Abstract

A high power passively Q-switched dual wavelength Yb fiber laser using a Cr4+:YAG saturable absorber has been realized. Two wavelengths centered at 1040 nm and 1070 nm are generated directly from the cladding pumped Yb doped fiber laser. The pulse trains exhibit regions of stability and instability dependent on the pump power. At a pump power of 7.8 W, 1040 nm and 1070 nm pulses are generated alternatively, with pulse durations of 105 ns, pulse-repetition rates of 32 KHz and average pulse energies of 56 μJ and 47 μJ, respectively. A theoretical model is developed to simulate the two-wavelength Q-switched operation, which gives qualitative agreement with the experimental observations.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2007 (6)

W. Guan and J. R. Marciante, "Dual-frequency operation in a short-cavity Ytterbium-doped fiber laser," IEEE Photon. Technol. Lett. 19, 261-263 (2007).
[CrossRef]

L. R. Chen and X. J. Gu, "Dual-wavelength Yb-doped fiber laser stabilized through four-wave mixing," Opt. Express 15, 5083-5088 (2007).
[CrossRef] [PubMed]

C. Tu, W. Guo, Y. Li, S. Zhang, and F. Lu, "Stable multiwavelength and passively mode-locked Yb-doped fiber laser based on nonlinear polarization rotation," Opt. Commun. 280, 448-452 (2007).
[CrossRef]

J. Y. Huang, H. C. Liang, K. W. Su, and Y. F. Chen, "High power passively Q-switched ytterbium fiber laser with Cr4+:YAG as a saturable absorber," Opt. Express 15, 473-479 (2007).
[CrossRef] [PubMed]

L. Pan, I. Utkin, and R. Fedosejevs, "Passively Q-switched ytterbium doped double-clad fiber laser with a Cr4+:YAG saturable absorber," IEEE Photon. Technol. Lett. 19, 1979-1981 (2007).
[CrossRef]

H. Ridderbusch and T. Graf, "Saturation of 1047- and 1064nm absorption in Cr4+:YAG crystals," IEEE J. Quantum Electron. 43, 168-173 (2007).
[CrossRef]

2006 (4)

2005 (1)

Y. M. Huo and P. K. Cheo, "Modeling of passively Q-switched Er3+/Yb3+-codoped clad-pumped fiber lasers," IEEE J. Sel. Top Quantum Electron. 11, 658-665 (2005).
[CrossRef]

2004 (3)

A. Piper, A. Malinowski, K. Furusawa, D. J. Richardson, "High-power, high- brightness, mJ Q-switched ytterbium-doped fibre laser," Electron. Lett. 40, 928-929 (2004).
[CrossRef]

X. Feng, Y. Liu, S. Fu, S. Yuan, and X. Dong, "Switchable dual-wavelength ytterbium-doped fiber laser based on a few-mode fiber grating," IEEE Photon. Technol. Lett. 16, 762-764 (2004).
[CrossRef]

M. D. Wei, C. H. Chen, and K. C. Tu, "Spatial and temporal instabilities in a passively Q-switched Nd:YAG laser with a Cr4+:YAG saturable absorber," Opt. Express 12, 3972-3980 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (1)

K. Lu and N. K. Dutta, "Spectroscopic properties of Yb-doped silica glass," J. Appl. Phys. 91, 576-581 (2002).
[CrossRef]

2001 (1)

C. C. Ranaud, H. L. Offerhaus, J. A. Alvarez-Chavez, C. J. Nilsson, W. A. Clarkson, P. W. Turner, D. J. Richardson, and A. B. Grudinin, "Characteristics of Q-switched cladding-pumped ytterbium- doped fiber lasers with different high-energy fiber designs," IEEE J. Quantum Electron. 37, 199-206 (2001).
[CrossRef]

2000 (1)

1997 (2)

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, L. K. Sun, and S. J. Zhang, "Optimization of Cr-doped saturable-absorber -switched lasers," IEEE J. Quantum Electron. 33, 2286-2294 (1997).
[CrossRef]

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, "Ytterbium- doped fiber amplifiers," IEEE J. Quantum Electron. 33, 1049-1056 (1997).
[CrossRef]

1994 (1)

Acco, S.

Aït-Ameur, K.

