Abstract

A simplified numerical approach to modeling of dissipative dispersion-managed fiber lasers is examined. We present a new numerical iteration algorithm for finding the periodic solutions of the system of nonlinear ordinary differential equations describing the intra-cavity dynamics of the dissipative soliton characteristics in dispersion-managed fiber lasers. We demonstrate that results obtained using simplified model are in good agreement with full numerical modeling based on the corresponding partial differential equations.

© 2014 Optical Society of America

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References

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  1. H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975).
    [Crossref]
  2. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992).
    [Crossref]
  3. S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099–2111 (1997).
    [Crossref]
  4. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48(4), 629–678 (2006).
    [Crossref]
  5. N. Akhmediev and A. Ankiewicz, eds. Dissipative Solitons: Lecture Notes in Physics (Springer, 2005), V. 661.
  6. S.K. Turitsyn, B. Bale, and M.P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
    [Crossref]
  7. T. Schreiber, B. Ortac, J. Limpert, and A. Tunnermann, “On the study of pulse evolution in ultra-short pulse mode-locked fiber lasers by numerical simulations,” Opt. Express 15(13), 8252–8262 (2007).
    [Crossref] [PubMed]
  8. O. Shtyrina, M. Fedoruk, S. Turitsyn, R. Herda, and O. Okhotnikov, “Evolution and stability of pulse regimes in SESAM-mode-locked femtosecond fiber lasers,” J. Opt. Soc. Am. B 26(2), 346–352 (2009).
    [Crossref]
  9. B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
    [Crossref]
  10. X. Tian, M. Tang, X. Cheng, P. P. Shum, Y. Gong, and C. Lin, “High-energy wave-breaking-free pulse from all-fiber mode-locked laser system,” Opt. Express 17(9), 7222–7227 (2009).
    [Crossref] [PubMed]
  11. B.G Bale, O.G. Okhotnikov, and S.K. Turitsyn, “Modeling and Technologies of Ultrafast Fiber Lasers,” in Fiber Lasers, O. G. Okhotnikov, ed. (Wiley-VCH Verlag GmbH Co., 2012).
  12. I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21(5), 327–329 (1996).
    [Crossref]
  13. S. K. Turitsyn, “Breathing self-similar dynamics and oscillatory tails of the chirped dispersion-managed soliton,” Phys. Rev. E. 58(2), R1256–R1259 (1998).
    [Crossref]
  14. S. K. Turitsyn and V. K. Mezentsev, “Dynamics of self-similar dispersion-managed soliton presented in the basis of chirped Gauss-Hermite functions,” JETP Lett. 67(9), 640–646 (1998).
    [Crossref]
  15. E. G. Shapiro and S. K. Turitsyn, “Theory of guiding-center breathing soliton propagation in optical communication systems with strong dispersion management,” Opt. Lett. 22(20), 1544–1546 (1997).
    [Crossref]
  16. S. K. Turitsyn, T. Schäfer, and V. K. Mezentsev, “Self-similar core and oscillatory tails of a path-averaged chirped dispersion-managed optical pulse,” Opt. Lett. 23(17), 1351–1353 (1998).
    [Crossref]
  17. S. Turitsyn and E. Shapiro, “Enhanced power breathing soliton in communication systems with dispersion management,” Phys. Rev. E. 56(5), R4951–R4955, (1997).
    [Crossref]
  18. J. Holt, “Numerical solution of nonlinear two-point boundary problems by finite difference methods,” Commun. ACM. 7(6), 366–373 (1964).
    [Crossref]
  19. H. Keller, Numerical Methods for Two-point Boundary Value Problem (Blaisdell Publishing Co., 1968).
  20. S. Roberts and S. J. Shipman, Two-point Boundary Value Problems: Shooting Methods (Elsevier, 1972).
  21. J. Nijhof, W. Forisiak, and N. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quant. 6(2), 330–336 (2000).
    [Crossref]
  22. I. A. Yarutkina and O. V. Shtyrina, “Mathematical modelling of dispersion-managed thulium/holmium fibre lasers,” Quantum Electron. 43(11), 1019–1023 (2013).
    [Crossref]
  23. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
    [Crossref]

