Abstract

Nonlinear propagation in slow-light states of high-index photonic crystal fibers (PCFs) is studied numerically. To avoid divergencies in dispersion and nonlinear parameters around the zero-velocity mode, a time-propagating generalized nonlinear Schrödinger equation is formulated. Calculated slow-light modes in a solid core chalcogenide PCF are used to parameterize the model, which is shown to support standing and moving spatial solitons. Inclusion of Raman scattering slows down moving solitons exponentially, so that the zero-velocity soliton becomes an attractor state. An analytical expression for the deceleration rate that compares favorably with the numerical results is derived. Collisions of successive solitons due to the Raman deceleration are studied numerically, and it is found that the soliton interaction is mostly repulsive, as expected from the established theory of fiber solitons.

© 2011 Optical Society of America

Full Article  |  PDF Article

Corrections

Jesper Lægsgaard, "Zero-velocity solitons in high-index photonic crystal fibers: erratum," J. Opt. Soc. Am. B 28, 432-432 (2011)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-28-3-432

References

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  1. T. Baba, “Slow light in photonic crystals,” Nat. Photon. 2, 465–473 (2008).
    [CrossRef]
  2. M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
    [CrossRef] [PubMed]
  3. A. F. Oskooi, J. D. Joannopoulos, and S. G. Johnson, “Zero-group-velocity modes in chalcogenide holey photonic-crystal fibers,” Opt. Express 17, 10082–10090 (2009).
    [CrossRef] [PubMed]
  4. Q. Coulombier, L. Brilland, P. Houizot, T. Chartier, T. N. N’Guyen, F. Smektala, G. Renversez, A. Monteville, D. Méchin, T. Pain, H. Orain, J.-C. Sangleboeuf, and J. Trolès, “Casting method for producing low-loss chalcogenide microstructured optical fibers,” Opt. Express 18, 9107–9112 (2010).
    [CrossRef] [PubMed]
  5. J. Grgić, S. Xiao, J. Mørk, A.-P. Jauho, and N. A. Mortensen, “Slow-light enhanced absorption in a hollow-core fiber,” Opt. Express 18, 14270–14279 (2010).
    [CrossRef] [PubMed]
  6. M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
    [CrossRef]
  7. S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318(2009).
    [CrossRef] [PubMed]
  8. M. D. Turner, T. M. Monro, and S. Afshar V., “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: stimulated Raman scattering,” Opt. Express 17, 11565–11581 (2009).
    [CrossRef] [PubMed]
  9. A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  21. A. Tuniz, G. Brawley, D. J. Moss, and B. J. Eggleton, “Two-photon absorption effects on Raman gain in single mode As2Se3 chalcogenide glass fiber,” Opt. Express 16, 18524–18534(2008).
    [CrossRef] [PubMed]

2010 (2)

2009 (3)

2008 (2)

2007 (1)

2005 (2)

A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[CrossRef]

Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect,” Phys. Rev. A 72, 063802 (2005).
[CrossRef]

2004 (5)

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[CrossRef]

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

M. Soljačić, E. Lidorikis, M. Ibanescu, S. Johnson, J. Joannopoulos, and Y. Fink, “Optical bistability and cutoff solitons in photonic bandgap fibers,” Opt. Express 12, 1518–1527(2004).
[CrossRef] [PubMed]

N. A. Mortensen and M. D. Nielsen, “Modeling of realistic cladding structures for air-core photonic bandgap fibers,” Opt. Lett. 29, 349–351 (2004).
[CrossRef] [PubMed]

2003 (2)

2001 (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]

Afshar V., S.

Agrawal, G. P.

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

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photon. 2, 465–473 (2008).
[CrossRef]

Binosi, D.

A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[CrossRef]

Bjarklev, A.

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]

Brawley, G.

Brilland, L.

Chartier, T.

Coulombier, Q.

de Cordoba, P.

A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[CrossRef]

Eggleton, B. J.

Ferrando, A.

A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[CrossRef]

Fink, Y.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

M. Soljačić, E. Lidorikis, M. Ibanescu, S. Johnson, J. Joannopoulos, and Y. Fink, “Optical bistability and cutoff solitons in photonic bandgap fibers,” Opt. Express 12, 1518–1527(2004).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

Grgic, J.

Houizot, P.

Ibanescu, M.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

M. Soljačić, E. Lidorikis, M. Ibanescu, S. Johnson, J. Joannopoulos, and Y. Fink, “Optical bistability and cutoff solitons in photonic bandgap fibers,” Opt. Express 12, 1518–1527(2004).
[CrossRef] [PubMed]

Jauho, A.-P.

