Abstract

A detailed formalism to achieve an analytical solution of a lossy high power Yb-doped silica fiber laser is introduced. The solutions for the lossless fiber laser is initially attained in detail. Next, the solution for the lossy fiber laser is obtained based on the lossless fiber laser solution. To examine the solutions for both the lossless and lossy fiber laser, two sets of values are compared with the exact numerical solutions, and the results are in a good agreement. Furthermore, steps and procedures for achieving the final solutions are explained clearly and precisely.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

M. Peysokhan, E. Mobini, A. Allahverdi, B. Abaie, and A. Mafi, “Characterization of Yb-doped ZBLAN fiber as a platform for radiation-balanced lasers,” Photonics Res. 8(2), 202–210 (2020).
[Crossref]

2019 (2)

2018 (1)

2017 (1)

2016 (3)

2014 (5)

Z. Mohammed, H. Saghafifar, and M. Soltanolkotabi, “An approximate analytical model for temperature and power distribution in high-power Yb-doped double-clad fiber lasers,” Laser Phys. 24(11), 115107 (2014).
[Crossref]

Z. Mohammed and H. Saghafifar, “Optimization of strongly pumped Yb-doped double-clad fiber lasers using a wide-scope approximate analytical model,” Opt. Laser Technol. 55, 50–57 (2014).
[Crossref]

M. N. Zervas and C. A. Codemard, “High power fiber lasers: A review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

W. Shi, Q. Fang, X. Zhu, R. A. Norwood, and N. Peyghambarian, “Fiber lasers and their applications,” Appl. Opt. 53(28), 6554–6568 (2014).
[Crossref]

M. N. Zervas and C. A. Codemard, “High power fiber lasers: a review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

2013 (2)

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

Z. Liu, P. Zhou, X. Xu, X. Wang, and Y. Ma, “Coherent beam combining of high power fiber lasers: Progress and prospect,” Sci. China: Technol. Sci. 56(7), 1597–1606 (2013).
[Crossref]

2012 (1)

2011 (1)

P. Yan, S. Yin, J. He, C. Fu, Y. Wang, and M. Gong, “1.1-kw Ytterbium monolithic fiber laser with assembled end-pump scheme to couple high brightness single emitters,” IEEE Photonics Technol. Lett. 23(11), 697–699 (2011).
[Crossref]

2010 (1)

2009 (3)

2008 (1)

2007 (2)

Z. Duan, L. Zhang, and J. Chen, “Analytical characterization of an end pumped rare-earth-doped double-clad fiber laser,” Opt. Fiber Technol. 13(2), 143–148 (2007).
[Crossref]

S. Klingebiel, F. Röser, B. Ortaç, J. Limpert, and A. Tünnermann, “Spectral beam combining of Yb-doped fiber lasers with high efficiency,” J. Opt. Soc. Am. B 24(8), 1716–1720 (2007).
[Crossref]

2004 (2)

J. Nilsson, W. Clarkson, R. Selvas, J. Sahu, P. Turner, S.-U. Alam, and A. Grudinin, “High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers,” Opt. Fiber Technol. 10(1), 5–30 (2004).
[Crossref]

L. Xiao, P. Yan, M. Gong, W. Wei, and P. Ou, “An approximate analytic solution of strongly pumped Yb-doped double-clad fiber lasers without neglecting the scattering loss,” Opt. Commun. 230(4-6), 401–410 (2004).
[Crossref]

1999 (2)

I. Kelson and A. Hardy, “Optimization of strongly pumped fiber lasers,” J. Lightwave Technol. 17(5), 891–897 (1999).
[Crossref]

S. R. Bowman, “Lasers without internal heat generation,” IEEE J. Quantum Electron. 35(1), 115–122 (1999).
[Crossref]

1998 (1)

I. Kelson and A. A. Hardy, “Strongly pumped fiber lasers,” IEEE J. Quantum Electron 34(9), 1570–1577 (1998).
[Crossref]

1995 (1)

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1-1.2 µm region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995).
[Crossref]

1991 (1)

J. Y. Allain, M. Monerie, and H. Poignant, “Tunable cw lasing around 610, 635, 695, 715, 885 and 910 nm in praseodymium-doped fluorozirconate fibre,” Electron. Lett. 27(2), 189–191 (1991).
[Crossref]

1964 (1)

Abaie, B.

