Abstract

The common practice is to derive in the time domain the equations for the nonlinear propagation of ultrashort pulses and to solve them by the split-step Fourier method. When fields at different frequencies propagate and interact in a fiber, it is also usual to write a distinct equation for each of these fields. This implicitly assumes that the spectra of these various fields do not overlap, which may no longer be the case for femtosecond pulses. It then becomes necessary to consider the propagation of the total field as a whole. Moreover, for such broad field spectra the dispersion of higher orders must be taken into account, which can be dealt with in the frequency domain without any difficulty. It is shown that writing the equations for the total field presents many advantages, and an alternative to the split-step Fourier method is proposed, based on an integration rule of higher order.

© 1991 Optical Society of America

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References

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  1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  2. N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
    [CrossRef]
  3. A. Hasegawa, “Numerical study of optical soliton transmission amplified periodically by the stimulated Raman process,” Appl. Opt. 23, 3302–3309 (1984).
    [CrossRef] [PubMed]
  4. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  5. W. Zhao and E. Bourkoff, “Femtosecond pulse propagation in optical fibers: higher order effects,” IEEE J. Quantum Electron. 24, 365–372 (1988).
    [CrossRef]
  6. R. H. Stolen and J. P. Gordon, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
    [CrossRef]
  7. K. Blow, paper delivered at the Workshop on Waveguide Theory, Arundel, UK (1988).
  8. C. Pask and A. Vatarescu, “Spectral approach to pulse propagation in a dispersive nonlinear medium,” J. Opt. Soc. Am. B 3, 1018–1024 (1986).
    [CrossRef]
  9. R. R. Alfano and P. P. Ho, “Self-, cross-, and induced-phase modulations of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
    [CrossRef]
  10. D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
    [CrossRef]
  11. E. Fehlberg, “Klassische Runge–Kutta–Formel vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Warmeleitungsprobleme,” Computing 6, 61–71 (1970).
    [CrossRef]
  12. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  13. D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers,” Electron. Lett. 20, 659–660 (1984).
    [CrossRef]
  14. P. L. Francois, “Zero dispersion in attenuation optimized doubly-clad fibers,” IEEE J. Lightwave Technol. LT-1, 26–37 (1983).
    [CrossRef]
  15. P. L. François, “Design of monomode quadruple-clad fibres,” Electron. Lett. 20, 688–689 (1984).
    [CrossRef]
  16. G. R. Boyer and M. A. Franco, “Numerical and experimental comparison of spectral broadening of femtosecond optical asymmetric pulses in a monomode fiber,” Opt. Lett. 14, 465–467 (1989).
    [CrossRef] [PubMed]
  17. C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. Inst. Electr. Eng. Part J 134, 145–151 (1987).
  18. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  19. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  20. P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1938–1946 (1987).
    [CrossRef]
  21. C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, “Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing,” Opt. Lett. 6, 493–495 (1981).
    [CrossRef] [PubMed]
  22. S. J. Garth and C. Pask, “Four-photon mixing and dispersion in single-mode fibers,” Opt. Lett. 11, 380–382 (1986).
    [CrossRef] [PubMed]
  23. C. Vassallo, Théorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985), Chap. 5.
  24. C. Vassallo, Electromagnétisme Classique dans la Matière (Dunod Université, Paris, 1980), Chap. 4, Eq. (4.C.6), p. 73.
  25. J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” App. Opt. 20, 1403–1406 (1981).
    [CrossRef]
  26. J. L. Ayral, J. P. Pocholle, J. Raffy, and M. Papuchon, “Optical Kerr coefficient measurement at 1.55 μ m in single mode optical fibers,” Opt. Commun. 49, 405–408 (1984).
    [CrossRef]
  27. B. Crosignani, S. Piazzolla, and P. Spano, “Direct measurement of the nonlinear phase shift between the orthogonally polarized states of a single-mode fiber,” Opt. Lett. 10, 89–91 (1985).
    [CrossRef] [PubMed]
  28. F. A. Hopf and G. I. Stegeman, Nonlinear Optics, Vol. II of Applied Classical Electrodynamics (Wiley, New York, 1986), Chap. 18.
  29. R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
    [CrossRef]

1989 (2)

1988 (2)

W. Zhao and E. Bourkoff, “Femtosecond pulse propagation in optical fibers: higher order effects,” IEEE J. Quantum Electron. 24, 365–372 (1988).
[CrossRef]

R. R. Alfano and P. P. Ho, “Self-, cross-, and induced-phase modulations of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
[CrossRef]

1987 (2)

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. Inst. Electr. Eng. Part J 134, 145–151 (1987).

