Abstract

We first investigate the polarization coexistence and switching between the orthogonal π- and σ-polarization states in a free running optical bistability Tm,Ho:LLF laser. The output performances of the optical bistability Tm,Ho:LLF laser is numerically simulated, based on a new rate equation theory model taking into account the influences of polarization-dependent gain and losses. The simulation results show the polarization coexistence and switching of output laser in the process of decreasing pump power. The physical mechanism of polarization coexistence and switching is clarified by analyzing the polarization-dependent net-gain coefficients. Furthermore, the polarization coexistence and switching are experimentally realized in the optical bistability Tm,Ho:LLF laser, which validates the theoretical analysis. The theoretical model can be applied to analyze the polarization coexistence and switching in other kinds of (quasi-) three-level optical bistable lasers.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  24. J. H. Liu and X. P. Tian, “Generalization of the modeling analysis of optical bistability in quasi-three-level laser,” IEEE J. Quantum Electron. 49(2), 247–251 (2013).
  25. X. L. Zhang, L. Li, Y. Zheng, and Y. Z. Wang, “Formation mechanism of optical bistability in end-pumped quasi-three-level Tm,Ho:YLF lasers,” J. Opt. Soc. Am. B 26(12), 2434–2439 (2009).
  26. M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).
  27. M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).
  28. P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).
  29. H. Zhao and A. Major, “Orthogonally polarized dual-wavelength Yb:KGW laser induced by thermal lensing,” Appl. Phys. B 122(6), 163 (2016).
  30. J. W. Kim, J. I. Mackenzie, D. Parisi, S. Veronesi, M. Tonelli, and W. A. Clarkson, “Efficient in-band pumped Ho:LuLiF4 2 µm laser,” Opt. Lett. 35(3), 420–422 (2010).
  31. X. L. Zhang, Y. Z. Wang, L. Li, and Y. L. Ju, “Heat generation and thermal lensing in end-pumped Tm,Ho:YLF laser crystals,” J. Phys. D Appl. Phys. 40(22), 6930–6935 (2007).
  32. Y. Wang and R. Zhang, “Optimizing the mode-to-pump ratio in end-pumped quasi-three-level Nd-doped lasers considering the energy-transfer upconversion,” J. Phys. At. Mol. Opt. Phys. 44(13), 135401 (2011).

2017 (1)

2016 (1)

H. Zhao and A. Major, “Orthogonally polarized dual-wavelength Yb:KGW laser induced by thermal lensing,” Appl. Phys. B 122(6), 163 (2016).

2014 (2)

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

X. L. Zhang, S. Zhang, Z. L. Xie, G. X. Li, L. Yu, J. H. Cui, J. Q. Zhao, and L. Li, “Polarization switching and optical bistability in the diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 11(10), 105804 (2014).

2013 (4)

X. Zhang, L. Yu, S. Zhang, L. Li, J. Zhao, and J. Cui, “Diode-pumped continuous wave and passively Q-switched Tm,Ho:LLF laser at 2 µm,” Opt. Express 21(10), 12629–12634 (2013).

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

X. L. Zhang, L. Yu, S. Zhang, L. Li, J. Q. Zhao, J. H. Cui, G. Z. Dong, and R. Wang, “Controlled optical bistability switching in a diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 10(12), 125801 (2013).

J. H. Liu and X. P. Tian, “Generalization of the modeling analysis of optical bistability in quasi-three-level laser,” IEEE J. Quantum Electron. 49(2), 247–251 (2013).

2012 (1)

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

2011 (2)

Y. Wang and R. Zhang, “Optimizing the mode-to-pump ratio in end-pumped quasi-three-level Nd-doped lasers considering the energy-transfer upconversion,” J. Phys. At. Mol. Opt. Phys. 44(13), 135401 (2011).

S. J. Shu, T. Yu, J. Y. Hou, R. T. Liu, M. J. Huang, and W. B. Chen, “End-pumped all solid-state high repetition rate Tm,Ho:LuLF laser,” Chin. Opt. Lett. 9(2), 021401 (2011).

2010 (2)

J. W. Kim, J. I. Mackenzie, D. Parisi, S. Veronesi, M. Tonelli, and W. A. Clarkson, “Efficient in-band pumped Ho:LuLiF4 2 µm laser,” Opt. Lett. 35(3), 420–422 (2010).

