Abstract

We demonstrate a master oscillator power amplifier (MOPA) architecture based on Yb:YAG amplifiers and adaptive optics (AO) systems with a high power and high beam quality laser output. With two conduction cooled, dual-end-pumped Yb:YAG zigzag-slab amplifiers at room temperature, the fiber laser of 300 W was scaled to 11.9 kW. Moreover, AO system positioned downstream was utilized to correct wavefront of amplified laser. The beam quality β at maximum output power was 2.8 times diffraction limited with closed-loop AO system.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. T. Gottwald, C. Stolzenburg, D. Bauer, J. Kleinbauer, V. Kuhn, T. Metzger, S. Schad, D. Sutter, and A. Killi, “Recent disk laser development at Trumpf,” Proceedings of SPIE - The International Society for Optical Engineering 8547, 85470C (2012).
    [Crossref]
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    [Crossref]
  3. G. Goodno, S. Palese, J. Harkenrider, and H. Injeyan, “High average power Yb:YAG end-pumped zig-zag slab laser,” Advanced Solid (2001).
  4. G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
    [Crossref] [PubMed]
  5. G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
    [Crossref]
  6. G. D. Goodno, S. Palese, J. Harkenrider, and H. Injeyan, “Yb:YAG power oscillator with high brightness and linear polarization,” Opt. Lett. 26(21), 1672–1674 (2001).
    [Crossref] [PubMed]
  7. X. Chen, L. Xu, H. Hu, T. Zhou, Y. Sun, H. Jiang, J. Lei, W. Lv, H. Su, Y. Shi, M. Li, Y. Wu, Z. Yao, N. Zhao, X. Xu, Q. Gao, X. Wang, and C. Tang, “High-efficiency, high-average-power, CW Yb:YAG zigzag slab master oscillator power amplifier at room temperature,” Opt. Express 24(21), 24517–24523 (2016).
    [Crossref] [PubMed]
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  10. M. Li, X. Jiang, and J. Zheng, “Modeling of the performance of a cryogenic gas cooled Yb:YAG multislab amplifier with a longitudinal doping gradient concentration,” Opt. Eng. 55, 046109 (2016).
  11. D. Albach, J. C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express 17(5), 3792–3801 (2009).
    [Crossref] [PubMed]
  12. Y. G. Guan, Q. Li, J. Z. He, X. Y. Gao, W. C. Zhou, and J. F. Wei, “Effects of laser facula detector array parameters on measurement accuracy of far-field beam quality,” Optik - International Journal for Light and Electron Optics 126(23), 3906–3911 (2015).
    [Crossref]

2016 (2)

2015 (1)

Y. G. Guan, Q. Li, J. Z. He, X. Y. Gao, W. C. Zhou, and J. F. Wei, “Effects of laser facula detector array parameters on measurement accuracy of far-field beam quality,” Optik - International Journal for Light and Electron Optics 126(23), 3906–3911 (2015).
[Crossref]

2010 (1)

2009 (1)

2007 (1)

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

2006 (2)

2001 (1)

2000 (1)

Albach, D.

Anderegg, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Asman, C. P.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Bourdet, G. L.

Brosnan, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Chanteloup, J. C.

Chen, X.

Cheung, E. C.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

Clarkson, W. A.

Epp, P.

Gao, Q.

Gao, X. Y.

Y. G. Guan, Q. Li, J. Z. He, X. Y. Gao, W. C. Zhou, and J. F. Wei, “Effects of laser facula detector array parameters on measurement accuracy of far-field beam quality,” Optik - International Journal for Light and Electron Optics 126(23), 3906–3911 (2015).
[Crossref]

Goodno, G. D.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

G. D. Goodno, S. Palese, J. Harkenrider, and H. Injeyan, “Yb:YAG power oscillator with high brightness and linear polarization,” Opt. Lett. 26(21), 1672–1674 (2001).
[Crossref] [PubMed]

Guan, Y. G.

