Abstract

Thermal effects and output power characteristics of kilowatt all-fiber master-oscillator power amplifier (MOPA) are investigated. Proper designs for cooling apparatus are proposed and demonstrated experimentally, for the purpose of minimizing splice heating which is critical for the reliability of high power operation. By using these optimized methods, a thermal damage-free, highly efficient ytterbium-doped double-clad fiber MOPA operating at 1080 nm with 1.17 kW output was obtained. The maximum surface temperature at the pump light launching end splice of the booster amplifier was 345 K, and the temperature rise for this key splice was 0.052 K/W.

© 2011 OSA

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References

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  1. S. Yin, P. Yan, and M. Gong, “End-pumped 300 W continuous-wave ytterbium-doped all-fiber laser with master oscillator multi-stage power amplifiers configuration,” Opt. Express 16(22), 17864–17869 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-22-17864 .
    [CrossRef] [PubMed]
  2. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=josab-27-11-B63 .
    [CrossRef]
  3. Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6088 .
    [CrossRef] [PubMed]
  4. E. Stiles, “New developments in IPG fiber laser technology,” in Proceedings of the 5th International Workshop on Fiber Lasers (2009).
  5. B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
    [CrossRef]
  6. C. Wirth, O. Schmidt, I. Tsybin, T. Schreiber, T. Peschel, F. Brückner, T. Clausnitzer, J. Limpert, R. Eberhardt, A. Tünnermann, M. Gowin, E. ten Have, K. Ludewigt, and M. Jung, “2 kW incoherent beam combining of four narrow-linewidth photonic crystal fiber amplifiers,” Opt. Express 17(3), 1178–1183 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-3-1178 .
    [CrossRef] [PubMed]
  7. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 37(2), 207–217 (2001).
    [CrossRef]
  8. Y. Wang, C. Q. Xu, and H. Po, “Thermal effects in kilowatt fiber lasers,” IEEE Photon. Technol. Lett. 16(1), 63–65 (2004).
    [CrossRef]
  9. N. A. Brilliant and K. Lagonik, “Thermal effects in a dual-clad ytterbium fiber laser,” Opt. Lett. 26(21), 1669–1671 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=ol-26-21-1669 .
    [CrossRef] [PubMed]
  10. M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power CW fiber lasers,” Proc. SPIE 7195, 719511 (2009).
  11. Y. Wang, “Heat dissipation in Kilowatt fiber power amplifiers,” IEEE J. Quantum Electron. 40(6), 731–740 (2004).
    [CrossRef]
  12. L. Li, H. Li, T. Qiu, V. L. Temyanko, M. M. Morrell, A. Schülzgen, A. Mafi, J. V. Moloney, and N. Peyghambarian, “3-Dimensional thermal analysis and active cooling of short-length high-power fiber lasers,” Opt. Express 13(9), 3420–3428 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-9-3420 .
    [CrossRef] [PubMed]
  13. P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
    [CrossRef]
  14. A. Hardy and R. Oron, “Signal amplification in strongly pumped fiber amplifiers,” IEEE J. Quantum Electron. 33(3), 307–313 (1997).
    [CrossRef]
  15. D. E. Gray, American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, 1972).
  16. P. Yan, A. Xu, and M. Gong, “Numerical analysis of temperature distributions in Yb-doped double-clad fiber lasers with consideration of radiative heat transfer,” Opt. Eng. 45(12), 124201 (2006).
    [CrossRef]
  17. B. Zintzen, T. Langer, J. Geiger, D. Hoffmann, and P. Loosen, “Heat transport in solid and air-clad fibers for high-power fiber lasers,” Opt. Express 15(25), 16787–16793 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-16787 .
    [CrossRef] [PubMed]
  18. J. P. Gwinn and R. L. Webb, “Performance and testing of thermal interface materials,” Microelectron. J. 34(3), 215–222 (2003).
    [CrossRef]

2010 (2)

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=josab-27-11-B63 .
[CrossRef]