M. Peysokhan, E. Mobini, A. Allahverdi, B. Abaie, and A. Mafi, “Characterization of Yb-doped ZBLAN fiber as a platform for radiation-balanced lasers,” Photonics Res. 8(2), 202–210 (2020).
[Crossref]

E. Mobini, M. Peysokhan, B. Abaie, and A. Mafi, “Thermal modeling, heat mitigation, and radiative cooling for double-clad fiber amplifiers,” J. Opt. Soc. Am. B 35(10), 2484–2493 (2018).
[Crossref]

Aghbolagh, F.

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, vol. 222 (John Wiley & Sons, 2012).

Alam, S.-U.

J. Nilsson, W. Clarkson, R. Selvas, J. Sahu, P. Turner, S.-U. Alam, and A. Grudinin, “High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers,” Opt. Fiber Technol. 10(1), 5–30 (2004).
[Crossref]

Allahverdi, A.

M. Peysokhan, E. Mobini, A. Allahverdi, B. Abaie, and A. Mafi, “Characterization of Yb-doped ZBLAN fiber as a platform for radiation-balanced lasers,” Photonics Res. 8(2), 202–210 (2020).
[Crossref]

Allain, J. Y.

J. Y. Allain, M. Monerie, and H. Poignant, “Tunable cw lasing around 610, 635, 695, 715, 885 and 910 nm in praseodymium-doped fluorozirconate fibre,” Electron. Lett. 27(2), 189–191 (1991).
[Crossref]

Barber, P. R.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1-1.2 µm region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995).
[Crossref]

Beier, F.

Benabid, F.

Bowman, S. R.

S. R. Bowman, “Lasers without internal heat generation,” IEEE J. Quantum Electron. 35(1), 115–122 (1999).
[Crossref]

Boyland, A. J.

Breitkopf, S.

Brunet, F.

Carman, R. J.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1-1.2 µm region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995).
[Crossref]

Carter, A. L.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron. 15(1), 85–92 (2009).
[Crossref]

Chen, J.

Z. Duan, L. Zhang, and J. Chen, “Analytical characterization of an end pumped rare-earth-doped double-clad fiber laser,” Opt. Fiber Technol. 13(2), 143–148 (2007).
[Crossref]

Chen, X.

Chung, S.-H.

Clarkson, W.

J. Nilsson, W. Clarkson, R. Selvas, J. Sahu, P. Turner, S.-U. Alam, and A. Grudinin, “High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers,” Opt. Fiber Technol. 10(1), 5–30 (2004).
[Crossref]

Clarkson, W. A.

Codemard, C. A.

M. N. Zervas and C. A. Codemard, “High power fiber lasers: A review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

M. N. Zervas and C. A. Codemard, “High power fiber lasers: a review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

Dawes, J. M.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1-1.2 µm region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995).
[Crossref]

Debord, B.

Dong, L.

Duan, Z.

Z. Duan, L. Zhang, and J. Chen, “Analytical characterization of an end pumped rare-earth-doped double-clad fiber laser,” Opt. Fiber Technol. 13(2), 143–148 (2007).
[Crossref]

Eberhardt, R.

Fang, Q.

Faucher, M.

Frith, G.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron. 15(1), 85–92 (2009).
[Crossref]

Fu, C.

P. Yan, S. Yin, J. He, C. Fu, Y. Wang, and M. Gong, “1.1-kw Ytterbium monolithic fiber laser with assembled end-pump scheme to couple high brightness single emitters,” IEEE Photonics Technol. Lett. 23(11), 697–699 (2011).
[Crossref]

Fu, L.

Gerome, F.

Gong, M.

P. Yan, S. Yin, J. He, C. Fu, Y. Wang, and M. Gong, “1.1-kw Ytterbium monolithic fiber laser with assembled end-pump scheme to couple high brightness single emitters,” IEEE Photonics Technol. Lett. 23(11), 697–699 (2011).
[Crossref]

L. Xiao, P. Yan, M. Gong, W. Wei, and P. Ou, “An approximate analytic solution of strongly pumped Yb-doped double-clad fiber lasers without neglecting the scattering loss,” Opt. Commun. 230(4-6), 401–410 (2004).
[Crossref]

Grudinin, A.

J. Nilsson, W. Clarkson, R. Selvas, J. Sahu, P. Turner, S.-U. Alam, and A. Grudinin, “High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers,” Opt. Fiber Technol. 10(1), 5–30 (2004).
[Crossref]

Gu, G.

Haarlammert, N.

Hanna, D. C.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1-1.2 µm region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995).
[Crossref]

Hardy, A.

Hardy, A. A.

Hawkins, T. W.

He, B.

He, J.

