Abstract

First-order nonlinear perturbations of several fiber spans may be summed up linearly to yield the total nonlinear response of a group of spans, if nonlinear effects are neglected beyond the first order. Such linear additivity of first-order nonlinear perturbations about the nonlinear coefficients enables nonlinear compensations between one span with stronger nonlinearity and a group of spans with weaker nonlinearity, provided that there is a scaled translation symmetry between the strongly and the weakly nonlinear spans, and the dispersion of each span is properly managed. Excellent nonlinear compensations are achieved both in systems with an optical phase conjugator in the middle and in systems without.

© 2006 Optical Society of America

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References

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  1. A. H. Gnauck and R. M. Jopson, "Dispersion compensation for optical fiber systems," in Optical Fiber Telecommunications IIIA, I.P.Kaminow and T.L.Koch, eds. (Academic, 1997).
  2. F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, "Fiber nonlinearities and their impact on transmission systems," in Optical Fiber Telecommunications IIIA, I.P.Kaminow and T.L.Koch, eds. (Academic, l997).
  3. M. E. Marhic, N. Kagi, T.-K. Chiang, and L. G. Kazovsky, "Cancellation of third-order nonlinear effects in amplified fiber links by dispersion compensation, phase conjugation, and alternating dispersion," Opt. Lett. 20, 863-865 (1995).
    [CrossRef] [PubMed]
  4. D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov, and S. K. Turitsyn, "Optimal schemes for dispersion compensation of standard monomode fiber based links," Opt. Commun. 140, 15-18 (1997).
    [CrossRef]
  5. D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, E. G. Shapiro, I. Gabitov, and S. K. Turitsyn, "Symmetrical dispersion compensation for standard monomode-fiber-based communication systems with large amplifier spacing," Opt. Lett. 22, 982-984 (1997).
    [CrossRef] [PubMed]
  6. P. Kaewplung, T. Angkaew, and K. Kikuchi, "Simultaneous suppression of third-order dispersion and sideband instability in single-channel optical fiber transmission by midway optical phase conjugation employing higher order dispersion management," J. Lightwave Technol. 21, 1465-1473 (2003).
    [CrossRef]
  7. H. Wei and D. V. Plant, "Two means of compensating fiber nonlinearity using optical phase conjugation," arXiv:physics/0307022.
  8. H. Wei and D. V. Plant, "Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation," Opt. Express 12, 1938-1958 (2004).
    [CrossRef] [PubMed]
  9. A. G. Striegler and B. Schmauss, "Compensation of intrachannel effects in symmetric dispersion-managed transmission systems," J. Lightwave Technol. 22, 1877-1882 (2004).
    [CrossRef]
  10. H. Wei and D. V. Plant, "Intra-channel nonlinearity compensation with scaled translational symmetry," Opt. Express l2, 4282-4296 (2004).
    [CrossRef]
  11. A. G. Striegler and B. Schmauss, "Fiber-based compensation of IXPM-induced timing jitter," IEEE Photon. Technol. Lett. 16, 2574-2576 (2004).
    [CrossRef]
  12. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995), Chap. 2.
  13. H. Wei and D. V. Plant, "Fundamental equations of nonlinear fiber optics," in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE 5178, 255-266 (2003).
    [CrossRef]
  14. E. E. Narimanov and P. Mitra, "The channel capacity of a fiber optics communication system: perturbation theory," J. Lightwave Technol. 20, 530-537 (2002).
    [CrossRef]
  15. R.-J. Essiambre, G. Raybon, and B. Mikkelson, "Pseudo-linear transmission of high-speed TDM signals: 40 and160Gb/s," in Optical Fiber Telecommunications IVB: Systems and Impairments, I.P.Kaminow and T.Li, eds. (Academic, 2002).
  16. R.-J. Essiambre, B. Mikkelsen, and G. Raybon, "Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems," Electron. Lett. 35, 1576-1578 (1999).
    [CrossRef]
  17. P. V. Mamyshev and N. A. Mamysheva, "Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing," Opt. Lett. 24, 1454-1456 (1999).
    [CrossRef]
  18. F. Merlaud and S. K. Turitsyn, "Intra-channel four wave mixing and ghost pulses generation: time domain approach," in 26th European Conference on Optical Communication (IEEE Press, 2000), paper 7.2.4.
  19. P. Johannisson, D. Anderson, A. Berntson, and J. Martensson, "Generation and dynamics of ghost pulses in strongly dispersion-managed fiber-optic communication systems," Opt. Lett. 26, 1227-1229 (2001).
    [CrossRef]
  20. H. Wei and D. V. Plant, "Reversing intrachannel ghost-pulse generation by midspan self-phase modulation," Opt. Lett. 30, 2366-2368 (2005).
    [CrossRef] [PubMed]
  21. S. N. Knudsen and T. Veng, "Large effective area dispersion compensating fiber for cabled compensation of standard single mode fiber," in Optical Fiber Communications Conference (OFC), Postconference Digest, Vol. 37 of OSA Trends in Optics and Photonics (Optical Society of America, 2000), paper TuG5.
  22. K. Mukasa, H. Moridaira, T. Yagi, and K. Kokura, "New type of dispersion management transmission line with MDFSD for long-haul 40Gb/s transmission," in Optical Fiber Communication Conference (OFC), Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper ThGG2.

