Abstract

We develop an improved moment method to model soliton propagation in optical fibers. We account for the full Raman gain spectrum of the material and derive a system of coupled differential equations describing the evolution of five moments of the pulse, valid for arbitrary soliton durations. By comparing with the numerical solution of the generalized nonlinear Schrödinger equation, the improved moment method is shown to accurately represent soliton self-frequency shift under complex dispersion, nonlinearity, and Raman gain spectra. Numerical examples are presented for a dispersion-shifted fused silica fiber and a non-uniform ZBLAN fluoride fiber taper. The latter demonstrates an enhanced soliton self-frequency shift through axial dispersion and nonlinearity engineering along the taper length.

© 2010 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  6. H. Lim, J. Buckley, A. Chong, and F. W. Wise, “Fibre-based source of femtosecond pulses tunable from 1.0 to 1.3 μm,” Electron. Lett. 40, 1523–1525 (2004).
    [CrossRef]
  7. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [CrossRef]
  8. Z. Chen, A. J. Taylor, and A. Efimov, “Coherent mid-infrared broadband continuum generation in non-uniform ZBLAN fiber taper,” Opt. Express 17, 5852–5860 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
    [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] [PubMed]
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    [CrossRef] [PubMed]
  24. A. Efimov, “Fundamental nonlinear-optical interactions in photonic fibers: time-spectral visualization,” Laser Phys. 18, 667–681 (2008).
    [CrossRef]
  25. D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function,” J. Opt. Soc. Am. B 19, 2886–2892 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  29. J. C. Travers, J. M. Stone, A. B. Rulkov, B. A. Cumberland, A. K. George, S. V. Popov, J. C. Knight, and J. R. Taylor, “Optical pulse compression in dispersion decreasing photonic crystal fiber,” Opt. Express 15, 13203–13211 (2007).
    [CrossRef] [PubMed]

2009 (2)

2008 (2)

A. A. Voronin and A. M. Zheltikov, “Soliton self-frequency shift decelerated by self-steepening,” Opt. Lett. 33, 1723–1725 (2008).
[CrossRef] [PubMed]

A. Efimov, “Fundamental nonlinear-optical interactions in photonic fibers: time-spectral visualization,” Laser Phys. 18, 667–681 (2008).
[CrossRef]

2007 (2)

2006 (3)

2004 (2)

H. Lim, J. Buckley, A. Chong, and F. W. Wise, “Fibre-based source of femtosecond pulses tunable from 1.0 to 1.3 μm,” Electron. Lett. 40, 1523–1525 (2004).
[CrossRef]

P. T. Dinda, A. Labruyère, and K. Nakkeeran, “Theory of Raman effect on solitons in optical fibre systems: impact and control processes for high-speed long-distance transmission lines,” Opt. Commun. 234, 137–151 (2004).
[CrossRef]

2003 (1)

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[CrossRef]

2002 (2)

N. Nishizawa, Y. Ito, and T. Goto, “0.78–0.90-μm wavelength-tunable femtosecond soliton pulse generation using photonic crystal fiber,” IEEE Photon. Technol. Lett. 14, 986–988 (2002).
[CrossRef]

D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function,” J. Opt. Soc. Am. B 19, 2886–2892 (2002).
[CrossRef]

2001 (1)

1995 (2)

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

1993 (1)

1992 (1)

1991 (1)

1989 (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

1987 (1)

1986 (3)

1985 (2)

A. Saissy, J. Botineau, L. Macon, and G. Maze, “Raman scattering in a fluorozirconate glass optical fiber,” J. Phys. (France) Lett. 46, 289–294 (1985).

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294–297 (1985).

Agrawal, G. P.

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, 1992).

Bang, O.

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

Botineau, J.

A. Saissy, J. Botineau, L. Macon, and G. Maze, “Raman scattering in a fluorozirconate glass optical fiber,” J. Phys. (France) Lett. 46, 289–294 (1985).

Brabec, T.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Buckley, J.

H. Lim, J. Buckley, A. Chong, and F. W. Wise, “Fibre-based source of femtosecond pulses tunable from 1.0 to 1.3 μm,” Electron. Lett. 40, 1523–1525 (2004).
[CrossRef]

Burgoyne, B.