J. Liu, W. Han, H. Zhang, H. Yang, and V. Petrov, “Study of the optical bistability in the laser oscillation of Yb:GdVO4 crystal,” Appl. Phys. B 98(1), 87–91 (2010).

2009 (1)

2008 (2)

J. Liu, H. Zhang, X. Mateos, W. Han, and V. Petrov, “Bistable laser operation of a Yb0.0054:Y0.3481Gd0.6465VO4 mixed crystal,” Opt. Lett. 33(16), 1810–1812 (2008).

F. Y. Wang, G. X. Li, H. L. Tan, K. W. Cheah, and S. N. Zhu, “Optical bistability and multistability in one-dimensional periodic metal-dielectric photonic crystal,” Appl. Phys. Lett. 92(21), 211109 (2008).

2007 (5)

M. Y. Vilensky, Y. Prior, and I. Sh. Averbukh, “Cooling in a bistable optical cavity,” Phys. Rev. Lett. 99(10), 103002 (2007).

E. Weidner, S. Combrié, A. De Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101118 (2007).

C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, “All-optical switching in quantum cascade lasers,” Appl. Phys. Lett. 90(5), 053505 (2007).

X. Zhang and Y. Wang, “Optical bistability effects in a Tm,Ho:YLF laser at room temperature,” Opt. Lett. 32(16), 2333–2335 (2007).

X. L. Zhang, Y. Z. Wang, L. Li, and Y. L. Ju, “Heat generation and thermal lensing in end-pumped Tm,Ho:YLF laser crystals,” J. Phys. D Appl. Phys. 40(22), 6930–6935 (2007).

2006 (2)

2004 (2)

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95(7), 3255–3271 (2004).

L. D. Bozano, B. W. Kean, V. R. Deline, J. R. Salem, and J. C. Scott, “Mechanism for bistability in organic memory elements,” Appl. Phys. Lett. 84(4), 607–609 (2004).

2003 (2)

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).

V. Sudesh and K. Asai, “Spectroscopic and dioed-pumped-laser properties of Tm,Ho:YLF; Tm,Ho:LuLF; and Tm,Ho:LuAG crystals: a comparative study,” J. Opt. Soc. Am. B 20(9), 1829–1837 (2003).

2001 (1)

X. Huang, A. Stintz, H. Li, A. Rice, G. T. Liu, L. F. Lester, J. Cheng, and K. L. Malloy, “Bistable operation of a two-section 1.3-μm InAs quantum dot laser-absorption saturation and the quantum confined Stark effect,” IEEE J. Quantum Electron. 37(3), 414–417 (2001).

2000 (1)

1998 (1)

1983 (1)

E. Arimondo and B. M. Dinelli, “Optical bistability of a CO2 laser with intracavity saturable absorber: experiment and model,” Opt. Commun. 44(4), 277–282 (1983).

Aguiló, M.

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

Akimov, A. V.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).

Arimondo, E.

E. Arimondo and B. M. Dinelli, “Optical bistability of a CO2 laser with intracavity saturable absorber: experiment and model,” Opt. Commun. 44(4), 277–282 (1983).

Asai, K.

Averbukh, I. Sh.

M. Y. Vilensky, Y. Prior, and I. Sh. Averbukh, “Cooling in a bistable optical cavity,” Phys. Rev. Lett. 99(10), 103002 (2007).

Bai, Y.

Barnes, N. P.

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95(7), 3255–3271 (2004).

Bozano, L. D.

L. D. Bozano, B. W. Kean, V. R. Deline, J. R. Salem, and J. C. Scott, “Mechanism for bistability in organic memory elements,” Appl. Phys. Lett. 84(4), 607–609 (2004).

Budni, P. A.

Carvajal, J. J.

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

Cassette, S.

E. Weidner, S. Combrié, A. De Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101118 (2007).

Cheah, K. W.

F. Y. Wang, G. X. Li, H. L. Tan, K. W. Cheah, and S. N. Zhu, “Optical bistability and multistability in one-dimensional periodic metal-dielectric photonic crystal,” Appl. Phys. Lett. 92(21), 211109 (2008).

Chen, S.

Chen, W. B.

Cheng, J.

X. Huang, A. Stintz, H. Li, A. Rice, G. T. Liu, L. F. Lester, J. Cheng, and K. L. Malloy, “Bistable operation of a two-section 1.3-μm InAs quantum dot laser-absorption saturation and the quantum confined Stark effect,” IEEE J. Quantum Electron. 37(3), 414–417 (2001).

Chicklis, E. P.

