Abstract

We investigate the effect of temperature gradients in high-power Yb-doped fiber amplifiers by a numerical beam propagation model, which takes thermal effects into account in a self-consistent way. The thermally induced change in the refractive index of the fiber leads to a thermal lensing effect, which decreases the effective mode area. Furthermore, it is demonstrated that the thermal lensing effect may lead to effective multi-mode behavior, even in single-mode designs, which could possibly lead to degradation of the output beam quality.

© 2011 Optical Society of America

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References

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  1. Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088–6092 (2004).
    [Crossref] [PubMed]
  2. D. Gapontsev, “6 kW CW single mode Ytterbium fiber laser in all-fiber format,” in Proc. Solid State and Diode Laser Technology Review (2008).
  3. J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
    [Crossref]
  4. J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Perschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express 11, 2982–2990 (2003).
    [Crossref] [PubMed]
  5. S. Hädrich, T. Schreiber, T. Pertsch, J. Limpert, T. Peschel, R. Eberhardt, and A. Tünnermann, “Thermo-optical behavior of rare-earth-doped low-NA fibers in high power operation,” Opt. Express 14, 6091–6097 (2006).
    [Crossref] [PubMed]
  6. J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
    [Crossref]
  7. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35, 94–96 (2010).
    [Crossref] [PubMed]
  8. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011).
    [Crossref] [PubMed]
  9. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011).
    [Crossref] [PubMed]
  10. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19, 3258–3271 (2011).
    [Crossref] [PubMed]
  11. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
    [Crossref] [PubMed]
  12. M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 29, 1739–1741 (2004).
    [Crossref] [PubMed]
  13. R. W. Boyd, Nonlinear Optics, 3rd. ed. (Elsevier2008).
  14. P. W. Milonni and J. H. Eberly, Lasers (John Wiley & Sons1988).
  15. D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136, A954–A957 (1964).
    [Crossref]
  16. G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16, 624–626 (1991).
    [Crossref] [PubMed]
  17. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University Press1989).
  18. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15, 15402–15409 (2007).
    [Crossref] [PubMed]

2011 (4)

2010 (1)

2009 (1)

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

2008 (1)

J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
[Crossref]

2007 (1)

2006 (1)

2004 (2)

2003 (1)

1991 (1)

1964 (1)

D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136, A954–A957 (1964).
[Crossref]

Andersen, T. V.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd. ed. (Elsevier2008).

Cao, X.

J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
[Crossref]

Duan, K.

J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
[Crossref]

Eberhardt, R.

Eberly, J. H.

P. W. Milonni and J. H. Eberly, Lasers (John Wiley & Sons1988).

Eidam, T.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University Press1989).

Gabler, T.

Gapontsev, D.

D. Gapontsev, “6 kW CW single mode Ytterbium fiber laser in all-fiber format,” in Proc. Solid State and Diode Laser Technology Review (2008).

Guo, Y.

J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
[Crossref]

Guyenot, V.

Hadley, G. R.

Hädrich, S.

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

S. Hädrich, T. Schreiber, T. Pertsch, J. Limpert, T. Peschel, R. Eberhardt, and A. Tünnermann, “Thermo-optical behavior of rare-earth-doped low-NA fibers in high power operation,” Opt. Express 14, 6091–6097 (2006).
[Crossref] [PubMed]

Hanf, S.

Jansen, F.

Jauregui, C.

Jeong, Y.

Koshiba, M.

Li, J.

J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
[Crossref]

Liem, A.

Limpert, J.

F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19, 3258–3271 (2011).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011).
[Crossref] [PubMed]

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35, 94–96 (2010).
[Crossref] [PubMed]

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

S. Hädrich, T. Schreiber, T. Pertsch, J. Limpert, T. Peschel, R. Eberhardt, and A. Tünnermann, “Thermo-optical behavior of rare-earth-doped low-NA fibers in high power operation,” Opt. Express 14, 6091–6097 (2006).
[Crossref] [PubMed]

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Perschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express 11, 2982–2990 (2003).
[Crossref] [PubMed]

Lin, X.

J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
[Crossref]

McCumber, D. E.