Cantrell, C. D.

Chandalia, J. K.

Chen, Z.

Chernikov, S. V.

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, 1992).

Chong, A.

H. Lim, J. Buckley, A. Chong, and F. W. Wise, “Fibre-based source of femtosecond pulses tunable from 1.0 to 1.3 μm,” Electron. Lett. 40, 1523–1525 (2004).
[CrossRef]

Chu, P. L.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Cumberland, B. A.

de Sterke, C. M.

Desem, C.

Dianov, E. M.

S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
[CrossRef] [PubMed]

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294–297 (1985).

Dinda, P. T.

P. T. Dinda, A. Labruyère, and K. Nakkeeran, “Theory of Raman effect on solitons in optical fibre systems: impact and control processes for high-speed long-distance transmission lines,” Opt. Commun. 234, 137–151 (2004).
[CrossRef]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Efimov, A.

Z. Chen, A. J. Taylor, and A. Efimov, “Coherent mid-infrared broadband continuum generation in non-uniform ZBLAN fiber taper,” Opt. Express 17, 5852–5860 (2009).
[CrossRef] [PubMed]

A. Efimov, “Fundamental nonlinear-optical interactions in photonic fibers: time-spectral visualization,” Laser Phys. 18, 667–681 (2008).
[CrossRef]

Eggleton, B. J.

Elgin, J. N.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Fomichev, A. A.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294–297 (1985).

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

George, A. K.

Godbout, N.

Gordon, J. P.

Goto, T.

N. Nishizawa, Y. Ito, and T. Goto, “0.78–0.90-μm wavelength-tunable femtosecond soliton pulse generation using photonic crystal fiber,” IEEE Photon. Technol. Lett. 14, 986–988 (2002).
[CrossRef]

He, F.

Hodel, W.

Hollenbeck, D.

Horak, P.

Ito, Y.

N. Nishizawa, Y. Ito, and T. Goto, “0.78–0.90-μm wavelength-tunable femtosecond soliton pulse generation using photonic crystal fiber,” IEEE Photon. Technol. Lett. 14, 986–988 (2002).
[CrossRef]

Judge, A. C.

Karasik, A. Y.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294–297 (1985).

Kath, W. L.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

Kelly, S. M. J.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Knight, J. C.

Knox, W. H.

Kosinski, S. G.

Kuhlmey, B. T.

Labruyère, A.

P. T. Dinda, A. Labruyère, and K. Nakkeeran, “Theory of Raman effect on solitons in optical fibre systems: impact and control processes for high-speed long-distance transmission lines,” Opt. Commun. 234, 137–151 (2004).
[CrossRef]

Lacroix, S.

Lim, H.

H. Lim, J. Buckley, A. Chong, and F. W. Wise, “Fibre-based source of femtosecond pulses tunable from 1.0 to 1.3 μm,” Electron. Lett. 40, 1523–1525 (2004).
[CrossRef]

Liu, X.

Macon, L.

A. Saissy, J. Botineau, L. Macon, and G. Maze, “Raman scattering in a fluorozirconate glass optical fiber,” J. Phys. (France) Lett. 46, 289–294 (1985).

Mägi, E. C.

Mamyshev, P. V.

S. V. Chernikov and P. V. Mamyshev, “Femtosecond soliton propagation in fibers with slowly decreasing dispersion,” J. Opt. Soc. Am. B 8, 1633–1641 (1991).
[CrossRef]

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294–297 (1985).

Martijn de Sterke, C.

Maze, G.

A. Saissy, J. Botineau, L. Macon, and G. Maze, “Raman scattering in a fluorozirconate glass optical fiber,” J. Phys. (France) Lett. 46, 289–294 (1985).

Mitschke, F. M.

Mollenauer, L. F.

Nakkeeran, K.

P. T. Dinda, A. Labruyère, and K. Nakkeeran, “Theory of Raman effect on solitons in optical fibre systems: impact and control processes for high-speed long-distance transmission lines,” Opt. Commun. 234, 137–151 (2004).
[CrossRef]

Nishizawa, N.

