Abstract

In this paper, we propose hybrid-pumped resonant gain-switched thulium fiber lasers to realize high-average-power and high-pulse-energy 2-μm laser emissions. Based on numerical simulation, laser dynamics (pulse peak power, pulse energy, pulse duration, etc.) of this kind of laser system are investigated in detail. By taking advantages of the 793 nm continuous wave pump and the 1900 nm pulsed pump, performance of the laser emission can be significantly improved, with the highest average power of 28 W, peak power of 3.5 kW, pulse energy of 281 μJ, and narrowest pulse duration of 92 ns, all of which can be further optimized through designing the cavity parameters and the pumping circumstance. Compared with the pump pulses, two times improvement in pulse energy and average power has been achieved. This hybrid resonant gain-switched system has an all-fiber configuration and high efficiency (low heat load), and can be steadily extended into the cladding pump scheme, thus paving a new way to realize high power (>100 W average power) and high pulse energy (>1 mJ) 2 μm thulium fiber lasers.

© 2015 Optical Society of America

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References

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  1. I. F. Elder and M. J. P. Payne, “Lasing in diode-pumped Tm: YAP, Tm, Ho: YAP and Tm, Ho: YLF,” Opt. Commun. 145(1–6), 329–339 (1998).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  7. Y. Tang and J. Xu, “Self-induced pulsing in Tm3+-doped fiber lasers with different output couplings,” Proc. SPIE 7276, 72760L (2008).
    [Crossref]
  8. Y. Tang and J. Xu, “Model and characteristics of self-pulsing in Tm3+-doped silica fiber lasers,” IEEE J. Quantum Electron. 47(2), 165–171 (2011).
    [Crossref]
  9. S. D. Jackson, “Cross relaxation and energy transfer upconversion processes relevant to the functioning of 2 μm Tm3+-doped silica fibre lasers,” Opt. Commun. 230(1–3), 197–203 (2004).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  13. J. Yang, H. Li, Y. Tang, and J. Xu, “Temporal characteristics of in-band-pumped gain-switched thulium-doped fiber lasers,” J. Opt. Soc. Am. B 31(1), 80–86 (2014).
    [Crossref]
  14. B. C. Dickinson, S. D. Jackson, and T. A. King, “10 mJ total output from a gain-switched Tm-doped fibre laser,” Opt. Commun. 182(1-3), 199–203 (2000).
    [Crossref]
  15. Y. Zhang, B. Q. Yao, Y. L. Ju, and Y. Z. Wang, “Gain-switched Tm3+-doped double-clad silica fiber laser,” Opt. Express 13(4), 1085–1089 (2005).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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  25. S. D. Jackson and T. A. King, “Theoretical Modeling of Tm-Doped Silica Fiber Lasers,” J. Lightwave Technol. 17(5), 948–956 (1999).
    [Crossref]
  26. J. Li and S. D. Jackson, “Numerical modeling and optimization of diode pumped heavily-Erbium-doped fluoride fiber lasers,” IEEE J. Quantum Electron. 48(4), 454–464 (2012).
    [Crossref]
  27. D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
    [Crossref]
  28. C. Huang, Y. Tang, H. Li, Y. Wang, J. Xu, and C. Du, “A versatile model for temperature-dependent effects in Tm-doped silica fiber lasers,” J. Lightwave Technol. 32(3), 421–428 (2014).
    [Crossref]
  29. S. D. Jackson and T. A. King, “Efficient gain-switched operation of a Tm-doped silica fiber laser,” IEEE J. Quantum Electron. 34, 770–789 (1998).
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    [Crossref] [PubMed]
  31. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, Elsevier, 2007), Chap. 8, p. 278.

