Abstract

A set of coupled rate equations for diode-pumped Q-switched and mode-locked laser with electro-optic (EO) modulator and middle semiconductor saturable absorber mirror under the Gaussian spatial distribution approximation are given. The numerically simulated results of these equations show that the pulse width of the Q-switched envelope are related to the repetition rate of EO, the stimulated emission section of the gain medium, the pump power and so on. When the pulse width of the Q-switched envelope is shorter than the cavity roundtrip transmit time, i.e., the interval of two neighboring mode-locking pulses, there is only one mode-locked pulse lying in a Q-switched envelope and its repetition rate depends on that of EO. This means that single mode-locking pulses with low repetition rate, subnanosecond duration, high peak power, and high stability are generated. The simulated results are consistent with the experimental values.

© 2013 Optical Society of America

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  1. S. Lévêque-Fort, D. N. Papadopoulos, S. Forget, F. Balembois, and P. Georges, “Fluorescence lifetime imaging with low-repetition-rate passively mode-locked diode-pumped Nd:YVO4 oscillator,” Opt. Lett. 30, 168–170 (2005).
    [CrossRef]
  2. P. G. Antal and R. Szipocs, “Tunable, low-repetition-rate, cost-efficient femtosecond Ti:sapphire laser for nonlinear microscopy,” Appl. Phys. B 107, 17–22 (2012).
    [CrossRef]
  3. H. Sayinc, D. Mortag, D. Wandt, J. Neumann, and D. Kracht, “Sub-100 fs pulses from a low repetition rate Yb-doped fiber laser,” Opt. Express 17, 5731–5735 (2009).
    [CrossRef]
  4. C. Theobald, M. Weitz, R. Knappe, R. Wallenstein, and J. A. L’Huillier, “Stable Q-switch mode-locking of Nd:YVO4lasers with a semiconductor saturable absorber,” Appl. Phys. B 92, 1–3 (2008).
    [CrossRef]
  5. X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
    [CrossRef]
  6. R. Lan, L. Pan, I. Utkin, Q. Ren, H. Zhang, Z. Wang, and R. Fedosejevs, “Passively Q-switched Yb3+:NaY(WO4)2 laser with GaAs saturable absorber,” Opt. Express 18, 4000–4005 (2010).
    [CrossRef]
  7. U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
    [CrossRef]
  8. A. Steinmetz, D. Nodop, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “2 MHz repetition rate, 200 ps pulse duration from a monolithic, passively Q-switched microchip laser,” Appl. Phys. B 97, 317–320 (2009).
    [CrossRef]
  9. A. Steinmetz, D. Nodop, A. Martin, J. Limpert, and A. Tünnermann, “Reduction of timing jitter in passively Q-switched microchip lasers using self-injection seeding,” Opt. Lett. 35, 2885–2887 (2010).
    [CrossRef]
  10. J.-H. Lin, H.-R. Chen, H.-H. Hsu, M.-D. Wei, K.-H. Lin, and W.-F. Hsie, “Stable Q-switched mode-locked Nd3+:LuVO4laser by Cr4+:YAG crystal,” Opt. Express 16, 16538–16545 (2008).
  11. J.-H. Lin, J.-L. Jhu, S.-S. Jyu, T.-C. Lin, and Y. Lai, “Characteristics of a low repetition rate passively mode-locked Yb-doped fiber laser in an all-normal dispersion cavity,” Laser Phys. 23, 025103 (2013).
    [CrossRef]
  12. K.-H. Lin, J.-H. Lin, and C.-C. Chen, “Switchable mode-locking states in an all-fiber all–normal-dispersion ytterbium-doped laser,” Laser Phys. 20, 1984–1989 (2010).
    [CrossRef]
  13. S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
    [CrossRef]
  14. T. Li, S. Zhao, Z. Zhuo, K. Yang, G. Li, and D. Li, “Dual-loss-modulated Q-switched and mode-locked YVO4/Nd:YVO4/KTP green laser with EO and Cr4+:YAG saturable absorber,” Opt. Express 18, 10315–10322 (2010).
    [CrossRef]
  15. P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
    [CrossRef]
  16. Y. F. Chen, J. L. Lee, H. D. Hsieh, and S. W. Tsai, “Analysis of passively Q-switched lasers with simultaneous mode-locking,” IEEE J. Quantum Electron. 38, 312–317 (2002).
    [CrossRef]
  17. K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
    [CrossRef]
  18. S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, G. Zhang, and K. Cheng, “Numerical simulation of dual-loss-modulated Q-switched and mode-locked laser with AO and Cr4+:YAG saturable absorber,” Appl. Opt. 49, 1802–1808 (2010).
    [CrossRef]
  19. Y. G. Wang, X. Y. Ma, Y. Liu, L. Q. Sun, and Q. Tian, “Passive mode-locked Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Chin. J. Laser 32, 1034–1035 (2005).
  20. Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).
  21. C. Hönninger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, “Q-switching stability limits of continuous-wave passive mode locking,” J. Opt. Soc. Am. B 16, 46–56(1999).
    [CrossRef]