2005

2004

H. Wei and D. V. Plant, "Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation," Opt. Express 12, 1938-1958 (2004).
[CrossRef] [PubMed]

A. G. Striegler and B. Schmauss, "Compensation of intrachannel effects in symmetric dispersion-managed transmission systems," J. Lightwave Technol. 22, 1877-1882 (2004).
[CrossRef]

H. Wei and D. V. Plant, "Intra-channel nonlinearity compensation with scaled translational symmetry," Opt. Express l2, 4282-4296 (2004).
[CrossRef]

A. G. Striegler and B. Schmauss, "Fiber-based compensation of IXPM-induced timing jitter," IEEE Photon. Technol. Lett. 16, 2574-2576 (2004).
[CrossRef]

2003

2002

2001

1999

R.-J. Essiambre, B. Mikkelsen, and G. Raybon, "Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems," Electron. Lett. 35, 1576-1578 (1999).
[CrossRef]

P. V. Mamyshev and N. A. Mamysheva, "Pulse-overlapped dispersion-managed data transmission and intrachannel four-wave mixing," Opt. Lett. 24, 1454-1456 (1999).
[CrossRef]

1997

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov, and S. K. Turitsyn, "Optimal schemes for dispersion compensation of standard monomode fiber based links," Opt. Commun. 140, 15-18 (1997).
[CrossRef]

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, E. G. Shapiro, I. Gabitov, and S. K. Turitsyn, "Symmetrical dispersion compensation for standard monomode-fiber-based communication systems with large amplifier spacing," Opt. Lett. 22, 982-984 (1997).
[CrossRef] [PubMed]

1995

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995), Chap. 2.

Anderson, D.

Angkaew, T.

Berntson, A.

Breuer, D.

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov, and S. K. Turitsyn, "Optimal schemes for dispersion compensation of standard monomode fiber based links," Opt. Commun. 140, 15-18 (1997).
[CrossRef]

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, E. G. Shapiro, I. Gabitov, and S. K. Turitsyn, "Symmetrical dispersion compensation for standard monomode-fiber-based communication systems with large amplifier spacing," Opt. Lett. 22, 982-984 (1997).
[CrossRef] [PubMed]

Chiang, T.-K.

Chraplyvy, A. R.

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, "Fiber nonlinearities and their impact on transmission systems," in Optical Fiber Telecommunications IIIA, I.P.Kaminow and T.L.Koch, eds. (Academic, l997).

Essiambre, R.-J.

R.-J. Essiambre, B. Mikkelsen, and G. Raybon, "Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems," Electron. Lett. 35, 1576-1578 (1999).
[CrossRef]

R.-J. Essiambre, G. Raybon, and B. Mikkelson, "Pseudo-linear transmission of high-speed TDM signals: 40 and160Gb/s," in Optical Fiber Telecommunications IVB: Systems and Impairments, I.P.Kaminow and T.Li, eds. (Academic, 2002).

Forghieri, F.