Clarkson, W. A.

Combrié, S.

E. Weidner, S. Combrié, A. De Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101118 (2007).

Cui, J.

Cui, J. H.

X. L. Zhang, S. Zhang, Z. L. Xie, G. X. Li, L. Yu, J. H. Cui, J. Q. Zhao, and L. Li, “Polarization switching and optical bistability in the diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 11(10), 105804 (2014).

X. L. Zhang, L. Yu, S. Zhang, L. Li, J. Q. Zhao, J. H. Cui, G. Z. Dong, and R. Wang, “Controlled optical bistability switching in a diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 10(12), 125801 (2013).

Dai, J.

De Rossi, A.

E. Weidner, S. Combrié, A. De Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101118 (2007).

Deline, V. R.

L. D. Bozano, B. W. Kean, V. R. Deline, J. R. Salem, and J. C. Scott, “Mechanism for bistability in organic memory elements,” Appl. Phys. Lett. 84(4), 607–609 (2004).

Díaz, F.

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

Dijkhuis, J. I.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).

Dinelli, B. M.

E. Arimondo and B. M. Dinelli, “Optical bistability of a CO2 laser with intracavity saturable absorber: experiment and model,” Opt. Commun. 44(4), 277–282 (1983).

Dong, G. Z.

X. L. Zhang, L. Yu, S. Zhang, L. Li, J. Q. Zhao, J. H. Cui, G. Z. Dong, and R. Wang, “Controlled optical bistability switching in a diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 10(12), 125801 (2013).

Frogley, M. D.

C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, “All-optical switching in quantum cascade lasers,” Appl. Phys. Lett. 90(5), 053505 (2007).

Golubev, V. G.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).

Griebner, U.

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

J. Liu, V. Petrov, U. Griebner, F. Noack, H. Zhang, J. Wang, and M. Jiang, “Optical bistability in the operation of a continuous-wave diode-pumped Yb:LuVO4 laser,” Opt. Express 14(25), 12183–12187 (2006).

Han, W.

J. Liu, W. Han, H. Zhang, H. Yang, and V. Petrov, “Study of the optical bistability in the laser oscillation of Yb:GdVO4 crystal,” Appl. Phys. B 98(1), 87–91 (2010).

J. Liu, H. Zhang, X. Mateos, W. Han, and V. Petrov, “Bistable laser operation of a Yb0.0054:Y0.3481Gd0.6465VO4 mixed crystal,” Opt. Lett. 33(16), 1810–1812 (2008).

Hopkinson, M.

C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, “All-optical switching in quantum cascade lasers,” Appl. Phys. Lett. 90(5), 053505 (2007).

Hou, J. Y.

Huang, J.

Huang, M. J.

Huang, X.

X. Huang, A. Stintz, H. Li, A. Rice, G. T. Liu, L. F. Lester, J. Cheng, and K. L. Malloy, “Bistable operation of a two-section 1.3-μm InAs quantum dot laser-absorption saturation and the quantum confined Stark effect,” IEEE J. Quantum Electron. 37(3), 414–417 (2001).

Jiang, M.

Ju, Y. L.

X. L. Zhang, Y. Z. Wang, L. Li, and Y. L. Ju, “Heat generation and thermal lensing in end-pumped Tm,Ho:YLF laser crystals,” J. Phys. D Appl. Phys. 40(22), 6930–6935 (2007).

Kadankov, M.

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

Kavaya, M. J.

Kean, B. W.

L. D. Bozano, B. W. Kean, V. R. Deline, J. R. Salem, and J. C. Scott, “Mechanism for bistability in organic memory elements,” Appl. Phys. Lett. 84(4), 607–609 (2004).

Kerst, R.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).

Killinger, D. K.

Kim, J. W.

Kuleshov, N. V.

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

Kundys, D. O.

C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, “All-optical switching in quantum cascade lasers,” Appl. Phys. Lett. 90(5), 053505 (2007).

Kurdyukov, D. A.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).

Lemons, M. L.

Lester, L. F.

X. Huang, A. Stintz, H. Li, A. Rice, G. T. Liu, L. F. Lester, J. Cheng, and K. L. Malloy, “Bistable operation of a two-section 1.3-μm InAs quantum dot laser-absorption saturation and the quantum confined Stark effect,” IEEE J. Quantum Electron. 37(3), 414–417 (2001).

Li, G. X.