2014 (3)

2013 (3)

2012 (4)

Y. Tang and J. Xu, “Hybrid-pumped gain-switched narrow-band thulium fiber laser,” Appl. Phys. Express 5(7), 072702 (2012).
[Crossref]

J. Li and S. D. Jackson, “Numerical modeling and optimization of diode pumped heavily-Erbium-doped fluoride fiber lasers,” IEEE J. Quantum Electron. 48(4), 454–464 (2012).
[Crossref]

J. Geng, Q. Wang, and S. Jiang, “High-spectral-flatness mid-infrared supercontinuum generated from a Tm-doped fiber amplifier,” Appl. Opt. 51(7), 834–840 (2012).
[Crossref] [PubMed]

Y. Tang, C. Huang, S. Wang, H. Li, and J. Xu, “High-power narrow-bandwidth thulium fiber laser with an all-fiber cavity,” Opt. Express 20(16), 17539–17544 (2012).
[Crossref] [PubMed]

2011 (4)

Y. Tang and J. Xu, “Model and characteristics of self-pulsing in Tm3+-doped silica fiber lasers,” IEEE J. Quantum Electron. 47(2), 165–171 (2011).
[Crossref]

Y. Tang, F. Li, and J. Xu, “High peak-power gain-switched Tm3+-doped fiber laser,” IEEE Photonics Technol. Lett. 23(13), 893–895 (2011).
[Crossref]

N. Simakov, A. Hemming, S. Bennetts, and J. Haub, “Efficient, polarised, gain-switched operation of a Tm-doped fibre laser,” Opt. Express 19(16), 14949–14954 (2011).
[Crossref] [PubMed]

F. Wang, D. Shen, H. Chen, D. Fan, and Q. Lu, “Model and optimization of stable gain-switched Tm-doped fiber lasers,” Opt. Rev. 18(4), 360–364 (2011).
[Crossref]

2010 (2)

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” Proc. SPIE 7580, 7580112 (2010).

Y. Tang, L. Xu, Y. Yang, and J. Xu, “High-power gain-switched Tm3+-doped fiber laser,” Opt. Express 18(22), 22964–22972 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

Y. Tang and J. Xu, “Self-induced pulsing in Tm3+-doped fiber lasers with different output couplings,” Proc. SPIE 7276, 72760L (2008).
[Crossref]

2007 (2)

2005 (2)

G. Frith, D. G. Lancaster, and S. D. Jackson, “85W Tm3+ -doped silica fiber laser,” Electron. Lett. 41(12), 687–688 (2005).
[Crossref]

Y. Zhang, B. Q. Yao, Y. L. Ju, and Y. Z. Wang, “Gain-switched Tm3+-doped double-clad silica fiber laser,” Opt. Express 13(4), 1085–1089 (2005).
[Crossref] [PubMed]

2004 (1)

S. D. Jackson, “Cross relaxation and energy transfer upconversion processes relevant to the functioning of 2 μm Tm3+-doped silica fibre lasers,” Opt. Commun. 230(1–3), 197–203 (2004).
[Crossref]

2002 (1)

T. J. Carrig, A. K. Hankla, G. J. Wagner, C. B. Rawle, and I. T. McKinnie, “Tunable infrared laser sources for DIAL,” Proc. SPIE 4723, 147–155 (2002).
[Crossref]

2001 (1)

2000 (1)

B. C. Dickinson, S. D. Jackson, and T. A. King, “10 mJ total output from a gain-switched Tm-doped fibre laser,” Opt. Commun. 182(1-3), 199–203 (2000).
[Crossref]

1999 (1)

1998 (2)

S. D. Jackson and T. A. King, “Efficient gain-switched operation of a Tm-doped silica fiber laser,” IEEE J. Quantum Electron. 34, 770–789 (1998).

I. F. Elder and M. J. P. Payne, “Lasing in diode-pumped Tm: YAP, Tm, Ho: YAP and Tm, Ho: YLF,” Opt. Commun. 145(1–6), 329–339 (1998).
[Crossref]

1977 (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

Alam, S. U.

Becker, M.

Bennetts, S.

Book, L. D.

Carrig, T. J.

T. J. Carrig, A. K. Hankla, G. J. Wagner, C. B. Rawle, and I. T. McKinnie, “Tunable infrared laser sources for DIAL,” Proc. SPIE 4723, 147–155 (2002).
[Crossref]

Chen, H.

F. Wang, D. Shen, H. Chen, D. Fan, and Q. Lu, “Model and optimization of stable gain-switched Tm-doped fiber lasers,” Opt. Rev. 18(4), 360–364 (2011).
[Crossref]

Dickinson, B. C.