M. Laroche, H. Gilles, S. Girard, N. Passilly, and K. Aït-Ameur, "Nanosecond pulse generation in a passively Q-switched Yb-doped fiber laser by Cr4+:YAG saturable absorber," IEEE Photon. Technol. Lett. 18, 764-766 (2006).
[CrossRef]

Alvarez-Chavez, J. A.

C. C. Ranaud, H. L. Offerhaus, J. A. Alvarez-Chavez, C. J. Nilsson, W. A. Clarkson, P. W. Turner, D. J. Richardson, and A. B. Grudinin, "Characteristics of Q-switched cladding-pumped ytterbium- doped fiber lasers with different high-energy fiber designs," IEEE J. Quantum Electron. 37, 199-206 (2001).
[CrossRef]

J. A. Alvarez-Chavez, H. L. Offerhaus, J. Nilsson, P. W. Turner, W. A. Clarkson, and D. J. Richardson, "High-energy, high-power ytterbium-doped Q-switched fiber laser," Opt. Lett. 25, 37-39 (2000).
[CrossRef]

Chen, C. H.

Chen, L. R.

Chen, S.

R. Chi, K. Lu, and S. Chen, "Multi-wavelength Yb-doped fiber ring laser," Microwave Opt. Technol. Lett. 36, 170-172 (2003).
[CrossRef]

Chen, Y. F.

Cheo, P. K.

Y. M. Huo and P. K. Cheo, "Modeling of passively Q-switched Er3+/Yb3+-codoped clad-pumped fiber lasers," IEEE J. Sel. Top Quantum Electron. 11, 658-665 (2005).
[CrossRef]

Chi, R.

R. Chi, K. Lu, and S. Chen, "Multi-wavelength Yb-doped fiber ring laser," Microwave Opt. Technol. Lett. 36, 170-172 (2003).
[CrossRef]

Clarkson, W. A.

C. C. Ranaud, H. L. Offerhaus, J. A. Alvarez-Chavez, C. J. Nilsson, W. A. Clarkson, P. W. Turner, D. J. Richardson, and A. B. Grudinin, "Characteristics of Q-switched cladding-pumped ytterbium- doped fiber lasers with different high-energy fiber designs," IEEE J. Quantum Electron. 37, 199-206 (2001).
[CrossRef]

J. A. Alvarez-Chavez, H. L. Offerhaus, J. Nilsson, P. W. Turner, W. A. Clarkson, and D. J. Richardson, "High-energy, high-power ytterbium-doped Q-switched fiber laser," Opt. Lett. 25, 37-39 (2000).
[CrossRef]

Dill, C.

Dong, X.

X. Feng, Y. Liu, S. Fu, S. Yuan, and X. Dong, "Switchable dual-wavelength ytterbium-doped fiber laser based on a few-mode fiber grating," IEEE Photon. Technol. Lett. 16, 762-764 (2004).
[CrossRef]

Dutta, N. K.

K. Lu and N. K. Dutta, "Spectroscopic properties of Yb-doped silica glass," J. Appl. Phys. 91, 576-581 (2002).
[CrossRef]

Englander, A.

Fedosejevs, R.

L. Pan, I. Utkin, and R. Fedosejevs, "Passively Q-switched ytterbium doped double-clad fiber laser with a Cr4+:YAG saturable absorber," IEEE Photon. Technol. Lett. 19, 1979-1981 (2007).
[CrossRef]

Feng, X.

X. Feng, Y. Liu, S. Fu, S. Yuan, and X. Dong, "Switchable dual-wavelength ytterbium-doped fiber laser based on a few-mode fiber grating," IEEE Photon. Technol. Lett. 16, 762-764 (2004).
[CrossRef]

Fu, S.

X. Feng, Y. Liu, S. Fu, S. Yuan, and X. Dong, "Switchable dual-wavelength ytterbium-doped fiber laser based on a few-mode fiber grating," IEEE Photon. Technol. Lett. 16, 762-764 (2004).
[CrossRef]

Furusawa, K.

A. Piper, A. Malinowski, K. Furusawa, D. J. Richardson, "High-power, high- brightness, mJ Q-switched ytterbium-doped fibre laser," Electron. Lett. 40, 928-929 (2004).
[CrossRef]

Gao, C. Q.