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, "Fiber nonlinearities and their impact on transmission systems," in Optical Fiber Telecommunications IIIA, I.P.Kaminow and T.L.Koch, eds. (Academic, l997).

Gabitov, I.

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov, and S. K. Turitsyn, "Optimal schemes for dispersion compensation of standard monomode fiber based links," Opt. Commun. 140, 15-18 (1997).
[CrossRef]

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, E. G. Shapiro, I. Gabitov, and S. K. Turitsyn, "Symmetrical dispersion compensation for standard monomode-fiber-based communication systems with large amplifier spacing," Opt. Lett. 22, 982-984 (1997).
[CrossRef] [PubMed]

Gnauck, A. H.

A. H. Gnauck and R. M. Jopson, "Dispersion compensation for optical fiber systems," in Optical Fiber Telecommunications IIIA, I.P.Kaminow and T.L.Koch, eds. (Academic, 1997).

Johannisson, P.

Jopson, R. M.

A. H. Gnauck and R. M. Jopson, "Dispersion compensation for optical fiber systems," in Optical Fiber Telecommunications IIIA, I.P.Kaminow and T.L.Koch, eds. (Academic, 1997).

Jürgensen, K.

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, E. G. Shapiro, I. Gabitov, and S. K. Turitsyn, "Symmetrical dispersion compensation for standard monomode-fiber-based communication systems with large amplifier spacing," Opt. Lett. 22, 982-984 (1997).
[CrossRef] [PubMed]

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov, and S. K. Turitsyn, "Optimal schemes for dispersion compensation of standard monomode fiber based links," Opt. Commun. 140, 15-18 (1997).
[CrossRef]

Kaewplung, P.

Kagi, N.

Kazovsky, L. G.

Kikuchi, K.

Knudsen, S. N.

S. N. Knudsen and T. Veng, "Large effective area dispersion compensating fiber for cabled compensation of standard single mode fiber," in Optical Fiber Communications Conference (OFC), Postconference Digest, Vol. 37 of OSA Trends in Optics and Photonics (Optical Society of America, 2000), paper TuG5.

Kokura, K.

K. Mukasa, H. Moridaira, T. Yagi, and K. Kokura, "New type of dispersion management transmission line with MDFSD for long-haul 40Gb/s transmission," in Optical Fiber Communication Conference (OFC), Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper ThGG2.

Küppers, F.

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov, and S. K. Turitsyn, "Optimal schemes for dispersion compensation of standard monomode fiber based links," Opt. Commun. 140, 15-18 (1997).
[CrossRef]

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, E. G. Shapiro, I. Gabitov, and S. K. Turitsyn, "Symmetrical dispersion compensation for standard monomode-fiber-based communication systems with large amplifier spacing," Opt. Lett. 22, 982-984 (1997).
[CrossRef] [PubMed]

Mamyshev, P. V.

Mamysheva, N. A.

Marhic, M. E.

Martensson, J.

Mattheus, A.

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, E. G. Shapiro, I. Gabitov, and S. K. Turitsyn, "Symmetrical dispersion compensation for standard monomode-fiber-based communication systems with large amplifier spacing," Opt. Lett. 22, 982-984 (1997).
[CrossRef] [PubMed]

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov, and S. K. Turitsyn, "Optimal schemes for dispersion compensation of standard monomode fiber based links," Opt. Commun. 140, 15-18 (1997).
[CrossRef]

Merlaud, F.

F. Merlaud and S. K. Turitsyn, "Intra-channel four wave mixing and ghost pulses generation: time domain approach," in 26th European Conference on Optical Communication (IEEE Press, 2000), paper 7.2.4.

Mikkelsen, B.

R.-J. Essiambre, B. Mikkelsen, and G. Raybon, "Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems," Electron. Lett. 35, 1576-1578 (1999).
[CrossRef]

Mikkelson, B.

R.-J. Essiambre, G. Raybon, and B. Mikkelson, "Pseudo-linear transmission of high-speed TDM signals: 40 and160Gb/s," in Optical Fiber Telecommunications IVB: Systems and Impairments, I.P.Kaminow and T.Li, eds. (Academic, 2002).

Mitra, P.

Moridaira, H.