2013

J.-H. Lin, J.-L. Jhu, S.-S. Jyu, T.-C. Lin, and Y. Lai, “Characteristics of a low repetition rate passively mode-locked Yb-doped fiber laser in an all-normal dispersion cavity,” Laser Phys. 23, 025103 (2013).
[CrossRef]

2012

P. G. Antal and R. Szipocs, “Tunable, low-repetition-rate, cost-efficient femtosecond Ti:sapphire laser for nonlinear microscopy,” Appl. Phys. B 107, 17–22 (2012).
[CrossRef]

2010

2009

A. Steinmetz, D. Nodop, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “2 MHz repetition rate, 200 ps pulse duration from a monolithic, passively Q-switched microchip laser,” Appl. Phys. B 97, 317–320 (2009).
[CrossRef]

H. Sayinc, D. Mortag, D. Wandt, J. Neumann, and D. Kracht, “Sub-100 fs pulses from a low repetition rate Yb-doped fiber laser,” Opt. Express 17, 5731–5735 (2009).
[CrossRef]

2008

C. Theobald, M. Weitz, R. Knappe, R. Wallenstein, and J. A. L’Huillier, “Stable Q-switch mode-locking of Nd:YVO4lasers with a semiconductor saturable absorber,” Appl. Phys. B 92, 1–3 (2008).
[CrossRef]

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

J.-H. Lin, H.-R. Chen, H.-H. Hsu, M.-D. Wei, K.-H. Lin, and W.-F. Hsie, “Stable Q-switched mode-locked Nd3+:LuVO4laser by Cr4+:YAG crystal,” Opt. Express 16, 16538–16545 (2008).

2007

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

2006

K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
[CrossRef]

2005

Y. G. Wang, X. Y. Ma, Y. Liu, L. Q. Sun, and Q. Tian, “Passive mode-locked Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Chin. J. Laser 32, 1034–1035 (2005).

S. Lévêque-Fort, D. N. Papadopoulos, S. Forget, F. Balembois, and P. Georges, “Fluorescence lifetime imaging with low-repetition-rate passively mode-locked diode-pumped Nd:YVO4 oscillator,” Opt. Lett. 30, 168–170 (2005).
[CrossRef]

2004

P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
[CrossRef]

2002

Y. F. Chen, J. L. Lee, H. D. Hsieh, and S. W. Tsai, “Analysis of passively Q-switched lasers with simultaneous mode-locking,” IEEE J. Quantum Electron. 38, 312–317 (2002).
[CrossRef]

1999

1996

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Alsousb, M. B.

P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
[CrossRef]

An, J.

K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
[CrossRef]

Antal, P. G.

P. G. Antal and R. Szipocs, “Tunable, low-repetition-rate, cost-efficient femtosecond Ti:sapphire laser for nonlinear microscopy,” Appl. Phys. B 107, 17–22 (2012).
[CrossRef]

Aus der Au, J.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Balembois, F.

Braun, B.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Chen, C.-C.

K.-H. Lin, J.-H. Lin, and C.-C. Chen, “Switchable mode-locking states in an all-fiber all–normal-dispersion ytterbium-doped laser,” Laser Phys. 20, 1984–1989 (2010).
[CrossRef]

Chen, H.-R.

Chen, Y. F.