2013 (1)

I. A. Yarutkina and O. V. Shtyrina, “Mathematical modelling of dispersion-managed thulium/holmium fibre lasers,” Quantum Electron. 43(11), 1019–1023 (2013).
[Crossref]

2012 (1)

S.K. Turitsyn, B. Bale, and M.P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
[Crossref]

2010 (1)

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

2009 (2)

2008 (1)

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
[Crossref]

2007 (1)

2006 (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48(4), 629–678 (2006).
[Crossref]

2000 (1)

J. Nijhof, W. Forisiak, and N. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quant. 6(2), 330–336 (2000).
[Crossref]

1998 (3)

S. K. Turitsyn, “Breathing self-similar dynamics and oscillatory tails of the chirped dispersion-managed soliton,” Phys. Rev. E. 58(2), R1256–R1259 (1998).
[Crossref]

S. K. Turitsyn and V. K. Mezentsev, “Dynamics of self-similar dispersion-managed soliton presented in the basis of chirped Gauss-Hermite functions,” JETP Lett. 67(9), 640–646 (1998).
[Crossref]

S. K. Turitsyn, T. Schäfer, and V. K. Mezentsev, “Self-similar core and oscillatory tails of a path-averaged chirped dispersion-managed optical pulse,” Opt. Lett. 23(17), 1351–1353 (1998).
[Crossref]

1997 (3)

1996 (1)

1992 (1)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992).
[Crossref]

1975 (1)

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975).
[Crossref]

1964 (1)

J. Holt, “Numerical solution of nonlinear two-point boundary problems by finite difference methods,” Commun. ACM. 7(6), 366–373 (1964).
[Crossref]

Bale, B.

S.K. Turitsyn, B. Bale, and M.P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
[Crossref]

Bale, B. G.

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

Bale, B.G

B.G Bale, O.G. Okhotnikov, and S.K. Turitsyn, “Modeling and Technologies of Ultrafast Fiber Lasers,” in Fiber Lasers, O. G. Okhotnikov, ed. (Wiley-VCH Verlag GmbH Co., 2012).

Boscolo, S.

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

Cheng, X.

Chong, A.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
[Crossref]

Doran, N.

J. Nijhof, W. Forisiak, and N. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quant. 6(2), 330–336 (2000).
[Crossref]

Fedoruk, M.

Fedoruk, M.P.

S.K. Turitsyn, B. Bale, and M.P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
[Crossref]

Forisiak, W.

J. Nijhof, W. Forisiak, and N. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quant. 6(2), 330–336 (2000).
[Crossref]

Fujimoto, J. G.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992).
[Crossref]

Gabitov, I.

Gong, Y.

Haus, H. A.

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099–2111 (1997).
[Crossref]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992).
[Crossref]

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975).
[Crossref]

Herda, R.

Holt, J.

J. Holt, “Numerical solution of nonlinear two-point boundary problems by finite difference methods,” Commun. ACM. 7(6), 366–373 (1964).
[Crossref]

Ippen, E. P.

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099–2111 (1997).
[Crossref]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992).
[Crossref]

Keller, H.

H. Keller, Numerical Methods for Two-point Boundary Value Problem (Blaisdell Publishing Co., 1968).

Kutz, J. N.

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48(4), 629–678 (2006).
[Crossref]

Limpert, J.

Lin, C.

Mezentsev, V. K.

S. K. Turitsyn, T. Schäfer, and V. K. Mezentsev, “Self-similar core and oscillatory tails of a path-averaged chirped dispersion-managed optical pulse,” Opt. Lett. 23(17), 1351–1353 (1998).
[Crossref]

S. K. Turitsyn and V. K. Mezentsev, “Dynamics of self-similar dispersion-managed soliton presented in the basis of chirped Gauss-Hermite functions,” JETP Lett. 67(9), 640–646 (1998).
[Crossref]

Namiki, S.

Nijhof, J.

J. Nijhof, W. Forisiak, and N. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quant. 6(2), 330–336 (2000).
[Crossref]

Okhotnikov, O.

Okhotnikov, O.G.