X. L. Zhang, S. Zhang, Z. L. Xie, G. X. Li, L. Yu, J. H. Cui, J. Q. Zhao, and L. Li, “Polarization switching and optical bistability in the diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 11(10), 105804 (2014).

F. Y. Wang, G. X. Li, H. L. Tan, K. W. Cheah, and S. N. Zhu, “Optical bistability and multistability in one-dimensional periodic metal-dielectric photonic crystal,” Appl. Phys. Lett. 92(21), 211109 (2008).

Li, H.

X. Huang, A. Stintz, H. Li, A. Rice, G. T. Liu, L. F. Lester, J. Cheng, and K. L. Malloy, “Bistable operation of a two-section 1.3-μm InAs quantum dot laser-absorption saturation and the quantum confined Stark effect,” IEEE J. Quantum Electron. 37(3), 414–417 (2001).

Li, J.

Li, L.

X. L. Zhang, S. Zhang, Z. L. Xie, G. X. Li, L. Yu, J. H. Cui, J. Q. Zhao, and L. Li, “Polarization switching and optical bistability in the diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 11(10), 105804 (2014).

X. L. Zhang, L. Yu, S. Zhang, L. Li, J. Q. Zhao, J. H. Cui, G. Z. Dong, and R. Wang, “Controlled optical bistability switching in a diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 10(12), 125801 (2013).

X. Zhang, L. Yu, S. Zhang, L. Li, J. Zhao, and J. Cui, “Diode-pumped continuous wave and passively Q-switched Tm,Ho:LLF laser at 2 µm,” Opt. Express 21(10), 12629–12634 (2013).

X. L. Zhang, L. Li, Y. Zheng, and Y. Z. Wang, “Formation mechanism of optical bistability in end-pumped quasi-three-level Tm,Ho:YLF lasers,” J. Opt. Soc. Am. B 26(12), 2434–2439 (2009).

X. L. Zhang, Y. Z. Wang, L. Li, and Y. L. Ju, “Heat generation and thermal lensing in end-pumped Tm,Ho:YLF laser crystals,” J. Phys. D Appl. Phys. 40(22), 6930–6935 (2007).

Liu, G. T.

X. Huang, A. Stintz, H. Li, A. Rice, G. T. Liu, L. F. Lester, J. Cheng, and K. L. Malloy, “Bistable operation of a two-section 1.3-μm InAs quantum dot laser-absorption saturation and the quantum confined Stark effect,” IEEE J. Quantum Electron. 37(3), 414–417 (2001).

Liu, J.

Liu, J. H.

J. H. Liu and X. P. Tian, “Generalization of the modeling analysis of optical bistability in quasi-three-level laser,” IEEE J. Quantum Electron. 49(2), 247–251 (2013).

Liu, R. T.

Loiko, P. A.

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

Luo, Y.

Mackenzie, J. I.

Major, A.

H. Zhao and A. Major, “Orthogonally polarized dual-wavelength Yb:KGW laser induced by thermal lensing,” Appl. Phys. B 122(6), 163 (2016).

Malloy, K. L.

X. Huang, A. Stintz, H. Li, A. Rice, G. T. Liu, L. F. Lester, J. Cheng, and K. L. Malloy, “Bistable operation of a two-section 1.3-μm InAs quantum dot laser-absorption saturation and the quantum confined Stark effect,” IEEE J. Quantum Electron. 37(3), 414–417 (2001).

Mateos, X.

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

J. Liu, H. Zhang, X. Mateos, W. Han, and V. Petrov, “Bistable laser operation of a Yb0.0054:Y0.3481Gd0.6465VO4 mixed crystal,” Opt. Lett. 33(16), 1810–1812 (2008).

Mazurenko, D. A.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).

Miller, C. A.

Modlin, E. A.

Mosto, J. R.

Noack, F.

Parisi, D.

Pavlyuk, A. A.

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

Petros, M.

J. Yu, B. C. Trieu, E. A. Modlin, U. N. Singh, M. J. Kavaya, S. Chen, Y. Bai, P. J. Petzar, and M. Petros, “1 J/pulse Q-switched 2 microm solid-state laser,” Opt. Lett. 31(4), 462–464 (2006).

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95(7), 3255–3271 (2004).

Petrov, V.

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

J. Liu, W. Han, H. Zhang, H. Yang, and V. Petrov, “Study of the optical bistability in the laser oscillation of Yb:GdVO4 crystal,” Appl. Phys. B 98(1), 87–91 (2010).