P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1938–1946 (1987).
[CrossRef]

1986 (4)

1985 (1)

1984 (4)

A. Hasegawa, “Numerical study of optical soliton transmission amplified periodically by the stimulated Raman process,” Appl. Opt. 23, 3302–3309 (1984).
[CrossRef] [PubMed]

J. L. Ayral, J. P. Pocholle, J. Raffy, and M. Papuchon, “Optical Kerr coefficient measurement at 1.55 μ m in single mode optical fibers,” Opt. Commun. 49, 405–408 (1984).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers,” Electron. Lett. 20, 659–660 (1984).
[CrossRef]

P. L. François, “Design of monomode quadruple-clad fibres,” Electron. Lett. 20, 688–689 (1984).
[CrossRef]

1983 (2)

P. L. Francois, “Zero dispersion in attenuation optimized doubly-clad fibers,” IEEE J. Lightwave Technol. LT-1, 26–37 (1983).
[CrossRef]

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[CrossRef]

1981 (3)

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[CrossRef]

C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, “Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing,” Opt. Lett. 6, 493–495 (1981).
[CrossRef] [PubMed]

J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” App. Opt. 20, 1403–1406 (1981).
[CrossRef]

1973 (2)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1970 (1)

E. Fehlberg, “Klassische Runge–Kutta–Formel vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Warmeleitungsprobleme,” Computing 6, 61–71 (1970).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

Alfano, R. R.

R. R. Alfano and P. P. Ho, “Self-, cross-, and induced-phase modulations of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
[CrossRef]

Anderson, D.

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[CrossRef]

Ashkin, A.

J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” App. Opt. 20, 1403–1406 (1981).
[CrossRef]

Ayral, J. L.

J. L. Ayral, J. P. Pocholle, J. Raffy, and M. Papuchon, “Optical Kerr coefficient measurement at 1.55 μ m in single mode optical fibers,” Opt. Commun. 49, 405–408 (1984).
[CrossRef]

Beaud, P.

P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1938–1946 (1987).
[CrossRef]

Blow, K.

K. Blow, paper delivered at the Workshop on Waveguide Theory, Arundel, UK (1988).

Bourkoff, E.

W. Zhao and E. Bourkoff, “Femtosecond pulse propagation in optical fibers: higher order effects,” IEEE J. Quantum Electron. 24, 365–372 (1988).
[CrossRef]

Boyer, G. R.

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers,” Electron. Lett. 20, 659–660 (1984).
[CrossRef]

Chu, P. L.

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. Inst. Electr. Eng. Part J 134, 145–151 (1987).

Crosignani, B.

Desem, C.

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. Inst. Electr. Eng. Part J 134, 145–151 (1987).

Dziedzic, J. M.

J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” App. Opt. 20, 1403–1406 (1981).
[CrossRef]

Fehlberg, E.

E. Fehlberg, “Klassische Runge–Kutta–Formel vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Warmeleitungsprobleme,” Computing 6, 61–71 (1970).
[CrossRef]

Franco, M. A.

Francois, P. L.

P. L. Francois, “Zero dispersion in attenuation optimized doubly-clad fibers,” IEEE J. Lightwave Technol. LT-1, 26–37 (1983).
[CrossRef]

François, P. L.

P. L. François, “Design of monomode quadruple-clad fibres,” Electron. Lett. 20, 688–689 (1984).
[CrossRef]

Garth, S. J.

Gordon, J. P.

Hasegawa, A.

A. Hasegawa, “Numerical study of optical soliton transmission amplified periodically by the stimulated Raman process,” Appl. Opt. 23, 3302–3309 (1984).
[CrossRef] [PubMed]

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Ho, P. P.

R. R. Alfano and P. P. Ho, “Self-, cross-, and induced-phase modulations of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
[CrossRef]

Hodel, W.

P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1938–1946 (1987).
[CrossRef]

Hopf, F. A.

F. A. Hopf and G. I. Stegeman, Nonlinear Optics, Vol. II of Applied Classical Electrodynamics (Wiley, New York, 1986), Chap. 18.

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

Jain, M.

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[CrossRef]

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers,” Electron. Lett. 20, 659–660 (1984).
[CrossRef]

Lin, C.

Lisak, M.

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[CrossRef]

Mitschke, F. M.

Mollenauer, L. F.

Papuchon, M.

J. L. Ayral, J. P. Pocholle, J. Raffy, and M. Papuchon, “Optical Kerr coefficient measurement at 1.55 μ m in single mode optical fibers,” Opt. Commun. 49, 405–408 (1984).
[CrossRef]

Pask, C.

Pearson, A. D.

Piazzolla, S.

Pocholle, J. P.

J. L. Ayral, J. P. Pocholle, J. Raffy, and M. Papuchon, “Optical Kerr coefficient measurement at 1.55 μ m in single mode optical fibers,” Opt. Commun. 49, 405–408 (1984).
[CrossRef]

Raffy, J.

