical modeling of mode formation in Tm3+:YVO4 microchip lasers,” Opt. Commun. 150136–140 (1998).
[CrossRef]

R. Borghi and M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

1994 (2)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

C. Palma and V. Bagnini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Arai, O.

T. Numazawa, O. Arai, Q. Hu, and T. Noda, “Thermal conductivity measurements for evaluation of crystal perfection at low temperatures,” Meas. Sci. Technol. 12, 2089–2094(2001).
[CrossRef]

Bagnini, V.

C. Palma and V. Bagnini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Bayramian, A.

A. Bayramian, “A high average power femto-petawatt laser for high intensity applications,” presented at the International Conference on Ultrahigh Intensity Lasers (ICUIL), Shanghai-Tongli, China, 27–31 October 2008.

Borghi, R.

Bourdet, G. L.

G. L. Bourdet and H. Yu, “Longitudinal temperature distribution in a end-pumped solid state amplifier medium: application to a high average power diode pumped Yb:YAG thin disk amplifier,” Appl. Opt. 46, 6033–6041(2007).
[CrossRef] [PubMed]

G. L. Bourdet, “Numerical simulation of a high average power diode pumped ytterbium doped YAG laser with an unstable cavity and a super Gaussian mirror,” Appl. Opt. 44, 1018–1027 (2005).
[CrossRef] [PubMed]

G. L. Bourdet, J.-C Chanteloup, A. Fülöp, Y. Julien, and A. Migus, “The LUCIA project: a high average power ytterbium diode pumped solid state laser chain,” Proc. SPIE 5478, 4–7 (2004).
[CrossRef]

G. L. Bourdet and G. Lescroart, “Theoretical modeling of mode formation in Tm3+:YVO4 microchip lasers,” Opt. Commun. 150136–140 (1998).
[CrossRef]

Chanteloup, J.-C

G. L. Bourdet, J.-C Chanteloup, A. Fülöp, Y. Julien, and A. Migus, “The LUCIA project: a high average power ytterbium diode pumped solid state laser chain,” Proc. SPIE 5478, 4–7 (2004).
[CrossRef]

Contag, K.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Fülöp, A.

G. L. Bourdet, J.-C Chanteloup, A. Fülöp, Y. Julien, and A. Migus, “The LUCIA project: a high average power ytterbium diode pumped solid state laser chain,” Proc. SPIE 5478, 4–7 (2004).
[CrossRef]

Giesen, A.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155-4), p. 5.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155-3), p. 5.

Hu, Q.

T. Numazawa, O. Arai, Q. Hu, and T. Noda, “Thermal conductivity measurements for evaluation of crystal perfection at low temperatures,” Meas. Sci. Technol. 12, 2089–2094(2001).
[CrossRef]

Hügel, H.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Julien, Y.

G. L. Bourdet, J.-C Chanteloup, A. Fülöp, Y. Julien, and A. Migus, “The LUCIA project: a high average power ytterbium diode pumped solid state laser chain,” Proc. SPIE 5478, 4–7 (2004).
[CrossRef]

Larionov, M.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Lescroart, G.

G. L. Bourdet and G. Lescroart, “Theoretical modeling of mode formation in Tm3+:YVO4 microchip lasers,” Opt. Commun. 150136–140 (1998).
[CrossRef]

Li, Y.

Y. Li, “New expansion for a flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

Migus, A.

G. L. Bourdet, J.-C Chanteloup, A. Fülöp, Y. Julien, and A. Migus, “The LUCIA project: a high average power ytterbium diode pumped solid state laser chain,” Proc. SPIE 5478, 4–7 (2004).
[CrossRef]

Noda, T.

T. Numazawa, O. Arai, Q. Hu, and T. Noda, “Thermal conductivity measurements for evaluation of crystal perfection at low temperatures,” Meas. Sci. Technol. 12, 2089–2094(2001).
[CrossRef]

Numazawa, T.

T. Numazawa, O. Arai, Q. Hu, and T. Noda, “Thermal conductivity measurements for evaluation of crystal perfection at low temperatures,” Meas. Sci. Technol. 12, 2089–2094(2001).
[CrossRef]

Palma, C.

C. Palma and V. Bagnini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155-3), p. 5.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155-4), p. 5.

