Abstract

A diode-pumped doubly passively Q-switched Cr,Nd:YAG/KTP green laser with GaAs saturable absorber is realized for the first time to our knowledge. This laser can generate a more symmetric and shorter pulse when compared with the self-Q-switched Cr,Nd:YAG/KTP green laser. A symmetry factor is defined to describe the temporal symmetry of the pulses. At an incident pump power of 4.1W, a pulse symmetry factor as high as 0.995 is obtained. A rate equation model is introduced to theoretically analyze the results obtained in the experiment, in which the spatial distributions of the intracavity photon density, the pump beam and the population-inversion density are taken into account. The numerical solutions of the rate equations are in good agreement with the experimental results.

© 2006 Optical Society of America

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  1. Y. F. Chen, T. M. Huang, and C. L. Wang, "Passively Q-switched diode-pumped Nd:YVO4/Cr4+:YAG single-frequency microchip laser," Electron. Lett. 33, 1880-1881 (1997).
    [CrossRef]
  2. Y. Bai, N. Wu, J. Zhang, J. Li, S. Li, J. Xu, and P. Deng, "Passively Q-switched Nd:YVO4 laser with a Cr4+:YAG crystal saturable absorbe," Appl. Opt. 36, 2468-2472 (1997).
    [CrossRef] [PubMed]
  3. M. I. Demebuk, V. P. Mikhailov, N. I. Zhavoronkov, N. V. Kuleshov, P. V. Prokoshin, K. V. Yumashev, M. G. Livshits, and B. I. Minkov, "Chromium-doped forsterite as a solid-state saturable absorber," Opt. Lett. 17, 929-930 (1992).
    [CrossRef]
  4. W. Chen, K. Spariosu, R. Stultz, Y. K. Kuo, M. Birnbaum, and A. V. Shestakov, "Cr4+:GSGG saturable absorber Q-switch for the ruby laser," Opt. Commun. 104, 71-74 (1993).
    [CrossRef]
  5. Y. K. Kuo, M. F. Huang, and M. Birnbaum, "Tunable Cr4+:YSO Q-switched Cr:LiCAF laser," IEEE J. Quantum Electron. 31, 657-663 (1995).
    [CrossRef]
  6. S. Zhou, K. K. Lee, and Y. C. Chen, "Monolithic self-Q-switched Cr,Nd:YAG laser," Opt. Lett. 18, 511-512 (1993).
    [CrossRef] [PubMed]
  7. Y. C. Chen, S. Li, K. K. Lee, and S. Zhou, "Self-stabilized single-longitudinal-mode operation in a self-Q-switched Cr,Nd:YAG laser," Opt. Lett. 18, 1418-1419 (1993).
    [CrossRef] [PubMed]
  8. J. Dong, P. Deng, Y. Lu, Y. Zhang, Y. Liu, J. Xu, and W. Chen, "Laser-diode-pumped Cr4+,Nd3+:YAG with self-Q-switched laser output of 1.4 W," Opt. Lett. 25, 1101-1103 (2000).
    [CrossRef]
  9. Z. Hong, H. Zheng, J. Chen, and J. Ge, "Laser-diode-pumped Cr4+,Nd3+:YAG self-Q-switched laser with high repetition rate and high stability," Appl. Phys. B 73, 205-207 (2001).
    [CrossRef]
  10. T. Yu, J. Cui, Y. Lu, and Q. Hu, "Diode-pumped solid-state Cr4+,Nd:YAG/KTP green laser," Chin. J. Lasers B10, 321-324 (2001).
  11. J. Dong, P. Deng, "Laser performance of monolithic Cr,Nd:YAG self-Q-switched laser," Opt. Commun. 220, 425-431 (2003).
    [CrossRef]
  12. 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]
  13. 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]
  14. S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, "Passively Q-switched self-frequency-doubling Nd3+:GdCa4O(BO3)3 laser with GaAs saturable absorber," Opt. Eng. 41, 559-560 (2002).
    [CrossRef]
  15. 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]
  16. G. C. Valley and A. L. Smirl, "Theory of transient energy transfer in gallium arsenide," IEEE J. Quantum Electron. 24, 304-310 (1988).
    [CrossRef]
  17. G. Li, S. Zhao, K. Yang, and W. Wu, "Pulse width reduction in diode-pumped Nd:GdVO4 laser with AO and GaAs double Q-switches," Jpn. J. Appl. Phys. 44, 3017-3021 (2005).
    [CrossRef]
  18. G. Li, S. Zhao, K. Yang, and P. Song, "Control of the pulse width in a diode-pumped passively Q-switched Nd:GdVO4/KTP green laser with a Cr4+:YAG saturable absorber," Appl. Opt. 44, 5990-5995 (2005).
    [CrossRef] [PubMed]

