Abstract

A perturbed nonlinear Schrödinger equation is used to model an optical laser fiber ring for generation of femtosecond optical pulses. The ring is built from a nonlinear laser fiber joint together with a passive nonlinear fiber. We assume that the optical pulses are solitons perturbed by loss and gain. Starting from the single-soliton solution of the nonlinear Schrödinger equation, we use a collective coordinate approach to derive a Lagrangian as a function of the collective coordinates. The evolution equations for the collective coordinates emerge from the Lagrange equations, including the associated generalized forces that result from the perturbations. Remarkably good agreement is obtained between the collective coordinate approach and the full numerical simulations of the fiber ring laser.

© 1995 Optical Society of America

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References

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  1. J. B. Schlager, Y. Yamabayashi, D. L. Franzen, and R. I. Juneau, “Mode-locked, long-cavity, erbium fiber lasers with subsequent soliton-like compression,” IEEE Photon. Technol. Lett. 1, 264–266 (1989).
    [CrossRef]
  2. I. N. Duling, “Subpicosecond all-fibre erbium laser,” Electron. Lett. 27, 544–545 (1991).
    [CrossRef]
  3. M. E. Fermann, M. Hofer, F. Haberl, A. J. Schmidt, and L. Turi, “Additive pulse compression mode locking of a neodymium fiber laser,” Opt. Lett. 16, 244–246 (1991).
    [CrossRef] [PubMed]
  4. J. Mark, L. Y. Liu, K. L. Hall, H. A. Haus, and E. P. Ippen, “Femtosecond pulse generation in a laser with a nonlinear external resonator,” Opt. Lett. 14, 48–50 (1989).
    [CrossRef] [PubMed]
  5. D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17, 1515–1517 (1992).
    [CrossRef] [PubMed]
  6. K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. 28, 2226–2228 (1992).
    [CrossRef]
  7. S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993).
    [CrossRef] [PubMed]
  8. C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Soliton fiber ring laser,” Opt. Lett. 17, 417–419 (1992).
    [CrossRef] [PubMed]
  9. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993).
    [CrossRef] [PubMed]
  10. O. Legrand, “Kink–antikink dissociation and annihilation: a collective-coordinate description,” Phys. Rev. A 36, 5068–5073 (1987).
    [CrossRef] [PubMed]
  11. A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
    [CrossRef]
  12. D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 1166–1174 (1993).
  13. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
    [CrossRef]
  14. L. Gagnon and P. A. Bélanger, “Adiabatic amplification of optical solitons,” Phys. Rev. A 43, 6187–6193 (1991).
    [CrossRef] [PubMed]
  15. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 24, 5689–5694 (1990).
    [CrossRef]
  16. D. J. Kaup, “Second-order perturbations for solitons,” in Nonlinearity with Disorder, F. Abdullaev, A. R. Bishop, and S. Pnevmatikos, eds., Vol. 67 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1990), pp. 14–22.
    [CrossRef]
  17. V. Petrov and W. Rudolph, “Femtosecond solitary waves in the presence of resonant absorption,” Opt. Commun. 76, 53–55 (1990).
    [CrossRef]
  18. 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]
  19. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  20. G. P. Agrawal, “Amplification of ultrashort solitons in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 2, 875–877 (1990).
    [CrossRef]
  21. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  22. T. Geisler, K. A. Shore, M. P. Soerensen, P. L. Christiansen, J. Mørk, and J. Mark, “Nonlinear fiber external cavity mode locking of erbium-doped fiber lasers,” J. Opt. Soc. Am. B 10, 1166–1174 (1993).
    [CrossRef]
  23. M. P. Soerensen, K. A. Shore, T. Geisler, P. L. Christiansen, J. Mørk, and J. Mark, “Dynamics of additive-pulse mode-locked fibre lasers,” Opt. Commun. 90, 65–69 (1992).
    [CrossRef]
  24. M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
    [CrossRef]
  25. M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
    [CrossRef]
  26. K. J. Blow, N. J. Doran, and D. Wood, “Generation and stabilization of short soliton pulses in the amplified nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 5, 381–391 (1988).
    [CrossRef]

1993 (4)

1992 (4)

M. P. Soerensen, K. A. Shore, T. Geisler, P. L. Christiansen, J. Mørk, and J. Mark, “Dynamics of additive-pulse mode-locked fibre lasers,” Opt. Commun. 90, 65–69 (1992).
[CrossRef]