Y. F. Chen, J. L. Lee, H. D. Hsieh, and S. W. Tsai, “Analysis of passively Q-switched lasers with simultaneous mode-locking,” IEEE J. Quantum Electron. 38, 312–317 (2002).
[CrossRef]

Cheng, K.

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, G. Zhang, and K. Cheng, “Numerical simulation of dual-loss-modulated Q-switched and mode-locked laser with AO and Cr4+:YAG saturable absorber,” Appl. Opt. 49, 1802–1808 (2010).
[CrossRef]

Du, K.

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

Fedosejevs, R.

Fluck, R.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Forget, S.

Ge, H.

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

George, J.

P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
[CrossRef]

Georges, P.

Gupta, P. K.

P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
[CrossRef]

H¨onninger, C.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Haas, C. R.

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

Hohmuth, R.

A. Steinmetz, D. Nodop, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “2 MHz repetition rate, 200 ps pulse duration from a monolithic, passively Q-switched microchip laser,” Appl. Phys. B 97, 317–320 (2009).
[CrossRef]

Hönninger, C.

Hsie, W.-F.

Hsieh, H. D.

Y. F. Chen, J. L. Lee, H. D. Hsieh, and S. W. Tsai, “Analysis of passively Q-switched lasers with simultaneous mode-locking,” IEEE J. Quantum Electron. 38, 312–317 (2002).
[CrossRef]

Hsu, H.-H.

Hu, P.

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

Jhu, J.-L.

J.-H. Lin, J.-L. Jhu, S.-S. Jyu, T.-C. Lin, and Y. Lai, “Characteristics of a low repetition rate passively mode-locked Yb-doped fiber laser in an all-normal dispersion cavity,” Laser Phys. 23, 025103 (2013).
[CrossRef]

Jung, I. D.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Jyu, S.-S.

J.-H. Lin, J.-L. Jhu, S.-S. Jyu, T.-C. Lin, and Y. Lai, “Characteristics of a low repetition rate passively mode-locked Yb-doped fiber laser in an all-normal dispersion cavity,” Laser Phys. 23, 025103 (2013).
[CrossRef]

Kärtner, F. X.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Keller, U.

C. Hönninger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, “Q-switching stability limits of continuous-wave passive mode locking,” J. Opt. Soc. Am. B 16, 46–56(1999).
[CrossRef]

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Knappe, R.

C. Theobald, M. Weitz, R. Knappe, R. Wallenstein, and J. A. L’Huillier, “Stable Q-switch mode-locking of Nd:YVO4lasers with a semiconductor saturable absorber,” Appl. Phys. B 92, 1–3 (2008).
[CrossRef]

Kopf, D.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Kracht, D.

L’Huillier, J. A.

C. Theobald, M. Weitz, R. Knappe, R. Wallenstein, and J. A. L’Huillier, “Stable Q-switch mode-locking of Nd:YVO4lasers with a semiconductor saturable absorber,” Appl. Phys. B 92, 1–3 (2008).
[CrossRef]

Lai, Y.

J.-H. Lin, J.-L. Jhu, S.-S. Jyu, T.-C. Lin, and Y. Lai, “Characteristics of a low repetition rate passively mode-locked Yb-doped fiber laser in an all-normal dispersion cavity,” Laser Phys. 23, 025103 (2013).
[CrossRef]

Lan, R.

Lee, J. L.

Y. F. Chen, J. L. Lee, H. D. Hsieh, and S. W. Tsai, “Analysis of passively Q-switched lasers with simultaneous mode-locking,” IEEE J. Quantum Electron. 38, 312–317 (2002).
[CrossRef]

Lévêque-Fort, S.

Li, D.

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

T. Li, S. Zhao, Z. Zhuo, K. Yang, G. Li, and D. Li, “Dual-loss-modulated Q-switched and mode-locked YVO4/Nd:YVO4/KTP green laser with EO and Cr4+:YAG saturable absorber,” Opt. Express 18, 10315–10322 (2010).
[CrossRef]

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, G. Zhang, and K. Cheng, “Numerical simulation of dual-loss-modulated Q-switched and mode-locked laser with AO and Cr4+:YAG saturable absorber,” Appl. Opt. 49, 1802–1808 (2010).
[CrossRef]

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
[CrossRef]

Li, G.