P. Yan, S. Yin, J. He, C. Fu, Y. Wang, and M. Gong, “1.1-kw Ytterbium monolithic fiber laser with assembled end-pump scheme to couple high brightness single emitters,” IEEE Photonics Technol. Lett. 23(11), 697–699 (2011).
[Crossref]

Hein, S.

Hess, O.

Holehouse, N.

Hu, M.

Hupel, C.

Jauregui, C.

Jeong, Y.-C.

Jetschke, S.

Kalichevsky-Dong, M.

Kanskar, M.

Kelson, I.

I. Kelson and A. Hardy, “Optimization of strongly pumped fiber lasers,” J. Lightwave Technol. 17(5), 891–897 (1999).
[Crossref]

I. Kelson and A. A. Hardy, “Strongly pumped fiber lasers,” IEEE J. Quantum Electron 34(9), 1570–1577 (1998).
[Crossref]

Kirchhof, J.

Klingebiel, S.

Koester, C. J.

Kuhn, S.

Leich, M.

Li, W.

Liem, A.

Limpert, J.

Liu, G.

Liu, K.

Liu, Z.

Z. Liu, P. Zhou, X. Xu, X. Wang, and Y. Ma, “Coherent beam combining of high power fiber lasers: Progress and prospect,” Sci. China: Technol. Sci. 56(7), 1597–1606 (2013).
[Crossref]

Ma, Y.

Z. Liu, P. Zhou, X. Xu, X. Wang, and Y. Ma, “Coherent beam combining of high power fiber lasers: Progress and prospect,” Sci. China: Technol. Sci. 56(7), 1597–1606 (2013).
[Crossref]

Mackechnie, C. J.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1-1.2 µm region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995).
[Crossref]

Mafi, A.

M. Peysokhan, E. Mobini, A. Allahverdi, B. Abaie, and A. Mafi, “Characterization of Yb-doped ZBLAN fiber as a platform for radiation-balanced lasers,” Photonics Res. 8(2), 202–210 (2020).
[Crossref]

E. Mobini, M. Peysokhan, B. Abaie, and A. Mafi, “Thermal modeling, heat mitigation, and radiative cooling for double-clad fiber amplifiers,” J. Opt. Soc. Am. B 35(10), 2484–2493 (2018).
[Crossref]

Matniyaz, T.

McKay, H. A.

Mobini, E.

M. Peysokhan, E. Mobini, A. Allahverdi, B. Abaie, and A. Mafi, “Characterization of Yb-doped ZBLAN fiber as a platform for radiation-balanced lasers,” Photonics Res. 8(2), 202–210 (2020).
[Crossref]

E. Mobini, M. Peysokhan, B. Abaie, and A. Mafi, “Thermal modeling, heat mitigation, and radiative cooling for double-clad fiber amplifiers,” J. Opt. Soc. Am. B 35(10), 2484–2493 (2018).
[Crossref]

Mohammed, Z.

Z. Mohammed, H. Saghafifar, and M. Soltanolkotabi, “An approximate analytical model for temperature and power distribution in high-power Yb-doped double-clad fiber lasers,” Laser Phys. 24(11), 115107 (2014).
[Crossref]

Z. Mohammed and H. Saghafifar, “Optimization of strongly pumped Yb-doped double-clad fiber lasers using a wide-scope approximate analytical model,” Opt. Laser Technol. 55, 50–57 (2014).
[Crossref]

Monerie, M.

J. Y. Allain, M. Monerie, and H. Poignant, “Tunable cw lasing around 610, 635, 695, 715, 885 and 910 nm in praseodymium-doped fluorozirconate fibre,” Electron. Lett. 27(2), 189–191 (1991).
[Crossref]

Moulton, P. F.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron. 15(1), 85–92 (2009).
[Crossref]

Nampoothiri, V.

Nilsson, J.

Nold, J.

Norwood, R. A.

Ortaç, B.

Otto, H.-J.

Ou, P.

L. Xiao, P. Yan, M. Gong, W. Wei, and P. Ou, “An approximate analytic solution of strongly pumped Yb-doped double-clad fiber lasers without neglecting the scattering loss,” Opt. Commun. 230(4-6), 401–410 (2004).
[Crossref]

Parsons, J.

Pask, H. M.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1-1.2 µm region,” IEEE J. Sel. Top. Quantum Electron. 1(1), 2–13 (1995).
[Crossref]

Payne, D. N.

Peng, X.

Peyghambarian, N.

Peysokhan, M.