Santarsiero, M.

Stewen, C.

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Yu, H.

Zayhowski, J. J.

J. J. Zayhowski, “Thermal guiding in microchip lasers,” in Advanced Solid State Lasers, G.Dube, ed., Vol. 6 of OSA Proceedings Series (Optical Society of America, 1990), pp. 9–13.

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

C. Stewen, K. Contag, M. Larionov, A. Giesen, and H. Hügel, “A 1kW CW thin disc laser,” IEEE J. Quantum Electron. 6, 650–657 (2000).
[CrossRef]

Meas. Sci. Technol. (1)

T. Numazawa, O. Arai, Q. Hu, and T. Noda, “Thermal conductivity measurements for evaluation of crystal perfection at low temperatures,” Meas. Sci. Technol. 12, 2089–2094(2001).
[CrossRef]

Opt. Commun. (4)

Y. Li, “New expansion for a flat-topped light beams,” Opt. Commun. 206, 225–234 (2002).
[CrossRef]

G. L. Bourdet and G. Lescroart, “Theoretical modeling of mode formation in Tm3+:YVO4 microchip lasers,” Opt. Commun. 150136–140 (1998).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

C. Palma and V. Bagnini, “Propagation of super-Gaussian beams,” Opt. Commun. 111, 6–10 (1994).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

G. L. Bourdet, J.-C Chanteloup, A. Fülöp, Y. Julien, and A. Migus, “The LUCIA project: a high average power ytterbium diode pumped solid state laser chain,” Proc. SPIE 5478, 4–7 (2004).
[CrossRef]

Other (6)

A. Bayramian, “A high average power femto-petawatt laser for high intensity applications,” presented at the International Conference on Ultrahigh Intensity Lasers (ICUIL), Shanghai-Tongli, China, 27–31 October 2008.

J. J. Zayhowski, “Thermal guiding in microchip lasers,” in Advanced Solid State Lasers, G.Dube, ed., Vol. 6 of OSA Proceedings Series (Optical Society of America, 1990), pp. 9–13.

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematic Functions (Dover, 1972), Eq. (5.1.1), p. 228.

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematic Functions, (Dover, 1972), Eq. (5.1.11), p. 229.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155-4), p. 5.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980), Eq. (0.155-3), p. 5.

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Figures (10)

Fig. 1
Fig. 1

Pump intensity profile for various values of M.

Fig. 2
Fig. 2

Temperature difference between the edge and the center when K does not depend on temperature.

Fig. 3
Fig. 3

Radial temperature distribution when K does not depend on temperature. Parabolic approximation in dot lines.

Fig. 4
Fig. 4

Temperature difference between the edge and the center when K depends on temperature and T > 160 K .

Fig. 5
Fig. 5

Radial temperature distribution center when K depends on temperature and T > 160 K . Parabolic approximation in dot lines.

Fig. 6
Fig. 6

Temperature difference between the edge and the center when K depends on temperature and 30 K < T < 160 K .

Fig. 7
Fig. 7

Radial temperature distribution when K depends on temperature and 30 K < T < 160 K . Parabolic approximation in dot lines.

Fig. 8
Fig. 8

OPD when the thermal conductivity does not depend on the temperature for M = 1 and 100. The dot and dash lines are the parabolic approximations for M = 1 and M = 100 , respectively.

Fig. 9
Fig. 9

OPD when the thermal conductivity depends on the temperature for M = 1 and 100 for T > 160 K . The dot and dash lines are the parabolic approximations for M = 1 and M = 100 , respectively.

Fig. 10
Fig. 10

OPD when the thermal conductivity depends on the temperature for M = 1 and 100 for 30 K < T < 160 K . The dot and dash lines are the parabolic approximations for M = 1 and M = 100 , respectively.