D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136, A954–A957 (1964).
[Crossref]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (John Wiley & Sons1988).

Misas, C. J.

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

Nilsson, J.

Nolte, S.

Otto, H.

Payne, D. N.

Perschel, T.

Pertsch, T.

Peschel, T.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University Press1989).

Röser, F.

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

Rothhardt, J.

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

Sahu, J. K.

Saitoh, K.

Schimpf, D. N.

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

Schmidt, O.

Schreiber, T.

Seise, E.

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35, 94–96 (2010).
[Crossref] [PubMed]

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

Smith, A. V.

Smith, J. J.

Steinmetz, A.

Stutzki, F.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University Press1989).

Tünnermann, A.

F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19, 3258–3271 (2011).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011).
[Crossref] [PubMed]

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35, 94–96 (2010).
[Crossref] [PubMed]

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

S. Hädrich, T. Schreiber, T. Pertsch, J. Limpert, T. Peschel, R. Eberhardt, and A. Tünnermann, “Thermo-optical behavior of rare-earth-doped low-NA fibers in high power operation,” Opt. Express 14, 6091–6097 (2006).
[Crossref] [PubMed]

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Perschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express 11, 2982–2990 (2003).
[Crossref] [PubMed]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University Press1989).

Wang, Y.

J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
[Crossref]

Wielandy, S.

Wirth, C.

Zellmer, H.

Zhao, W.

J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
[Crossref]

IEEE J. Sel. Topics Quantum Electron. (1)

J. Limpert, F. Röser, D. N. Schimpf, E. Seise, T. Eidam, S. Hädrich, J. Rothhardt, C. J. Misas, and A. Tünnermann, “High Repetition Rate Gigawatt Peak Power Fiber Laser Systems: Challenges, Design, and Experiment,” IEEE J. Sel. Topics Quantum Electron. 15, 159–169 (2009).
[Crossref]

J. Mod. Opt. (1)

J. Li, K. Duan, Y. Wang, X. Cao, W. Zhao, Y. Guo, and X. Lin, “Theoretical analysis of the heat dissipation mechanism in Yb3+-doped double-clad fiber lasers,” J. Mod. Opt. 55, 459–471 (2008).
[Crossref]

Opt. Express (7)

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Perschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express 11, 2982–2990 (2003).
[Crossref] [PubMed]

S. Hädrich, T. Schreiber, T. Pertsch, J. Limpert, T. Peschel, R. Eberhardt, and A. Tünnermann, “Thermo-optical behavior of rare-earth-doped low-NA fibers in high power operation,” Opt. Express 14, 6091–6097 (2006).
[Crossref] [PubMed]

Y. Jeong, J. K. Sahu, D. N. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12, 6088–6092 (2004).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19, 3258–3271 (2011).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
[Crossref] [PubMed]

S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15, 15402–15409 (2007).
[Crossref] [PubMed]

Opt. Lett. (4)

Phys. Rev. (1)

D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136, A954–A957 (1964).
[Crossref]

Other (4)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University Press1989).

R. W. Boyd, Nonlinear Optics, 3rd. ed. (Elsevier2008).

P. W. Milonni and J. H. Eberly, Lasers (John Wiley & Sons1988).

D. Gapontsev, “6 kW CW single mode Ytterbium fiber laser in all-fiber format,” in Proc. Solid State and Diode Laser Technology Review (2008).

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Figures (18)

Fig. 1
Fig. 1 Simplified energy level diagram for Yb3+.

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Fig. 2
Fig. 2 Signal and pump power as a function of z for (a) forward pumping and (b) backward pumping of Fiber A.

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Fig. 3
Fig. 3 Excited state population as a function of z and r for (a) forward pumping and (b) backward pumping of Fiber A at 1 kW pump power and 1 W input signal power.

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Fig. 4
Fig. 4 Temperature increment as a function of z and r for (a) forward pumping and (b) backward pumping of Fiber A at 1 kW pump power and 1 W input signal power.

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Fig. 5
Fig. 5 Beam effective area as a function of z for (a) forward pumping and (b) backward pumping of Fiber A at 1 kW pump power and 1 W input signal power.