B. C. Dickinson, S. D. Jackson, and T. A. King, “10 mJ total output from a gain-switched Tm-doped fibre laser,” Opt. Commun. 182(1-3), 199–203 (2000).
[Crossref]

Du, C.

Ehrenreich, T.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” Proc. SPIE 7580, 7580112 (2010).

Elder, I. F.

I. F. Elder and M. J. P. Payne, “Lasing in diode-pumped Tm: YAP, Tm, Ho: YAP and Tm, Ho: YLF,” Opt. Commun. 145(1–6), 329–339 (1998).
[Crossref]

Fan, D.

F. Wang, D. Shen, H. Chen, D. Fan, and Q. Lu, “Model and optimization of stable gain-switched Tm-doped fiber lasers,” Opt. Rev. 18(4), 360–364 (2011).
[Crossref]

Frith, G.

G. Frith, D. G. Lancaster, and S. D. Jackson, “85W Tm3+ -doped silica fiber laser,” Electron. Lett. 41(12), 687–688 (2005).
[Crossref]

Geng, J.

Goodno, G. D.

Hankla, A. K.

T. J. Carrig, A. K. Hankla, G. J. Wagner, C. B. Rawle, and I. T. McKinnie, “Tunable infrared laser sources for DIAL,” Proc. SPIE 4723, 147–155 (2002).
[Crossref]

Haub, J.

Heidt, A. M.

Hemming, A.

Hou, Y.

Huang, C.

Ibsen, M.

Jackson, S. D.

J. Li and S. D. Jackson, “Numerical modeling and optimization of diode pumped heavily-Erbium-doped fluoride fiber lasers,” IEEE J. Quantum Electron. 48(4), 454–464 (2012).
[Crossref]

G. Frith, D. G. Lancaster, and S. D. Jackson, “85W Tm3+ -doped silica fiber laser,” Electron. Lett. 41(12), 687–688 (2005).
[Crossref]

S. D. Jackson, “Cross relaxation and energy transfer upconversion processes relevant to the functioning of 2 μm Tm3+-doped silica fibre lasers,” Opt. Commun. 230(1–3), 197–203 (2004).
[Crossref]

B. C. Dickinson, S. D. Jackson, and T. A. King, “10 mJ total output from a gain-switched Tm-doped fibre laser,” Opt. Commun. 182(1-3), 199–203 (2000).
[Crossref]

S. D. Jackson and T. A. King, “Theoretical Modeling of Tm-Doped Silica Fiber Lasers,” J. Lightwave Technol. 17(5), 948–956 (1999).
[Crossref]

S. D. Jackson and T. A. King, “Efficient gain-switched operation of a Tm-doped silica fiber laser,” IEEE J. Quantum Electron. 34, 770–789 (1998).

Jiang, M.

Jiang, S.

Ju, Y. L.

Kelly, B.

King, T. A.

B. C. Dickinson, S. D. Jackson, and T. A. King, “10 mJ total output from a gain-switched Tm-doped fibre laser,” Opt. Commun. 182(1-3), 199–203 (2000).
[Crossref]

S. D. Jackson and T. A. King, “Theoretical Modeling of Tm-Doped Silica Fiber Lasers,” J. Lightwave Technol. 17(5), 948–956 (1999).
[Crossref]

S. D. Jackson and T. A. King, “Efficient gain-switched operation of a Tm-doped silica fiber laser,” IEEE J. Quantum Electron. 34, 770–789 (1998).

Lancaster, D. G.

G. Frith, D. G. Lancaster, and S. D. Jackson, “85W Tm3+ -doped silica fiber laser,” Electron. Lett. 41(12), 687–688 (2005).
[Crossref]

Leveille, R.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” Proc. SPIE 7580, 7580112 (2010).

Li, F.

Y. Tang, F. Li, and J. Xu, “High peak-power gain-switched Tm3+-doped fiber laser,” IEEE Photonics Technol. Lett. 23(13), 893–895 (2011).
[Crossref]

Li, H.

Li, J.

J. Li and S. D. Jackson, “Numerical modeling and optimization of diode pumped heavily-Erbium-doped fluoride fiber lasers,” IEEE J. Quantum Electron. 48(4), 454–464 (2012).
[Crossref]

Li, Z.

Liu, J.