Joannopoulos, J.

Joannopoulos, J. D.

Johnson, S.

Johnson, S. G.

Kolesik, M.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[CrossRef]

Lægsgaard, J.

Libori, S. E. B.

Lidorikis, E.

Luo, C.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Méchin, D.

Mizuta, Y.

Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect,” Phys. Rev. A 72, 063802 (2005).
[CrossRef]

Moloney, J. V.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[CrossRef]

Monro, T. M.

Montero, A.

A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[CrossRef]

Monteville, A.

Mørk, J.

Mortensen, N. A.

Moss, D. J.

N’Guyen, T. N.

Nagasawa, M.

Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect,” Phys. Rev. A 72, 063802 (2005).
[CrossRef]

Nielsen, M. D.

Ohtani, M.

Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect,” Phys. Rev. A 72, 063802 (2005).
[CrossRef]

Orain, H.

Oskooi, A. F.

Pain, T.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

Renversez, G.

Roundy, D.

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Sangleboeuf, J.-C.

Smektala, F.

Soljacic, M.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

Trolès, J.

Tuniz, A.

Turner, M. D.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

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]

Wright, E. M.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[CrossRef]

Xiao, S.

Yamashita, M.

Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect,” Phys. Rev. A 72, 063802 (2005).
[CrossRef]

Zacares, M.

A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[CrossRef]

Appl. Phys. B (1)

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (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]

J. Opt. Soc. Am. B (2)

Nat. Photon. (1)

T. Baba, “Slow light in photonic crystals,” Nat. Photon. 2, 465–473 (2008).
[CrossRef]

Opt. Express (9)

A. F. Oskooi, J. D. Joannopoulos, and S. G. Johnson, “Zero-group-velocity modes in chalcogenide holey photonic-crystal fibers,” Opt. Express 17, 10082–10090 (2009).
[CrossRef] [PubMed]

Q. Coulombier, L. Brilland, P. Houizot, T. Chartier, T. N. N’Guyen, F. Smektala, G. Renversez, A. Monteville, D. Méchin, T. Pain, H. Orain, J.-C. Sangleboeuf, and J. Trolès, “Casting method for producing low-loss chalcogenide microstructured optical fibers,” Opt. Express 18, 9107–9112 (2010).
[CrossRef] [PubMed]

J. Grgić, S. Xiao, J. Mørk, A.-P. Jauho, and N. A. Mortensen, “Slow-light enhanced absorption in a hollow-core fiber,” Opt. Express 18, 14270–14279 (2010).
[CrossRef] [PubMed]

S. Afshar V. and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318(2009).
[CrossRef] [PubMed]

M. D. Turner, T. M. Monro, and S. Afshar V., “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: stimulated Raman scattering,” Opt. Express 17, 11565–11581 (2009).
[CrossRef] [PubMed]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schro¨dinger equation,” Opt. Express 15, 16110–16123(2007).
[CrossRef] [PubMed]

M. Soljačić, E. Lidorikis, M. Ibanescu, S. Johnson, J. Joannopoulos, and Y. Fink, “Optical bistability and cutoff solitons in photonic bandgap fibers,” Opt. Express 12, 1518–1527(2004).
[CrossRef] [PubMed]

A. Tuniz, G. Brawley, D. J. Moss, and B. J. Eggleton, “Two-photon absorption effects on Raman gain in single mode As2Se3 chalcogenide glass fiber,” Opt. Express 16, 18524–18534(2008).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (1)

Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect,” Phys. Rev. A 72, 063802 (2005).
[CrossRef]

Phys. Rev. E (2)

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

M. Ibanescu, S. G. Johnson, D. Roundy, C. Luo, Y. Fink, and J. D. Joannopoulos, “Anomalous dispersion relations by symmetry breaking in axially uniform waveguides,” Phys. Rev. Lett. 92, 063903 (2004).
[CrossRef] [PubMed]

Other (2)

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

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

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Figures (4)

Fig. 1
Fig. 1

Generic fiber structure investigated in this work. The defining parameters are the pitch, Λ, the core and cladding airhole curvature radii, d c , d cl , and the minimal glass bridge width, w b . The core center is at the lower left corner of the figure.