N. Nishizawa, Y. Ito, and T. Goto, “0.78–0.90-μm wavelength-tunable femtosecond soliton pulse generation using photonic crystal fiber,” IEEE Photon. Technol. Lett. 14, 986–988 (2002).
[CrossRef]

Pant, R.

Payne, D. N.

Poletti, F.

Popov, S. V.

Price, J. H. V.

Prokhorov, A. M.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294–297 (1985).

Richardson, D. J.

Rulkov, A. B.

Saissy, A.

A. Saissy, J. Botineau, L. Macon, and G. Maze, “Raman scattering in a fluorozirconate glass optical fiber,” J. Phys. (France) Lett. 46, 289–294 (1985).

Santhanam, J.

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[CrossRef]

Serkin, V. N.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294–297 (1985).

Smyth, N. F.

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

Stel’makh, M. F.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294–297 (1985).

Stone, J. M.

Taylor, A. J.

Taylor, J. R.

Travers, J. C.

Tse, M. L. V.

Tsoy, E. N.

Voronin, A. A.

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, 1992).

Webber, H. P.

Windeler, R. S.

Wise, F. W.

H. Lim, J. Buckley, A. Chong, and F. W. Wise, “Fibre-based source of femtosecond pulses tunable from 1.0 to 1.3 μm,” Electron. Lett. 40, 1523–1525 (2004).
[CrossRef]

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

Xu, C.

Zheltikov, A. M.

Electron. Lett. (1)

H. Lim, J. Buckley, A. Chong, and F. W. Wise, “Fibre-based source of femtosecond pulses tunable from 1.0 to 1.3 μm,” Electron. Lett. 40, 1523–1525 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

N. Nishizawa, Y. Ito, and T. Goto, “0.78–0.90-μm wavelength-tunable femtosecond soliton pulse generation using photonic crystal fiber,” IEEE Photon. Technol. Lett. 14, 986–988 (2002).
[CrossRef]

J. Opt. Soc. Am. B (5)

J. Phys. (France) Lett. (1)

A. Saissy, J. Botineau, L. Macon, and G. Maze, “Raman scattering in a fluorozirconate glass optical fiber,” J. Phys. (France) Lett. 46, 289–294 (1985).

JETP Lett. (1)

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel’makh, and A. A. Fomichev, “Stimulated-Raman conversion of multisoliton pulses in quartz optical fibers,” JETP Lett. 41, 294–297 (1985).

Laser Phys. (1)

A. Efimov, “Fundamental nonlinear-optical interactions in photonic fibers: time-spectral visualization,” Laser Phys. 18, 667–681 (2008).
[CrossRef]

Opt. Commun. (3)

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

P. T. Dinda, A. Labruyère, and K. Nakkeeran, “Theory of Raman effect on solitons in optical fibre systems: impact and control processes for high-speed long-distance transmission lines,” Opt. Commun. 234, 137–151 (2004).
[CrossRef]

J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. 222, 413–420 (2003).
[CrossRef]

Opt. Express (3)

Opt. Lett. (8)

Phys. Rev. E (1)

W. L. Kath and N. F. Smyth, “Soliton evolution and radiation loss for the nonlinear Schrödinger equation,” Phys. Rev. E 51, 1484–1492 (1995).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, 1992).

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Figures (3)

Fig. 1
Fig. 1

(a) Calculated effective Raman parameter and (b) D ( T p ) [as defined in Eq. (23)] as functions of pulse width ( T p ) for fused silica (solid line) and ZBLAN (dashed line).

Fig. 2
Fig. 2

(a) SSFS ( Ω p / 2 π ) and (b) pulse width ( T p ) as functions of propagation distance in a dispersion-shifted silica fiber. Solid, dashed, dotted, and dashed-dotted lines correspond to the simulation results from improved MM, GNLS simulation, adiabatic soliton approximation, and standard MM, respectively.