Gilles, H.

M. Laroche, H. Gilles, S. Girard, N. Passilly, and K. Aït-Ameur, "Nanosecond pulse generation in a passively Q-switched Yb-doped fiber laser by Cr4+:YAG saturable absorber," IEEE Photon. Technol. Lett. 18, 764-766 (2006).
[CrossRef]

Girard, S.

M. Laroche, H. Gilles, S. Girard, N. Passilly, and K. Aït-Ameur, "Nanosecond pulse generation in a passively Q-switched Yb-doped fiber laser by Cr4+:YAG saturable absorber," IEEE Photon. Technol. Lett. 18, 764-766 (2006).
[CrossRef]

Glick, Y.

Graf, T.

H. Ridderbusch and T. Graf, "Saturation of 1047- and 1064nm absorption in Cr4+:YAG crystals," IEEE J. Quantum Electron. 43, 168-173 (2007).
[CrossRef]

Grudinin, A. B.

C. C. Ranaud, H. L. Offerhaus, J. A. Alvarez-Chavez, C. J. Nilsson, W. A. Clarkson, P. W. Turner, D. J. Richardson, and A. B. Grudinin, "Characteristics of Q-switched cladding-pumped ytterbium- doped fiber lasers with different high-energy fiber designs," IEEE J. Quantum Electron. 37, 199-206 (2001).
[CrossRef]

Gu, X. J.

Guan, W.

W. Guan and J. R. Marciante, "Dual-frequency operation in a short-cavity Ytterbium-doped fiber laser," IEEE Photon. Technol. Lett. 19, 261-263 (2007).
[CrossRef]

Guo, W.

C. Tu, W. Guo, Y. Li, S. Zhang, and F. Lu, "Stable multiwavelength and passively mode-locked Yb-doped fiber laser based on nonlinear polarization rotation," Opt. Commun. 280, 448-452 (2007).
[CrossRef]

Hanna, D. C.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, "Ytterbium- doped fiber amplifiers," IEEE J. Quantum Electron. 33, 1049-1056 (1997).
[CrossRef]

Hu, S. L.

Huang, J. Y.

Huo, Y. M.

Y. M. Huo and P. K. Cheo, "Modeling of passively Q-switched Er3+/Yb3+-codoped clad-pumped fiber lasers," IEEE J. Sel. Top Quantum Electron. 11, 658-665 (2005).
[CrossRef]

Katz, O.

Laroche, M.

M. Laroche, H. Gilles, S. Girard, N. Passilly, and K. Aït-Ameur, "Nanosecond pulse generation in a passively Q-switched Yb-doped fiber laser by Cr4+:YAG saturable absorber," IEEE Photon. Technol. Lett. 18, 764-766 (2006).
[CrossRef]

Lavi, R.

Li, Y.

C. Tu, W. Guo, Y. Li, S. Zhang, and F. Lu, "Stable multiwavelength and passively mode-locked Yb-doped fiber laser based on nonlinear polarization rotation," Opt. Commun. 280, 448-452 (2007).
[CrossRef]

Liang, H. C.

Liu, Y.

X. Feng, Y. Liu, S. Fu, S. Yuan, and X. Dong, "Switchable dual-wavelength ytterbium-doped fiber laser based on a few-mode fiber grating," IEEE Photon. Technol. Lett. 16, 762-764 (2004).
[CrossRef]

Lu, F.

C. Tu, W. Guo, Y. Li, S. Zhang, and F. Lu, "Stable multiwavelength and passively mode-locked Yb-doped fiber laser based on nonlinear polarization rotation," Opt. Commun. 280, 448-452 (2007).
[CrossRef]

Lu, K.

R. Chi, K. Lu, and S. Chen, "Multi-wavelength Yb-doped fiber ring laser," Microwave Opt. Technol. Lett. 36, 170-172 (2003).
[CrossRef]

K. Lu and N. K. Dutta, "Spectroscopic properties of Yb-doped silica glass," J. Appl. Phys. 91, 576-581 (2002).
[CrossRef]

Lü, F. Y.

Malinowski, A.