Tables (1)

Tables Icon

Table 1 Thermo-Optical Parameters of Yb:YAG

Equations (61)

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I p ( r ) = I p ( 0 ) exp [ 2 ( r w p ) 2 ] .
K 2 T = Q ,
1 r r ( r T r ) = q K π w p 2 2 l exp [ 2 ( r w p ) 2 ] ,
T ( r ) = 2 A { 1 2 Ei [ 2 ( r w p ) 2 ] + C 1 ln ( r ) + C 2 } , A = q 4 π K l ,
Ei ( x ) = γ ln ( x ) n = 1 ( 1 ) n x n n n ! ( | arg x | < π ) ,
γ = lim ( n ) [ k = 1 n 1 k ln ( n ) ] = 0.577 . ,
T ( r ) = 2 A { 1 2 γ + 1 2 ln [ 2 ( r w p ) 2 ] + C 1 ln ( r ) + C 2 } .
r 0 T ( r ) = q 2 π K l { 1 2 γ + ln ( 2 w p r ( C 1 + 1 ) ) + C 2 } .
C 2 = 1 2 A T ( a ) + 1 2 { Ei [ 2 ( a w p ) 2 ] + ln ( a ) } ,
T ( r ) T ( a ) = A { Ei [ 2 ( a w p ) 2 ] Ei [ 2 ( r w p ) 2 ] + ln [ ( a r ) 2 ] } .
ρ = r w p , α = a w p .
T ( ρ ) T ( α ) = A { Ei ( 2 α 2 ) Ei ( 2 ρ 2 ) + ln [ ( α ρ ) 2 ] } .
T ( ρ ) T ( α ) = A { Ei ( 2 α 2 ) + ln ( 2 α 2 ) + γ + ln ( 2 ρ 2 ) + n = 1 ( 2 ρ 2 ) n n n ! ln ( 2 ρ 2 ) } ,
T ( ρ ) T ( α ) = A { Ei ( 2 α 2 ) + ln ( 2 α 2 ) + γ + n = 1 ( 2 ρ 2 ) n n n ! } .
T ( 0 ) T ( α ) = A [ Ei ( 2 α 2 ) + ln ( α 2 ) + γ ] .
G M ( r ) = 1 ( 1 exp ( 2 ( r w M ) 2 ) ) M .
β = ( w p w M ) 2 .
G M ( r ) = m = 1 M α m exp ( 2 m β ( r w p ) 2 ) ,
α m = ( 1 ) ( m + 1 ) M ( M 1 ) ( M m + 1 ) m ! = ( 1 ) ( m + 1 ) M ! m ! ( M m ) ! .
0 2 π 0 G M ( r ) r d r = 2 π 0 exp ( 2 ( r w ) 2 ) r d r = π w 2 2 .
β M = m = 1 M 1 m .
T ( ρ ) T ( α ) = A { { m = 1 M α m m β M [ Ei ( 2 m β M α 2 ) Ei ( 2 m β M ρ 2 ) ] } + ln ( α 2 ) ln ( ρ 2 ) } .
m = 1 M α m m = β M , m = 1 M α m m β M = 1.
T ( ρ ) T ( α ) = A { { m = 1 M α m m β M [ Ei ( 2 m β M α 2 ) + ln ( m β M ) + n = 1 ( 1 ) n n n ! ( 2 m β M ρ 2 ) n ] } + ln ( 2 α 2 ) + γ } .
m = 1 M α m m β M [ n = 1 ( 1 ) n ( 2 m β M ρ 2 ) n n n ! ] = n = 1 β M n 1 [ m = 1 M α m m n 1 ] ( 1 ) n ( 2 ρ 2 ) n n n ! .
m = 0 M ( 1 ) m M ! m ! ( M m ) ! m n 1 = 0 , M n 1 , 0 0 1.
m = 1 M ( 1 ) ( m + 1 ) M ! m ! ( M m ) ! m n 1 = 0 M n > 1 m = 1 M ( 1 ) ( m + 1 ) M ! m ! ( M m ) ! = 1 n = 1 .
T ( ρ ) T ( α ) = A { { m = 1 M α m m β M [ Ei ( 2 m β M α 2 ) + ln ( m β M ) ] } + ln ( 2 α 2 ) + γ 2 ρ 2 + n = M + 1 β M n 1 [ m = 1 M α m m n 1 ] ( 1 ) n ( 2 ρ 2 ) n n n ! } .