B.G Bale, O.G. Okhotnikov, and S.K. Turitsyn, “Modeling and Technologies of Ultrafast Fiber Lasers,” in Fiber Lasers, O. G. Okhotnikov, ed. (Wiley-VCH Verlag GmbH Co., 2012).

Ortac, B.

Renninger, W. H.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
[Crossref]

Roberts, S.

S. Roberts and S. J. Shipman, Two-point Boundary Value Problems: Shooting Methods (Elsevier, 1972).

Schäfer, T.

Schreiber, T.

Shapiro, E.

S. Turitsyn and E. Shapiro, “Enhanced power breathing soliton in communication systems with dispersion management,” Phys. Rev. E. 56(5), R4951–R4955, (1997).
[Crossref]

Shapiro, E. G.

Shipman, S. J.

S. Roberts and S. J. Shipman, Two-point Boundary Value Problems: Shooting Methods (Elsevier, 1972).

Shtyrina, O.

Shtyrina, O. V.

I. A. Yarutkina and O. V. Shtyrina, “Mathematical modelling of dispersion-managed thulium/holmium fibre lasers,” Quantum Electron. 43(11), 1019–1023 (2013).
[Crossref]

Shum, P. P.

Tang, M.

Tian, X.

Tunnermann, A.

Turitsyn, S.

O. Shtyrina, M. Fedoruk, S. Turitsyn, R. Herda, and O. Okhotnikov, “Evolution and stability of pulse regimes in SESAM-mode-locked femtosecond fiber lasers,” J. Opt. Soc. Am. B 26(2), 346–352 (2009).
[Crossref]

S. Turitsyn and E. Shapiro, “Enhanced power breathing soliton in communication systems with dispersion management,” Phys. Rev. E. 56(5), R4951–R4955, (1997).
[Crossref]

Turitsyn, S. K.

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

S. K. Turitsyn, “Breathing self-similar dynamics and oscillatory tails of the chirped dispersion-managed soliton,” Phys. Rev. E. 58(2), R1256–R1259 (1998).
[Crossref]

S. K. Turitsyn and V. K. Mezentsev, “Dynamics of self-similar dispersion-managed soliton presented in the basis of chirped Gauss-Hermite functions,” JETP Lett. 67(9), 640–646 (1998).
[Crossref]

S. K. Turitsyn, T. Schäfer, and V. K. Mezentsev, “Self-similar core and oscillatory tails of a path-averaged chirped dispersion-managed optical pulse,” Opt. Lett. 23(17), 1351–1353 (1998).
[Crossref]

E. G. Shapiro and S. K. Turitsyn, “Theory of guiding-center breathing soliton propagation in optical communication systems with strong dispersion management,” Opt. Lett. 22(20), 1544–1546 (1997).
[Crossref]

I. Gabitov and S. K. Turitsyn, “Averaged pulse dynamics in a cascaded transmission system with passive dispersion compensation,” Opt. Lett. 21(5), 327–329 (1996).
[Crossref]

Turitsyn, S.K.

S.K. Turitsyn, B. Bale, and M.P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
[Crossref]

B.G Bale, O.G. Okhotnikov, and S.K. Turitsyn, “Modeling and Technologies of Ultrafast Fiber Lasers,” in Fiber Lasers, O. G. Okhotnikov, ed. (Wiley-VCH Verlag GmbH Co., 2012).

Wise, F. W.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
[Crossref]

Yarutkina, I. A.

I. A. Yarutkina and O. V. Shtyrina, “Mathematical modelling of dispersion-managed thulium/holmium fibre lasers,” Quantum Electron. 43(11), 1019–1023 (2013).
[Crossref]

Yu, C. X.