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, G. Zhang, and K. Cheng, “Numerical simulation of dual-loss-modulated Q-switched and mode-locked laser with AO and Cr4+:YAG saturable absorber,” Appl. Opt. 49, 1802–1808 (2010).
[CrossRef]

T. Li, S. Zhao, Z. Zhuo, K. Yang, G. Li, and D. Li, “Dual-loss-modulated Q-switched and mode-locked YVO4/Nd:YVO4/KTP green laser with EO and Cr4+:YAG saturable absorber,” Opt. Express 18, 10315–10322 (2010).
[CrossRef]

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
[CrossRef]

Li, M.

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, G. Zhang, and K. Cheng, “Numerical simulation of dual-loss-modulated Q-switched and mode-locked laser with AO and Cr4+:YAG saturable absorber,” Appl. Opt. 49, 1802–1808 (2010).
[CrossRef]

K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
[CrossRef]

Li, T.

Li, Y.

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, G. Zhang, and K. Cheng, “Numerical simulation of dual-loss-modulated Q-switched and mode-locked laser with AO and Cr4+:YAG saturable absorber,” Appl. Opt. 49, 1802–1808 (2010).
[CrossRef]

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

Limpert, J.

A. Steinmetz, D. Nodop, A. Martin, J. Limpert, and A. Tünnermann, “Reduction of timing jitter in passively Q-switched microchip lasers using self-injection seeding,” Opt. Lett. 35, 2885–2887 (2010).
[CrossRef]

A. Steinmetz, D. Nodop, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “2 MHz repetition rate, 200 ps pulse duration from a monolithic, passively Q-switched microchip laser,” Appl. Phys. B 97, 317–320 (2009).
[CrossRef]

Lin, J.-H.

J.-H. Lin, J.-L. Jhu, S.-S. Jyu, T.-C. Lin, and Y. Lai, “Characteristics of a low repetition rate passively mode-locked Yb-doped fiber laser in an all-normal dispersion cavity,” Laser Phys. 23, 025103 (2013).
[CrossRef]

K.-H. Lin, J.-H. Lin, and C.-C. Chen, “Switchable mode-locking states in an all-fiber all–normal-dispersion ytterbium-doped laser,” Laser Phys. 20, 1984–1989 (2010).
[CrossRef]

J.-H. Lin, H.-R. Chen, H.-H. Hsu, M.-D. Wei, K.-H. Lin, and W.-F. Hsie, “Stable Q-switched mode-locked Nd3+:LuVO4laser by Cr4+:YAG crystal,” Opt. Express 16, 16538–16545 (2008).

Lin, K.-H.

K.-H. Lin, J.-H. Lin, and C.-C. Chen, “Switchable mode-locking states in an all-fiber all–normal-dispersion ytterbium-doped laser,” Laser Phys. 20, 1984–1989 (2010).
[CrossRef]

J.-H. Lin, H.-R. Chen, H.-H. Hsu, M.-D. Wei, K.-H. Lin, and W.-F. Hsie, “Stable Q-switched mode-locked Nd3+:LuVO4laser by Cr4+:YAG crystal,” Opt. Express 16, 16538–16545 (2008).

Lin, T.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Lin, T.-C.

J.-H. Lin, J.-L. Jhu, S.-S. Jyu, T.-C. Lin, and Y. Lai, “Characteristics of a low repetition rate passively mode-locked Yb-doped fiber laser in an all-normal dispersion cavity,” Laser Phys. 23, 025103 (2013).
[CrossRef]

Liu, X.

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

Liu, Y.

Y. G. Wang, X. Y. Ma, Y. Liu, L. Q. Sun, and Q. Tian, “Passive mode-locked Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Chin. J. Laser 32, 1034–1035 (2005).

Ma, X. Y.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Y. G. Wang, X. Y. Ma, Y. Liu, L. Q. Sun, and Q. Tian, “Passive mode-locked Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Chin. J. Laser 32, 1034–1035 (2005).

Martin, A.

Matuschek, N.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Morier-Genoud, F.

Mortag, D.

Moser, M.

Mukhopadhyay, P. K.

P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
[CrossRef]

Nathan, T. P. S.

P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
[CrossRef]

Neumann, J.

Nodop, D.

