Abstract

A diode end-pumped doubly Q-switched YVO4/Nd:YVO4 laser has been realized for the first time to our knowledge by using both an electro-optic (EO) modulator and a Cr4+:YAG saturable absorber. A 3.8ns pulse width is generated by this laser under a pump power of 15W at 2kHz, which is obviously compressed in comparison with that of 8.8ns from a single actively EO Q-switched laser. Under the same conditions, peak power values of 174.7 and 93kW are also obtained. A coupled equation is given to theoretically analyze the experimental data. The experimental and theoretical results show that the doubly Q-switched laser has the advantages of a shorter pulse width and higher pulse peak power in contrast with a singly Q-switched laser.

© 2009 Optical Society of America

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    [CrossRef]
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2008 (1)

2007 (3)

2006 (2)

S. P. Ng, D. Y. Tang, A. Q. Liu, L. J. Qin, and X. L. Meng, “Short pulse passively Q-switched Nd:GdYVO4 laser using a GaAs mirror,” Opt. Commun. 259, 256-260 (2006).
[CrossRef]

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

2005 (2)

2004 (5)

Z. Li, Z. Xiong, N. Moore, G. C. Lim, W. L. Huang, and D. X. Huang, “Pulse width reduction in AO Q-switched diode-pumped Nd:YVO4 laser with GaAs coupler,” Opt. Commun. 237, 411-416 (2004).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26, 449-458 (2004).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

K. J. Yang, S. Zhao, G. Li, and H. Zhao, “A new model of laser-diode end-pumped actively Q-switched intracavity frequency doubling laser,” IEEE J. Quantum Electron. 40, 1252-1257(2004).
[CrossRef]

2003 (4)

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]

P. K. Mukhopadhyay, M. B. Aisous, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Simultaneous Q-switching and mode-locking in an intracavity frequency doubled diode-pumped Nd:YVO4/KTP green laser with Cr4+:YAG,” Opt. Commun. 222, 399-404 (2003).
[CrossRef]

H. L. Zhang, P. Shi, D. J. Li, and K. M. Du, “Diode-end-pumped electro-optically Q-switched Nd:YVO4 slab laser and second-harmonic generation,” Appl. Opt. 42, 1681-1684 (2003).
[CrossRef] [PubMed]

Y. H. Chen and Y. C. Huang, “Actively Q-switched Nd:YVO4 laser using an electro-optic periodically poled lithium niobate crystal as a laser Q-switch,” Opt. Lett. 28, 1460-1462 (2003).
[CrossRef] [PubMed]

2000 (1)

K. Chah, M. Aillerie, M. D. Fontana, and G. Malovichko, “Electro-optic properties in Fe-doped LiNbO3 crystals as a function of composition,” Opt. Commun. 176, 261-265 (2000).
[CrossRef]

1989 (1)

J. J. Degnan, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25, 214-220 (1989).
[CrossRef]

1988 (1)

1980 (1)

W. E. Schmidt, “Pulse stretching in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 16, 790-794 (1980).
[CrossRef]

Abarkan, M.

J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26, 449-458 (2004).
[CrossRef]

Aillerie, M.

K. Chah, M. Aillerie, M. D. Fontana, and G. Malovichko, “Electro-optic properties in Fe-doped LiNbO3 crystals as a function of composition,” Opt. Commun. 176, 261-265 (2000).
[CrossRef]

Aisous, M. B.

P. K. Mukhopadhyay, M. B. Aisous, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Simultaneous Q-switching and mode-locking in an intracavity frequency doubled diode-pumped Nd:YVO4/KTP green laser with Cr4+:YAG,” Opt. Commun. 222, 399-404 (2003).
[CrossRef]

An, J.

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

Andreani, L. C.

Brosi, J.

Chah, K.

K. Chah, M. Aillerie, M. D. Fontana, and G. Malovichko, “Electro-optic properties in Fe-doped LiNbO3 crystals as a function of composition,” Opt. Commun. 176, 261-265 (2000).
[CrossRef]

Chen, W.

Chen, Y. F.

Chen, Y. H.

Degnan, J. J.

J. J. Degnan, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25, 214-220 (1989).
[CrossRef]

Ding, S.

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

Dong, J.

Du, K. M.

Fontana, M. D.