Fig. 2
Fig. 2

Bandgap boundaries (solid black curves) and guided-mode dispersion curves for two different core designs with (a) d c = 0.82 and (b) d c = 0.5 . Insets show modal field energy distributions. The dashed black curve in (b) is a quadratic approximation to the hybrid-mode dispersion curve (red circles).

Fig. 3
Fig. 3

Soliton velocity versus time for three different pulse energies. The curves were determined by direct numerical differentiation of the soliton peak position as a function of time.

Fig. 4
Fig. 4

Collisions between 144 pJ solitons launched with a time interval of 2 ns . The time is measured relative to the launch of the second soliton. For a relative input phase ϕ = 0.87 radians (upper left panel), a strong interaction is observed, and the first soliton eventually carries away about 63% of the total energy. For ϕ = 0.9 radians (upper right), the interaction is already significantly reduced, and the first soliton eventually carries 53% of the energy. For ϕ = 1.0 radians (lower panel), the interaction is still further reduced, although notable oscillations are seen after the collision.

Equations (41)

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× E = μ 0 H t ,
× H = ε 0 ε ( r ) E t + P NL t .
H ( r , t ) = 1 2 π m d β [ ( A m ( t , β ) + δ m ( t , β ) ) h m ( r , β ) e i ( ω m ( β ) t β z ) + ( A m * ( t , β ) + δ m * ( t , β ) ) h m * ( r , β ) e i ( ω m ( β ) t + β z ) ] ,
E ( r , t ) = 1 2 π m d β [ A m ( t , β ) e m ( r , β ) e i ( ω m ( β ) t β z ) + A m * ( t , β ) e m * ( r , β ) e i ( ω m ( β ) t + β z ) ] ,
h m ( r , t ; β ) = h m ( r , β ) e i ( ω m ( β ) t β z ) , e m ( r , t ; β ) = e m ( r , β ) e i ( ω m ( β ) t β z ) ,
× e m ( r , t ; β ) = μ 0 h m ( r , t ; β ) t , × h m ( r , t ; β ) = ε 0 ε ( r ) e m ( r , t ; β ) t ,
A m ( t , β ) t = i ω m ( β ) δ m ( t , β ) δ m ( t , β ) t .
i ω m ( β ) δ m ( t , β ) = A m ( t , β ) t ,
ε 0 d r ε ( r ) e m * ( r , β ) · e n ( r , β ) = μ 0 d r h m * ( r , β ) · h n ( r , β ) = 1 2 δ m n ,
d rE · D t = m d β A m ( t , β ) 2 .
e m * ( r , β ) = e m ( r , β ) , h m * ( r , β ) = h m ( r , β ) ,
d r e m * ( r , t ; β ) · [ P NL t + ε 0 ε E t ] = d r e m * ( r , t ; β ) · × H = d rH · × e m * ( r , t ; β ) = i μ 0 ω m ( β ) d rH · h m * ( r , t ; β ) .
1 2 π d r e m * ( r , t ; β ) · P NL t + [ i ω m ( β ) A m ( t , β ) + A m ( t , β ) t + ( A * ( t , β ) t i ω m ( β ) A m * ( t , β ) ) e 2 i ω m ( β ) t ] ε 0 d r ε ( r ) e m ( r , β ) 2 = i ω m ( β ) μ 0 [ A m ( t , β ) + δ m ( t , β ) ( A m * ( t , β ) + δ m * ( t , β ) ) e 2 i ω m ( β ) t ] d r h m ( r , β ) 2 , A m ( t , β ) t + A * ( t , β ) t e 2 i ω m ( β ) t = 1 2 π d r e m * ( r , t ; β ) · P NL t .