J. L. Ayral, J. P. Pocholle, J. Raffy, and M. Papuchon, “Optical Kerr coefficient measurement at 1.55 μ m in single mode optical fibers,” Opt. Commun. 49, 405–408 (1984).
[CrossRef]

Reed, W. A.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Shang, H. T.

Spano, P.

Stegeman, G. I.

F. A. Hopf and G. I. Stegeman, Nonlinear Optics, Vol. II of Applied Classical Electrodynamics (Wiley, New York, 1986), Chap. 18.

Stolen, R. H.

R. H. Stolen and J. P. Gordon, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[CrossRef]

J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” App. Opt. 20, 1403–1406 (1981).
[CrossRef]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Tzoar, N.

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[CrossRef]

Vassallo, C.

C. Vassallo, Théorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985), Chap. 5.

C. Vassallo, Electromagnétisme Classique dans la Matière (Dunod Université, Paris, 1980), Chap. 4, Eq. (4.C.6), p. 73.

Vatarescu, A.

Weber, H. P.

P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1938–1946 (1987).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Zhao, W.

W. Zhao and E. Bourkoff, “Femtosecond pulse propagation in optical fibers: higher order effects,” IEEE J. Quantum Electron. 24, 365–372 (1988).
[CrossRef]

Zysset, B.

P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1938–1946 (1987).
[CrossRef]

App. Opt. (1)

J. M. Dziedzic, R. H. Stolen, and A. Ashkin, “Optical Kerr effect in long fibers,” App. Opt. 20, 1403–1406 (1981).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

Computing (1)

E. Fehlberg, “Klassische Runge–Kutta–Formel vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Warmeleitungsprobleme,” Computing 6, 61–71 (1970).
[CrossRef]

Electron. Lett. (2)

D. N. Christodoulides and R. I. Joseph, “Femtosecond solitary waves in optical fibers,” Electron. Lett. 20, 659–660 (1984).
[CrossRef]

P. L. François, “Design of monomode quadruple-clad fibres,” Electron. Lett. 20, 688–689 (1984).
[CrossRef]

IEEE J. Lightwave Technol. (1)

P. L. Francois, “Zero dispersion in attenuation optimized doubly-clad fibers,” IEEE J. Lightwave Technol. LT-1, 26–37 (1983).
[CrossRef]

IEEE J. Quantum Electron. (3)

W. Zhao and E. Bourkoff, “Femtosecond pulse propagation in optical fibers: higher order effects,” IEEE J. Quantum Electron. 24, 365–372 (1988).
[CrossRef]

R. R. Alfano and P. P. Ho, “Self-, cross-, and induced-phase modulations of ultrashort laser pulse propagation,” IEEE J. Quantum Electron. 24, 351–364 (1988).
[CrossRef]

P. Beaud, W. Hodel, B. Zysset, and H. P. Weber, “Ultrashort pulse propagation, pulse breakup, and fundamental soliton formation in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1938–1946 (1987).
[CrossRef]

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

Opt. Commun. (1)

J. L. Ayral, J. P. Pocholle, J. Raffy, and M. Papuchon, “Optical Kerr coefficient measurement at 1.55 μ m in single mode optical fibers,” Opt. Commun. 49, 405–408 (1984).
[CrossRef]

Opt. Lett. (6)

Phys. Rev. A (2)

D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides,” Phys. Rev. A 27, 1393–1398 (1983).
[CrossRef]

N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides,” Phys. Rev. A 23, 1266–1270 (1981).
[CrossRef]

Proc. Inst. Electr. Eng. Part J (1)

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. Inst. Electr. Eng. Part J 134, 145–151 (1987).

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Other (5)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

K. Blow, paper delivered at the Workshop on Waveguide Theory, Arundel, UK (1988).

C. Vassallo, Théorie des Guides d’Ondes Électromagnétiques (Eyrolles, Paris, 1985), Chap. 5.

C. Vassallo, Electromagnétisme Classique dans la Matière (Dunod Université, Paris, 1980), Chap. 4, Eq. (4.C.6), p. 73.

F. A. Hopf and G. I. Stegeman, Nonlinear Optics, Vol. II of Applied Classical Electrodynamics (Wiley, New York, 1986), Chap. 18.

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Figures (15)

Fig. 1
Fig. 1

Raman susceptibility function SRaman{Ω} normalized as max[Im(SRaman{Ω})] = i. The imaginary part (solid curve) is proportional to the experimental gain curve, and the real part (dotted curve) is obtained from the imaginary part through the Kramers–Kronig relations.

Fig. 2
Fig. 2

Evolution with propagation of the spectrum of a Gaussian input pulse exp[(−1/2) (τ/τ0)2] with τ0 = 0.1 psec and a 10-W peak power. The linear dispersion is neglected at the operating wavelength (D = 0, dD/dλ = 0 at λ = 1.55 μm). (a) The shock term is neglected: the spectrum remains perfectly symmetrical. (b) The shock term is taken into account: a strong asymmetry in the spectrum occurs.