Commun. ACM. (1)

J. Holt, “Numerical solution of nonlinear two-point boundary problems by finite difference methods,” Commun. ACM. 7(6), 366–373 (1964).
[Crossref]

IEEE J. Quantum Electron. (2)

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975).
[Crossref]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992).
[Crossref]

IEEE J. Sel. Top. Quant. (1)

J. Nijhof, W. Forisiak, and N. Doran, “The averaging method for finding exactly periodic dispersion-managed solitons,” IEEE J. Sel. Top. Quant. 6(2), 330–336 (2000).
[Crossref]

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

JETP Lett. (1)

S. K. Turitsyn and V. K. Mezentsev, “Dynamics of self-similar dispersion-managed soliton presented in the basis of chirped Gauss-Hermite functions,” JETP Lett. 67(9), 640–646 (1998).
[Crossref]

Laser Photonics Rev. (1)

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rep. (1)

S.K. Turitsyn, B. Bale, and M.P. Fedoruk, “Dispersion-managed solitons in fibre systems and lasers,” Phys. Rep. 521(4), 135–203 (2012).
[Crossref]

Phys. Rev. A (1)

B. G. Bale, S. Boscolo, J. N. Kutz, and S. K. Turitsyn, “Intracavity dynamics in high-power mode-locked fiber lasers,” Phys. Rev. A 81(3), 033828 (2010).
[Crossref]

Phys. Rev. E. (2)

S. Turitsyn and E. Shapiro, “Enhanced power breathing soliton in communication systems with dispersion management,” Phys. Rev. E. 56(5), R4951–R4955, (1997).
[Crossref]

S. K. Turitsyn, “Breathing self-similar dynamics and oscillatory tails of the chirped dispersion-managed soliton,” Phys. Rev. E. 58(2), R1256–R1259 (1998).
[Crossref]

Quantum Electron. (1)

I. A. Yarutkina and O. V. Shtyrina, “Mathematical modelling of dispersion-managed thulium/holmium fibre lasers,” Quantum Electron. 43(11), 1019–1023 (2013).
[Crossref]

SIAM Rev. (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48(4), 629–678 (2006).
[Crossref]

Other (4)

N. Akhmediev and A. Ankiewicz, eds. Dissipative Solitons: Lecture Notes in Physics (Springer, 2005), V. 661.

B.G Bale, O.G. Okhotnikov, and S.K. Turitsyn, “Modeling and Technologies of Ultrafast Fiber Lasers,” in Fiber Lasers, O. G. Okhotnikov, ed. (Wiley-VCH Verlag GmbH Co., 2012).

H. Keller, Numerical Methods for Two-point Boundary Value Problem (Blaisdell Publishing Co., 1968).

S. Roberts and S. J. Shipman, Two-point Boundary Value Problems: Shooting Methods (Elsevier, 1972).

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Figures (7)

Fig. 1
Fig. 1 The dispersion map scheme defined by Eq. (11). The green and red lines correspond to the fiber segments with anomalous and normal dispersion, respectively. Grey areas denote the corresponding dispersion depth.
Fig. 2
Fig. 2 a) Dynamics of the energy stabilization for the following set of parameters: g0 = 1, l0 = 0.5, ν = 0.1, ε0 = 10, ε = 1, and ε̃ = 10−8; b) Dynamics of characteristics of the pulse along z for the first step of the iteration process.
Fig. 3
Fig. 3 Contour plots of the pulse energy E, pulse width τ and peak power P obtained via a) NLSE and b) ODEs in the (<D>, D) plane for anomalous regimes.
Fig. 4
Fig. 4 Comparison of results obtained via NLSE and ODE models for anomalous regime. a) 3D pulse shape dynamics; b) Intracavity peak-power dynamics; c) 3D pulse spectrum dynamics; d) Comparison of spectral shapes at output. The green sections correspond to the part of the cavity with anomalous dispersion, and the red section corresponds to that with normal dispersion.
Fig. 5
Fig. 5 Pulse energy (a) and peak power (b) dependence on the cavity-accumulated dispersion as a function of passive fiber length. The curves from top to bottom correspond to the total cavity lengths of 27, 25, 22, 20, 17, 15, and 12 m. Red curves correspond to the modeling results obtained via ODEs while the black curves correspond to those obtained via NLSE.
Fig. 6
Fig. 6 Average pulse-power dependence on the total cavity length. Red curves correspond to the modeling results obtained via ODEs while the black ones correspond to those obtained via NLSE. a) The curves from left to right correspond dispersion compensation fiber lengths of 8, 10, 12, 15, and 17 m; b) The curves from top to bottom correspond to SMF-28 fiber lengths of 1, 2, 3, 4, 6, and 7 m.
Fig. 7
Fig. 7 a) and b) show the energy contour plots obtained via NLSE and ODEs, respectively, in the (<D>, D) plane for normal regimes. c) Pulse energy dependence on the cavity-accumulated dispersion <D>. Here, red curves correspond to the results obtained via ODEs while the black curves correspond to those obtained via NLSE. The curves from the top to bottom correspond to constant total cavity lengths of 85 m, 75 m, 65 m, and 55 m. The cavity-accumulated dispersion varies with variation in the passive fiber length. The termination point of each curve corresponds to the end of the range of generation. d) Pulse shape comparison for normal regimes. e) Pulse spectrum comparison for normal regimes.