2009 (2)

2008 (2)

P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
[CrossRef]

S. Yin, P. Yan, and M. Gong, “End-pumped 300 W continuous-wave ytterbium-doped all-fiber laser with master oscillator multi-stage power amplifiers configuration,” Opt. Express 16(22), 17864–17869 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-22-17864 .
[CrossRef] [PubMed]

2007 (1)

2006 (1)

P. Yan, A. Xu, and M. Gong, “Numerical analysis of temperature distributions in Yb-doped double-clad fiber lasers with consideration of radiative heat transfer,” Opt. Eng. 45(12), 124201 (2006).
[CrossRef]

2005 (1)

2004 (3)

Y. Wang, C. Q. Xu, and H. Po, “Thermal effects in kilowatt fiber lasers,” IEEE Photon. Technol. Lett. 16(1), 63–65 (2004).
[CrossRef]

Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6088 .
[CrossRef] [PubMed]

Y. Wang, “Heat dissipation in Kilowatt fiber power amplifiers,” IEEE J. Quantum Electron. 40(6), 731–740 (2004).
[CrossRef]

2003 (1)

J. P. Gwinn and R. L. Webb, “Performance and testing of thermal interface materials,” Microelectron. J. 34(3), 215–222 (2003).
[CrossRef]

2001 (2)

N. A. Brilliant and K. Lagonik, “Thermal effects in a dual-clad ytterbium fiber laser,” Opt. Lett. 26(21), 1669–1671 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=ol-26-21-1669 .
[CrossRef] [PubMed]

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

1997 (1)

A. Hardy and R. Oron, “Signal amplification in strongly pumped fiber amplifiers,” IEEE J. Quantum Electron. 33(3), 307–313 (1997).
[CrossRef]

Brilliant, N. A.

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

Brückner, F.

Cain-Skaff, M.

M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power CW fiber lasers,” Proc. SPIE 7195, 719511 (2009).

Chatigny, S.

M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power CW fiber lasers,” Proc. SPIE 7195, 719511 (2009).

Chen, M.

P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
[CrossRef]

Chen, W.

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Clarkson, W. A.

Clausnitzer, T.

Dong, J.

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Eberhardt, R.

Geiger, J.

Gong, M.

Gowin, M.

Gwinn, J. P.

J. P. Gwinn and R. L. Webb, “Performance and testing of thermal interface materials,” Microelectron. J. 34(3), 215–222 (2003).
[CrossRef]

Hardy, A.

A. Hardy and R. Oron, “Signal amplification in strongly pumped fiber amplifiers,” IEEE J. Quantum Electron. 33(3), 307–313 (1997).
[CrossRef]

He, B.

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

Hoffmann, D.

Jeong, Y.

Jiang, D.

P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
[CrossRef]

Jung, M.

Lagonik, K.

Langer, T.

Lapointe, M.-A.

M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power CW fiber lasers,” Proc. SPIE 7195, 719511 (2009).

Li, G.

P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
[CrossRef]

Li, H.

Li, L.

Li, P.

P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
[CrossRef]

Li, Z.

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Limpert, J.

Loosen, P.

Lou, Q.

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Ludewigt, K.

Mafi, A.

Maran, J.-N.

M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power CW fiber lasers,” Proc. SPIE 7195, 719511 (2009).

Moloney, J. V.

Morrell, M. M.

Nilsson, J.

Oron, R.

A. Hardy and R. Oron, “Signal amplification in strongly pumped fiber amplifiers,” IEEE J. Quantum Electron. 33(3), 307–313 (1997).
[CrossRef]

Payne, D.

Peschel, T.

Peyghambarian, N.

Piché, M.

M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power CW fiber lasers,” Proc. SPIE 7195, 719511 (2009).

Po, H.

Y. Wang, C. Q. Xu, and H. Po, “Thermal effects in kilowatt fiber lasers,” IEEE Photon. Technol. Lett. 16(1), 63–65 (2004).
[CrossRef]

Qiu, T.