Q M ( α ) = { m = 1 M α m m β M [ Ei ( 2 m β M α 2 ) + ln ( m β M ) ] } + ln ( 2 α 2 ) + γ ,
P M ( ρ ) = n = M + 1 β M n 1 [ m = 1 M α m m n 1 ] ( 2 ρ 2 ) n n n ! .
T ( 0 ) T ( α ) = A Q M ( α ) .
T ( ρ ) T ( α ) = A { Q M ( α ) 2 ρ 2 + P M ( ρ ) } .
K = W 1 T .
1 r r ( r T T r ) = 1 r r { r [ ln ( T ) ] r } = q W 1 π w p 2 2 l exp [ 2 ( r w p ) 2 ] .
B = q 4 π W 1 l .
ln [ T ( r ) T ( a ) ] = B { Ei [ 2 ( r w p ) 2 ] Ei [ 2 ( a w p ) 2 ] + ln ( r 2 ) ln ( a 2 ) } .
T ( ρ ) = T ( α ) exp { B [ Ei ( 2 α 2 ) + ln ( α 2 ) Ei ( 2 ρ 2 ) ln ( ρ 2 ) ] } .
T ( 0 ) = T ( α ) exp { B [ Ei ( 2 α 2 ) + ln ( α 2 ) + γ ] } .
T ( ρ ) = T ( 0 ) exp { B [ 2 ρ 2 + n = 2 ( 2 ρ 2 ) n n n ! ] } .
T ( ρ ) T ( 0 ) = T ( 0 ) { exp ( 2 B ρ 2 ) 1 } = 2 B T ( 0 ) ρ 2 .
1 r r { r [ ln ( T ) ] r } = q W 1 π w p 2 2 l m = 1 M α m exp ( 2 m β M ( r w p ) 2 ) .
T ( ρ ) = T ( α ) exp { B [ Q M ( α ) 2 ρ 2 + P M ( ρ ) ] } ,
T ( 0 ) = T ( α ) exp [ B Q M ( α ) ] .
K = W 2 2 T 2 .
1 r r { r [ 1 T ] r } = q W 2 2 π w p 2 2 l m = 1 M α m exp ( 2 m β M ( r w p ) 2 ) .
C = q 4 π W 2 2 l ,
1 T ( α ) 1 T ( ρ ) = C [ m = 1 M α m m β M [ Ei ( 2 m β M α 2 ) Ei ( 2 m β M ρ 2 ) ] + ln ( α 2 ) ln ( ρ 2 ) ] .
1 T ( α ) 1 T ( ρ ) = C [ Q M ( α ) 2 ρ 2 + P M ( ρ ) ] ,
T ( ρ ) = T ( α ) 1 T ( α ) C [ Q M ( α ) + P M ( ρ ) ] 1 1 + 2 T ( α ) C ρ 2 1 T ( α ) C [ Q M ( α ) + P M ( ρ ) ] .
T ( 0 ) = T ( α ) 1 T ( α ) C Q M ( α ) .
M , P M ( ρ ) 0 , ρ [ 0 , α ] .
T ( ρ ) T ( α ) = A { Q M ( α ) 2 ρ 2 } , T ( ρ ) = T ( 0 ) 2 A ρ 2 .
T ( ρ ) = T ( α ) exp { B [ Q M ( α ) 2 ρ 2 ] } .
T ( ρ ) = T ( α ) exp [ B Q M ( α ) ] ( 1 2 B ρ 2 ) , T ( ρ ) = T ( 0 ) 2 T ( 0 ) B ρ 2 .
T ( ρ ) = 1 1 C T ( α ) Q M ( α ) T ( α ) 1 + 2 C T ( α ) 1 C T ( α ) Q M ( α ) ρ 2 , T ( ρ ) = T ( 0 ) 1 1 + 2 C T ( 0 ) ρ 2 .
T ( ρ ) = T ( α ) 1 C T ( α ) Q M ( α ) [ 1 2 C T ( α ) 1 C T ( α ) Q M ( α ) ρ 2 ] , T ( ρ ) = T ( 0 ) 2 C T ( 0 ) 2 ρ 2 .
q = η Q ( 1 T r p ) D c P p ,
Δ ( ρ ) = ( d n d T + α 0 n ) l Δ T ( ρ ) ,
H = ( d n d T + α 0 n ) l = 3.56 × 10 8 m / K .
Δ ( ρ ) = 2 H A ρ 2 .
R c w p 2 = 1 4 H A .

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