Fig. 3
Fig. 3

(a) SSFS ( Ω p / 2 π ) and (b) pulse width ( T p ) as functions of propagation distance in a non-uniform ZBLAN fiber taper. Solid, dashed, dotted, and dashed-dotted lines correspond to the simulation results from improved MM, GNLS simulation, adiabatic soliton approximation, and standard MM, respectively. (c) GVD and nonlinear coefficient experienced at the local soliton carrier frequency as the soliton propagates along the ZBLAN taper length, calculated from MM (solid line) and GNLS simulation (dashed line).

Equations (28)

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

N = γ E T 0 2 | β 2 | = 1 ,
A z + α 2 A + m 2 i m 1 m ! β m m A T m = m 0 i m + 1 m ! γ m m T m ( A + R ( T ) | A ( z , T T ) | 2 d T ) .
A z + α 2 A + i β 2 2 2 A T 2 β 3 6 3 A T 3 = i γ 0 A | A | 2 i γ 0 T R A | A | 2 T γ 1 T ( A | A | 2 ) + γ 1 T R T ( A | A | 2 T ) .
E p = | A | 2 d T ,
q p = 1 E p T | A | 2 d T ,
Ω p = i 2 E p ( A A T A A T ) d T ,
σ p 2 = 1 E p ( T q p ) 2 | A | 2 d T ,
C p = i 2 E p ( T q p ) ( A A T A A T ) d T .
A ( z , T ) = E p 2 T p sech ( T q p T p ) exp [ i Ω p ( T q p ) i C p ( T q p ) 2 2 T p 2 ] .
d E p d z = α E p 4 γ 1 T R E p 2 15 T p 3 ,
d q p d z = β 2 Ω p + β 3 2 [ Ω p 2 + ( 1 + π 2 C p 2 4 ) 1 3 T p 2 ] + γ 1 E p 2 T p ,
d Ω p d z = 4 ( γ 0 + γ 1 Ω p ) T R E p 15 T p 3 + γ 1 C p E p 3 T p 3 ,
d T p d z = ( β 2 + β 3 Ω p ) C p T p + 4 γ 1 T R E p π 2 T p 2 ,
d C p d z = ( 4 π 2 + C p 2 ) ( β 2 + β 3 Ω p ) T p 2 + 2 ( γ 0 + γ 1 Ω p ) E p π 2 T p + ( 150 4 π 2 ) γ 1 T R E p C p 15 π 2 T p 3 .
d E p d z = α E p 4 γ 1 T R E p 2 15 T p 3 ,
d q p d z = β 3 6 T p 2 ( 1 + π 2 C p 2 4 ) + γ 1 E p 2 T p ,
d Ω p d z = 4 γ 0 T R E p 15 T p 3 + γ 1 C p E p 3 T p 3 ,
d T p d z = β 2 C p T p + 4 γ 1 T R E p π 2 T p 2 ,
d C p d z = ( 4 π 2 + C p 2 ) β 2 T p 2 + 2 γ 0 E p π 2 T p + ( 150 4 π 2 ) γ 1 T R E p C p 15 π 2 T p 3 .
T R ( T p ) = 0 B ( t , T p ) h R ( t ) d t ,
B ( t , T p ) = 15 8 f R csch 4 ( t T p ) [ 4 t + 2 t   cosh ( 2 t T p ) 3 T p   sinh ( 2 t T p ) ] .
d C p d z ( 4 π 2 + C p 2 ) β 2 T p 2 + 2 γ 0 E p π 2 T p + ( 150 4 π 2 ) γ 1 T R E p C p 15 π 2 T p 3 + 12 γ 0 f R E p π 2 T p D ( T p ) ,
D ( T p ) = 0 h R ( t ) csch 4 ( t T p ) [ 1 4 t 2 T p 2 ( t 2 2 T p 2 + 1 4 ) cosh ( 2 t T p ) + t T p sinh ( 2 t T p ) ] d t 1 6 .
d E p d z = α E p 4 γ 1 T R E p 2 15 T p 3 ,
d q p d z = β 3 6 T p 2 + γ 1 E p 2 T p ,
d Ω p d z = 4 γ 0 T R E p 15 T p 3 ,
T p = 2 | β 2 | γ 0 E p .
d Ω p d z = γ 0 4 T R E p 4 30 | β 2 | 3 .

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