J. Liu, H. Zhang, X. Mateos, W. Han, and V. Petrov, “Bistable laser operation of a Yb0.0054:Y0.3481Gd0.6465VO4 mixed crystal,” Opt. Lett. 33(16), 1810–1812 (2008).

J. Liu, V. Petrov, U. Griebner, F. Noack, H. Zhang, J. Wang, and M. Jiang, “Optical bistability in the operation of a continuous-wave diode-pumped Yb:LuVO4 laser,” Opt. Express 14(25), 12183–12187 (2006).

Petzar, P. J.

Pevtsov, A. B.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).

Phillips, C. C.

C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, “All-optical switching in quantum cascade lasers,” Appl. Phys. Lett. 90(5), 053505 (2007).

Pomeranz, L. A.

Prior, Y.

M. Y. Vilensky, Y. Prior, and I. Sh. Averbukh, “Cooling in a bistable optical cavity,” Phys. Rev. Lett. 99(10), 103002 (2007).

Pujol, M. C.

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

Rice, A.

X. Huang, A. Stintz, H. Li, A. Rice, G. T. Liu, L. F. Lester, J. Cheng, and K. L. Malloy, “Bistable operation of a two-section 1.3-μm InAs quantum dot laser-absorption saturation and the quantum confined Stark effect,” IEEE J. Quantum Electron. 37(3), 414–417 (2001).

Salem, J. R.

L. D. Bozano, B. W. Kean, V. R. Deline, J. R. Salem, and J. C. Scott, “Mechanism for bistability in organic memory elements,” Appl. Phys. Lett. 84(4), 607–609 (2004).

Scott, J. C.

L. D. Bozano, B. W. Kean, V. R. Deline, J. R. Salem, and J. C. Scott, “Mechanism for bistability in organic memory elements,” Appl. Phys. Lett. 84(4), 607–609 (2004).

Segura, M.

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

M. Segura, M. Kadankov, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Polarization switching in the 2-μm Tm:KLu (WO4)2 laser,” Laser Phys. Lett. 9(2), 104–109 (2012).

Sel’kin, A. V.

D. A. Mazurenko, R. Kerst, J. I. Dijkhuis, A. V. Akimov, V. G. Golubev, D. A. Kurdyukov, A. B. Pevtsov, and A. V. Sel’kin, “Ultrafast optical switching in three-dimensional photonic crystals,” Phys. Rev. Lett. 91(21), 213903 (2003).

Shu, S. J.

Singh, U. N.

J. Yu, B. C. Trieu, E. A. Modlin, U. N. Singh, M. J. Kavaya, S. Chen, Y. Bai, P. J. Petzar, and M. Petros, “1 J/pulse Q-switched 2 microm solid-state laser,” Opt. Lett. 31(4), 462–464 (2006).

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95(7), 3255–3271 (2004).

Skolnick, M. S.

C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, “All-optical switching in quantum cascade lasers,” Appl. Phys. Lett. 90(5), 053505 (2007).

Stintz, A.

X. Huang, A. Stintz, H. Li, A. Rice, G. T. Liu, L. F. Lester, J. Cheng, and K. L. Malloy, “Bistable operation of a two-section 1.3-μm InAs quantum dot laser-absorption saturation and the quantum confined Stark effect,” IEEE J. Quantum Electron. 37(3), 414–417 (2001).

Sudesh, V.

Taczak, T. M.

Tan, H. L.

F. Y. Wang, G. X. Li, H. L. Tan, K. W. Cheah, and S. N. Zhu, “Optical bistability and multistability in one-dimensional periodic metal-dielectric photonic crystal,” Appl. Phys. Lett. 92(21), 211109 (2008).

Tian, X. P.

J. H. Liu and X. P. Tian, “Generalization of the modeling analysis of optical bistability in quasi-three-level laser,” IEEE J. Quantum Electron. 49(2), 247–251 (2013).

Tonelli, M.

Tran, N.

E. Weidner, S. Combrié, A. De Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101118 (2007).

Trieu, B. C.

Veronesi, S.

Vilensky, M. Y.

M. Y. Vilensky, Y. Prior, and I. Sh. Averbukh, “Cooling in a bistable optical cavity,” Phys. Rev. Lett. 99(10), 103002 (2007).

Walsh, B. M.

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95(7), 3255–3271 (2004).

Wang, F. Y.

