Abstract

We present a set of modified rate equations to describe the photon density in a passively Q-switched laser by taking into account the amplified spontaneous emission. Experiments and numerical simulations were carried out on a longitudinally diode-pumped Nd:YVO4 laser that was passively Q switched using a bulk GaAs wafer as both saturable-absorber Q switch and output coupler. Both sets of results were found to agree with each other very well. Introducing amplified spontaneous emission into the rate equations has proved to be necessary to model accurately the dynamic behavior of passively Q-switched lasers.

© 2004 Optical Society of America

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References

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  1. G. J. Spuhler, R. Paschotta, R. Fluck, B. Braun, M. Moser, G. Zhang, E. Gini, and U. Keller, “Experimentally confirmed design guidelines for passively Q-switched microchip lasers using semiconductor saturable absorbers,” J. Opt. Soc. Am. B 16, 376–388 (1999).
    [CrossRef]
  2. B. Braun, F. X. Kartner, G. Zhang, M. Moser, and U. Keller, “56-ps passively Q-switched diode-pumped microchip laser,” Opt. Lett. 22, 381–383 (1997).
    [CrossRef] [PubMed]
  3. U. Keller, D. A. B. Miller, G. D. Boyd, T. H. Chiu, J. F. Ferguson, and M. T. Asom, “Solid-state low-loss intracavity saturable absorber for Nd:YLF lasers: an antiresonant semiconductor Fabry–Perot saturable absorber,” Opt. Lett. 17, 505–507 (1992).
    [CrossRef] [PubMed]
  4. T. T. Kajava and A. L. Gaeta, “Q-switching of a diode-pumped Nd:YAG laser with GaAs,” Opt. Lett. 21, 1244–1246 (1996).
    [CrossRef] [PubMed]
  5. T. T. Kajava and A. L. Gaeta, “Intra-cavity frequency-doubling of a Nd:YAG laser passively Q-switched with GaAs,” Opt. Commun. 137, 93–97 (1997).
    [CrossRef]
  6. P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
    [CrossRef]
  7. Jianhui Gu, Feng Zhou, Siu Chung Tam, Wenjue Xie, Yee Loy Lam, and Yihong Chen, “Passive Q-switching of a Nd:YAG laser with a GaAs output coupler,” Opt. Eng. 38, 1785–1788 (1999).
    [CrossRef]
  8. R. Fluck, B. Braun, E. Gini, H. Melchior, and U. Keller, “Passively Q-switched 1.34-μm Nd:YVO4 microchip laser with semiconductor saturable-absorber mirrors,” Opt. Lett. 22, 991–993 (1997).
    [CrossRef] [PubMed]
  9. R. Fluck, R. Haring, R. Paschotta, E. Gini, H. Melchior, and U. Keller, “Eyesafe pulsed microchip laser using semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 72, 3273–3275 (1998).
    [CrossRef]
  10. M. A. Jaspan, D. Welford, G. Xiao, and M. Bass, “Atypical behavior of Cr:YAG passively Q-switched Nd:YVO 4 microlasers at high-pumping rates,” Conference on Lasers and Electro-Optics, Vol. 39 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), p. 454.
  11. W. G. Wagner and B. A. Lengyel, “Evolution of the giant pulse in a laser,” J. Appl. Phys. 34, 2040–2046 (1963).
    [CrossRef]
  12. A. Szabo and R. A. Stein, “Theory of laser giant pulsing by a saturable absorber,” J. Appl. Phys. 36, 1562–1566 (1965).
    [CrossRef]
  13. W. Koechner, Solid State Laser Engineering, 5th ed. (Springer-Verlag, New York, 1999), p. 64.
  14. J. J. Zayhowski and P. L. Kelly, “Optimization of Q-switched lasers,” IEEE J. Quantum Electron. 27, 2220–2225 (1991).
    [CrossRef]
  15. J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31, 1890–1901 (1995).
    [CrossRef]
  16. Yen-Kuang Kuo, Man-Fang Huang, and M. Birnbaum, “Tunable Cr4+:YSO Q-switched Cr:LiCAF laser,” IEEE J. Quantum Electron. 31, 657–663 (1995).
    [CrossRef]
  17. G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
    [CrossRef]
  18. A. L. Smirl, G. C. Valley, K. M. Bohnert, and T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
    [CrossRef]
  19. W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381–2395 (2001).
    [CrossRef]
  20. Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986 (2003).
    [CrossRef]
  21. www.itieo.com/data/NdYVO4datasheet.pdf.