M. Peysokhan, E. Mobini, A. Allahverdi, B. Abaie, and A. Mafi, “Characterization of Yb-doped ZBLAN fiber as a platform for radiation-balanced lasers,” Photonics Res. 8(2), 202–210 (2020).
[Crossref]

E. Mobini, M. Peysokhan, B. Abaie, and A. Mafi, “Thermal modeling, heat mitigation, and radiative cooling for double-clad fiber amplifiers,” J. Opt. Soc. Am. B 35(10), 2484–2493 (2018).
[Crossref]

Poignant, H.

J. Y. Allain, M. Monerie, and H. Poignant, “Tunable cw lasing around 610, 635, 695, 715, 885 and 910 nm in praseodymium-doped fluorozirconate fibre,” Electron. Lett. 27(2), 189–191 (1991).
[Crossref]

Proske, F.

Richardson, D. J.

Rines, G. A.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron. 15(1), 85–92 (2009).
[Crossref]

Röser, F.

Rudolph, W.

Saghafifar, H.

Z. Mohammed, H. Saghafifar, and M. Soltanolkotabi, “An approximate analytical model for temperature and power distribution in high-power Yb-doped double-clad fiber lasers,” Laser Phys. 24(11), 115107 (2014).
[Crossref]

Z. Mohammed and H. Saghafifar, “Optimization of strongly pumped Yb-doped double-clad fiber lasers using a wide-scope approximate analytical model,” Opt. Laser Technol. 55, 50–57 (2014).
[Crossref]

Sahu, J.

J. Nilsson, W. Clarkson, R. Selvas, J. Sahu, P. Turner, S.-U. Alam, and A. Grudinin, “High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers,” Opt. Fiber Technol. 10(1), 5–30 (2004).
[Crossref]

Sahu, J. K.

Samson, B.

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron. 15(1), 85–92 (2009).
[Crossref]

Sattler, B.

Schreiber, T.

Schwuchow, A.

Selvas, R.

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J. Nilsson, W. Clarkson, R. Selvas, J. Sahu, P. Turner, S.-U. Alam, and A. Grudinin, “High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers,” Opt. Fiber Technol. 10(1), 5–30 (2004).
[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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Z. Mohammed, H. Saghafifar, and M. Soltanolkotabi, “An approximate analytical model for temperature and power distribution in high-power Yb-doped double-clad fiber lasers,” Laser Phys. 24(11), 115107 (2014).
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L. Xiao, P. Yan, M. Gong, W. Wei, and P. Ou, “An approximate analytic solution of strongly pumped Yb-doped double-clad fiber lasers without neglecting the scattering loss,” Opt. Commun. 230(4-6), 401–410 (2004).
[Crossref]

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Z. Duan, L. Zhang, and J. Chen, “Analytical characterization of an end pumped rare-earth-doped double-clad fiber laser,” Opt. Fiber Technol. 13(2), 143–148 (2007).
[Crossref]

J. Nilsson, W. Clarkson, R. Selvas, J. Sahu, P. Turner, S.-U. Alam, and A. Grudinin, “High-power wavelength-tunable cladding-pumped rare-earth-doped silica fiber lasers,” Opt. Fiber Technol. 10(1), 5–30 (2004).
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Z. Liu, P. Zhou, X. Xu, X. Wang, and Y. Ma, “Coherent beam combining of high power fiber lasers: Progress and prospect,” Sci. China: Technol. Sci. 56(7), 1597–1606 (2013).
[Crossref]

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G. P. Agrawal, Fiber-Optic Communication Systems, vol. 222 (John Wiley & Sons, 2012).

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Figures (4)

Fig. 1.
Fig. 1. Schematic of the laser system and propagation of the pump power and signal in the double-cladding fiber laser. Pump power is launched at z = 0 and the output signal is calculated at z = L at the power delivery port. $R_1$ and $R_2$ are the distributed Bragg reflectors at $z = 0$ and $z = L$.
Fig. 2.
Fig. 2. a) Comparison of the propagation of the analytical forward pump (Anlt. FW pump), analytical forward signal (Anlt. FW signal), and analytical backward signal (Anlt. BW signal) with their exact numerical counterpart solutions which are the exact numerical forward pump (Num. FW pump), exact numerical forward signal (Num. FW signal), and exact numerical backward signal (Num. BW signal) for the set of Value 1 which are represented in Table 1. b) A similar graph for the set of Value 2.
Fig. 3.
Fig. 3. a) Results of the analytical solution of a lossy fiber lasers which is a comparison of the propagation of the analytical forward pump (Anlt. FW pump), analytical forward signal (Anlt. FW signal), and analytical backward signal (Anlt. BW signal) with their exact numerical counterpart solutions which are the exact numerical forward pump (Num. FW pump), exact numerical forward signal (Num. FW signal), and exact numerical backward signal (Num. BW signal) for the set of Value 1 which are represented in Table 1. b) A similar graph for the set of Value 2.
Fig. 4.
Fig. 4. Comparison of the $\Pi (z)$, $\Sigma (z)$, and $N_2(z)/N$ along the doped fiber for the Value 1 that are listed in Table 1