J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26, 449-458 (2004).
[CrossRef]

K. Chah, M. Aillerie, M. D. Fontana, and G. Malovichko, “Electro-optic properties in Fe-doped LiNbO3 crystals as a function of composition,” Opt. Commun. 176, 261-265 (2000).
[CrossRef]

Freude, W.

Gan, A.

George, J.

P. K. Mukhopadhyay, M. B. Aisous, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Simultaneous Q-switching and mode-locking in an intracavity frequency doubled diode-pumped Nd:YVO4/KTP green laser with Cr4+:YAG,” Opt. Commun. 222, 399-404 (2003).
[CrossRef]

Gupta, P. K.

P. K. Mukhopadhyay, M. B. Aisous, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Simultaneous Q-switching and mode-locking in an intracavity frequency doubled diode-pumped Nd:YVO4/KTP green laser with Cr4+:YAG,” Opt. Commun. 222, 399-404 (2003).
[CrossRef]

He, W.

Huang, D. X.

Z. Li, Z. Xiong, N. Moore, G. C. Lim, W. L. Huang, and D. X. Huang, “Pulse width reduction in AO Q-switched diode-pumped Nd:YVO4 laser with GaAs coupler,” Opt. Commun. 237, 411-416 (2004).
[CrossRef]

Huang, K. F.

Huang, W. L.

Z. Li, Z. Xiong, N. Moore, G. C. Lim, W. L. Huang, and D. X. Huang, “Pulse width reduction in AO Q-switched diode-pumped Nd:YVO4 laser with GaAs coupler,” Opt. Commun. 237, 411-416 (2004).
[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]

Huang, Y. C.

Ju, Y.

Kaminskii, A. A.

Koos, C.

Ku, M. L.

Lai, H. C.

Leuthold, J.

Li, A.

Li, D.

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

Li, D. J.

Li, G.

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

K. J. Yang, S. Zhao, G. Li, and H. Zhao, “A new model of laser-diode end-pumped actively Q-switched intracavity frequency doubling laser,” IEEE J. Quantum Electron. 40, 1252-1257(2004).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

Li, L.

Li, M.

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

Li, T.

Z. Zhuo, T. Li, X. Li, and H. Yang, “Investigation of Nd:YVO4/YVO4 composite crystal and its laser performance pumped by a fiber coupled diode laser,” Opt. Commun. 274, 176-181 (2007).
[CrossRef]

Li, X.

Z. Zhuo, T. Li, X. Li, and H. Yang, “Investigation of Nd:YVO4/YVO4 composite crystal and its laser performance pumped by a fiber coupled diode laser,” Opt. Commun. 274, 176-181 (2007).
[CrossRef]

Li, Z.

Z. Li, Z. Xiong, N. Moore, G. C. Lim, W. L. Huang, and D. X. Huang, “Pulse width reduction in AO Q-switched diode-pumped Nd:YVO4 laser with GaAs coupler,” Opt. Commun. 237, 411-416 (2004).
[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. Li, Z. Xiong, N. Moore, G. C. Lim, W. L. Huang, and D. X. Huang, “Pulse width reduction in AO Q-switched diode-pumped Nd:YVO4 laser with GaAs coupler,” Opt. Commun. 237, 411-416 (2004).
[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]

Liu, A. Q.

S. P. Ng, D. Y. Tang, A. Q. Liu, L. J. Qin, and X. L. Meng, “Short pulse passively Q-switched Nd:GdYVO4 laser using a GaAs mirror,” Opt. Commun. 259, 256-260 (2006).
[CrossRef]

Malovichko, G.

K. Chah, M. Aillerie, M. D. Fontana, and G. Malovichko, “Electro-optic properties in Fe-doped LiNbO3 crystals as a function of composition,” Opt. Commun. 176, 261-265 (2000).
[CrossRef]

Meng, X. L.

S. P. Ng, D. Y. Tang, A. Q. Liu, L. J. Qin, and X. L. Meng, “Short pulse passively Q-switched Nd:GdYVO4 laser using a GaAs mirror,” Opt. Commun. 259, 256-260 (2006).
[CrossRef]

Moore, N.

Z. Li, Z. Xiong, N. Moore, G. C. Lim, W. L. Huang, and D. X. Huang, “Pulse width reduction in AO Q-switched diode-pumped Nd:YVO4 laser with GaAs coupler,” Opt. Commun. 237, 411-416 (2004).
[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]

Mukhopadhyay, P. K.