Liu, K.

Lu, J.

Lu, Q.

F. Wang, D. Shen, H. Chen, D. Fan, and Q. Lu, “Model and optimization of stable gain-switched Tm-doped fiber lasers,” Opt. Rev. 18(4), 360–364 (2011).
[Crossref]

Majid, I.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” Proc. SPIE 7580, 7580112 (2010).

Marcuse, D.

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

McKinnie, I. T.

T. J. Carrig, A. K. Hankla, G. J. Wagner, C. B. Rawle, and I. T. McKinnie, “Tunable infrared laser sources for DIAL,” Proc. SPIE 4723, 147–155 (2002).
[Crossref]

Michalska, M.

Moulton, P.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” Proc. SPIE 7580, 7580112 (2010).

Payne, M. J. P.

I. F. Elder and M. J. P. Payne, “Lasing in diode-pumped Tm: YAP, Tm, Ho: YAP and Tm, Ho: YLF,” Opt. Commun. 145(1–6), 329–339 (1998).
[Crossref]

Phelan, R.

Prabhu, M.

Rawle, C. B.

T. J. Carrig, A. K. Hankla, G. J. Wagner, C. B. Rawle, and I. T. McKinnie, “Tunable infrared laser sources for DIAL,” Proc. SPIE 4723, 147–155 (2002).
[Crossref]

Richardson, D. J.

Rines, G.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” Proc. SPIE 7580, 7580112 (2010).

Rothenberg, J. E.

Rothhardt, M.

Sahu, J.

Shardlow, P. C.

Shen, D.

F. Wang, D. Shen, H. Chen, D. Fan, and Q. Lu, “Model and optimization of stable gain-switched Tm-doped fiber lasers,” Opt. Rev. 18(4), 360–364 (2011).
[Crossref]

Shi, H.

Simakov, N.

Swiderski, J.

Tang, Y.

Tankala, K.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” Proc. SPIE 7580, 7580112 (2010).

Tayebati, P.

Ueda, K.

Wagner, G. J.

T. J. Carrig, A. K. Hankla, G. J. Wagner, C. B. Rawle, and I. T. McKinnie, “Tunable infrared laser sources for DIAL,” Proc. SPIE 4723, 147–155 (2002).
[Crossref]

Wang, F.

F. Wang, D. Shen, H. Chen, D. Fan, and Q. Lu, “Model and optimization of stable gain-switched Tm-doped fiber lasers,” Opt. Rev. 18(4), 360–364 (2011).
[Crossref]

Wang, P.

Wang, Q.

Wang, S.

Wang, Y.

Wang, Y. Z.

Wu, J.

Xing, D.

Xu, J.

C. Huang, Y. Tang, H. Li, Y. Wang, J. Xu, and C. Du, “A versatile model for temperature-dependent effects in Tm-doped silica fiber lasers,” J. Lightwave Technol. 32(3), 421–428 (2014).
[Crossref]

J. Yang, H. Li, Y. Tang, and J. Xu, “Temporal characteristics of in-band-pumped gain-switched thulium-doped fiber lasers,” J. Opt. Soc. Am. B 31(1), 80–86 (2014).
[Crossref]

J. Yang, Y. Tang, and J. Xu, “Development and applications of gain-switched fiber lasers,” Photon. Res. 1(1), 52–57 (2013).
[Crossref]

Y. Tang, C. Huang, S. Wang, H. Li, and J. Xu, “High-power narrow-bandwidth thulium fiber laser with an all-fiber cavity,” Opt. Express 20(16), 17539–17544 (2012).
[Crossref] [PubMed]

Y. Tang and J. Xu, “Hybrid-pumped gain-switched narrow-band thulium fiber laser,” Appl. Phys. Express 5(7), 072702 (2012).
[Crossref]

Y. Tang, F. Li, and J. Xu, “High peak-power gain-switched Tm3+-doped fiber laser,” IEEE Photonics Technol. Lett. 23(13), 893–895 (2011).
[Crossref]

Y. Tang and J. Xu, “Model and characteristics of self-pulsing in Tm3+-doped silica fiber lasers,” IEEE J. Quantum Electron. 47(2), 165–171 (2011).
[Crossref]