Tables (1)

Tables Icon

Table 1. YDCFL parameters

Equations (48)

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N 2 ( z ) N = Γ p σ p a P p ~ ( z ) h ν p A + Γ s σ s a P s   ~ ( z ) h ν s A Γ p σ p a e P p ~ ( z ) h ν p A + 1 τ + Γ s σ s a e P s   ~ ( z ) h ν s A ,
P p ~ ( z ) := P p + ( z ) + P p ( z ) , σ p a e := σ p a + σ p e ,
P s   ~ ( z ) := P s + ( z ) + P s ( z ) , σ s a e := σ s a + σ s e .
+ d P p + ( z ) d z = Γ p ( σ p a e N 2 ( z ) σ p a N ) P p + ( z ) α p P p + ( z ) ,
d P p ( z ) d z = Γ p ( σ p a e N 2 ( z ) σ p a N ) P p ( z ) α p P p ( z ) .
+ d P s + ( z ) d z = Γ s ( σ s a e N 2 ( z ) σ s a N ) P s + ( z ) α s P s + ( z ) ,
d P s ( z ) d z = Γ s ( σ s a e N 2 ( z ) σ s a N ) P s ( z ) α s P s ( z ) .
+ d P p + ( z ) d z ( Γ p σ p a N + α p ) P p + ( z ) ,
P p + ( z ) = P p + ( 0 ) e α z ,
α := ( Γ p σ p a N + α p ) .
P s + ( 0 ) = R 1 P s ( 0 ) ,
P s ( L ) = R 2 P s + ( L ) .
d P s + ( z ) P s + ( z ) + d P s ( z ) P s ( z ) = 0 ,
P s + ( 0 ) P s ( 0 ) = P s + ( L ) P s ( L ) = R 1 P s ( 0 ) 2 .
τ h ν p A ( Γ p ( σ p a e N 2 ( z ) σ p a N ) ) P p ~ + τ h ν s A ( Γ s ( σ s a e N 2 ( z ) σ s a N ) ) P s   ~ + N 2 ¯ = 0.
( d P p + d z d P p d z ) + α p P p ~ = Γ p ( σ p a e N 2 ( z ) σ p a N ) P p ~   ,
( d P s + d z d P s d z ) + α s P s   ~ = Γ s ( σ s a e N 2 ( z ) σ s a N ) P s   ~ .
τ h ν s A ( ( d P s + d z d P s d z ) + α s P s   ~ ) + τ h ν p A ( ( d P p + d z d P p d z ) + α p P p ~ ) + N 2 ¯ = 0.
( d P s + d z d P s d z ) + α s ( P s + + P s ) + ν s ν p e α z P p + ( 0 ) ( α p α ) + h ν s A N 2 ¯ τ = 0.
P s + ( z ) P s + ( 0 ) P s ( z ) + P s ( 0 ) + ν s ν p P p + ( 0 ) ( α p α ) ( e α z 1 α ) + h ν s A N 2 ¯ z τ = 0.
P s + ( z ) 2 + ( ( 1 R 1 ) P s ( 0 ) ν s ν p P p + ( 0 ) ( 1 α p α ) ( 1 e α z ) + h ν s A N 2 ¯ z τ ) P s + ( z ) R 1 P s ( 0 ) 2 = 0.
X 2 + 2 b X c = 0 , X = P s + ( z ) ,
b := ( 1 R 1 ) P s ( 0 ) 2 ν s 2 ν p P p + ( 0 ) ( 1 α p α ) ( 1 e α z ) + h ν s A N 2 ¯ z 2 τ , c := R 1 P s ( 0 ) 2 .
X P s + ( z ) = b + b 2 + c .
X 2 2 b X c = 0.
X P s ( z ) = b + b 2 + c .
{ P s + ( z ) = b + b 2 + R 1 P s ( 0 ) 2 , P s ( z ) = + b + b 2 + R 1 P s ( 0 ) 2 .