2005 (2)

G. Li, S. Zhao, K. Yang, and W. Wu, "Pulse width reduction in diode-pumped Nd:GdVO4 laser with AO and GaAs double Q-switches," Jpn. J. Appl. Phys. 44, 3017-3021 (2005).
[CrossRef]

G. Li, S. Zhao, K. Yang, and P. Song, "Control of the pulse width in a diode-pumped passively Q-switched Nd:GdVO4/KTP green laser with a Cr4+:YAG saturable absorber," Appl. Opt. 44, 5990-5995 (2005).
[CrossRef] [PubMed]

2003 (1)

J. Dong, P. Deng, "Laser performance of monolithic Cr,Nd:YAG self-Q-switched laser," Opt. Commun. 220, 425-431 (2003).
[CrossRef]

2002 (1)

S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, "Passively Q-switched self-frequency-doubling Nd3+:GdCa4O(BO3)3 laser with GaAs saturable absorber," Opt. Eng. 41, 559-560 (2002).
[CrossRef]

2001 (2)

Z. Hong, H. Zheng, J. Chen, and J. Ge, "Laser-diode-pumped Cr4+,Nd3+:YAG self-Q-switched laser with high repetition rate and high stability," Appl. Phys. B 73, 205-207 (2001).
[CrossRef]

T. Yu, J. Cui, Y. Lu, and Q. Hu, "Diode-pumped solid-state Cr4+,Nd:YAG/KTP green laser," Chin. J. Lasers B10, 321-324 (2001).

2000 (1)

1997 (3)

Y. F. Chen, T. M. Huang, and C. L. Wang, "Passively Q-switched diode-pumped Nd:YVO4/Cr4+:YAG single-frequency microchip laser," Electron. Lett. 33, 1880-1881 (1997).
[CrossRef]

Y. Bai, N. Wu, J. Zhang, J. Li, S. Li, J. Xu, and P. Deng, "Passively Q-switched Nd:YVO4 laser with a Cr4+:YAG crystal saturable absorbe," Appl. Opt. 36, 2468-2472 (1997).
[CrossRef] [PubMed]

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]

1996 (1)

1995 (1)

Y. K. Kuo, M. F. Huang, and M. Birnbaum, "Tunable Cr4+:YSO Q-switched Cr:LiCAF laser," IEEE J. Quantum Electron. 31, 657-663 (1995).
[CrossRef]

1993 (3)

1992 (1)

1988 (2)

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]

Bai, Y.

Birnbaum, M.

Y. K. Kuo, M. F. Huang, and M. Birnbaum, "Tunable Cr4+:YSO Q-switched Cr:LiCAF laser," IEEE J. Quantum Electron. 31, 657-663 (1995).
[CrossRef]

W. Chen, K. Spariosu, R. Stultz, Y. K. Kuo, M. Birnbaum, and A. V. Shestakov, "Cr4+:GSGG saturable absorber Q-switch for the ruby laser," Opt. Commun. 104, 71-74 (1993).
[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]

Chen, J.

Z. Hong, H. Zheng, J. Chen, and J. Ge, "Laser-diode-pumped Cr4+,Nd3+:YAG self-Q-switched laser with high repetition rate and high stability," Appl. Phys. B 73, 205-207 (2001).
[CrossRef]

Chen, L.

S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, "Passively Q-switched self-frequency-doubling Nd3+:GdCa4O(BO3)3 laser with GaAs saturable absorber," Opt. Eng. 41, 559-560 (2002).
[CrossRef]

Chen, W.

J. Dong, P. Deng, Y. Lu, Y. Zhang, Y. Liu, J. Xu, and W. Chen, "Laser-diode-pumped Cr4+,Nd3+:YAG with self-Q-switched laser output of 1.4 W," Opt. Lett. 25, 1101-1103 (2000).
[CrossRef]

W. Chen, K. Spariosu, R. Stultz, Y. K. Kuo, M. Birnbaum, and A. V. Shestakov, "Cr4+:GSGG saturable absorber Q-switch for the ruby laser," Opt. Commun. 104, 71-74 (1993).
[CrossRef]

Chen, Y. C.