A. Steinmetz, D. Nodop, A. Martin, J. Limpert, and A. Tünnermann, “Reduction of timing jitter in passively Q-switched microchip lasers using self-injection seeding,” Opt. Lett. 35, 2885–2887 (2010).
[CrossRef]

A. Steinmetz, D. Nodop, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “2 MHz repetition rate, 200 ps pulse duration from a monolithic, passively Q-switched microchip laser,” Appl. Phys. B 97, 317–320 (2009).
[CrossRef]

Pan, L.

Papadopoulos, D. N.

Paschotta, R.

Peng, J. Y.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Qian, L. S.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Ranganathan, K.

P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
[CrossRef]

Ren, Q.

Richter, W.

A. Steinmetz, D. Nodop, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “2 MHz repetition rate, 200 ps pulse duration from a monolithic, passively Q-switched microchip laser,” Appl. Phys. B 97, 317–320 (2009).
[CrossRef]

Sayinc, H.

Schell, A.

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

Sharma, S. K.

P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
[CrossRef]

Shi, P.

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

Steinmetz, A.

A. Steinmetz, D. Nodop, A. Martin, J. Limpert, and A. Tünnermann, “Reduction of timing jitter in passively Q-switched microchip lasers using self-injection seeding,” Opt. Lett. 35, 2885–2887 (2010).
[CrossRef]

A. Steinmetz, D. Nodop, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “2 MHz repetition rate, 200 ps pulse duration from a monolithic, passively Q-switched microchip laser,” Appl. Phys. B 97, 317–320 (2009).
[CrossRef]

Sun, L. Q.

Y. G. Wang, X. Y. Ma, Y. Liu, L. Q. Sun, and Q. Tian, “Passive mode-locked Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Chin. J. Laser 32, 1034–1035 (2005).

Szipocs, R.

P. G. Antal and R. Szipocs, “Tunable, low-repetition-rate, cost-efficient femtosecond Ti:sapphire laser for nonlinear microscopy,” Appl. Phys. B 107, 17–22 (2012).
[CrossRef]

Tan, H. M.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Theobald, C.

C. Theobald, M. Weitz, R. Knappe, R. Wallenstein, and J. A. L’Huillier, “Stable Q-switch mode-locking of Nd:YVO4lasers with a semiconductor saturable absorber,” Appl. Phys. B 92, 1–3 (2008).
[CrossRef]

Tian, Q.

Y. G. Wang, X. Y. Ma, Y. Liu, L. Q. Sun, and Q. Tian, “Passive mode-locked Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Chin. J. Laser 32, 1034–1035 (2005).

Tsai, S. W.

Y. F. Chen, J. L. Lee, H. D. Hsieh, and S. W. Tsai, “Analysis of passively Q-switched lasers with simultaneous mode-locking,” IEEE J. Quantum Electron. 38, 312–317 (2002).
[CrossRef]

Tünnermann, A.

A. Steinmetz, D. Nodop, A. Martin, J. Limpert, and A. Tünnermann, “Reduction of timing jitter in passively Q-switched microchip lasers using self-injection seeding,” Opt. Lett. 35, 2885–2887 (2010).
[CrossRef]

A. Steinmetz, D. Nodop, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “2 MHz repetition rate, 200 ps pulse duration from a monolithic, passively Q-switched microchip laser,” Appl. Phys. B 97, 317–320 (2009).
[CrossRef]

Utkin, I.

Wallenstein, R.

C. Theobald, M. Weitz, R. Knappe, R. Wallenstein, and J. A. L’Huillier, “Stable Q-switch mode-locking of Nd:YVO4lasers with a semiconductor saturable absorber,” Appl. Phys. B 92, 1–3 (2008).
[CrossRef]

Wandt, D.

Wang, J.

K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
[CrossRef]

Wang, Q. Y.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Wang, Y. G.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Y. G. Wang, X. Y. Ma, Y. Liu, L. Q. Sun, and Q. Tian, “Passive mode-locked Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Chin. J. Laser 32, 1034–1035 (2005).

Wang, Z.

Wei, M.-D.

Weingarten, K. J.

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

Weitz, M.

C. Theobald, M. Weitz, R. Knappe, R. Wallenstein, and J. A. L’Huillier, “Stable Q-switch mode-locking of Nd:YVO4lasers with a semiconductor saturable absorber,” Appl. Phys. B 92, 1–3 (2008).
[CrossRef]

Wu, N.