Tables (1)

Tables Icon

Table 1 Values of fiber laser parameters

Equations (25)

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

i U Z 1 2 β 2 U T T + γ | U | 2 U = i ( G Γ ) U + i G Ω g 2 U T T ,
G = G ( z ) = G 0 1 + E ( Z ) / ( P sat G T R ) ,
i u z + d ( z ) u T T + ε | u | 2 u = i ( g l 0 ) u + i ν g u T T ,
u ( z , t ) = P ( z ) exp [ t 2 2 τ 2 ( z ) ( 1 i C ( z ) ) + i φ ( z ) ] .
τ z = 2 d ( z ) C τ ν g 1 τ ( C 2 1 ) ,
C z = ( 2 d ( z ) 2 ν g C ) 1 + C 2 τ 2 ε 2 P ,
P z = 2 d ( z ) C P τ 2 + 2 ( g l 0 ) P 2 ν g P τ 2 ,
g = g ( τ , P ) = g 0 1 + P τ π / ε 0 ,
τ ( 0 ) = τ ( 1 ) , C ( 0 ) = C ( 1 ) , P ( 0 ) = P ( 1 ) .
d V ( z ) d z = F ( z , V ( z ) ) , z ( 0 , 1 ) ;
V ( 0 ) = V ( 1 ) ,
< D > = d 1 + d 2 ,
D = d 1 d 2
d 1 = D + < D > , d 2 = D + < D > .
d ( z ) = { d 1 , 0 z < 0.25 , d 2 , 0.25 z < 0.75 , d 1 , 0.75 z < 1 .
d V ( z ) d z = F ( z , V ( z ) ) , z > 0
V ( 0 ) = W ,
d V j ( z ) d z = F ( z , V j ( z ) ) , z > 0
V j ( 0 ) = V j ,
u ( V j ( z ) , t ) = P j ( z ) exp [ t 2 2 τ j , 2 ( z ) ( 1 i C j ( z ) ) + i φ ( z ) ] ,
u ( V j + 1 , t ) = α ϕ ( u ( V j ( z max ) , t ) ) + ( 1 α ) ϕ ( u ( V j ( z min ) , t ) ) | α ϕ ( u ( V j ( z max ) , t ) ) + ( 1 α ) ϕ ( u ( V j ( z min ) , t ) ) | 2 d t | u ( V 0 , t ) | 2 d t ,
ϕ ( u ( V j ( z ˜ ) , t ) ) = u ( V j ( z ˜ ) , 0 ) | u ( V j ( z ˜ ) , 0 ) | u ( V j ( z ˜ ) , t ) = P j ( z ˜ ) exp [ t 2 2 τ j 2 ( z ˜ ) ( 1 i C j ( z ˜ ) ) ] .
V j + 1 = ( τ i + 1 , C j + 1 , P j + 1 ) T = ( τ j + 1 , ( τ j + 1 ) 2 d 2 arg ( u ( V j + 1 , t ) ) d t 2 , | u ( V j + 1 , 0 ) | 2 ) T ,
i u Z d ( Z ) u T T + ε | u | 2 u = i ( g ^ ( T ) l 0 ) u .
g ^ ( ω ) = g 1 + E / ε 0 × 1 1 + ( ω ω 0 Ω g ) 2 .

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