Richardson, D. J.

Sahu, J.

Schmidt, O.

Schreiber, T.

Schülzgen, A.

Temyanko, V. L.

ten Have, E.

Tsybin, I.

Tünnermann, A.

Wang, W.

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Wang, Y.

Y. Wang, “Heat dissipation in Kilowatt fiber power amplifiers,” IEEE J. Quantum Electron. 40(6), 731–740 (2004).
[CrossRef]

Y. Wang, C. Q. Xu, and H. Po, “Thermal effects in kilowatt fiber lasers,” IEEE Photon. Technol. Lett. 16(1), 63–65 (2004).
[CrossRef]

Webb, R. L.

J. P. Gwinn and R. L. Webb, “Performance and testing of thermal interface materials,” Microelectron. J. 34(3), 215–222 (2003).
[CrossRef]

Wei, Y.

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Wirth, C.

Xu, A.

P. Yan, A. Xu, and M. Gong, “Numerical analysis of temperature distributions in Yb-doped double-clad fiber lasers with consideration of radiative heat transfer,” Opt. Eng. 45(12), 124201 (2006).
[CrossRef]

Xu, C. Q.

Y. Wang, C. Q. Xu, and H. Po, “Thermal effects in kilowatt fiber lasers,” IEEE Photon. Technol. Lett. 16(1), 63–65 (2004).
[CrossRef]

Xue, Y.

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Yan, P.

Yin, S.

Zhao, H.

P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
[CrossRef]

Zhou, J.

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Zhu, C.

P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
[CrossRef]

Zintzen, B.

Zou, S.

P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
[CrossRef]

IEEE J. Quantum Electron. (2)

Y. Wang, “Heat dissipation in Kilowatt fiber power amplifiers,” IEEE J. Quantum Electron. 40(6), 731–740 (2004).
[CrossRef]

A. Hardy and R. Oron, “Signal amplification in strongly pumped fiber amplifiers,” IEEE J. Quantum Electron. 33(3), 307–313 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 37(2), 207–217 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Y. Wang, C. Q. Xu, and H. Po, “Thermal effects in kilowatt fiber lasers,” IEEE Photon. Technol. Lett. 16(1), 63–65 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

Microelectron. J. (1)

J. P. Gwinn and R. L. Webb, “Performance and testing of thermal interface materials,” Microelectron. J. 34(3), 215–222 (2003).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

B. He, J. Zhou, Q. Lou, Y. Xue, Z. Li, W. Wang, J. Dong, Y. Wei, and W. Chen, “1.75 killowatt continuous-wave output fiber laser using homemade ytterbium-doped large-core fiber,” Microw. Opt. Technol. Lett. 52(7), 1668–1671 (2010).
[CrossRef]

Opt. Eng. (1)

P. Yan, A. Xu, and M. Gong, “Numerical analysis of temperature distributions in Yb-doped double-clad fiber lasers with consideration of radiative heat transfer,” Opt. Eng. 45(12), 124201 (2006).
[CrossRef]

Opt. Express (5)

Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6088 .
[CrossRef] [PubMed]

L. Li, H. Li, T. Qiu, V. L. Temyanko, M. M. Morrell, A. Schülzgen, A. Mafi, J. V. Moloney, and N. Peyghambarian, “3-Dimensional thermal analysis and active cooling of short-length high-power fiber lasers,” Opt. Express 13(9), 3420–3428 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-9-3420 .
[CrossRef] [PubMed]

B. Zintzen, T. Langer, J. Geiger, D. Hoffmann, and P. Loosen, “Heat transport in solid and air-clad fibers for high-power fiber lasers,” Opt. Express 15(25), 16787–16793 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-16787 .
[CrossRef] [PubMed]

S. Yin, P. Yan, and M. Gong, “End-pumped 300 W continuous-wave ytterbium-doped all-fiber laser with master oscillator multi-stage power amplifiers configuration,” Opt. Express 16(22), 17864–17869 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-22-17864 .
[CrossRef] [PubMed]