Y. Tang, L. Xu, Y. Yang, and J. Xu, “High-power gain-switched Tm3+-doped fiber laser,” Opt. Express 18(22), 22964–22972 (2010).
[Crossref] [PubMed]

Y. Tang and J. Xu, “Self-induced pulsing in Tm3+-doped fiber lasers with different output couplings,” Proc. SPIE 7276, 72760L (2008).
[Crossref]

J. Xu, M. Prabhu, J. Lu, K. Ueda, and D. Xing, “Efficient double-clad thulium-doped fiber laser with a ring cavity,” Appl. Opt. 40(12), 1983–1988 (2001).
[Crossref] [PubMed]

Xu, L.

Yang, J.

Yang, Y.

Yao, B. Q.

Yao, Z.

Zhang, Y.

Zong, J.

Appl. Opt. (2)

Appl. Phys. Express (1)

Y. Tang and J. Xu, “Hybrid-pumped gain-switched narrow-band thulium fiber laser,” Appl. Phys. Express 5(7), 072702 (2012).
[Crossref]

Bell Syst. Tech. J. (1)

D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56(5), 703–718 (1977).
[Crossref]

Electron. Lett. (1)

G. Frith, D. G. Lancaster, and S. D. Jackson, “85W Tm3+ -doped silica fiber laser,” Electron. Lett. 41(12), 687–688 (2005).
[Crossref]

IEEE J. Quantum Electron. (3)

Y. Tang and J. Xu, “Model and characteristics of self-pulsing in Tm3+-doped silica fiber lasers,” IEEE J. Quantum Electron. 47(2), 165–171 (2011).
[Crossref]

J. Li and S. D. Jackson, “Numerical modeling and optimization of diode pumped heavily-Erbium-doped fluoride fiber lasers,” IEEE J. Quantum Electron. 48(4), 454–464 (2012).
[Crossref]

S. D. Jackson and T. A. King, “Efficient gain-switched operation of a Tm-doped silica fiber laser,” IEEE J. Quantum Electron. 34, 770–789 (1998).

IEEE Photonics Technol. Lett. (1)

Y. Tang, F. Li, and J. Xu, “High peak-power gain-switched Tm3+-doped fiber laser,” IEEE Photonics Technol. Lett. 23(13), 893–895 (2011).
[Crossref]

J. Lightwave Technol. (2)

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

Opt. Commun. (3)

B. C. Dickinson, S. D. Jackson, and T. A. King, “10 mJ total output from a gain-switched Tm-doped fibre laser,” Opt. Commun. 182(1-3), 199–203 (2000).
[Crossref]

S. D. Jackson, “Cross relaxation and energy transfer upconversion processes relevant to the functioning of 2 μm Tm3+-doped silica fibre lasers,” Opt. Commun. 230(1–3), 197–203 (2004).
[Crossref]

I. F. Elder and M. J. P. Payne, “Lasing in diode-pumped Tm: YAP, Tm, Ho: YAP and Tm, Ho: YLF,” Opt. Commun. 145(1–6), 329–339 (1998).
[Crossref]

Opt. Express (5)

Opt. Lett. (5)

Opt. Rev. (1)

F. Wang, D. Shen, H. Chen, D. Fan, and Q. Lu, “Model and optimization of stable gain-switched Tm-doped fiber lasers,” Opt. Rev. 18(4), 360–364 (2011).
[Crossref]

Photon. Res. (1)

Proc. SPIE (3)

Y. Tang and J. Xu, “Self-induced pulsing in Tm3+-doped fiber lasers with different output couplings,” Proc. SPIE 7276, 72760L (2008).
[Crossref]

T. J. Carrig, A. K. Hankla, G. J. Wagner, C. B. Rawle, and I. T. McKinnie, “Tunable infrared laser sources for DIAL,” Proc. SPIE 4723, 147–155 (2002).
[Crossref]

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” Proc. SPIE 7580, 7580112 (2010).

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, Elsevier, 2007), Chap. 8, p. 278.