F. Y. Wang, G. X. Li, H. L. Tan, K. W. Cheah, and S. N. Zhu, “Optical bistability and multistability in one-dimensional periodic metal-dielectric photonic crystal,” Appl. Phys. Lett. 92(21), 211109 (2008).

Wang, J.

Wang, R.

X. L. Zhang, L. Yu, S. Zhang, L. Li, J. Q. Zhao, J. H. Cui, G. Z. Dong, and R. Wang, “Controlled optical bistability switching in a diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 10(12), 125801 (2013).

Wang, T.

Wang, Y.

Y. Wang and R. Zhang, “Optimizing the mode-to-pump ratio in end-pumped quasi-three-level Nd-doped lasers considering the energy-transfer upconversion,” J. Phys. At. Mol. Opt. Phys. 44(13), 135401 (2011).

X. Zhang and Y. Wang, “Optical bistability effects in a Tm,Ho:YLF laser at room temperature,” Opt. Lett. 32(16), 2333–2335 (2007).

Wang, Y. Z.

X. L. Zhang, L. Li, Y. Zheng, and Y. Z. Wang, “Formation mechanism of optical bistability in end-pumped quasi-three-level Tm,Ho:YLF lasers,” J. Opt. Soc. Am. B 26(12), 2434–2439 (2009).

X. L. Zhang, Y. Z. Wang, L. Li, and Y. L. Ju, “Heat generation and thermal lensing in end-pumped Tm,Ho:YLF laser crystals,” J. Phys. D Appl. Phys. 40(22), 6930–6935 (2007).

Weidner, E.

E. Weidner, S. Combrié, A. De Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101118 (2007).

Wilson, L. R.

C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, “All-optical switching in quantum cascade lasers,” Appl. Phys. Lett. 90(5), 053505 (2007).

Xie, Z. L.

X. L. Zhang, S. Zhang, Z. L. Xie, G. X. Li, L. Yu, J. H. Cui, J. Q. Zhao, and L. Li, “Polarization switching and optical bistability in the diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 11(10), 105804 (2014).

Yang, H.

J. Liu, W. Han, H. Zhang, H. Yang, and V. Petrov, “Study of the optical bistability in the laser oscillation of Yb:GdVO4 crystal,” Appl. Phys. B 98(1), 87–91 (2010).

Yu, J.

J. Yu, B. C. Trieu, E. A. Modlin, U. N. Singh, M. J. Kavaya, S. Chen, Y. Bai, P. J. Petzar, and M. Petros, “1 J/pulse Q-switched 2 microm solid-state laser,” Opt. Lett. 31(4), 462–464 (2006).

B. M. Walsh, N. P. Barnes, M. Petros, J. Yu, and U. N. Singh, “Spectroscopy and modeling of solid state lanthanide lasers: Application to trivalent Tm3+ and Ho3+ in YLiF4 and LuLiF4,” J. Appl. Phys. 95(7), 3255–3271 (2004).

Yu, L.

X. L. Zhang, S. Zhang, Z. L. Xie, G. X. Li, L. Yu, J. H. Cui, J. Q. Zhao, and L. Li, “Polarization switching and optical bistability in the diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 11(10), 105804 (2014).

X. L. Zhang, L. Yu, S. Zhang, L. Li, J. Q. Zhao, J. H. Cui, G. Z. Dong, and R. Wang, “Controlled optical bistability switching in a diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 10(12), 125801 (2013).

X. Zhang, L. Yu, S. Zhang, L. Li, J. Zhao, and J. Cui, “Diode-pumped continuous wave and passively Q-switched Tm,Ho:LLF laser at 2 µm,” Opt. Express 21(10), 12629–12634 (2013).

Yu, T.

Yumashev, K. V.

P. A. Loiko, X. Mateos, N. V. Kuleshov, A. A. Pavlyuk, K. V. Yumashev, V. Petrov, U. Griebner, M. Aguiló, and F. Díaz, “Thermal-lens-driven effects in Ng-cut Yb- and Tm-doped monoclinic KLu(WO4)2 crystals,” IEEE J. Quantum Electron. 50(8), 669–676 (2014).

Zervos, C.

C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, “All-optical switching in quantum cascade lasers,” Appl. Phys. Lett. 90(5), 053505 (2007).

Zhang, H.

Zhang, J.

Zhang, R.

Y. Wang and R. Zhang, “Optimizing the mode-to-pump ratio in end-pumped quasi-three-level Nd-doped lasers considering the energy-transfer upconversion,” J. Phys. At. Mol. Opt. Phys. 44(13), 135401 (2011).