K. Mukasa, H. Moridaira, T. Yagi, and K. Kokura, "New type of dispersion management transmission line with MDFSD for long-haul 40Gb/s transmission," in Optical Fiber Communication Conference (OFC), Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper ThGG2.

Mukasa, K.

K. Mukasa, H. Moridaira, T. Yagi, and K. Kokura, "New type of dispersion management transmission line with MDFSD for long-haul 40Gb/s transmission," in Optical Fiber Communication Conference (OFC), Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper ThGG2.

Narimanov, E. E.

Plant, D. V.

H. Wei and D. V. Plant, "Reversing intrachannel ghost-pulse generation by midspan self-phase modulation," Opt. Lett. 30, 2366-2368 (2005).
[CrossRef] [PubMed]

H. Wei and D. V. Plant, "Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation," Opt. Express 12, 1938-1958 (2004).
[CrossRef] [PubMed]

H. Wei and D. V. Plant, "Intra-channel nonlinearity compensation with scaled translational symmetry," Opt. Express l2, 4282-4296 (2004).
[CrossRef]

H. Wei and D. V. Plant, "Fundamental equations of nonlinear fiber optics," in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE 5178, 255-266 (2003).
[CrossRef]

H. Wei and D. V. Plant, "Two means of compensating fiber nonlinearity using optical phase conjugation," arXiv:physics/0307022.

Raybon, G.

R.-J. Essiambre, B. Mikkelsen, and G. Raybon, "Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems," Electron. Lett. 35, 1576-1578 (1999).
[CrossRef]

R.-J. Essiambre, G. Raybon, and B. Mikkelson, "Pseudo-linear transmission of high-speed TDM signals: 40 and160Gb/s," in Optical Fiber Telecommunications IVB: Systems and Impairments, I.P.Kaminow and T.Li, eds. (Academic, 2002).

Schmauss, B.

A. G. Striegler and B. Schmauss, "Fiber-based compensation of IXPM-induced timing jitter," IEEE Photon. Technol. Lett. 16, 2574-2576 (2004).
[CrossRef]

A. G. Striegler and B. Schmauss, "Compensation of intrachannel effects in symmetric dispersion-managed transmission systems," J. Lightwave Technol. 22, 1877-1882 (2004).
[CrossRef]

Shapiro, E. G.

Striegler, A. G.

A. G. Striegler and B. Schmauss, "Fiber-based compensation of IXPM-induced timing jitter," IEEE Photon. Technol. Lett. 16, 2574-2576 (2004).
[CrossRef]

A. G. Striegler and B. Schmauss, "Compensation of intrachannel effects in symmetric dispersion-managed transmission systems," J. Lightwave Technol. 22, 1877-1882 (2004).
[CrossRef]

Tkach, R. W.

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, "Fiber nonlinearities and their impact on transmission systems," in Optical Fiber Telecommunications IIIA, I.P.Kaminow and T.L.Koch, eds. (Academic, l997).

Turitsyn, S. K.

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, E. G. Shapiro, I. Gabitov, and S. K. Turitsyn, "Symmetrical dispersion compensation for standard monomode-fiber-based communication systems with large amplifier spacing," Opt. Lett. 22, 982-984 (1997).
[CrossRef] [PubMed]

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov, and S. K. Turitsyn, "Optimal schemes for dispersion compensation of standard monomode fiber based links," Opt. Commun. 140, 15-18 (1997).
[CrossRef]

F. Merlaud and S. K. Turitsyn, "Intra-channel four wave mixing and ghost pulses generation: time domain approach," in 26th European Conference on Optical Communication (IEEE Press, 2000), paper 7.2.4.

Veng, T.

S. N. Knudsen and T. Veng, "Large effective area dispersion compensating fiber for cabled compensation of standard single mode fiber," in Optical Fiber Communications Conference (OFC), Postconference Digest, Vol. 37 of OSA Trends in Optics and Photonics (Optical Society of America, 2000), paper TuG5.

Wei, H.