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Figures (9)

Fig. 1
Fig. 1 Schematic experimental setup of hybrid pumped the laser.
Fig. 2
Fig. 2 Energy diagrams of the Tm3+ ions.
Fig. 3
Fig. 3 Temporal dynamics of the 2-μm laser (blue line), 1900 nm pump (green line) and the population density of the upper laser level (red line) for 793 nm pump power of (a) 0 W, (b) 10 W and (c) 30 W. The pulsed pump at 1900 nm has an average power level of 5 W, repetition rate of 100 kHz, and pulse width of 100 ns.
Fig. 4
Fig. 4 Single pulsing regime of the hybrid gain-switched fiber laser.
Fig. 5
Fig. 5 Average out power of the hybrid gain-switched fiber laser under various combinations of the 793 nm CW pump and the 1900 nm pulse pump.
Fig. 6
Fig. 6 Output laser pulse characteristics of the hybrid gain-switched fiber laser versus the 793 nm CW pump under several 1900 nm pump levels.
Fig. 7
Fig. 7 Laser output characteristics of the hybrid gain-switched fiber laser under 10 W 793 nm pump and with 1900 nm pump power of 5W (100 kHz) in pulse width of (a) 10 ns, (b) 100 ns, (c) 2 μs, (d) 4 μs and (e) CW mode.
Fig. 8
Fig. 8 Output characteristics of the hybrid gain-switched fiber laser when the 1900 nm pump source has repetition rate of 200 kHz.
Fig. 9
Fig. 9 Tolerable pump ranges of the hybrid gain-switched fiber laser

Tables (1)

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Table 1 Constants Employed in the Numerical Simulation

Equations (20)

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N 4 (z,t) t = W 14 (z,t) N 1 (z,t)+ W 34 (z,t) N 3 (z,t)CR1 N 4 (z,t) τ 4
N 3 (z,t) t = W 34 (z,t) N 3 (z,t)CR2+ β 43 N 4 (z,t) τ 4 N 3 (z,t) τ 3
N 2 (z,t) t = W 12 (z,t) N 1 (z,t) W 21 (z,t) N 2 (z,t)+2CR1+2CR2 + β 42 N 4 (z,t) τ 4 + β 32 N 3 (z,t) τ 3 N 2 (z,t) τ 2
N 1 (z,t)=N N 2 (z,t) N 3 (z,t) N 4 (z,t)
CR1= k 1242 N 1 (z,t) N 4 (z,t) k 2124 N 2 (z,t) 2
CR2= k 1232 N 1 (z,t) N 3 (z,t) k 2123 N 2 (z,t) 2
W 14 = σ 14 λ p 1 Γ p clad ( P f 1 + P r 1 ) hc A core
W 34 = σ 34 λ s Γ s ( S f + S r ) hc A core
W 12 = σ 12 ( λ p 2 ) Γ p core ( P f 2 + P r 2 ) λ p 2 hc A core
W 21 = σ 21 ( λ s ) Γ s ( S f + S r ) λ s hc A core + σ 21 ( λ p 2 ) Γ p core ( P f 2 + P r 2 ) λ p 2 hc A core
V p,s = 2π r c A core λ p,s
ω p,s = r c (0.65+ 1.619 V p,s 1.5 + 2.876 V p,s 6 )
Γ p,s =1exp( 2 r c 2 ω p,s 2 )
± P f,r 1 z + P f,r 1 v g t = Γ p clad σ 14 ( λ p 1 ) Ν 1 (z,t) P f,r 1 (z,t) α p 1 P f,r 1 (z,t)
± P f,r 2 z + P f,r 2 v g t = Γ p core [ σ 12 ( λ p 2 ) Ν 1 (z,t)+ σ 21 ( λ p 2 ) Ν 2 (z,t)] P f,r 2 (z,t) α p 2 P f,r 2 (z,t)
± S f,r z + S f,r v g t = Γ s [ σ 21 ( λ s ) Ν 2 (z,t) σ 34 ( λ s ) Ν 3 (z,t)] S f,r (z,t) α s S f,r (z,t)+ Γ s σ 21 ( λ s ) Ν 2 (z,t) 2h c 2 ( λ s ) 3 Δ λ ASE
P f i (0)=P f i + R p1 i P r i (0) i=1,2
P r i (L)= R p2 i P f i (L) i=1,2
S f (0)= R s1 S r (0)
S r (L)= R s2 S f (L)

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