Zhang, S.

X. L. Zhang, S. Zhang, Z. L. Xie, G. X. Li, L. Yu, J. H. Cui, J. Q. Zhao, and L. Li, “Polarization switching and optical bistability in the diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 11(10), 105804 (2014).

X. L. Zhang, L. Yu, S. Zhang, L. Li, J. Q. Zhao, J. H. Cui, G. Z. Dong, and R. Wang, “Controlled optical bistability switching in a diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 10(12), 125801 (2013).

X. Zhang, L. Yu, S. Zhang, L. Li, J. Zhao, and J. Cui, “Diode-pumped continuous wave and passively Q-switched Tm,Ho:LLF laser at 2 µm,” Opt. Express 21(10), 12629–12634 (2013).

Zhang, X.

Zhang, X. L.

X. L. Zhang, S. Zhang, Z. L. Xie, G. X. Li, L. Yu, J. H. Cui, J. Q. Zhao, and L. Li, “Polarization switching and optical bistability in the diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 11(10), 105804 (2014).

X. L. Zhang, L. Yu, S. Zhang, L. Li, J. Q. Zhao, J. H. Cui, G. Z. Dong, and R. Wang, “Controlled optical bistability switching in a diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 10(12), 125801 (2013).

X. L. Zhang, L. Li, Y. Zheng, and Y. Z. Wang, “Formation mechanism of optical bistability in end-pumped quasi-three-level Tm,Ho:YLF lasers,” J. Opt. Soc. Am. B 26(12), 2434–2439 (2009).

X. L. Zhang, Y. Z. Wang, L. Li, and Y. L. Ju, “Heat generation and thermal lensing in end-pumped Tm,Ho:YLF laser crystals,” J. Phys. D Appl. Phys. 40(22), 6930–6935 (2007).

Zhao, H.

H. Zhao and A. Major, “Orthogonally polarized dual-wavelength Yb:KGW laser induced by thermal lensing,” Appl. Phys. B 122(6), 163 (2016).

Zhao, J.

Zhao, J. Q.

X. L. Zhang, S. Zhang, Z. L. Xie, G. X. Li, L. Yu, J. H. Cui, J. Q. Zhao, and L. Li, “Polarization switching and optical bistability in the diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 11(10), 105804 (2014).

X. L. Zhang, L. Yu, S. Zhang, L. Li, J. Q. Zhao, J. H. Cui, G. Z. Dong, and R. Wang, “Controlled optical bistability switching in a diode-pumped Tm,Ho:LLF laser,” Laser Phys. Lett. 10(12), 125801 (2013).

Zheng, Y.

Zhu, S. N.

F. Y. Wang, G. X. Li, H. L. Tan, K. W. Cheah, and S. N. Zhu, “Optical bistability and multistability in one-dimensional periodic metal-dielectric photonic crystal,” Appl. Phys. Lett. 92(21), 211109 (2008).

Appl. Opt. (2)

Appl. Phys. B (3)

J. Liu, W. Han, H. Zhang, H. Yang, and V. Petrov, “Study of the optical bistability in the laser oscillation of Yb:GdVO4 crystal,” Appl. Phys. B 98(1), 87–91 (2010).

M. Segura, X. Mateos, M. C. Pujol, J. J. Carvajal, M. Aguiló, F. Díaz, U. Griebner, and V. Petrov, “Dual-wavelength diode-pumped laser operation of Np-cut and Ng-cut Tm:KLu(WO4)2, crystals,” Appl. Phys. B 113(1), 125–131 (2013).

H. Zhao and A. Major, “Orthogonally polarized dual-wavelength Yb:KGW laser induced by thermal lensing,” Appl. Phys. B 122(6), 163 (2016).

Appl. Phys. Lett. (4)

E. Weidner, S. Combrié, A. De Rossi, N. Tran, and S. Cassette, “Nonlinear and bistable behavior of an ultrahigh-Q GaAs photonic crystal nanocavity,” Appl. Phys. Lett. 90(10), 101118 (2007).

L. D. Bozano, B. W. Kean, V. R. Deline, J. R. Salem, and J. C. Scott, “Mechanism for bistability in organic memory elements,” Appl. Phys. Lett. 84(4), 607–609 (2004).

F. Y. Wang, G. X. Li, H. L. Tan, K. W. Cheah, and S. N. Zhu, “Optical bistability and multistability in one-dimensional periodic metal-dielectric photonic crystal,” Appl. Phys. Lett. 92(21), 211109 (2008).