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

Yang, K.

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, G. Zhang, and K. Cheng, “Numerical simulation of dual-loss-modulated Q-switched and mode-locked laser with AO and Cr4+:YAG saturable absorber,” Appl. Opt. 49, 1802–1808 (2010).
[CrossRef]

T. Li, S. Zhao, Z. Zhuo, K. Yang, G. Li, and D. Li, “Dual-loss-modulated Q-switched and mode-locked YVO4/Nd:YVO4/KTP green laser with EO and Cr4+:YAG saturable absorber,” Opt. Express 18, 10315–10322 (2010).
[CrossRef]

K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
[CrossRef]

Zhai, L.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Zhang, G.

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, G. Zhang, and K. Cheng, “Numerical simulation of dual-loss-modulated Q-switched and mode-locked laser with AO and Cr4+:YAG saturable absorber,” Appl. Opt. 49, 1802–1808 (2010).
[CrossRef]

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

Zhang, H.

Zhang, Z. G.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Zhao, S.

T. Li, S. Zhao, Z. Zhuo, K. Yang, G. Li, and D. Li, “Dual-loss-modulated Q-switched and mode-locked YVO4/Nd:YVO4/KTP green laser with EO and Cr4+:YAG saturable absorber,” Opt. Express 18, 10315–10322 (2010).
[CrossRef]

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, G. Zhang, and K. Cheng, “Numerical simulation of dual-loss-modulated Q-switched and mode-locked laser with AO and Cr4+:YAG saturable absorber,” Appl. Opt. 49, 1802–1808 (2010).
[CrossRef]

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
[CrossRef]

Zhuo, Z.

Acta Photonica Sin.

Y. G. Wang, J. Y. Peng, H. M. Tan, L. S. Qian, L. Zhai, Z. G. Zhang, Q. Y. Wang, T. Lin, and X. Y. Ma, “Study for characteristics of passively Q—switched Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Acta Photonica Sin. 36, 401–404 (2007).

Appl. Opt.

Appl. Phys. B

P. K. Mukhopadhyay, M. B. Alsousb, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Analysis of laser-diode end-pumped intra-cavity frequency-doubled, passively Q-switched and mode-locked Nd:YVO4 laser,” Appl. Phys. B 79, 713–720 (2004).
[CrossRef]

P. G. Antal and R. Szipocs, “Tunable, low-repetition-rate, cost-efficient femtosecond Ti:sapphire laser for nonlinear microscopy,” Appl. Phys. B 107, 17–22 (2012).
[CrossRef]

C. Theobald, M. Weitz, R. Knappe, R. Wallenstein, and J. A. L’Huillier, “Stable Q-switch mode-locking of Nd:YVO4lasers with a semiconductor saturable absorber,” Appl. Phys. B 92, 1–3 (2008).
[CrossRef]

A. Steinmetz, D. Nodop, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “2 MHz repetition rate, 200 ps pulse duration from a monolithic, passively Q-switched microchip laser,” Appl. Phys. B 97, 317–320 (2009).
[CrossRef]

Chin. J. Laser

Y. G. Wang, X. Y. Ma, Y. Liu, L. Q. Sun, and Q. Tian, “Passive mode-locked Nd:YVO4 laser with central semiconductor saturable absorption mirror,” Chin. J. Laser 32, 1034–1035 (2005).

IEEE J. Quantum Electron.

Y. F. Chen, J. L. Lee, H. D. Hsieh, and S. W. Tsai, “Analysis of passively Q-switched lasers with simultaneous mode-locking,” IEEE J. Quantum Electron. 38, 312–317 (2002).
[CrossRef]

K. Yang, S. Zhao, G. Li, M. Li, D. Li, J. Wang, and J. An, “Diode pumped passively Q-switched mode-locked c-cut Nd:GdVO4/KTP green laser with a GaAs wafer,” IEEE J. Quantum Electron. 42, 683–689 (2006).
[CrossRef]

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. H¨onninger, N. Matuschek, and J. Aus der Au, “Semiconductor saturable absorber mirrors (SESAM’s) for femtosecond to nanosecond pulse generation in solid-state lasers,” IEEE J. Quantum Electron. 2, 435–453 (1996).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys.