2003 (1)

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986 (2003).
[CrossRef]

2001 (2)

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381–2395 (2001).
[CrossRef]

P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
[CrossRef]

1999 (2)

1998 (1)

R. Fluck, R. Haring, R. Paschotta, E. Gini, H. Melchior, and U. Keller, “Eyesafe pulsed microchip laser using semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 72, 3273–3275 (1998).
[CrossRef]

1997 (3)

1996 (1)

1995 (2)

J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31, 1890–1901 (1995).
[CrossRef]

Yen-Kuang Kuo, Man-Fang Huang, and M. Birnbaum, “Tunable Cr4+:YSO Q-switched Cr:LiCAF laser,” IEEE J. Quantum Electron. 31, 657–663 (1995).
[CrossRef]

1992 (1)

1991 (1)

J. J. Zayhowski and P. L. Kelly, “Optimization of Q-switched lasers,” IEEE J. Quantum Electron. 27, 2220–2225 (1991).
[CrossRef]

1988 (2)

G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
[CrossRef]

A. L. Smirl, G. C. Valley, K. M. Bohnert, and T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

1965 (1)

A. Szabo and R. A. Stein, “Theory of laser giant pulsing by a saturable absorber,” J. Appl. Phys. 36, 1562–1566 (1965).
[CrossRef]

1963 (1)

W. G. Wagner and B. A. Lengyel, “Evolution of the giant pulse in a laser,” J. Appl. Phys. 34, 2040–2046 (1963).
[CrossRef]

Asom, M. T.

Birnbaum, M.

Yen-Kuang Kuo, Man-Fang Huang, and M. Birnbaum, “Tunable Cr4+:YSO Q-switched Cr:LiCAF laser,” IEEE J. Quantum Electron. 31, 657–663 (1995).
[CrossRef]

Boggess, T. F.

A. L. Smirl, G. C. Valley, K. M. Bohnert, and T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

Bohnert, K. M.

A. L. Smirl, G. C. Valley, K. M. Bohnert, and T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

Boyd, G. D.

Braun, B.

Chen, Yihong

Jianhui Gu, Feng Zhou, Siu Chung Tam, Wenjue Xie, Yee Loy Lam, and Yihong Chen, “Passive Q-switching of a Nd:YAG laser with a GaAs output coupler,” Opt. Eng. 38, 1785–1788 (1999).
[CrossRef]

Chiu, T. H.

Clarkson, W. A.

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381–2395 (2001).
[CrossRef]

Degnan, J. J.

J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31, 1890–1901 (1995).
[CrossRef]

Ferguson, J. F.

Fluck, R.

Gaeta, A. L.

T. T. Kajava and A. L. Gaeta, “Intra-cavity frequency-doubling of a Nd:YAG laser passively Q-switched with GaAs,” Opt. Commun. 137, 93–97 (1997).
[CrossRef]

T. T. Kajava and A. L. Gaeta, “Q-switching of a diode-pumped Nd:YAG laser with GaAs,” Opt. Lett. 21, 1244–1246 (1996).
[CrossRef] [PubMed]

Gini, E.

Gu, Jianhui

Jianhui Gu, Feng Zhou, Siu Chung Tam, Wenjue Xie, Yee Loy Lam, and Yihong Chen, “Passive Q-switching of a Nd:YAG laser with a GaAs output coupler,” Opt. Eng. 38, 1785–1788 (1999).
[CrossRef]

Haring, R.

R. Fluck, R. Haring, R. Paschotta, E. Gini, H. Melchior, and U. Keller, “Eyesafe pulsed microchip laser using semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 72, 3273–3275 (1998).
[CrossRef]

He, J.

P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
[CrossRef]

Huang, Man-Fang

Yen-Kuang Kuo, Man-Fang Huang, and M. Birnbaum, “Tunable Cr4+:YSO Q-switched Cr:LiCAF laser,” IEEE J. Quantum Electron. 31, 657–663 (1995).
[CrossRef]

Huang, W. L.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986 (2003).
[CrossRef]

Kajava, T. T.

T. T. Kajava and A. L. Gaeta, “Intra-cavity frequency-doubling of a Nd:YAG laser passively Q-switched with GaAs,” Opt. Commun. 137, 93–97 (1997).
[CrossRef]

T. T. Kajava and A. L. Gaeta, “Q-switching of a diode-pumped Nd:YAG laser with GaAs,” Opt. Lett. 21, 1244–1246 (1996).
[CrossRef] [PubMed]

Kartner, F. X.

Keller, U.

Kelly, P. L.

J. J. Zayhowski and P. L. Kelly, “Optimization of Q-switched lasers,” IEEE J. Quantum Electron. 27, 2220–2225 (1991).
[CrossRef]

Kuo, Yen-Kuang

Yen-Kuang Kuo, Man-Fang Huang, and M. Birnbaum, “Tunable Cr4+:YSO Q-switched Cr:LiCAF laser,” IEEE J. Quantum Electron. 31, 657–663 (1995).
[CrossRef]

Lam, Yee Loy

Jianhui Gu, Feng Zhou, Siu Chung Tam, Wenjue Xie, Yee Loy Lam, and Yihong Chen, “Passive Q-switching of a Nd:YAG laser with a GaAs output coupler,” Opt. Eng. 38, 1785–1788 (1999).
[CrossRef]

Lengyel, B. A.