C. Wirth, O. Schmidt, I. Tsybin, T. Schreiber, T. Peschel, F. Brückner, T. Clausnitzer, J. Limpert, R. Eberhardt, A. Tünnermann, M. Gowin, E. ten Have, K. Ludewigt, and M. Jung, “2 kW incoherent beam combining of four narrow-linewidth photonic crystal fiber amplifiers,” Opt. Express 17(3), 1178–1183 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-3-1178 .
[CrossRef] [PubMed]

Opt. Laser Technol. (1)

P. Li, C. Zhu, S. Zou, H. Zhao, D. Jiang, G. Li, and M. Chen, “Theoretical and experimental investigation of thermal effects in a high power Yb3+-doped double-clad fiber laser,” Opt. Laser Technol. 40(2), 360–364 (2008).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

M.-A. Lapointe, S. Chatigny, M. Piché, M. Cain-Skaff, and J.-N. Maran, “Thermal effects in high power CW fiber lasers,” Proc. SPIE 7195, 719511 (2009).

Other (2)

E. Stiles, “New developments in IPG fiber laser technology,” in Proceedings of the 5th International Workshop on Fiber Lasers (2009).

D. E. Gray, American Institute of Physics Handbook, 3rd ed. (McGraw-Hill, 1972).

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Figures (14)

Fig. 1
Fig. 1

Model for solving the temperature distribution of DCF.

Fig. 2
Fig. 2

Schematic of a single-stage MOPA.

Fig. 3
Fig. 3

Center temperatures along the fiber.

Fig. 4
Fig. 4

Temperature distribution as a function of r for three different h values at z=0 m.

Fig. 5
Fig. 5

Fiber center and coating surface temperatures as a function of h.

Fig. 6
Fig. 6

Schematic representation of the experimental setup (a) fiber was simply dropped in the V-groove without any external pressure; (b) fiber was ideally fixed by copper tape.

Fig. 7
Fig. 7

(a) Equivalent geometry of the heat sink for Fig. 6(a); (b) equivalent lower half-space geometry for Fig. 6(b); (c) equivalent upper half-space geometry for Fig. 6(b).

Fig. 8
Fig. 8

Surface temperature rises under different cooling methods.

Fig. 9
Fig. 9

Exploded view of thermal contact resistance with real TIM.

Fig. 10
Fig. 10

Experimental arrangement of all-fiber MOPA.

Fig. 11
Fig. 11

A micrograph of a good fusion splice.

Fig. 12
Fig. 12

Output power and splice surface temperature versus pump power.

Fig. 13
Fig. 13

Stability of the output powers and splice surface temperatures within 10 minutes running time.

Fig. 14
Fig. 14

Laser spectrum of the MOPA at maximum output power.

Tables (1)

Tables Icon

Table 1 Thermal Contact Resistances

Equations (6)

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1 r r [ r T ( r ) r ] = Q k ,
T r | r = 0 = 0 , k 3 T 3 r | r = c = h [ T c T 3 ( r = c ) ] ,
T 1 ( r ) = T 0 q 1 r 2 4 k 1 , T 2 ( r ) = T 0 q 1 a 2 4 k 1 q 1 a 2 2 k 2 ln ( r a ) , T 3 ( r ) = T 0 q 1 a 2 4 k 1 q 1 a 2 2 k 2 ln ( b a ) q 1 a 2 2 k 3 ln ( r b ) , T 0 = T c + q 1 a 2 2 h c + q 1 a 2 4 k 1 + q 1 a 2 2 k 2 ln ( b a ) + q 1 a 2 2 k 3 ln ( c b ) ,
R t c " = T s T q " ,
q 0 = ( 1 10 α d L / 10 ) P ( 1 λ p λ s ) π a 2 = q d L π a 2 ,
R t c " = R contact 1 + R cond-TIM + R contact2 , R cond-TIM = L k TIM A ,

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