P NL ( r , t ) = ε 0 [ χ K ( 3 ) ( r ) E ( r , t ) E ( r , t ) E ( r , t ) + χ R ( 3 ) ( r ) E ( r , t ) t d t 1 R ( t t 1 ) E ( r , t 1 ) E ( r , t 1 ) ] ,
P NL ( r , t ) = n p q ε 0 ( 2 π ) 3 / 2 d β 1 3 e i ( β 1 + β 2 + β 3 ) z { [ 2 A ˜ n ( t , β 1 ) A ˜ p ( t , β 2 ) A ˜ q * ( t , β 3 ) + A ˜ n * ( t , β 1 ) A ˜ p ( t , β 2 ) A ˜ q ( t , β 3 ) ] χ K ( 3 ) e n ( r , β 1 ) e p ( r , β 2 ) e q * ( r , β 3 ) + [ 2 A ˜ n ( t , β 1 ) t d t 1 R ( t t 1 ) A ˜ p ( t 1 , β 2 ) A ˜ q * ( t 1 , β 3 ) + A ˜ n * ( t , β 1 ) t d t 1 R ( t t 1 ) A ˜ p ( t 1 , β 2 ) A ˜ q ( t 1 , β 3 ) ] χ R ( 3 ) e n ( r , β 1 ) e p ( r , β 2 ) e q * ( r , β 3 ) } ,
A ˜ m ( t , β ) = A m ( t , β ) e i ω m ( β ) t .
P NL ( r , t ) t = n p q ε 0 ( 2 π ) 3 / 2 d β 1 3 e i ( β 1 + β 2 + β 3 ) z { i [ 2 ( ω n ( β 1 ) + ω p ( β 2 ) ω q ( β 3 ) ) × A ˜ n ( t , β 1 ) A ˜ p ( t , β 2 ) A ˜ q * ( t , β 3 ) + ( ω p ( β 2 ) + ω q ( β 3 ) ω n ( β 1 ) ) A ˜ n * ( t , β 1 ) A ˜ p ( t , β 2 ) A ˜ q ( t , β 3 ) ] × χ K ( 3 ) e n ( r , β 1 ) e p ( r , β 2 ) e q * ( r , β 3 ) + 2 A ˜ n ( t , β 1 ) t d t 1 ( i ω n ( β 1 ) R ( t t 1 ) + R ( t t 1 ) ) A ˜ p ( t 1 , β 2 ) A ˜ q * ( t 1 , β 3 ) × χ R ( 3 ) e n ( r , β 1 ) e p ( r , β 2 ) e q * ( r , β 3 ) } .
A m ( t , β ) t = e i ω m ( β ) t ε 0 2 π n p q d β 1 d β 2 { K m n p q ( β , β 1 , β 2 ) × [ 2 A ˜ n ( t , β 1 ) A ˜ p ( t , β 2 ) A ˜ q * ( t , β 1 + β 2 β ) i ( ω n ( β 1 ) + ω p ( β 2 ) ω q ( β 1 + β 2 β ) ) + A ˜ n * ( t , β 1 ) A ˜ p ( t , β 2 ) A ˜ q ( t , β β 1 β 2 ) i ( ω p ( β 2 ) + ω q ( β 1 + β 2 β ) ω n ( β 1 ) ) ] + R m n p q ( β , β 1 , β 2 ) 2 A ˜ n ( t , β 1 ) t d t 1 A ˜ p ( t 1 , β 2 ) A ˜ q * ( t 1 , β 1 + β 2 β ) ( R ( t t 1 ) + i ω n ( β 1 ) R ( t t 1 ) ) ,
K m n p q ( β , β 1 , β 2 ) = d r e m * ( r , β ) · χ K ( 3 ) ( r ) e n ( r , β 1 ) e p ( r , β 2 ) e q * ( r , β 1 + β 2 β ) ,
R m n p q ( β , β 1 , β 2 ) = d r e m * ( r , β ) · χ R ( 3 ) ( r ) e n ( r , β 1 ) e p ( r , β 2 ) e q * ( r , β 1 + β 2 β ) .
K = χ K x x x x ( 3 ) m d r e ( r , 0 ) 4 , R = χ R x x x x ( 3 ) m d r e ( r , 0 ) 4 ,
K = χ K x x x x ( 3 ) m d r ε 2 ( r ) e ( r , 0 ) 4 4 ε 0 2 ε m 2 [ d r ε ( r ) e ( r , 0 ) 2 ] 2 χ K x x x x ( 3 ) 4 ε 0 2 ε m 2 A eff ( β = 0 ) ,
A ˜ eff ( β ) = ε m ( v g c ) 2 [ d r ε ( r ) e ( r , β ) 2 ] 2 m d r ε 2 ( r ) e ( r , β ) 4 = ε m ( v g c ) 2 A eff ( β ) ,