C. Zervos, M. D. Frogley, C. C. Phillips, D. O. Kundys, L. R. Wilson, M. Hopkinson, and M. S. Skolnick, “All-optical switching in quantum cascade lasers,” Appl. Phys. Lett. 90(5), 053505 (2007).

Chin. Opt. Lett. (1)

IEEE J. Quantum Electron. (3)

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Figures (7)

Fig. 1
Fig. 1 Gain spectra of Tm,Ho:LLF for (a) π-polarization and (b) σ-polarization for inversion parameters β = 0.2, 0.25, 0.3 and 0.4.
Fig. 2
Fig. 2 Calculated diffraction losses as a function of pump power for the π- and σ-polarizations.
Fig. 3
Fig. 3 Calculated output power as a function of pump power for the Tm,Ho:LLF laser with (a) increasing and (b) decreasing pump power, showing polarization coexistence and switching, and optical bistability.
Fig. 4
Fig. 4 Calculated (a) total, (b) π-polarization, and (c) σ-polarization net gain coefficients as a function of pump power.
Fig. 5
Fig. 5 Experimental setup of the optical bistability Tm,Ho:LLF laser.
Fig. 6
Fig. 6 Output power as a function of pump power for the optical bistability Tm,Ho:LLF laser with (a) increasing and (b) decreasing pump power.
Fig. 7
Fig. 7 Output spectra for (a) 2069 nm π-polarized laser and (b) 2066 nm σ-polarized laser.

Equations (21)

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G=( N Ho l) σ em (λ)[β(1β) Z exc Z gnd exp(( E ZL 1 λ )hc/ k b T)]
d N 1 dt = η p R r p (r,z) N 1 τ Q N 1 2 σ π c n [ f Ho ( f uπ + f lπ ) N 1 f lπ N Ho ] Φ π φ l (r,z) σ σ c n [ f Ho ( f uσ + f lσ ) N 1 f lσ N Ho ] Φ σ φ l (r,z)
d N 2 dt = η p R r p (r,z) N 2 τ Q N 2 2 σ π c n [ f Ho ( f uπ + f lπ ) N 2 f lπ N Ho ] Φ π φ l (r,z) σ σ c n [ f Ho ( f uσ + f lσ ) N 2 f lσ N Ho ] Φ σ φ l (r,z)
dΦ dt = σ π c Φ π n V 1 Δ N 1π ϕ l (r,z)dV+ σ π c Φ π n V 2 Δ N 2π ϕ l (r,z)dV- Φ π τ cπ + σ σ c Φ σ n V 1 Δ N 1σ ϕ l (r,z)dV+ σ σ c Φ σ n V 2 Δ N 2σ ϕ l (r,z)dV- Φ σ τ cσ
r p (r,z)= 2α η α π ω p 2 (z) exp[ 2 r 2 ω p 2 (z) ]exp(αz)
ω p 2 (z)= ω p0 2 + [ λ p (zd) π ω p0 2 ] 2
ϕ l (r,z)= 2 π ω l 2 l eff exp( 2 r 2 ω l 2 )
τ cπ/σ = 2 l eff c δ π/σ
δ π/σ = δ fπ/σ + δ dπ/σ +ln( 1 1 T OC )
δ d =1 | 0 r b e iΔφ(r) e 2 r 2 / ω l 2 rdr 0 r b e 2 r 2 / ω l 2 rdr | 2
Δφ(r)= dn dT ξ P in η α K c λ (1+ln r b 2 ω p 2 ) (r ω P )
Δφ(r)= dn dT ξ P in η α K c λ ( r 2 ω p 2 +ln r b 2 r 2 ) (r ω P )
G Total = Φ π G π + Φ σ G σ Φ = Φ π G π + Φ σ G σ Φ π + Φ σ
G π = σ π n V 1 Δ N 1π φ l (r,z)dV
G σ = σ σ n V 1 Δ N 1σ φ l (r,z)dV
L Total = Φ π L π + Φ σ L σ Φ = Φ π L π + Φ σ L σ Φ π + Φ σ
L π = σ π n V 2 Δ N 2π φ l (r,z)dV+ δ π 2 l eff
L σ = σ σ n V 2 Δ N 2σ φ l (r,z)dV+ δ σ 2 l eff
G N = G Total L Total
G Nπ = G π L π
G Nσ = G σ L σ