C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Soliton fiber ring laser,” Opt. Lett. 17, 417–419 (1992).
[CrossRef] [PubMed]

D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17, 1515–1517 (1992).
[CrossRef] [PubMed]

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. 28, 2226–2228 (1992).
[CrossRef]

1991 (4)

1990 (4)

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 24, 5689–5694 (1990).
[CrossRef]

V. Petrov and W. Rudolph, “Femtosecond solitary waves in the presence of resonant absorption,” Opt. Commun. 76, 53–55 (1990).
[CrossRef]

G. P. Agrawal, “Amplification of ultrashort solitons in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 2, 875–877 (1990).
[CrossRef]

1989 (3)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

J. Mark, L. Y. Liu, K. L. Hall, H. A. Haus, and E. P. Ippen, “Femtosecond pulse generation in a laser with a nonlinear external resonator,” Opt. Lett. 14, 48–50 (1989).
[CrossRef] [PubMed]

J. B. Schlager, Y. Yamabayashi, D. L. Franzen, and R. I. Juneau, “Mode-locked, long-cavity, erbium fiber lasers with subsequent soliton-like compression,” IEEE Photon. Technol. Lett. 1, 264–266 (1989).
[CrossRef]

1988 (1)

1987 (1)

O. Legrand, “Kink–antikink dissociation and annihilation: a collective-coordinate description,” Phys. Rev. A 36, 5068–5073 (1987).
[CrossRef] [PubMed]

1986 (1)

1979 (1)

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

1973 (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]

Agrawal, G. P.

G. P. Agrawal, “Amplification of ultrashort solitons in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 2, 875–877 (1990).
[CrossRef]

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

Anderson, D.

D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 1166–1174 (1993).

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Bélanger, P. A.

L. Gagnon and P. A. Bélanger, “Adiabatic amplification of optical solitons,” Phys. Rev. A 43, 6187–6193 (1991).
[CrossRef] [PubMed]

Blow, K. J.

Bondeson, A.

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Chen, C.-J.

Chernikov, S. V.

Christiansen, P. L.

T. Geisler, K. A. Shore, M. P. Soerensen, P. L. Christiansen, J. Mørk, and J. Mark, “Nonlinear fiber external cavity mode locking of erbium-doped fiber lasers,” J. Opt. Soc. Am. B 10, 1166–1174 (1993).
[CrossRef]

M. P. Soerensen, K. A. Shore, T. Geisler, P. L. Christiansen, J. Mørk, and J. Mark, “Dynamics of additive-pulse mode-locked fibre lasers,” Opt. Commun. 90, 65–69 (1992).
[CrossRef]

Dianov, E. M.

Doran, N. J.

Duling, I. N.

I. N. Duling, “Subpicosecond all-fibre erbium laser,” Electron. Lett. 27, 544–545 (1991).
[CrossRef]

Fermann, M. E.

Franzen, D. L.

J. B. Schlager, Y. Yamabayashi, D. L. Franzen, and R. I. Juneau, “Mode-locked, long-cavity, erbium fiber lasers with subsequent soliton-like compression,” IEEE Photon. Technol. Lett. 1, 264–266 (1989).
[CrossRef]

Gagnon, L.

L. Gagnon and P. A. Bélanger, “Adiabatic amplification of optical solitons,” Phys. Rev. A 43, 6187–6193 (1991).
[CrossRef] [PubMed]

Geisler, T.

T. Geisler, K. A. Shore, M. P. Soerensen, P. L. Christiansen, J. Mørk, and J. Mark, “Nonlinear fiber external cavity mode locking of erbium-doped fiber lasers,” J. Opt. Soc. Am. B 10, 1166–1174 (1993).
[CrossRef]

M. P. Soerensen, K. A. Shore, T. Geisler, P. L. Christiansen, J. Mørk, and J. Mark, “Dynamics of additive-pulse mode-locked fibre lasers,” Opt. Commun. 90, 65–69 (1992).
[CrossRef]

Haberl, F.

Hall, K. L.

Hasegawa, A.

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]

Haus, H. A.

Hofer, M.

Ippen, E. P.

Juneau, R. I.