J.-H. Lin, J.-L. Jhu, S.-S. Jyu, T.-C. Lin, and Y. Lai, “Characteristics of a low repetition rate passively mode-locked Yb-doped fiber laser in an all-normal dispersion cavity,” Laser Phys. 23, 025103 (2013).
[CrossRef]

K.-H. Lin, J.-H. Lin, and C.-C. Chen, “Switchable mode-locking states in an all-fiber all–normal-dispersion ytterbium-doped laser,” Laser Phys. 20, 1984–1989 (2010).
[CrossRef]

Laser Phys. Lett.

S. Zhao, G. Li, D. Li, K. Yang, Y. Li, M. Li, T. Li, K. Cheng, G. Zhang, and H. Ge, “Dual-loss-modulated Q-switched and mode-locked Nd:GdVO4 laser with AO and Cr4+:YAG saturable absorber,” Laser Phys. Lett. 7, 29–33 (2010).
[CrossRef]

Opt. Commun.

X. Liu, P. Hu, D. Li, A. Schell, P. Shi, C. R. Haas, N. Wu, and K. Du, “Electro-optically Q-switched lasers with polarization output coupling and their pulse control,” Opt. Commun. 281, 303–309 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

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Figures (7)

Fig. 1.
Fig. 1.

Pulse width of the Q-switched envelope versus pump power for different fp of EO.

Fig. 2.
Fig. 2.

Number of the mode-locking pulse in a Q-switched envelope at fp=1KHz. (a) Pin=5.31W; (b) Pin=7.69W; (c) Pin=9.81W.

Fig. 3.
Fig. 3.

Pulse width of the Q-switched envelope versus pump power for two different R.

Fig. 4.
Fig. 4.

Pulse widths of the Q-switched envelope versus pump power for different values of σ.

Fig. 5.
Fig. 5.

Experimental setup of doubly QML YVO4/NdYVO4 laser.

Fig. 6.
Fig. 6.

Pulse width of the Q-switched envelope versus pump power. Scattered points: experimental results. Solid line: simulated values.

Fig. 7.
Fig. 7.

(a) Temporal profile of the QML pulse in a Q-switched envelope. (b) Oscilloscope traces of the pulse trains.

Tables (1)

Tables Icon

Table 1. Parameters of Numerical Simulation

Equations (10)

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

f(t)=12σcτpsech2(tτp);cσf(t)dt=1,
ϕ(r,t)=k=0ϕk(r,t)=k=0Φkf(ttk)exp(2r2wl2)=ϕ(0,t)exp(2r2wl2),
δkp=ΔR(1eEkp/Esat)(EsatEkp)=ΔR(1eSk)Sk,
ni=RinτG[1exp(1fpτG)],
Φk=Φk1exp{2πwl20[2σn(r,tk)lwl2wG2exp(2r2wG2)2σ+n+(r,tk)lAwl2wA2exp(2r2wA2)2σ0[n0n+(r,tk)]lAwl2wA2exp(2r2wA2)BlAwl4wA4exp(4r2wA2)Φk1δkpwl2wA2exp(2r2wA2)[δe+L+ln(1R)]exp(2r2wl2)]2πrdr},
dn(r,t)dt=Rin(r)σcn(r,t)ϕG(r,t)n(r,t)τa,
dn+(r,t)dt=σ0c[n0n+(r,t)]ϕA(r,t)σ+cn+(r,t)ϕA(r,t),
n(r,tk)=exp(tkτa)0k1exp[wl2wG2exp(2r2wG2)Φm]×{Rin(r)exp(tkτa)0tkm=0k1exp[wl2wG2exp(2r2wG2)Φm]dt+niexp(2r2wp2)},
n+(r,tk)=[m=0k1exp[wl2wA2exp(2r2wA2)Φm]](σ++σ0)/σ×{0k[m=0k1exp[wl2wA2exp(2r2wA2)Φm]](σ++σ0)/σ×σ0n02στp×m=0k1Φmwl2wA2exp(2r2wA2)sech2(tmtrτp)dt+n0}.
P(t)=1A(ln1R)0[chνAϕ(r,t)2]2πrdr=hvA4στp(ln1R)k=0Φksech2(ttkτp),

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