{ P s + ( L ) P s ( L ) = P s + ( 0 ) P s ( 0 ) , P s ( L ) = R 2 P s + ( L ) , P s + ( 0 ) = R 1 P s ( 0 ) ,
{ R 2 P s + ( L ) 2 = R 1 P s ( 0 ) 2 , P s + ( L ) = R 1 R 2 1 P s ( 0 ) ,
P s + ( L ) 2 + 2 b L P s + ( L ) c = 0.
b ( L ) ( 1 R 1 ) P s ( 0 ) 2 + f ( L ) ,
f ( L ) ν s 2 ν p P p + ( 0 ) ( 1 α p α ) ( 1 e α L ) + h ν s A N 2 ¯ L 2 τ .
R 1 R 2 P s ( 0 ) 2 + 2 ( 1 R 1 2 P s ( 0 ) + f ( L ) ) R 1 R 2 P s ( 0 ) R 1 P s + ( 0 ) 2 = 0.
( R 1 R 2 + ( 1 R 1 ) R 1 R 2 R 1 ) P s ( 0 ) + 2 f ( L ) R 1 R 2 = 0 , ( R 1 R 2 + ( R 1 1 ) R 1 R 2 ) P s ( 0 ) = 2 f ( L ) , ( R 1 R 2 + R 2 R 1 R 1 R 2 ) P s ( 0 ) = 2 R 2 f ( L ) .
P s ( 0 ) = R 2 [ R 1 ( 1 R 2 ) + R 2 ( 1 R 1 ) ] × [ ( ν s ν p ) P p + ( 0 ) ( 1 α p α ) ( 1 e α L ) h ν s A N 2 ¯ L τ ] .
P s ( 0 ) = R 2 [ R 1 ( 1 R 2 ) + R 2 ( 1 R 1 ) ] × [ ( ν s ν p ) P p + ( 0 ) ( 1 α p α ) ( 1 e α L ) h ν s A τ ( ln 1 R 1 R 2 + ( Γ s σ s a N + α s ) L Γ s ( σ s e + σ s a ) ) ] .
P s + ( z ) P s + ( 0 ) P s ( z ) + P s ( 0 ) + ν s ν p P p + ( 0 ) ( α p α ) ( e α z 1 α ) + h ν s A N 2 ¯ z τ + α s 0 z ( P s + ( z ) + P s ( z ) ) d z = 0.
δ = ( α s 2 ) 0 z ( P s + ( z ) + P s ( z ) ) d z ,
{ P s δ + ( z ) = ( b + δ ) + ( b + δ ) 2 + R 1 P s δ ( 0 ) 2 , P s δ ( z ) = + ( b + δ ) + ( b + δ ) 2 + R 1 P s δ ( 0 ) 2 .
δ = α s 0 z b ( z ) 2 + R 1 P s ( 0 ) 2 d z .
δ ( z ) α s b ( z / 2 ) 2 + R 1 P s ( 0 ) 2 z ,
P p ~ ( z ) ( τ Γ p h ν p A ) ( σ p a ( N 2 ( z ) / N ) σ p a e ) = N 2 ( z ) N + P s   ~ ( τ Γ s h ν s A ) ( ( N 2 ( z ) / N ) σ s a e σ s a ) ,
Π ( z ) := P p ~ ( z ) ( τ Γ p h ν p A ) ( σ p a ( N 2 ( z ) / N ) σ p a e ) , Σ ( z ) := P s   ~ ( z ) ( τ Γ s h ν s A ) ( ( N 2 ( z ) / N ) σ s a e σ s a ) .
N 2 ¯ = 1 L 0 L N 2 ( z ) d z .
G = 0 L d P s + ( z ) P s + ( z ) + L 0 d P s ( z ) P s ( z ) = 0 L ( Γ s ( σ s a e N 2 ( z ) σ s a N ) α s ) d z L 0 ( Γ s ( σ s a e N 2 ( z ) σ s a N ) α s ) d z = 2 Γ s σ s a e 0 L N 2 ( z ) d z 2 ( Γ s σ s a N + α ) L ,
1 L 0 L N 2 ( z ) d z = N 2 ¯ = G 2 L + ( Γ s σ s a N + α s ) Γ s σ s a e .
R 1 R 2 e G = 1 G 2 L = 1 L ln 1 R 1 R 2 .
N 2 ¯ = 1 L ln 1 R 1 R 2 + ( Γ s σ s a N + α s ) Γ s ( σ s e + σ s a ) ,