Fig. 3
Fig. 3

Evolution in time corresponding to the spectra displayed in Fig. 2(b). Owing to the shock term, the pulse maximum suffers a 0.1-psec retardation after propagation over 1 km.

Fig. 4
Fig. 4

Time plots and spectra of the pulse considered in Figs. 2 and 3 but in the presence of chromatic dispersion. With retention of a zero dispersion at λ = 1.55 μm, a 0.05 psec/km-nm2 dispersion slope was taken into account, leading to β0(2) = 0 but β0(3) > 0. The shock term was neglected. A dispersive tail appears on the trailing edge of the pulse after propagation over only 20–50 m. The asymmetry induced by the shock term, as seen in Fig. 3, is thus much weaker than asymmetries due to other processes, for example, the third-order dispersion.

Fig. 5
Fig. 5

Linear propagation of Gaussian pulses exp[(−1/2) (τ/τ0)2] with τ0 = 0.125 psec in the presence of second- and third-order dispersion. The dispersion slope at λ = 1.55 μm is equal to dD/dλ = 0.05 psec/km-nm2 for all the plots. Augmenting the dispersion value D0 decreases the asymmetry because the importance of β0(3) becomes smaller than that of β0(2).

Fig. 6
Fig. 6

The side of the pulse where the dispersive tail appears depends on the sign of β0(3). The same input pulse as in Fig. 5 is considered here but for the case of a fiber with a negative dispersion slope equal to dD/dλ = −0.05 psec/km-nm2, leading to β0(3) < 0. Unlike the situation in Fig. 5, the dispersive tail appears on the leading edge of the pulse. The sign of β0(2) and of the B factor does not modify the side where the asymmetry appears.

Fig. 7
Fig. 7

Interaction of two sech(τ/τ0) solitons with τ0 = 0.25 psec, separated by an 8τ0 interval. The fiber chromatic dispersion is 3 psec/km-nm at λ = 1.55 μm. (a) If the third-order chromatic dispersion β0(3) is neglected, the two solitons undergo a periodic attraction. (b) In the presence of third-order chromatic dispersion β0(3) ≠ 0 (dD/dλ = 0.05 psec/km-nm2), the attraction is suppressed.

Fig. 8
Fig. 8

The relative influence of the third-order dispersion is decreased by augmenting the dispersion values D0 as indicated. The two input solitons of Fig. 7 are also considered here. For B = 0.006 a slight asymmetry still appears. Only for 0 ≤ B < 0.003 does it become possible to neglect the third-order dispersion in the equations.

Fig. 9
Fig. 9

Action of the Raman-induced self-pumping on the spectrum of a third-order soliton (τ0 = 0.25 psec, D0 = 15 psec/km-nm, and dD/dλ = 0.05 psec/km-nm2 at λ = 1.55 μm). (a) The Raman effect and the third-order dispersion are neglected. No energy transfer appears, and the result is the usual third-order soliton. (b) The third-order dispersion is still neglected, but the Raman effect is taken into account. The self-pumping induces an energy transfer from the high-frequency toward the low-frequency part of the spectrum, leading to a frequency shift of a fraction of the spectrum. The numbers to the left of the curves are the mean wavelengths (in micrometers) of this part of the spectrum. (c) The third-order dispersion is also taken into account, and the result is a smaller frequency shift.

Fig. 10
Fig. 10

Time evolutions corresponding to the spectra in Fig. 9. (a) Standard third-order soliton. (b) Owing to the chromatic dispersion, the energy transferred by the frequency shift toward lower frequencies has a smaller group velocity than the input soliton. This energy corresponds to the narrow pulse behind what remains of the input pulse. (c) The retardation of the narrow pulse is reduced, as a consequence of the smaller frequency shift, owing to the presence of third-order dispersion.

Fig. 11
Fig. 11

Interaction of a Gaussian pump pulse (peak power 100 W, 2τ0 = 5 psec) and a signal pulse (peak power 10 W, 2τ0 = 1 psec) at the Stokes frequency. The four-wave mixing and the Raman effect are in competition. Starting from a dispersion-free fiber (D = 0, dD/dλ = 0), the phase matching is deteriorated by increasing the dispersion slope to 0.002 and 0.05 psec/km-nm2, which decreases the generation of fields at new frequencies through four-wave mixing. In the plot at the far right (D = 1 psec/km-nm) only the Raman effect generation subsists.

Fig. 12
Fig. 12

Spectrum showing the interaction of a Gaussian pump pulse Ap (peak power 50 W, 2τ0 = 10 psec) and a Gaussian signal pulse As (peak power 5 W, 2τ0 = 1 psec) at the Stokes frequency. At λ = 1.3 μm the fiber dispersion and the dispersion slope are 1 psec/km-nm and 0.05 psec/km-nm2, respectively. With this dispersion, no unwanted four-wave mixing occurs because the phase-matching condition is not satisfied.