Y. G. Guan, Q. Li, J. Z. He, X. Y. Gao, W. C. Zhou, and J. F. Wei, “Effects of laser facula detector array parameters on measurement accuracy of far-field beam quality,” Optik - International Journal for Light and Electron Optics 126(23), 3906–3911 (2015).
[Crossref]

Hammons, D.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Harkenrider, J.

He, J. Z.

Y. G. Guan, Q. Li, J. Z. He, X. Y. Gao, W. C. Zhou, and J. F. Wei, “Effects of laser facula detector array parameters on measurement accuracy of far-field beam quality,” Optik - International Journal for Light and Electron Optics 126(23), 3906–3911 (2015).
[Crossref]

Howland, D.

Hu, H.

Injeyan, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

G. D. Goodno, S. Palese, J. Harkenrider, and H. Injeyan, “Yb:YAG power oscillator with high brightness and linear polarization,” Opt. Lett. 26(21), 1672–1674 (2001).
[Crossref] [PubMed]

Jiang, H.

Jiang, X.

M. Li, X. Jiang, and J. Zheng, “Modeling of the performance of a cryogenic gas cooled Yb:YAG multislab amplifier with a longitudinal doping gradient concentration,” Opt. Eng. 55, 046109 (2016).

Komine, H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

Lei, J.

Li, M.

Li, Q.

Y. G. Guan, Q. Li, J. Z. He, X. Y. Gao, W. C. Zhou, and J. F. Wei, “Effects of laser facula detector array parameters on measurement accuracy of far-field beam quality,” Optik - International Journal for Light and Electron Optics 126(23), 3906–3911 (2015).
[Crossref]

Long, W.

Long, W. H.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Lv, W.

Mcclellan, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

McNaught, S. J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

Nilsson, J.

Palese, S.

Redmond, S.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

Richardson, D. J.

Shi, Y.

Simpson, R.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

Sollee, J.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

Su, H.

Sun, Y.

Tang, C.

Touzé, G.

Wang, X.

Weber, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

Wei, J. F.

Y. G. Guan, Q. Li, J. Z. He, X. Y. Gao, W. C. Zhou, and J. F. Wei, “Effects of laser facula detector array parameters on measurement accuracy of far-field beam quality,” Optik - International Journal for Light and Electron Optics 126(23), 3906–3911 (2015).
[Crossref]

Weiss, S. B.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006).
[Crossref] [PubMed]

Wickham, M.

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

Wu, Y.

Xu, L.

Xu, X.

Yao, Z.

Zhao, N.

Zheng, J.

M. Li, X. Jiang, and J. Zheng, “Modeling of the performance of a cryogenic gas cooled Yb:YAG multislab amplifier with a longitudinal doping gradient concentration,” Opt. Eng. 55, 046109 (2016).

Zhou, T.

Zhou, W. C.

Y. G. Guan, Q. Li, J. Z. He, X. Y. Gao, W. C. Zhou, and J. F. Wei, “Effects of laser facula detector array parameters on measurement accuracy of far-field beam quality,” Optik - International Journal for Light and Electron Optics 126(23), 3906–3911 (2015).
[Crossref]

Appl. Opt. (2)

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

G. D. Goodno, C. P. Asman, J. Anderegg, S. Brosnan, E. C. Cheung, D. Hammons, H. Injeyan, H. Komine, W. H. Long, M. Mcclellan, S. J. McNaught, S. Redmond, R. Simpson, J. Sollee, M. Weber, S. B. Weiss, and M. Wickham, “Brightness-Scaling Potential of Actively Phase-Locked Solid-State Laser Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 460–472 (2007).
[Crossref]

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

Opt. Eng. (1)

M. Li, X. Jiang, and J. Zheng, “Modeling of the performance of a cryogenic gas cooled Yb:YAG multislab amplifier with a longitudinal doping gradient concentration,” Opt. Eng. 55, 046109 (2016).