A. Piper, A. Malinowski, K. Furusawa, D. J. Richardson, "High-power, high- brightness, mJ Q-switched ytterbium-doped fibre laser," Electron. Lett. 40, 928-929 (2004).
[CrossRef]

Marciante, J. R.

W. Guan and J. R. Marciante, "Dual-frequency operation in a short-cavity Ytterbium-doped fiber laser," IEEE Photon. Technol. Lett. 19, 261-263 (2007).
[CrossRef]

Meng, X. L.

Nafcha, Y.

Ng, S. P.

Nilsson, C. J.

C. C. Ranaud, H. L. Offerhaus, J. A. Alvarez-Chavez, C. J. Nilsson, W. A. Clarkson, P. W. Turner, D. J. Richardson, and A. B. Grudinin, "Characteristics of Q-switched cladding-pumped ytterbium- doped fiber lasers with different high-energy fiber designs," IEEE J. Quantum Electron. 37, 199-206 (2001).
[CrossRef]

Nilsson, J.

Offerhaus, H. L.

C. C. Ranaud, H. L. Offerhaus, J. A. Alvarez-Chavez, C. J. Nilsson, W. A. Clarkson, P. W. Turner, D. J. Richardson, and A. B. Grudinin, "Characteristics of Q-switched cladding-pumped ytterbium- doped fiber lasers with different high-energy fiber designs," IEEE J. Quantum Electron. 37, 199-206 (2001).
[CrossRef]

J. A. Alvarez-Chavez, H. L. Offerhaus, J. Nilsson, P. W. Turner, W. A. Clarkson, and D. J. Richardson, "High-energy, high-power ytterbium-doped Q-switched fiber laser," Opt. Lett. 25, 37-39 (2000).
[CrossRef]

Pan, L.

L. Pan, I. Utkin, and R. Fedosejevs, "Passively Q-switched ytterbium doped double-clad fiber laser with a Cr4+:YAG saturable absorber," IEEE Photon. Technol. Lett. 19, 1979-1981 (2007).
[CrossRef]

Paschotta, R.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, "Ytterbium- doped fiber amplifiers," IEEE J. Quantum Electron. 33, 1049-1056 (1997).
[CrossRef]

Passilly, N.

M. Laroche, H. Gilles, S. Girard, N. Passilly, and K. Aït-Ameur, "Nanosecond pulse generation in a passively Q-switched Yb-doped fiber laser by Cr4+:YAG saturable absorber," IEEE Photon. Technol. Lett. 18, 764-766 (2006).
[CrossRef]

Piper, A.

A. Piper, A. Malinowski, K. Furusawa, D. J. Richardson, "High-power, high- brightness, mJ Q-switched ytterbium-doped fibre laser," Electron. Lett. 40, 928-929 (2004).
[CrossRef]

Qin, L. J.

Ranaud, C. C.

C. C. Ranaud, H. L. Offerhaus, J. A. Alvarez-Chavez, C. J. Nilsson, W. A. Clarkson, P. W. Turner, D. J. Richardson, and A. B. Grudinin, "Characteristics of Q-switched cladding-pumped ytterbium- doped fiber lasers with different high-energy fiber designs," IEEE J. Quantum Electron. 37, 199-206 (2001).
[CrossRef]

Richardson, D. J.

A. Piper, A. Malinowski, K. Furusawa, D. J. Richardson, "High-power, high- brightness, mJ Q-switched ytterbium-doped fibre laser," Electron. Lett. 40, 928-929 (2004).
[CrossRef]

C. C. Ranaud, H. L. Offerhaus, J. A. Alvarez-Chavez, C. J. Nilsson, W. A. Clarkson, P. W. Turner, D. J. Richardson, and A. B. Grudinin, "Characteristics of Q-switched cladding-pumped ytterbium- doped fiber lasers with different high-energy fiber designs," IEEE J. Quantum Electron. 37, 199-206 (2001).
[CrossRef]

J. A. Alvarez-Chavez, H. L. Offerhaus, J. Nilsson, P. W. Turner, W. A. Clarkson, and D. J. Richardson, "High-energy, high-power ytterbium-doped Q-switched fiber laser," Opt. Lett. 25, 37-39 (2000).
[CrossRef]

Ridderbusch, H.