H. Wei and D. V. Plant, "Reversing intrachannel ghost-pulse generation by midspan self-phase modulation," Opt. Lett. 30, 2366-2368 (2005).
[CrossRef] [PubMed]

H. Wei and D. V. Plant, "Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation," Opt. Express 12, 1938-1958 (2004).
[CrossRef] [PubMed]

H. Wei and D. V. Plant, "Intra-channel nonlinearity compensation with scaled translational symmetry," Opt. Express l2, 4282-4296 (2004).
[CrossRef]

H. Wei and D. V. Plant, "Fundamental equations of nonlinear fiber optics," in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE 5178, 255-266 (2003).
[CrossRef]

H. Wei and D. V. Plant, "Two means of compensating fiber nonlinearity using optical phase conjugation," arXiv:physics/0307022.

Yagi, T.

K. Mukasa, H. Moridaira, T. Yagi, and K. Kokura, "New type of dispersion management transmission line with MDFSD for long-haul 40Gb/s transmission," in Optical Fiber Communication Conference (OFC), Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper ThGG2.

Electron. Lett.

R.-J. Essiambre, B. Mikkelsen, and G. Raybon, "Intra-channel cross-phase modulation and four-wave mixing in high-speed TDM systems," Electron. Lett. 35, 1576-1578 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

A. G. Striegler and B. Schmauss, "Fiber-based compensation of IXPM-induced timing jitter," IEEE Photon. Technol. Lett. 16, 2574-2576 (2004).
[CrossRef]

J. Lightwave Technol.

Opt. Commun.

D. Breuer, K. Jürgensen, F. Küppers, A. Mattheus, I. Gabitov, and S. K. Turitsyn, "Optimal schemes for dispersion compensation of standard monomode fiber based links," Opt. Commun. 140, 15-18 (1997).
[CrossRef]

Opt. Express

H. Wei and D. V. Plant, "Simultaneous nonlinearity suppression and wide-band dispersion compensation using optical phase conjugation," Opt. Express 12, 1938-1958 (2004).
[CrossRef] [PubMed]

H. Wei and D. V. Plant, "Intra-channel nonlinearity compensation with scaled translational symmetry," Opt. Express l2, 4282-4296 (2004).
[CrossRef]

Opt. Lett.

Proc. SPIE

H. Wei and D. V. Plant, "Fundamental equations of nonlinear fiber optics," in Optical Modeling and Performance Predictions, M. A. Kahan, ed., Proc. SPIE 5178, 255-266 (2003).
[CrossRef]

Other

S. N. Knudsen and T. Veng, "Large effective area dispersion compensating fiber for cabled compensation of standard single mode fiber," in Optical Fiber Communications Conference (OFC), Postconference Digest, Vol. 37 of OSA Trends in Optics and Photonics (Optical Society of America, 2000), paper TuG5.

K. Mukasa, H. Moridaira, T. Yagi, and K. Kokura, "New type of dispersion management transmission line with MDFSD for long-haul 40Gb/s transmission," in Optical Fiber Communication Conference (OFC), Vol. 70 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), paper ThGG2.

A. H. Gnauck and R. M. Jopson, "Dispersion compensation for optical fiber systems," in Optical Fiber Telecommunications IIIA, I.P.Kaminow and T.L.Koch, eds. (Academic, 1997).

F. Forghieri, R. W. Tkach, and A. R. Chraplyvy, "Fiber nonlinearities and their impact on transmission systems," in Optical Fiber Telecommunications IIIA, I.P.Kaminow and T.L.Koch, eds. (Academic, l997).

H. Wei and D. V. Plant, "Two means of compensating fiber nonlinearity using optical phase conjugation," arXiv:physics/0307022.

F. Merlaud and S. K. Turitsyn, "Intra-channel four wave mixing and ghost pulses generation: time domain approach," in 26th European Conference on Optical Communication (IEEE Press, 2000), paper 7.2.4.

R.-J. Essiambre, G. Raybon, and B. Mikkelson, "Pseudo-linear transmission of high-speed TDM signals: 40 and160Gb/s," in Optical Fiber Telecommunications IVB: Systems and Impairments, I.P.Kaminow and T.Li, eds. (Academic, 2002).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995), Chap. 2.

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Figures (6)

Fig. 1
Fig. 1

System in one-for-many STS between ( RDF + SMF ) and ( SMF + RDF ) spans with OPC in the middle. TX, transmitter; RX, receiver.