Fig. 13
Fig. 13

Evolution in time of the two Gaussian pulses of the spectrum in Fig. 12. In the total field plot, the chromatic dispersion induces a retardation of the signal with respect to the pump. After the spectrum region corresponding to the pump in Fig. 12 is set to zero, an inverse Fourier transform yields the time evolution of the Stokes field alone.

Fig. 14
Fig. 14

Spectrum showing the generation through four-wave mixing of a field at λ = 1.55 μm from a Gaussian pump at λ = 1.3 μm (peak power 50 W, 2τ0 = 10 psec) and a Gaussian signal at λ = 1.12 μm (peak power 1 W, 2τ0 = 1 psec). The phase-matching condition is ensured by a dispersion and a dispersion slope equal to −0.83 psec/km-nm and 0.05 psec/km-nm2, respectively, at λ = 1.3 μm.

Fig. 15
Fig. 15

Evolution in time of the total field corresponding to the spectrum in Fig. 14. After the two spectrum regions corresponding to the pump and the signal in Fig. 14 are set to zero, an inverse Fourier transform yields the time evolution of the field at λ = 1.55 μm alone, created through four-wave mixing. As is seen, no initial seed at this wavelength is needed to initiate the process.

Equations (70)

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E total ( z , t , r ) = 2 κ ψ ( r ) p ^ Re ( A total ( z , τ ) exp { i [ ω 0 t - β ( ω 0 ) z ] } ) ,
A ˜ total ( z , Ω ) z - i Δ ( Ω ) A ˜ total ( z , Ω )             ( chromatic dispersion ) - Γ ( Ω ) A ˜ total ( z , Ω )             ( linear attenuation ) - i ( 1 + Ω / ω 0 ) Q Kerr FT Ω [ A total ( z , τ ) A total ( z , τ ) 2 ]             ( Kerr effect ) - i ( 1 + Ω / ω 0 ) Q Raman FT Ω ( A total ( z , τ ) × FT τ - 1 { S Raman FT Ω [ A total ( z , τ ) 2 ] } )             ( Raman effect ) ,
Δ ( Ω ) = k = 2 β 0 ( k ) Ω k k ! ,             β 0 ( k ) = d k β d ω k ( ω 0 ) , Q Kerr = ω 0 c 3 χ 1111 ( 3 ) n 0 c μ 0 4 n 0 P scale A eff 0.17 λ 0 ( μ m ) P scale ( W ) A eff ( μ m 2 )             ( m - 1 ) , Q Raman = ω 0 c G Raman P scale A eff 0.05 λ 0 ( μ m ) P scale ( W ) A eff ( μ m 2 )             ( m - 1 ) .
A total ( z , τ ) z k = 2 ( - i ) k + 1 β 0 ( k ) k ! k A total ( z , τ ) τ k             ( chromatic dispersion ) - Γ ( ω 0 ) A total ( z , τ )             ( linear attenuation ) - ( i + 1 ω 0 τ ) Q Kerr A total ( z , τ ) A total ( z , τ ) 2             ( Kerr effect ) - ( i + 1 ω 0 τ ) Q Raman A total ( z , τ ) × FT τ - 1 { S Raman FT Ω [ A total ( z , τ ) 2 ] }             ( Raman effect ) ,