H. Ridderbusch and T. Graf, "Saturation of 1047- and 1064nm absorption in Cr4+:YAG crystals," IEEE J. Quantum Electron. 43, 168-173 (2007).
[CrossRef]

Sintov, Y.

Su, K. W.

Sun, L. K.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, L. K. Sun, and S. J. Zhang, "Optimization of Cr-doped saturable-absorber -switched lasers," IEEE J. Quantum Electron. 33, 2286-2294 (1997).
[CrossRef]

Tang, D. Y.

Tropper, A. C.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, "Ytterbium- doped fiber amplifiers," IEEE J. Quantum Electron. 33, 1049-1056 (1997).
[CrossRef]

Tu, C.

C. Tu, W. Guo, Y. Li, S. Zhang, and F. Lu, "Stable multiwavelength and passively mode-locked Yb-doped fiber laser based on nonlinear polarization rotation," Opt. Commun. 280, 448-452 (2007).
[CrossRef]

Tu, K. C.

Turner, P. W.

C. C. Ranaud, H. L. Offerhaus, J. A. Alvarez-Chavez, C. J. Nilsson, W. A. Clarkson, P. W. Turner, D. J. Richardson, and A. B. Grudinin, "Characteristics of Q-switched cladding-pumped ytterbium- doped fiber lasers with different high-energy fiber designs," IEEE J. Quantum Electron. 37, 199-206 (2001).
[CrossRef]

J. A. Alvarez-Chavez, H. L. Offerhaus, J. Nilsson, P. W. Turner, W. A. Clarkson, and D. J. Richardson, "High-energy, high-power ytterbium-doped Q-switched fiber laser," Opt. Lett. 25, 37-39 (2000).
[CrossRef]

Utkin, I.

L. Pan, I. Utkin, and R. Fedosejevs, "Passively Q-switched ytterbium doped double-clad fiber laser with a Cr4+:YAG saturable absorber," IEEE Photon. Technol. Lett. 19, 1979-1981 (2007).
[CrossRef]

Wang, Q. P.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, L. K. Sun, and S. J. Zhang, "Optimization of Cr-doped saturable-absorber -switched lasers," IEEE J. Quantum Electron. 33, 2286-2294 (1997).
[CrossRef]

Wang, Y.

Wei, G. H.

Wei, M. D.

Xu, C. Q.

Yu, J.

Yuan, S.

X. Feng, Y. Liu, S. Fu, S. Yuan, and X. Dong, "Switchable dual-wavelength ytterbium-doped fiber laser based on a few-mode fiber grating," IEEE Photon. Technol. Lett. 16, 762-764 (2004).
[CrossRef]

Zayhowski, J. J.

Zhang, Q. D.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, L. K. Sun, and S. J. Zhang, "Optimization of Cr-doped saturable-absorber -switched lasers," IEEE J. Quantum Electron. 33, 2286-2294 (1997).
[CrossRef]

Zhang, S.

C. Tu, W. Guo, Y. Li, S. Zhang, and F. Lu, "Stable multiwavelength and passively mode-locked Yb-doped fiber laser based on nonlinear polarization rotation," Opt. Commun. 280, 448-452 (2007).
[CrossRef]

Zhang, S. J.

X. Y. Zhang, S. Z. Zhao, Q. P. Wang, Q. D. Zhang, L. K. Sun, and S. J. Zhang, "Optimization of Cr-doped saturable-absorber -switched lasers," IEEE J. Quantum Electron. 33, 2286-2294 (1997).
[CrossRef]

Zhang, X. Y.

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Figures (11)

Fig. 1.
Fig. 1.

Experiment setup; HR-high reflection mirror; SA - saturable absorber T0=30%; Yb DC fiber - Yb doped single-mode double-clad fiber; DM - dichroic mirror; L1, L2, L3 and L4 are focusing or imaging lenses; LD - fiber coupled pump laser diode; BS - beam splitter; PD1, PD2 – photodiodes; OC – output coupling mirror 4% reflection; the arrow (→) marks the direction convention for distance x in the theoretical simulation.

Fig. 2.
Fig. 2.

Absorption and emission cross section of Yb doped fiber, provided by the fiber manufacturer LIEKKI.

Fig. 3.
Fig. 3.