W. G. Wagner and B. A. Lengyel, “Evolution of the giant pulse in a laser,” J. Appl. Phys. 34, 2040–2046 (1963).
[CrossRef]

Li, P.

P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
[CrossRef]

Li, S.

P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
[CrossRef]

Li, Z. G.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986 (2003).
[CrossRef]

Lim, G. C.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986 (2003).
[CrossRef]

Lu, X.

P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
[CrossRef]

Melchior, H.

R. Fluck, R. Haring, R. Paschotta, E. Gini, H. Melchior, and U. Keller, “Eyesafe pulsed microchip laser using semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 72, 3273–3275 (1998).
[CrossRef]

R. Fluck, B. Braun, E. Gini, H. Melchior, and U. Keller, “Passively Q-switched 1.34-μm Nd:YVO4 microchip laser with semiconductor saturable-absorber mirrors,” Opt. Lett. 22, 991–993 (1997).
[CrossRef] [PubMed]

Miller, D. A. B.

Moore, N.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986 (2003).
[CrossRef]

Moser, M.

Paschotta, R.

G. J. Spuhler, R. Paschotta, R. Fluck, B. Braun, M. Moser, G. Zhang, E. Gini, and U. Keller, “Experimentally confirmed design guidelines for passively Q-switched microchip lasers using semiconductor saturable absorbers,” J. Opt. Soc. Am. B 16, 376–388 (1999).
[CrossRef]

R. Fluck, R. Haring, R. Paschotta, E. Gini, H. Melchior, and U. Keller, “Eyesafe pulsed microchip laser using semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 72, 3273–3275 (1998).
[CrossRef]

Smirl, A. L.

G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
[CrossRef]

A. L. Smirl, G. C. Valley, K. M. Bohnert, and T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

Spuhler, G. J.

Stein, R. A.

A. Szabo and R. A. Stein, “Theory of laser giant pulsing by a saturable absorber,” J. Appl. Phys. 36, 1562–1566 (1965).
[CrossRef]

Szabo, A.

A. Szabo and R. A. Stein, “Theory of laser giant pulsing by a saturable absorber,” J. Appl. Phys. 36, 1562–1566 (1965).
[CrossRef]

Tam, Siu Chung

Jianhui Gu, Feng Zhou, Siu Chung Tam, Wenjue Xie, Yee Loy Lam, and Yihong Chen, “Passive Q-switching of a Nd:YAG laser with a GaAs output coupler,” Opt. Eng. 38, 1785–1788 (1999).
[CrossRef]

Valley, G. C.

A. L. Smirl, G. C. Valley, K. M. Bohnert, and T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
[CrossRef]

Wagner, W. G.

W. G. Wagner and B. A. Lengyel, “Evolution of the giant pulse in a laser,” J. Appl. Phys. 34, 2040–2046 (1963).
[CrossRef]

Wang, Q.

P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
[CrossRef]

Wang, Y.

P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
[CrossRef]

Xie, Wenjue

Jianhui Gu, Feng Zhou, Siu Chung Tam, Wenjue Xie, Yee Loy Lam, and Yihong Chen, “Passive Q-switching of a Nd:YAG laser with a GaAs output coupler,” Opt. Eng. 38, 1785–1788 (1999).
[CrossRef]

Xiong, Z.

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986 (2003).
[CrossRef]

Zayhowski, J. J.

J. J. Zayhowski and P. L. Kelly, “Optimization of Q-switched lasers,” IEEE J. Quantum Electron. 27, 2220–2225 (1991).
[CrossRef]

Zhang, G.

Zhang, X.

P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
[CrossRef]

Zhou, Feng

Jianhui Gu, Feng Zhou, Siu Chung Tam, Wenjue Xie, Yee Loy Lam, and Yihong Chen, “Passive Q-switching of a Nd:YAG laser with a GaAs output coupler,” Opt. Eng. 38, 1785–1788 (1999).
[CrossRef]

Appl. Phys. Lett. (1)

R. Fluck, R. Haring, R. Paschotta, E. Gini, H. Melchior, and U. Keller, “Eyesafe pulsed microchip laser using semiconductor saturable absorber mirrors,” Appl. Phys. Lett. 72, 3273–3275 (1998).
[CrossRef]

IEEE J. Quantum Electron. (6)

J. J. Zayhowski and P. L. Kelly, “Optimization of Q-switched lasers,” IEEE J. Quantum Electron. 27, 2220–2225 (1991).
[CrossRef]

J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31, 1890–1901 (1995).
[CrossRef]