Fig. 2
Fig. 2

Typical eye diagram of optical signals received at the end of the one-for-many scaled translation-symmetric system shown in Fig. 1.

Fig. 3
Fig. 3

Typical received optical eye diagrams of the two comparative systems. Left: the system with no one-for-many STS but with OPC in the middle. Right: the system with neither one-for-many STS nor OPC.

Fig. 4
Fig. 4

Optical eye diagrams at the midpoint of transmissions. Left: the optimized system. Right: the comparative system using all SMF + RDF spans without one-for-many STS.

Fig. 5
Fig. 5

Optical eye diagrams at the end of transmissions. Left: the optimized system. Right: the comparative system using all SMF + RDF spans without one-for-many STS.

Fig. 6
Fig. 6

Typical eye diagram of optical signals received at the end of the optimized system with one-for-many STS and midway SPM.

Equations (19)

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

α ( z ) α ( z ) = β 2 ( z ) β 2 ( z ) = β 3 ( z ) β 3 ( z ) = γ ( z ) P 0 γ ( z ) P 0 = g ( z , t ) P 0 g ( z , t ) P 0 = z z = 1 R ,
α ( z ) α ( z ) = β 2 ( z ) β 2 ( z ) = β 3 ( z ) β 3 ( z ) = γ ( z ) P 0 γ ( z ) P 0 = z z = 1 R ,
[ α ( z ) , β 2 ( z ) , ± β 3 ( z ) ] = R [ α ( z ) , β 2 ( z ) , β 3 ( z ) ] , R > 0 ,
E ( z , t ) = A ( z , t ) exp [ i z β 0 ( ζ ) d ζ i ω 0 t ] ,
A ( z , t ) z + k = 2 + i k 1 β k ( z ) k ! ( t ) k A ( z , t ) + α ( z ) 2 A ( z , t ) = i γ ( z ) A ( z , t ) 2 A ( z , t ) + i [ g ( z , t ) A ( z , t ) 2 ] A ( z , t ) ,
β k ( z ) = def 1 2 β 0 ( z ) k [ β 2 ( ω , z ) ] ω k ω = ω 0 , k 2 ,
A ( z , t ) z + k = 2 + i k 1 β k ( z ) k ! ( t ) k A ( z , t ) + α ( z ) 2 A ( z , t ) = 0 ,
H ( z 1 , z 2 , ω ) = def exp [ i k = 2 + ω k k ! z 1 z 2 β k ( z ) d z 1 2 z 1 z 2 α ( z ) d z ] .
P ( z 1 , z 2 ) = def F 1 H ( z 1 , z 2 , ω ) F .
h ( z 1 , z 2 , t ) = def F 1 [ H ( z 1 , z 2 , ω ) ] .
P ( z 1 , z 2 ) = h ( z 1 , z 2 , t ) .
A 0 ( z 2 , t ) = P ( z 1 , z 2 ) A ( z 1 , t ) ,
A 1 ( z 2 , t ) = z 1 z 2 P ( z , z 2 ) { i γ ( z ) A 0 ( z , t ) 2 A 0 ( z , t ) + i [ g ( z , t ) A 0 ( z , t ) 2 ] A 0 ( z , t ) } d z ,
[ α ( n ) ( z ( n ) ) , β 2 ( n ) ( z ( n ) ) , ± β 3 ( n ) ( z ( n ) ) ] = R ( n ) [ α ( z ) , β 2 ( z ) , β 3 ( z ) ] , R ( n ) > 0 ,
[ γ ( n ) ( z ( n ) ) , g ( n ) ( z ( n ) , t ) ] = Q ( n ) [ γ ( z ) , g ( z , t ) ] , Q ( n ) > 0 ,
z ( n ) = z R ( n ) , 0 z L ,
A 1 ( n ) ( z ( n ) = L R ( n ) , t ) = [ Q ( n ) P 0 ( n ) R ( n ) P 0 ] A 1 * ( z = L , t ) , n [ 1 , N ] ,
n = 1 N A 1 ( n ) ( z ( n ) = L R ( n ) , t ) = A 1 * ( z = L , t ) ,
n = 1 N [ Q ( n ) R ( n ) ] P 0 ( n ) = P 0 ,

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