Laser output spectrum measured by a compact spectrometer (Ocean-Optics with resolution ~1.5 nm) at a pump power of 7.8 W; the low peak height at 1070 nm is due to the rapidly decreasing responsivity of the silicon based spectrometer CCD at this wavelength.

Fig. 4.
Fig. 4.

Dependences of total average power, 1070 nm, and 1040 nm power on pump power.

Fig. 5.
Fig. 5.

Synchronized (a) pulse trains and (b) pulse shapes of the output at the two wavelengths for a pump power of 4.5 W, the dashed line marks the peak of 1040 nm pulse. The intensity of 1070 nm pulse has been amplified for convenience of viewing.

Fig. 6.
Fig. 6.

Synchronized pulse trains at a pump power of (a) 7.8 W (b) 9.9 W.

Fig. 7.
Fig. 7.

Simulated Q-switched pulse trains and shapes at different pump powers P0, fiber length L= 190 cm; (b) the dashed line marks the peak of 1040 nm pulse; (f) t1 to t6 mark different times through the pulse at a pump power of 18 W, which are explained in Fig. 8. Note each plot uses different vertical scale.

Fig. 8.
Fig. 8.

Inversion number distribution along fiber at different times which are shown in Fig.7(f); t1 and t6 are the times when the inversion number is maximum and minimum; t3 and t4 mark the peaks of the two pulses when the inversion number drops to threshold for 1040 nm and 1070 nm wavelengths respectively; t2 is the time when the gain at 1070 nm becomes higher than at 1040 nm; t5 marks 10% intensity tail of the 1040 nm pulse.

Fig. 9.
Fig. 9.

Simulated dynamics of g1040(t), g1070(t), N̲2(t) and N̲2 sa with 1040 nm or 1070 nm pulses at a pump power of 11 W. The pulse train is also shown in Fig. 7(d). Cavity linear loss (~3.56) includes output coupler loss, cavity loss η, and fiber attenuation αk.

Fig. 10.
Fig. 10.

Expanded views in time of g1040(t) and g1070(t) for the pulses P1 and P2 shown in Fig. 9.The vertical dashed lines mark the time when g1070(t) becomes higher than g1040(t) during the pulses.

Fig. 11.
Fig. 11.

Yb energy levels [21]; 974nm represents pump absorption transition; 1040nm and 1068nm represent two lasing transitions in this experiment.

Tables (1)

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Table 1. Values of parameters in simulation

Equations (13)

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N 0 = N 1 + N 2
N 2 ( x , t ) t = Γ p λ p hcA co [ σ αp N 1 ( x , t ) σ ep N 2 ( x , t ) ] P p ( x , t )
+ k Γ k λ k hcA co [ σ αk N 1 ( x , t ) σ ek N 2 ( x , t ) ] ( P k + ( x , t ) + P k ( x , t ) ) N 2 ( x , t ) τ
P p ( x , t ) x + 1 v p P p ( x , t ) t = Γ p [ σ ep N 2 ( x , t ) σ αp N 1 ( x , t ) ] P p ( x , t ) α p . P p ( x , t )
± P k ± ( x , t ) x + 1 v k P k ± ( x , t ) t = Γ k [ σ ek N 2 ( x , t ) σ αk N 1 ( x , t ) ] P k ± ( x , t ) α k P k ± ( x , t ) + 2 Γ k σ ek N 2 ( x , t ) hc 2 λ k 3 Δλ k
N sa = N 1 sa ( t ) + N 2 sa ( t )
T k ( t ) = exp [ 2 L sa ( σ gska N 1 sa + σ esak N 2 sa ) ]
dN 2 sa dt = k P k + ( 0 , t ) + P k ( 0 , t ) 2 λ k A sa hc σ gsak N 1 sa N 1 sa τ sa
P p ( L ) = P 0
P k ( L , t ) = P k + ( L , t ) R oc
P k + ( 0 , t ) = P k ( 0 , t ) ( 1 η ) 2 T k ( t )
N 2 ( t ) = 0 L N 2 ( x , t ) dx L
gk ( t ) = 2 . L . Γ k . [ σ ek . N 2 ( t ) σ αk . ( N 0 N 2 ( t ) ) ] ln [ 1 T k ( t ) ]

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