Λ 1 2 π A ˜ total ( z , Ω ) z i [ ( τ 0 Ω ) 2 - B ( τ 0 Ω ) 3 ] A ˜ total ( a , Ω )             ( chromatic dispersion ) - 2 τ 0 2 Γ ( Ω ) - β 0 ( 2 ) A ˜ total ( z , Ω )             ( linear attenuation ) - 2 i ( 1 + Ω / ω 0 ) × FT Ω [ A total ( z , τ ) A total ( z , τ ) 2 ]             ( Kerr effect ) - 2 i ( 1 + Ω / ω 0 ) 0.29 FT Ω ( A total ( z , τ ) × FT τ - 1 { S Raman FT Ω [ A total ( z , τ ) 2 ] } )             ( Raman effect ) ,
B = β 0 ( 3 ) - 3 β 0 ( 2 ) τ 0 = λ 0 6 π c τ 0 ( λ D d D d λ + 2 ) λ = λ 0 , 2 π Λ 1 = 1 2 0.17 λ 0 ( μ m ) P 1 ( W ) A eff ( μ m 2 )             ( m - 1 ) = λ 0 2 ( μ m 2 ) D 0 ( psec / km - nm ) ( 1200 π ) τ 0 2 ( psec 2 )             ( m - 1 )
A ˜ total ( z + d z , Ω ) exp { - [ i Δ ( Ω ) + Γ ( Ω ) ] d z } × FT Ω [ A total ( z , τ ) exp [ - i Q Kerr A total ( z , τ ) 2 d z ] × exp ( - i Q Raman FT τ - 1 { S Raman FT Ω [ A total ( z , τ ) 2 ] } d z ) ] ,
A ˜ total ( z , τ ) z = A ˜ total ( z + d z , τ ) - A ˜ total ( z , τ ) d z + o ( d z 2 ) ,
from the Kerr term : A p 2 A s *             ( four - wave mixing ) , from the Raman term : A p 2 FT τ - 1 ( FT Ω [ A s * ] S Raman { Ω + Ω R } ) ;
from the Kerr term : A p ( 2 A s 2 + A p 2 )             ( phase modulation ) , from the Raman term : A p A s FT τ - 1 ( FT Ω [ A s * ] S Raman { Ω + Ω R } ) , A p FT τ - 1 ( FT Ω [ A s 2 ] S Raman { Ω } ) , A p A p 2 S Raman { 0 } ;
from the Kerr term : A s ( A s 2 + 2 A p 2 )             ( phase modulation ) , from the Raman term : A p 2 FT τ - 1 [ A ˜ s S Raman { Ω - Ω R } ] , A s FT τ - 1 ( FT Ω [ A s 2 ] S Raman { Ω } )             ( self - pumping ) , A s A p 2 S Raman { 0 } ;
from the Kerr term : A s 2 A p *             ( four - wave mixing ) , from the Raman term : A s A p * FT τ - 1 ( A ˜ s S Raman { Ω - Ω R } ) .
β ( ω 0 + Ω R ) = 2 β ( ω 0 ) - β ( ω 0 - Ω R ) ,
β ( ω 0 + Ω ) = β ( ω 0 ) + Ω V 0 + β 0 ( 2 ) Ω 2 2 + β 0 ( 3 ) Ω 3 6 + β 0 ( 4 ) Ω 4 24 + .
β ( ω 0 - Ω FWM ) = 2 β ( ω 0 ) - β ( ω 0 + Ω FWM ) ,
β 0 ( 2 ) = - β 0 ( 4 ) ( Ω FWM 2 / 12 ) ,
- 2 = ( λ 0 λ s - 1 ) 2 ( λ D d D d λ + 1 ) λ = λ 0
τ ( ω 0 + Ω ) = 1 V 0 + β 0 ( 2 ) Ω + β 0 ( 3 ) Ω 2 2 + β 0 ( 4 ) Ω 3 6 + ,
τ ( ω 0 - Ω FWM ) - τ ( ω 0 + Ω FWM ) = - 2 β 0 ( 2 ) Ω FWM - 2 β 0 ( 4 ) Ω FWM 3 6 0.23 psec m > 0.
rot E ˜ total = - i ω μ 0 H ˜ total , rot H ˜ total = i ω ɛ ( ω ) E ˜ total + i ω P ˜ NL ,
{ e 0 ( z , ω , r ) h 0 ( z , ω , r ) } = { e ( ω , r ) + u ^ z e z ( ω , r ) h ( ω , r ) + u ^ z h z ( ω , r ) } exp [ - i b 0 ( ω ) z ]
{ E ˜ total , H ˜ total , } = S ˜ 0 ( z , ω ) { e ( ω , r ) h ( ω , r ) } exp [ - i b 0 ( ω ) z ] + other modal contributions ,
b 0 ( ω ) = β ( ω ) - i Γ ( ω ) ,
div ( E ˜ total × h 0 t - e 0 t × H ˜ total ) = i ω P ˜ NL · e 0 t ,