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Fig. 6
Fig. 6 B integral as a function of pump power for (a) forward pumping and (b) backward pumping of Fiber A and B with 1 W input signal power. The dashed lines show the results with the thermal sensitivity η set to zero.

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Fig. 7
Fig. 7 Signal and pump power as a function of z for (a) forward pumping and (b) backward pumping of Fiber C.

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Fig. 8
Fig. 8 Beam effective area as a function of z for (a) forward pumping and (b) backward pumping of Fiber C at 5 kW pump power and 50 W input signal power.

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Fig. 9
Fig. 9 Temperature increment as a function of z and r for (a) forward pumping and (b) backward pumping of Fiber C at 5 kW pump power and 50 W input signal power.

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Fig. 10
Fig. 10 Excited state population as a function of z and r for (a) forward pumping and (b) backward pumping of Fiber C at 5 kW pump power and 50 W input signal power.

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Fig. 11
Fig. 11 Thermally perturbed refractive index profiles at various z for (a) forward pumping and (b) backward pumping of Fiber C at 5 kW pump power and 50 W input signal power.

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Fig. 12
Fig. 12 Initial (z = 0) local modes for (a) forward pumping and (b) backward pumping of Fiber C at 5 kW pump power and 50 W input signal power.

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Fig. 13
Fig. 13 Final (z = 1 m) local modes for (a) forward pumping and (b) backward pumping of Fiber C at 5 kW pump power and 50 W input signal power.

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Fig. 14
Fig. 14 Local mode content as a function of z for (a) forward pumping and (b) backward pumping of Fiber C at 5 kW pump power and 50 W input signal power.

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Fig. 15
Fig. 15 Signal and pump power as a function of z for (a) forward pumping and (b) backward pumping of Fiber D.

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Fig. 16
Fig. 16 Local mode content as a function of z for (a) forward pumping and (b) backward pumping of Fiber D at 2 kW pump power and 50 W input signal power.

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Fig. 17
Fig. 17 Temperature increment as a function of z and r for (a) forward pumping and (b) backward pumping of Fiber D at 2 kW pump power and 50 W input signal power.

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Fig. 18
Fig. 18 Local mode content at the output as a function of pump power for Fiber C on (a) linear scale and (b) log scale.

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Equations (33)