Opt. Express (2)

Opt. Lett. (2)

Optik - International Journal for Light and Electron Optics (1)

Y. G. Guan, Q. Li, J. Z. He, X. Y. Gao, W. C. Zhou, and J. F. Wei, “Effects of laser facula detector array parameters on measurement accuracy of far-field beam quality,” Optik - International Journal for Light and Electron Optics 126(23), 3906–3911 (2015).
[Crossref]

Other (2)

T. Gottwald, C. Stolzenburg, D. Bauer, J. Kleinbauer, V. Kuhn, T. Metzger, S. Schad, D. Sutter, and A. Killi, “Recent disk laser development at Trumpf,” Proceedings of SPIE - The International Society for Optical Engineering 8547, 85470C (2012).
[Crossref]

G. Goodno, S. Palese, J. Harkenrider, and H. Injeyan, “High average power Yb:YAG end-pumped zig-zag slab laser,” Advanced Solid (2001).

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Figures (5)

Fig. 1
Fig. 1 The schematic of the laser system. (a) The schematic of MOPA chain with AO system positioned downstream, (b) the schematic of single gain module, (c) the results of simulation and measured fluorescence distribution of the detailed coupling system on the objective plane.
Fig. 2
Fig. 2 Output results of laser power. (a) Output power versus pump power of the triple-passed preamplifier, (b) output power versus pump power of the double-passed amplifier, (c) lasing process and power stability of amplified laser within the duration of 110 s at maximum output, (d) the measured spectrum of amplified laser.
Fig. 3
Fig. 3 Simulation results of temperature distribution profile at x-z plane and inversion ratio in two different slabs under different pump powers. (a) and (b) are the temperature distribution profile of preamplifier with 300 W injected under pump powers of 13.4 kW and 16.8 kW, respectively; (c) and (d) are the temperature distribution profile of amplifier with 3.6 kW injected under pump powers of 26.8 kW and 33.6 kW, respectively; (e) and (f) show the population inversion ratios Xu at the centers of two slabs as a function of position along length (z) direction under different pump powers, and the influence on Xu caused by ASE effect is also illustrated.
Fig. 4
Fig. 4 Laser wavefront comparison between open-loop and closed-loop system. (a) Wavefront of laser beam with open-loop AO system (b) laser beam wavefront with closed-loop AO system.
Fig. 5
Fig. 5 Laser beam profile in the far field at the maximum output power. (a) far-field beam profile with the open-loop AO system, (b) far-field beam profile with closed-loop AO system

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

d n 2 dt =c ϕ p σ p ( f ap n 1 f bp n 2 ) dλc ϕ l σ l ( f bl n 2 f al n 1 ) dλ n 2 τ u M ase ,
n t = n 1 + n 2 ,
f p = f ap f ap + f bp , f l = f al f al + f bl , X u = n 2 n t ,
I sat l = h ν l ( f al + f bl ) σ l τ u , I sat p = h ν p ( f ap + f bp ) σ p τ u ,
I k = I k / I sat k .
X u (z)= f p (z) i=1 n I pi (z,λ) Δλ+ f l (z) i=1 m I li (z,λ)Δλ M ase + i=1 n I pi (z,λ)Δλ + i=1 m I li (z,λ)Δλ ,
I li (z,λ)= I li + (z,λ)+ I li (z,λ), I pi (z,λ)= I pi + (z,λ)+ I pi (z,λ).
d I l ε (z,λ) I l ε (z,λ) =εg(z,λ)dz, d I p ε ' (z,λ) I p ε ' (z,λ) = ε ' α(z,λ)dz,
g(z,λ)= σ l n t ( f al + f bl )( X u f l ), α(z,λ)= σ p n t ( f ap + f bp )( f p X u ).
Tt(x,z)= T c + Qg(x,z)t 2H +( Qg(x,z) t 2 8 k c )(1 x 2 )
β= θ θ 0

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