{ e 0 t h 0 t } = { e ( ω , r ) - u ^ z e z ( ω , r ) - h ( ω , r ) + u ^ z h z ( ω , r ) } exp [ + i b 0 ( ω ) z ] .
C ˜ total ( z , ω ) z = - Γ ( ω ) C ˜ total ( z , ω ) - i ω 2 P ˜ NL · e - P ˜ z NL e z e × h · u ^ z exp [ i β ( ω ) z ] .
e z e h z h ( n core - n cladding n core ) 1 / 2 Δ core 0.
C ˜ total ( z , ω ) z = - Γ ( ω ) C ˜ total ( z , ω ) - i ω c 1 2 n 0 ɛ 0 P ˜ NL · e e · e exp [ i β ( ω ) z ] .
E total ( z , t , r ) = 1 2 π ω 0 - δ Ω ω 0 + δ Ω E ˜ total ( z , ω , r ) exp ( i ω t ) d ω + c . c . = 1 2 π - 8 Ω + δ Ω C ˜ total ( z , ω 0 + Ω ) ψ ( ω 0 + Ω , r ) p ^ × exp { i [ ( ω 0 + Ω ) t - β ( ω 0 + Ω ) z ] } d Ω + c . c .
β ( ω 0 + Ω ) = β ( ω 0 ) + Ω / V 0 + Δ ( Ω ) ,
C ˜ total ( z , ω 0 + Ω ) exp [ - i Δ ( Ω ) z ] = κ A ˜ total ( z , Ω ) .
E total ( z , t , r ) = 1 2 π exp { i [ ω 0 t - β ( ω 0 ) z ] } - κ A ˜ total ( z , Ω ) × ψ ( ω 0 + Ω , r ) p ^ exp ( i Ω τ ) d Ω + c . c .
E total ( z , t , r ) = 2 κ ψ ( r ) p ^ Re ( A total ( z , τ ) × exp { i [ ω 0 t - β ( ω 0 ) z ] } ) ,
κ = [ P scale ( 2 n 0 / c μ 0 ) ψ 2 ] 1 / 2 .
ψ ( ω 0 + Ω , r ) ψ ( ω 0 , r ) + Ω ( ψ / ω ) ω = ω 0 .
ψ ( ω 0 + Ω , r ) ( 1 + Ω ω 0 r 2 a 2 ) ψ ( ω 0 , r ) ,
E total ( z , t , r ) = 2 κ ψ ( r ) p ^ Re ( exp { i [ ω 0 t - β ( ω 0 ) z ] } × ( 1 - i ω 0 r 2 a 2 τ ) A total ( z , τ ) ) .
A ˜ total ( z , Ω ) z [ - i Δ ( Ω ) - Γ ( Ω ) ] A ˜ total ( z , Ω ) - i [ 1 + Ω ω 0 ] F ˜ NL ( z , Ω ) ,
F ˜ NL ( z , Ω ) = ω 0 c 1 2 n 0 ɛ 0 ψ ( r ) P ˜ NL ( ω 0 + Ω ) · p ^ κ ψ 2 × exp { i [ β ( ω 0 ) + Ω / V 0 ] z } .
P ˜ Kerr ( z , ω 0 + Ω , r ) = ɛ 0 ( 1 2 π ) 2 - - d ω 1 d ω 2 × [ χ ( 3 ) E ˜ total ( z , ω 1 , r ) E ˜ total ( z , ω 2 , r ) E ˜ total ( z , ω 3 , r ) ] ,
P ˜ Kerr ( z , ω 0 + Ω , r ) = 3 ɛ 0 ( 1 2 π ) 2 × - δ Ω δ Ω - δ Ω δ Ω d u d v × [ χ ( 3 ) E ˜ total ( z , ω 0 + u - v , r ) E ˜ total * ( z , ω 0 + u , r ) × E ˜ total ( z , ω 0 + Ω + v , r ) ] ,
P ˜ Kerr ( z , ω 0 + Ω , r ) = 3 ɛ 0 χ 1111 ( 3 ) ψ 3 κ 3 × exp { - i [ β ( ω 0 ) + Ω / V 0 ] z } p ^ × ( 1 2 π ) 2 - d v × A ˜ total ( z , Ω + v ) - d u A ˜ total * ( z , u ) A ˜ total ( z , u - v ) .
F ˜ Kerr ( z , Ω ) = ω 0 c 3 χ 111 ( 3 ) n 0 c μ 0 4 n 0 P scale A eff FT Ω [ A total ( z , τ ) 2 A total ( z , τ ) ] ,
Δ ϕ biref = ω 0 c 4 χ 1111 ( 3 ) n 0 c μ 0 4 n 0 P pump A eff L fiber .
P Raman ( z , t , r ) = N ɛ 0 α q q ( z , t , r ) E total ( z , t , r ) ,
q ¨ + Γ R q ˙ + Ω R 2 q = ɛ 0 2 m α q E total ( z , t , r ) · E total ( z , t , r ) ,
P ˜ Raman ( z , ω 0 + Ω , r ) = N ɛ 0 α q 1 2 π - d Θ × q ˜ ( z , Θ , r ) E ˜ total ( z , ω 0 + Ω - Θ , r ) ,