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

E s ( r , t ) = u s E s ( r , z ) e i ( β z − ω s t ) ,
∂ E s ∂ z = i 2 β [ ∂ 2 E s ∂ r 2 + 1 r ∂ E s ∂ r + ( k 0 2 [ ɛ ( r ) + Δ ɛ ( r , z ) ] − β 2 ) E s + μ 0 ω s 2 p ( r , z ) ] .
P Y b ( r , t ) = u s p ( r , z ) e i ( β z − ω s t ) .
σ ˙ 11 = i h ¯ ( μ 12 E p * σ 21 − c . c . ) + γ 21 σ 22 + γ ¯ 31 σ 33 − γ ¯ 13 σ 11
σ ˙ 22 = − i h ¯ ( μ 12 E p * σ 21 + μ 32 E s * σ 23 − c . c . ) − ( γ 23 + γ 21 ) σ 22
σ ˙ 33 = i h ¯ ( μ 32 E s * σ 23 − c . c . ) + γ 23 σ 22 − γ ¯ 31 σ 33 + γ ¯ 13 σ 11
σ ˙ 21 = i h ¯ ( μ 12 * E p ( σ 11 − σ 22 ) + μ 32 * E s σ 31 ) − Γ ˜ 21 σ 21
σ ˙ 23 = i h ¯ ( μ 32 * E s ( σ 33 − σ 22 ) + μ 12 * E p σ 31 * ) − Γ ˜ 23 σ 23
σ ˙ 31 = i h ¯ ( μ 32 E s * σ 21 − μ 12 * E p σ 23 * ) − Γ ˜ 31 σ 31 ,
Γ ˜ 21 = γ ˜ 21 − i Δ p
Γ ˜ 23 = γ ˜ 23 − i Δ s
Γ ˜ 31 = γ ˜ 31 − i Δ 2 ,
σ ˙ 22 = B p ( Δ p ) | E p | 2 ( σ 11 − σ 22 ) + B s ( Δ s ) | E s | 2 ( σ 33 − σ 22 ) − ( γ 23 + γ 21 ) σ 22 ,
B p = 2 γ ˜ 21 | μ 12 | 2 h ¯ 2 ( γ ˜ 21 2 + Δ p 2 )               and                     B s = 2 γ ˜ 23 | μ 32 | 2 h ¯ 2 ( γ ˜ 23 2 + Δ s 2 ) .
ρ 2 ( r , z ) = σ ap Φ p ( z ) + σ as Φ s ( r , z ) ( σ ap + σ e p ) Φ p ( z ) + ( σ as + σ es ) Φ s ( r , z ) + γ ,
σ a s ( ω s ) = e − Δ E / k B T 1 + e − Δ E / k B T h ¯ ω s B s 2 ɛ 0 c n c             and                   σ e s ( ω s ) = h ¯ ω s B s 2 ɛ 0 c n c ,
σ a p ( ω p ) = 1 1 + e − Δ E / k B T h ¯ ω p B p 2 ɛ 0 c n c           and           σ e p ( ω p ) = h ¯ ω p B p 2 ɛ 0 c n c .
p ( r , z ) = ɛ 0 n c ρ Y b ( r ) k 0 ( i + Δ s γ ˜ 23 ) [ σ a s ( ω s ) − [ σ a s ( ω s ) + σ e s ( ω s ) ] ρ 2 ( r , z ) ] E s ( r , z ) ,
∂ E s ∂ z = i 2 β ( ∂ 2 E s ∂ r 2 + 1 r ∂ E s ∂ r + ( k 0 2 ( ɛ + Δ ɛ ) − β 2 ) E s ) + ɛ k 0 ρ Y b 2 β ( σ e s ρ 2 − ( 1 − ρ 2 ) σ a s ) E s .
E s ( z + Δ z , r ) ≈ exp ( Δ z 2 R ^ ) exp ( Δ z N ^ ) exp ( Δ z 2 R ^ ) E z ( z , r ) ,
R ^ E s = i 2 β ( ∂ 2 E s ∂ r 2 + 1 r ∂ E s ∂ r ) ,
N ^ E s = [ − ρ Y b 2 ɛ ɛ eff ( σ a s − ( σ a s + σ e s ) ρ 2 ) + i k 0 2 2 β ( ɛ + Δ ɛ − ɛ eff ) ] E s ,
E s ( 0 , r ) = P s ( 0 ) π ɛ 0 c n c w 2 e − r 2 / w 2 ,
∂ 2 Δ T ∂ r 2 + 1 r ∂ Δ T ∂ r = − Q κ .
Q = ρ Y b h ¯ ( ω p − ω s ) ( Φ s ( σ e s ρ 2 − σ a s ρ 1 ) + γ 23 ρ 2 ) ,
d P p d z = ± 2 π A cl P p ( z ) ∫ 0 R c ρ Y b ( r ) [ σ ap − ( σ ap + σ ep ) ρ 2 ( r , z ) ] r d r ,
P p ( z + Δ z ) ≈ P p ( z ) ± 2 π A cl P p ( z ) Δ z ∫ 0 R c ρ Y b ( r ) [ σ a p − ( σ a p + σ e p ) ρ 2 ( r , z ) ] r d r .
B = 2 π n 2 λ s ∫ 0 L P s ( z ) A eff ( z ) d z ,
A eff ( z ) = 2 π ( ∫ 0 ∞ | E ( z , r ) | 2 r d r ) 2 ∫ 0 ∞ | E ( z , r ) | 4 r d r .
∂ 2 Ψ ∂ r 2 + 1 r ∂ Ψ ∂ r + k 0 2 [ ɛ ( r ) + Δ ɛ ( r , z ) ] Ψ ( r , z ) = β 2 ( z ) Ψ ( r , z ) ,
a i ( z ) = 〈 Ψ i | E s 〉 ,
〈 Ψ | Φ 〉 = ∫ 0 ∞ Ψ * ( r ) Φ ( r ) r d r .
A i ( z ) = | a i ( z ) | 2 〈 E s | E s 〉

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