P. K. Mukhopadhyay, M. B. Aisous, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Simultaneous Q-switching and mode-locking in an intracavity frequency doubled diode-pumped Nd:YVO4/KTP green laser with Cr4+:YAG,” Opt. Commun. 222, 399-404 (2003).
[CrossRef]

Nathan, T. P. S.

P. K. Mukhopadhyay, M. B. Aisous, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Simultaneous Q-switching and mode-locking in an intracavity frequency doubled diode-pumped Nd:YVO4/KTP green laser with Cr4+:YAG,” Opt. Commun. 222, 399-404 (2003).
[CrossRef]

Ng, S. P.

S. P. Ng, D. Y. Tang, A. Q. Liu, L. J. Qin, and X. L. Meng, “Short pulse passively Q-switched Nd:GdYVO4 laser using a GaAs mirror,” Opt. Commun. 259, 256-260 (2006).
[CrossRef]

Qin, L. J.

S. P. Ng, D. Y. Tang, A. Q. Liu, L. J. Qin, and X. L. Meng, “Short pulse passively Q-switched Nd:GdYVO4 laser using a GaAs mirror,” Opt. Commun. 259, 256-260 (2006).
[CrossRef]

Ranganathan, K.

P. K. Mukhopadhyay, M. B. Aisous, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Simultaneous Q-switching and mode-locking in an intracavity frequency doubled diode-pumped Nd:YVO4/KTP green laser with Cr4+:YAG,” Opt. Commun. 222, 399-404 (2003).
[CrossRef]

Risk, W. P.

Salvestrini, J. P.

J. P. Salvestrini, M. Abarkan, and M. D. Fontana, “Comparative study of nonlinear optical crystals for electro-optic Q-switching of laser resonators,” Opt. Mater. 26, 449-458 (2004).
[CrossRef]

Schmidt, W. E.

W. E. Schmidt, “Pulse stretching in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 16, 790-794 (1980).
[CrossRef]

Sharma, S. K.

P. K. Mukhopadhyay, M. B. Aisous, K. Ranganathan, S. K. Sharma, P. K. Gupta, J. George, and T. P. S. Nathan, “Simultaneous Q-switching and mode-locking in an intracavity frequency doubled diode-pumped Nd:YVO4/KTP green laser with Cr4+:YAG,” Opt. Commun. 222, 399-404 (2003).
[CrossRef]

Shi, P.

Su, K. W.

Sun, Y.

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

Tang, D. Y.

S. P. Ng, D. Y. Tang, A. Q. Liu, L. J. Qin, and X. L. Meng, “Short pulse passively Q-switched Nd:GdYVO4 laser using a GaAs mirror,” Opt. Commun. 259, 256-260 (2006).
[CrossRef]

Ueda, K.

Waldow, M.

Wang, J.

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

Wang, Y.

Xiong, Z.

Z. Li, Z. Xiong, N. Moore, G. C. Lim, W. L. Huang, and D. X. Huang, “Pulse width reduction in AO Q-switched diode-pumped Nd:YVO4 laser with GaAs coupler,” Opt. Commun. 237, 411-416 (2004).
[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]

Yang, H.

Z. Zhuo, T. Li, X. Li, and H. Yang, “Investigation of Nd:YVO4/YVO4 composite crystal and its laser performance pumped by a fiber coupled diode laser,” Opt. Commun. 274, 176-181 (2007).
[CrossRef]

Yang, K.

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

Yang, K. J.

K. J. Yang, S. Zhao, G. Li, and H. Zhao, “A new model of laser-diode end-pumped actively Q-switched intracavity frequency doubling laser,” IEEE J. Quantum Electron. 40, 1252-1257(2004).
[CrossRef]

Yao, B.

Zhang, H. L.

Zhao, H.

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

K. J. Yang, S. Zhao, G. Li, and H. Zhao, “A new model of laser-diode end-pumped actively Q-switched intracavity frequency doubling laser,” IEEE J. Quantum Electron. 40, 1252-1257(2004).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

Zhao, J.

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

Zhao, S.