q ˜ ( z , Θ , r ) = ɛ 0 2 m α q 1 Ω R 2 - Θ 2 + i Γ R Θ × 1 2 π - d ω E ˜ total ( z , ω , r ) · E ˜ total * ( z , ω - Θ , r ) .
q ˜ ( z , Θ , r ) = ɛ 0 2 m α q 2 ψ 2 κ 2 exp [ - i ( Θ / V 0 ) z ] Ω R 2 - Θ 2 + i Γ r Θ × FT Θ [ A total ( z , τ ) 2 ] ,
F ˜ Raman ( z , Ω ) = ω 0 c ( α q ) 2 N ɛ 0 m n 0 c μ 0 4 n 0 P scale A eff × FT Ω ( A total ( z , τ ) FT τ - 1 { FT Ω [ A total ( z , τ ) 2 ] Ω R 2 - Ω 2 + i Γ R Ω } ) .
F ˜ Raman ( z , Ω ) = ω 0 c G Raman P scale A eff × FT Ω ( A total ( z , τ ) FT τ - 1 { S Raman FT Ω [ A total ( z , τ ) 2 ] } ) ,
d d z A signal ( z ) 2 = [ - 2 Γ signal + 2 ω 0 c G Raman P scale A pump ( z ) 2 A eff ] A signal ( z ) 2 , d d z A pump ( z ) 2 = [ - 2 Γ pump - 2 ω 0 c G Raman P scale A signal ( z ) 2 A eff ] A pump ( z ) 2 ,
A signal ( z ) 2 = A signal ( 0 ) 2 exp ( - 2 Γ signal z ) × exp [ 2 ω 0 c G Raman P scale A pump ( 0 ) 2 A eff L eff ] , A pump ( z ) 2 = A pump ( 0 ) 2 exp ( - 2 Γ pump z ) ,
L eff = 0 z exp ( - 2 Γ pump u ) d u = effective length .
A ˜ ( z , Ω ) z = - β 0 ( 2 ) Ω 2 2 A ˜ ( z , Ω ) - i Q Kerr FT Ω [ A ( z , τ ) 2 A ( z , τ ) ] ,
A ˜ = π τ 0 sech ( π 2 Ω τ 0 ) exp ( - i ζ z ) , FT Ω [ A 2 A ] = π τ 0 2 [ ( Ω τ 0 ) 2 + 1 ] sech ( π 2 Ω τ 0 ) exp ( - i ζ z )
ζ = Q Kerr / 2 ,             - β 0 ( 2 ) = ( λ 0 / ω 0 ) D ( λ 0 ) = Q Kerr τ 0 2 ,
P 1 ( W ) τ 0 2 ( psec ) D ( λ 0 ) λ 0 3 ( μ m ) A eff ( μ m 2 ) / 102 π ,
2 π Λ = ζ = Q Kerr 2 .
2 π Λ 1 = Q Kerr 2 = - β 0 ( 2 ) 2 τ 0 2             ( - β 0 ( 2 ) > 0 ) ,
A ˜ ( z , Ω ) z = - i [ β 0 ( 2 ) Ω 2 2 + β 0 ( 3 ) Ω 3 6 ] A ˜ ( z , Ω ) - i ( 1 + Ω ω 0 ) Q Kerr FT Ω [ A ( z , τ ) 2 A ( z , τ ) ] ,
A ˜ = π τ 0 sech ( π Ω + ξ 2 τ 0 ) exp ( - i ζ z ) , FT Ω [ A 2 A ] = π τ 0 2 [ ( Ω + ξ ) 2 τ 0 2 + 1 ] sech ( π Ω + ξ 2 τ 0 ) × exp ( - i ζ z )
ξ = ω 0 2 ( 1 - η ) ,             η = 3 β 0 ( 2 ) ω 0 β 0 ( 3 ) = - 3 ( γ D d D d λ + 2 ) λ = λ 0 , τ 0 2 = 1 ω 0 2 4 ( η + 3 ) ( 1 - η ) , Q Kerr = - ω 0 3 β 0 ( 3 ) τ 0 2 = - 1 3 τ 0 2 λ 0 ω 0 ( λ d D d λ + 2 D ) λ = λ 0 , ζ = Q Kerr 2 ( 1 + ξ 2 τ 0 2 ) .
Ω = - Ω A ˜ 2 d Ω - A ˜ 2 d Ω ,
d d z - Ω n A ˜ 2 d Ω = - Ω n ( A ˜ A ˜ * z + A ˜ * A ˜ z ) d Ω = 2 Re ( - Ω n A ˜ * A ˜ z d Ω )
d d z - A ˜ 2 d Ω = - 2 - Γ ( Ω ) A ˜ 2 d Ω , d d z - Ω A ˜ 2 d Ω = - 2 - Γ ( Ω ) Ω A ˜ 2 d Ω + Q Raman - Ω b Raman { Ω } FT [ A 2 ] ( Ω ) FT [ A 2 ] ( - Ω ) d Ω ,
Im ( S Raman { Ω } ) = i b Raman { Ω } .
d Ω d z = Q Raman × - Ω b Raman { Ω } FT [ A 2 ] ( Ω ) FT [ A 2 ] ( - Ω ) d Ω - A ˜ 2 d Ω .
d Ω d z = - Q Raman Γ R 2 Ω R 3 8 15 1 τ 0 2
i b Raman { Ω } = Im ( Γ R Ω R Ω R 2 - Ω 2 + i Γ R Ω ) - i Γ R 2 Ω R 3 Ω .

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