Chen, Y. F.

Y. F. Chen, T. M. Huang, and C. L. Wang, "Passively Q-switched diode-pumped Nd:YVO4/Cr4+:YAG single-frequency microchip laser," Electron. Lett. 33, 1880-1881 (1997).
[CrossRef]

Cheng, H.

S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, "Passively Q-switched self-frequency-doubling Nd3+:GdCa4O(BO3)3 laser with GaAs saturable absorber," Opt. Eng. 41, 559-560 (2002).
[CrossRef]

Cheng, Z.

S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, "Passively Q-switched self-frequency-doubling Nd3+:GdCa4O(BO3)3 laser with GaAs saturable absorber," Opt. Eng. 41, 559-560 (2002).
[CrossRef]

Cui, J.

T. Yu, J. Cui, Y. Lu, and Q. Hu, "Diode-pumped solid-state Cr4+,Nd:YAG/KTP green laser," Chin. J. Lasers B10, 321-324 (2001).

Demebuk, M. I.

Deng, P.

Dong, J.

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]

Ge, J.

Z. Hong, H. Zheng, J. Chen, and J. Ge, "Laser-diode-pumped Cr4+,Nd3+:YAG self-Q-switched laser with high repetition rate and high stability," Appl. Phys. B 73, 205-207 (2001).
[CrossRef]

Hong, Z.

Z. Hong, H. Zheng, J. Chen, and J. Ge, "Laser-diode-pumped Cr4+,Nd3+:YAG self-Q-switched laser with high repetition rate and high stability," Appl. Phys. B 73, 205-207 (2001).
[CrossRef]

Hu, Q.

T. Yu, J. Cui, Y. Lu, and Q. Hu, "Diode-pumped solid-state Cr4+,Nd:YAG/KTP green laser," Chin. J. Lasers B10, 321-324 (2001).

Huang, M. F.

Y. K. Kuo, M. F. Huang, and M. Birnbaum, "Tunable Cr4+:YSO Q-switched Cr:LiCAF laser," IEEE J. Quantum Electron. 31, 657-663 (1995).
[CrossRef]

Huang, T. M.

Y. F. Chen, T. M. Huang, and C. L. Wang, "Passively Q-switched diode-pumped Nd:YVO4/Cr4+:YAG single-frequency microchip laser," Electron. Lett. 33, 1880-1881 (1997).
[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]

Kuleshov, N. V.

Kuo, Y. K.

Y. K. Kuo, M. F. Huang, and M. Birnbaum, "Tunable Cr4+:YSO Q-switched Cr:LiCAF laser," IEEE J. Quantum Electron. 31, 657-663 (1995).
[CrossRef]

W. Chen, K. Spariosu, R. Stultz, Y. K. Kuo, M. Birnbaum, and A. V. Shestakov, "Cr4+:GSGG saturable absorber Q-switch for the ruby laser," Opt. Commun. 104, 71-74 (1993).
[CrossRef]

Lee, K. K.

Li, G.

G. Li, S. Zhao, K. Yang, and W. Wu, "Pulse width reduction in diode-pumped Nd:GdVO4 laser with AO and GaAs double Q-switches," Jpn. J. Appl. Phys. 44, 3017-3021 (2005).
[CrossRef]

G. Li, S. Zhao, K. Yang, and P. Song, "Control of the pulse width in a diode-pumped passively Q-switched Nd:GdVO4/KTP green laser with a Cr4+:YAG saturable absorber," Appl. Opt. 44, 5990-5995 (2005).
[CrossRef] [PubMed]

Li, J.

Li, S.

Liu, Y.

Livshits, M. G.

Lu, Y.

T. Yu, J. Cui, Y. Lu, and Q. Hu, "Diode-pumped solid-state Cr4+,Nd:YAG/KTP green laser," Chin. J. Lasers B10, 321-324 (2001).

J. Dong, P. Deng, Y. Lu, Y. Zhang, Y. Liu, J. Xu, and W. Chen, "Laser-diode-pumped Cr4+,Nd3+:YAG with self-Q-switched laser output of 1.4 W," Opt. Lett. 25, 1101-1103 (2000).
[CrossRef]

Mikhailov, V. P.