S. Zhao, J. Zhao, G. Li, K. Yang, Y. Sun, D. Li, J. An, J. Wang, and M. Li, “Doubly Q-switched laser with electric-optic modulator and GaAs saturable absorber,” Laser Phys. Lett. 3, 471-473 (2006).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

K. J. Yang, S. Zhao, G. Li, and H. Zhao, “A new model of laser-diode end-pumped actively Q-switched intracavity frequency doubling laser,” IEEE J. Quantum Electron. 40, 1252-1257(2004).
[CrossRef]

Zhuo, Z.

Z. Zhuo, T. Li, X. Li, and H. Yang, “Investigation of Nd:YVO4/YVO4 composite crystal and its laser performance pumped by a fiber coupled diode laser,” Opt. Commun. 274, 176-181 (2007).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (4)

J. J. Degnan, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25, 214-220 (1989).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

G. Li, S. Zhao, H. Zhao, K. Yang, and S. Ding, “Rate equations and solutions of a laser-diode end-pumped passively Q-switched intracavity doubling laser by taking into account intracavity laser spatial distribution,” Opt. Commun. 234, 321-328 (2004).
[CrossRef]

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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Cavity schematic diagram of doubly Q-switched laser.

Fig. 2
Fig. 2

Beam size versus absorbed pump power.

Fig. 3
Fig. 3

Average output power versus incident pump power.

Fig. 4
Fig. 4

Pulse width as a function of pump power with different repetitive rate. Solid curves, calculated results; symbols, experimental data.

Fig. 5
Fig. 5

Temporal profile for two kinds of lasers. Solid curves, oscilloscope traces; dashed curves, calculated results.

Fig. 6
Fig. 6

Peak power versus incident pump power. Solid curves, calculated results; symbols, experimental data.

Fig. 7
Fig. 7

Dependence of the thermally induced diffraction losses on the absorbed pump power.

Tables (1)

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Table 1 Parameters of the Theoretical Calculation

Equations (15)

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ϕ ( r , t ) = ϕ ( 0 , t ) exp ( 2 r 2 w l 2 ) ,
ϕ i ( r , t ) = w l 2 w i 2 ϕ ( 0 , t ) exp ( 2 r 2 w i 2 ) ( i = g , e , s ) .
R in ( r , z , t ) = 2 α P in h ν p π w p 2 ( z ) exp [ 2 r 2 w P 2 ( z ) ] exp [ α z ] ,
w P ( z ) = w P 0 { 1 + [ λ ( z z 0 ) π w P 0 ] 2 } 1 / 2 ,
0 d ϕ ( r , t ) d t 2 π r d r = 1 τ r { 0 2 σ ϕ g ( r , t ) 0 l c n ( r , z , t ) d z 2 σ g n s 1 ( r , t ) l s ϕ s ( r , t ) 2 σ e [ n s 0 n s 1 ( r , t ) ] l s ϕ s ( r , t ) ln ( 1 1 δ e ( t ) ) ϕ e ( r , t ) ( L + δ T ) ϕ ( r , t ) } 2 π r d r ,
d n ( r , z , t ) d t = R in σ c n ( r , z , t ) ϕ g ( r , t ) n ( r , z , t ) τ ,
d n s 1 ( r , t ) d t = n s 0 n s 1 ( r , t ) τ s σ g c n s 1 ( r , t ) ϕ s ( r , t ) ,
δ e ( t ) = cos 2 ( π 2 V ( t ) V λ / 4 ) ,
V ( t ) = { 0 t 0 V λ / 4 t > 0 .
n ( r , z , 0 ) = n ( 0 , 0 , 0 ) exp ( 2 r 2 w P 2 ( z ) ) exp ( α z ) ,
n s 1 = n s 0 ,
n ( 0 , 0 , 0 ) = 2 α P in h ν P π w P 2 ( 0 ) [ f e + 1 / ( f e τ ) ] ,
0 d ϕ ( r , t ) d t 2 π r d r = 0 1 t r [ 2 σ ϕ g ( r , t ) 0 l n ( r , z , t ) d z ln ( 1 1 δ e ) ϕ e ( r , t ) ( L + δ T ) ϕ ( r , t ) ] 2 π r d r ,
d n ( r , z , t ) d t = R in σ c n ( r , t ) ϕ g ( r , t ) n ( r , z , t ) τ .
P = π w l 2 h ν P 4 σ τ r [ ln ( 1 R ) + L ] ln ( 1 R ) ϕ m ,

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