Yen-Kuang Kuo, Man-Fang Huang, and M. Birnbaum, “Tunable Cr4+:YSO Q-switched Cr:LiCAF laser,” IEEE J. Quantum Electron. 31, 657–663 (1995).
[CrossRef]

G. C. Valley and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304–310 (1988).
[CrossRef]

A. L. Smirl, G. C. Valley, K. M. Bohnert, and T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μm,” IEEE J. Quantum Electron. 24, 289–303 (1988).
[CrossRef]

Z. Xiong, Z. G. Li, N. Moore, W. L. Huang, and G. C. Lim, “Detailed investigation of thermal effects in longitudinally diode-pumped Nd:YVO4 lasers,” IEEE J. Quantum Electron. 39, 979–986 (2003).
[CrossRef]

J. Appl. Phys. (2)

W. G. Wagner and B. A. Lengyel, “Evolution of the giant pulse in a laser,” J. Appl. Phys. 34, 2040–2046 (1963).
[CrossRef]

A. Szabo and R. A. Stein, “Theory of laser giant pulsing by a saturable absorber,” J. Appl. Phys. 36, 1562–1566 (1965).
[CrossRef]

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

J. Phys. D (1)

W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381–2395 (2001).
[CrossRef]

Opt. Commun. (1)

T. T. Kajava and A. L. Gaeta, “Intra-cavity frequency-doubling of a Nd:YAG laser passively Q-switched with GaAs,” Opt. Commun. 137, 93–97 (1997).
[CrossRef]

Opt. Eng. (1)

Jianhui Gu, Feng Zhou, Siu Chung Tam, Wenjue Xie, Yee Loy Lam, and Yihong Chen, “Passive Q-switching of a Nd:YAG laser with a GaAs output coupler,” Opt. Eng. 38, 1785–1788 (1999).
[CrossRef]

Opt. Laser Technol. (1)

P. Li, Q. Wang, X. Zhang, Y. Wang, S. Li, J. He, and X. Lu, “Analysis of a diode-pumped Nd:YVO4 laser passively Q switched with GaAs,” Opt. Laser Technol. 33, 383–387 (2001).
[CrossRef]

Opt. Lett. (4)

Other (3)

W. Koechner, Solid State Laser Engineering, 5th ed. (Springer-Verlag, New York, 1999), p. 64.

M. A. Jaspan, D. Welford, G. Xiao, and M. Bass, “Atypical behavior of Cr:YAG passively Q-switched Nd:YVO 4 microlasers at high-pumping rates,” Conference on Lasers and Electro-Optics, Vol. 39 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), p. 454.

www.itieo.com/data/NdYVO4datasheet.pdf.

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Figures (8)

Fig. 1
Fig. 1

Energy-level diagram used to describe the absorption and relaxation processes of undoped semi-insulating GaAs.

Fig. 2
Fig. 2

Simplified energy-level diagram of a four-level laser, including the spontaneous emission that depletes the inversion population.

Fig. 3
Fig. 3

Beam sizes at both the laser rod and GaAs wafer with a separation of 120 mm as a function of incident pump power.

Fig. 4
Fig. 4

Schematic diagram of the experimental setup.

Fig. 5
Fig. 5

Typical set of Q-switched pulses from both (a) experiment and (b) simulation; also shown are (c) the corresponding gain and (d) loss coefficients as function of time.

Fig. 6
Fig. 6

Dependence on pump power of (a) pulse width, (b) repetition rate, (c) pulse energy, and (d) average output power. The points are experimental results, and the solid curves are simulation results.

Fig. 7
Fig. 7

Typical pulse temporal profile with a pump power of 8 W.

Fig. 8
Fig. 8

Evolution in photon density, gain, and loss in different cases: (a) 5 W with ASE, (b) 5 W without ASE.

Tables (1)

Tables Icon

Table 1 Parameters and Values in Simulations

Equations (11)

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

dN+dt=cϕq[σe(N0-N+)-σhN+]-γetnN++γhdp(N0-N+),
dndt=cϕqσe(N0-N+)+Bcϕq2/12-γetnN+-γehnp,
dpdt=cϕqσhN++Bcϕq2/12-γhdp(N0-N+)-γehnp,
α=σe(N0-N+)+σhN++σfcn+Bϕ/2,
dϕdt=1τR 2σ(N2-N1)l-2αlq-ln1R-Lϕ+N2-N1τ21ξ,
dN2dt=PN0-N2τf-σcϕ(N2-N1)-N2-N1τ21ξ,
dN1dt=N1τ10+N2τ21+σcϕ(N2-N1),
dN0dt=N2τ20+N1τ10-PN0,
-N2-N1τ21ξ
fth(r)=2r2fth(0)kωp2[1-exp(-2r2/ωp2)],
fth=πKcωp2Ppγhηabs(dn/dT),

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