Minkov, B. I.

Prokoshin, P. V.

Shestakov, A. V.

W. Chen, K. Spariosu, R. Stultz, Y. K. Kuo, M. Birnbaum, and A. V. Shestakov, "Cr4+:GSGG saturable absorber Q-switch for the ruby laser," Opt. Commun. 104, 71-74 (1993).
[CrossRef]

Smirl, A. L.

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]

Song, P.

Spariosu, K.

W. Chen, K. Spariosu, R. Stultz, Y. K. Kuo, M. Birnbaum, and A. V. Shestakov, "Cr4+:GSGG saturable absorber Q-switch for the ruby laser," Opt. Commun. 104, 71-74 (1993).
[CrossRef]

Stultz, R.

W. Chen, K. Spariosu, R. Stultz, Y. K. Kuo, M. Birnbaum, and A. V. Shestakov, "Cr4+:GSGG saturable absorber Q-switch for the ruby laser," Opt. Commun. 104, 71-74 (1993).
[CrossRef]

Valley, G. C.

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]

Wang, C. L.

Y. F. Chen, T. M. Huang, and C. L. Wang, "Passively Q-switched diode-pumped Nd:YVO4/Cr4+:YAG single-frequency microchip laser," Electron. Lett. 33, 1880-1881 (1997).
[CrossRef]

Wu, N.

Wu, W.

G. Li, S. Zhao, K. Yang, and W. Wu, "Pulse width reduction in diode-pumped Nd:GdVO4 laser with AO and GaAs double Q-switches," Jpn. J. Appl. Phys. 44, 3017-3021 (2005).
[CrossRef]

Xu, J.

Yang, K.

G. Li, S. Zhao, K. Yang, and W. Wu, "Pulse width reduction in diode-pumped Nd:GdVO4 laser with AO and GaAs double Q-switches," Jpn. J. Appl. Phys. 44, 3017-3021 (2005).
[CrossRef]

G. Li, S. Zhao, K. Yang, and P. Song, "Control of the pulse width in a diode-pumped passively Q-switched Nd:GdVO4/KTP green laser with a Cr4+:YAG saturable absorber," Appl. Opt. 44, 5990-5995 (2005).
[CrossRef] [PubMed]

Yu, T.

T. Yu, J. Cui, Y. Lu, and Q. Hu, "Diode-pumped solid-state Cr4+,Nd:YAG/KTP green laser," Chin. J. Lasers B10, 321-324 (2001).

Yumashev, K. V.

Zhang, J.

Zhang, X.

S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, "Passively Q-switched self-frequency-doubling Nd3+:GdCa4O(BO3)3 laser with GaAs saturable absorber," Opt. Eng. 41, 559-560 (2002).
[CrossRef]

Zhang, Y.

Zhao, S.

G. Li, S. Zhao, K. Yang, and P. Song, "Control of the pulse width in a diode-pumped passively Q-switched Nd:GdVO4/KTP green laser with a Cr4+:YAG saturable absorber," Appl. Opt. 44, 5990-5995 (2005).
[CrossRef] [PubMed]

G. Li, S. Zhao, K. Yang, and W. Wu, "Pulse width reduction in diode-pumped Nd:GdVO4 laser with AO and GaAs double Q-switches," Jpn. J. Appl. Phys. 44, 3017-3021 (2005).
[CrossRef]

S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, "Passively Q-switched self-frequency-doubling Nd3+:GdCa4O(BO3)3 laser with GaAs saturable absorber," Opt. Eng. 41, 559-560 (2002).
[CrossRef]

Zhavoronkov, N. I.

Zheng, H.

Z. Hong, H. Zheng, J. Chen, and J. Ge, "Laser-diode-pumped Cr4+,Nd3+:YAG self-Q-switched laser with high repetition rate and high stability," Appl. Phys. B 73, 205-207 (2001).
[CrossRef]

Zheng, J.

S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, "Passively Q-switched self-frequency-doubling Nd3+:GdCa4O(BO3)3 laser with GaAs saturable absorber," Opt. Eng. 41, 559-560 (2002).
[CrossRef]

Zhou, S.