B ( t , β ) = i ω ( β ) A ˜ m ( t , β ) , A ˜ ( t , z ) = 1 2 π d β e i β z A ˜ m ( t , β ) , B ( t , z ) = 1 2 π d β e i β z B ( t , β ) ,
A ( t , β ) t = e i ω ( β ) t N 2 A eff 1 2 π d z e i β z [ ( 1 f R ) ( 2 A ˜ ( t , z ) 2 B ( t , z ) + A ˜ 2 ( t , z ) B * ( t , z ) ) + f R ( A ˜ ( t , z ) G ( t , z ) + B ( t , z ) F ( t , z ) ) ] ,
F ( t , z ) = t d t 1 R ( t t 1 ) A ˜ ( t 1 , z ) 2 , G ( t , z ) = t d t 1 R ( t t 1 ) A ˜ ( t 1 , z ) 2 ,
N 2 = 3 χ s ( 3 ) 4 ε 0 ε m 2 , χ s ( 3 ) = χ K x x x x ( 3 ) + 2 3 χ R x x x x ( 3 ) , f R = 2 χ R x x x x ( 3 ) 3 χ s ( 3 ) .
A ˜ m ( t , β ) t = i ω ( β ) A ˜ m ( t , β ) i ω 0 N 2 A eff 1 2 π d z e i β z A ˜ ( t , z ) A ˜ ( t , z ) 2 .
A ˜ ( t , z ) t = i [ ω 0 ω 2 2 2 z 2 ] A ˜ ( t , z ) i ω 0 N 2 A eff A ˜ ( t , z ) A ˜ ( t , z ) 2 .
A ( t , z ) = ξ 0 sech ( z v g t z 0 ) e i t ( ω ( β 0 ) 1 / 2 T NL ) e i β 0 z , z 0 2 = ω 2 Γ ξ 0 , Γ = ω 0 N 2 A eff , T NL = 1 Γ ξ 0 ,
R ( t ) = τ 1 2 + τ 2 2 τ 1 τ 2 2 sin ( t τ 1 ) e t τ 2 ,
G ( t , z ) = τ 1 2 + τ 2 2 2 i τ 1 τ 2 2 t d t 1 [ ( i τ 1 1 τ 2 ) e ( i τ 1 1 τ 2 ) ( t t 1 ) + ( i τ 1 + 1 τ 2 ) e ( i τ 1 + 1 τ 2 ) ( t t 1 ) ] × A ˜ ( t 1 , z ) 2 G + ( t , z ) + G ( t , z ) ,
G ± ( t , z ) t = τ 1 2 + τ 2 2 2 i τ 1 τ 2 2 ( i τ 1 1 τ 2 ) A ˜ ( t , z ) 2 + ( ± i τ 1 1 τ 2 ) G ± ( t , z ) ,
F ( t , z ) t = G ( t , z ) .
β ¯ = d β A ( t , β ) 2 β d β A ( t , β ) 2 ,
d β ¯ d t = d β [ A * ( t , β ) A ( t , β ) t + A ( t , β ) A * ( t , β ) t ] β d β A ( t , β ) 2 β ¯ d β [ A * ( t , β ) A ( t , β ) t + A ( t , β ) A * ( t , β ) t ] d β A ( t , β ) 2 = d β [ A * ( t , β ) A ( t , β ) t + A ( t , β ) A * ( t , β ) t ] ( β β ¯ ) E p .
F ( t , z ) = t d t 1 R ( t t 1 ) A ˜ ( t 1 , z ) 2 A ˜ ( t , z ) 2 2 τ 1 ( τ 1 τ 2 + τ 2 τ 1 ) A ( t , z ) 2 t = A ˜ ( t , z ) 2 4 ξ 0 v g z 0 sech 3 ( z v g t z 0 ) sinh ( z v g t z 0 ) τ 1 ( τ 1 τ 2 + τ 2 τ 1 ) .
A * ( t , β ) A ( t , β ) t = 4 ω τ 1 ( τ 1 τ 2 + τ 2 τ 1 ) z 0 ξ 0 2 f R N 2 A eff v g sech ( π ( β β ¯ ) z 0 2 ) × 0 d z sin ( β z ) sech 4 ( z v g t z 0 ) sinh ( z v g t z 0 ) ,
d β ¯ d t = 4 ω τ 1 ( τ 1 τ 2 + τ 2 τ 1 ) ξ 0 z 0 2 f R N 2 A eff v g d u u sech ( π u 2 ) 0 d v sin ( u v ) sech 4 ( v ) sinh ( v ) = 16 15 ω τ 1 ( τ 1 τ 2 + τ 2 τ 1 ) ξ 0 z 0 2 f R N 2 A eff v g .
v g ( t ) = v g ( t = 0 ) e t / t 0 , t 0 = 15 16 ( τ 1 τ 2 + τ 2 τ 1 ) ω τ 1 z 0 2 ξ 0 A eff f R N 2 ω 2 .
z = 0 d t v g ( t ) = v g ( t = 0 ) t 0 .

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