Appl. Opt. (2)

Appl. Phys. B (1)

Z. Hong, H. Zheng, J. Chen, and J. Ge, "Laser-diode-pumped Cr4+,Nd3+:YAG self-Q-switched laser with high repetition rate and high stability," Appl. Phys. B 73, 205-207 (2001).
[CrossRef]

Chin. J. Lasers (1)

T. Yu, J. Cui, Y. Lu, and Q. Hu, "Diode-pumped solid-state Cr4+,Nd:YAG/KTP green laser," Chin. J. Lasers B10, 321-324 (2001).

Electron. Lett. (1)

Y. F. Chen, T. M. Huang, and C. L. Wang, "Passively Q-switched diode-pumped Nd:YVO4/Cr4+:YAG single-frequency microchip laser," Electron. Lett. 33, 1880-1881 (1997).
[CrossRef]

IEEE J. Quantum Electron. (3)

Y. K. Kuo, M. F. Huang, and M. Birnbaum, "Tunable Cr4+:YSO Q-switched Cr:LiCAF laser," IEEE J. Quantum Electron. 31, 657-663 (1995).
[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]

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

Jpn. J. Appl. Phys. (1)

G. Li, S. Zhao, K. Yang, and W. Wu, "Pulse width reduction in diode-pumped Nd:GdVO4 laser with AO and GaAs double Q-switches," Jpn. J. Appl. Phys. 44, 3017-3021 (2005).
[CrossRef]

Opt. Commun. (3)

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]

J. Dong, P. Deng, "Laser performance of monolithic Cr,Nd:YAG self-Q-switched laser," Opt. Commun. 220, 425-431 (2003).
[CrossRef]

W. Chen, K. Spariosu, R. Stultz, Y. K. Kuo, M. Birnbaum, and A. V. Shestakov, "Cr4+:GSGG saturable absorber Q-switch for the ruby laser," Opt. Commun. 104, 71-74 (1993).
[CrossRef]

Opt. Eng. (1)

S. Zhao, X. Zhang, J. Zheng, L. Chen, Z. Cheng, and H. Cheng, "Passively Q-switched self-frequency-doubling Nd3+:GdCa4O(BO3)3 laser with GaAs saturable absorber," Opt. Eng. 41, 559-560 (2002).
[CrossRef]

Opt. Lett. (5)

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Figures (8)

Fig. 1.
Fig. 1.

Schematic of the experimental setup.

Fig. 2.
Fig. 2.

Temporal profile of single pulse: (a) double Q-switching; (b) self-Q-switching. Solid lines, oscilloscope traces; dotted lines, calculated results.

Fig. 3.
Fig. 3.

Pulse symmetry factor versus pump power.

Fig. 4.
Fig. 4.

Pulse width versus pump power.

Fig. 5.
Fig. 5.

Pulse repetition rate versus pump power.

Fig. 6.
Fig. 6.

Single-pulse energy versus pump power.

Fig. 7.
Fig. 7.

Pulse peak power versus pump power.

Fig. 8.
Fig. 8.

Beam size versus pump power.

Tables (1)

Tables Icon

Table 1. The parameters of the theoretical calculation.

Equations (8)

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

0 d ϕ ( r , t ) d t 2 π r d r = 0 1 t r { 2 σ n ( r , t ) l ϕ g ( r , t ) 2 σ g n s 1 ( r , t ) l ϕ g ( r , t )
2 σ e [ n s 0 n s 1 ( r , t ) ] l ϕ g ( r , t ) 2 σ + n + ( r , t ) l s ϕ s ( r , t )
2 σ 0 n 0 n + ( r , t ) l s ϕ s ( r , t ) B l s ϕ s 2 ( r , t )
δ k ϕ k 2 ( r , t ) L ϕ ( r , t ) } 2 π r d r ,
d n ( r , t ) d t = R in ( r ) σ cn ( r , t ) ϕ g ( r , t ) n ( r , t ) τ ,
d n s 1 ( r , t ) d t = n s 0 n s 1 ( r , t ) τ s σ g cn s 1 ( r , t ) ϕ g ( r , t ) ,
d n + ( r , t ) d t = c ϕ s ( r , t ) { σ 0 [ n 0 n + ( r , t ) ] σ + n + ( r , t ) } ,
δ k = ω 3 d eff 2 l k 2 c 2 ε 0 n e 2 ω n o ω n e ω ,

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