J. B. Schlager, Y. Yamabayashi, D. L. Franzen, and R. I. Juneau, “Mode-locked, long-cavity, erbium fiber lasers with subsequent soliton-like compression,” IEEE Photon. Technol. Lett. 1, 264–266 (1989).
[CrossRef]

Kaup, D. J.

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 24, 5689–5694 (1990).
[CrossRef]

D. J. Kaup, “Second-order perturbations for solitons,” in Nonlinearity with Disorder, F. Abdullaev, A. R. Bishop, and S. Pnevmatikos, eds., Vol. 67 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1990), pp. 14–22.
[CrossRef]

Kimura, Y.

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Kubota, H.

M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
[CrossRef]

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Kurokawa, K.

M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
[CrossRef]

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Legrand, O.

O. Legrand, “Kink–antikink dissociation and annihilation: a collective-coordinate description,” Phys. Rev. A 36, 5068–5073 (1987).
[CrossRef] [PubMed]

Lisak, M.

D. Anderson, M. Lisak, and T. Reichel, “Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system,” J. Opt. Soc. Am. B 5, 1166–1174 (1993).

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Liu, L. Y.

Malomed, B. A.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Mark, J.

Menyuk, C. R.

Mitschke, F. M.

Mollenauer, L. F.

Mørk, J.

T. Geisler, K. A. Shore, M. P. Soerensen, P. L. Christiansen, J. Mørk, and J. Mark, “Nonlinear fiber external cavity mode locking of erbium-doped fiber lasers,” J. Opt. Soc. Am. B 10, 1166–1174 (1993).
[CrossRef]

M. P. Soerensen, K. A. Shore, T. Geisler, P. L. Christiansen, J. Mørk, and J. Mark, “Dynamics of additive-pulse mode-locked fibre lasers,” Opt. Commun. 90, 65–69 (1992).
[CrossRef]

Nakazawa, M.

M. Nakazawa, H. Kubota, K. Kurokawa, and E. Yamada, “Femtosecond optical soliton transmission over long distances using adiabatic trapping and soliton standardization,” J. Opt. Soc. Am. B 8, 1811–1817 (1991).
[CrossRef]

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Nelson, L. E.

Noske, D. U.

Pandit, N.

Payne, D. N.

Petrov, V.

V. Petrov and W. Rudolph, “Femtosecond solitary waves in the presence of resonant absorption,” Opt. Commun. 76, 53–55 (1990).
[CrossRef]

Reichel, T.

Richardson, D. J.

Rudolph, W.

V. Petrov and W. Rudolph, “Femtosecond solitary waves in the presence of resonant absorption,” Opt. Commun. 76, 53–55 (1990).
[CrossRef]

Schlager, J. B.

J. B. Schlager, Y. Yamabayashi, D. L. Franzen, and R. I. Juneau, “Mode-locked, long-cavity, erbium fiber lasers with subsequent soliton-like compression,” IEEE Photon. Technol. Lett. 1, 264–266 (1989).
[CrossRef]

Schmidt, A. J.

Shore, K. A.

T. Geisler, K. A. Shore, M. P. Soerensen, P. L. Christiansen, J. Mørk, and J. Mark, “Nonlinear fiber external cavity mode locking of erbium-doped fiber lasers,” J. Opt. Soc. Am. B 10, 1166–1174 (1993).
[CrossRef]

M. P. Soerensen, K. A. Shore, T. Geisler, P. L. Christiansen, J. Mørk, and J. Mark, “Dynamics of additive-pulse mode-locked fibre lasers,” Opt. Commun. 90, 65–69 (1992).
[CrossRef]

Soerensen, M. P.

T. Geisler, K. A. Shore, M. P. Soerensen, P. L. Christiansen, J. Mørk, and J. Mark, “Nonlinear fiber external cavity mode locking of erbium-doped fiber lasers,” J. Opt. Soc. Am. B 10, 1166–1174 (1993).
[CrossRef]

M. P. Soerensen, K. A. Shore, T. Geisler, P. L. Christiansen, J. Mørk, and J. Mark, “Dynamics of additive-pulse mode-locked fibre lasers,” Opt. Commun. 90, 65–69 (1992).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Tamura, K.

K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993).
[CrossRef] [PubMed]

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. 28, 2226–2228 (1992).
[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]

Taylor, J. R.

Turi, L.

Wai, P. K. A.

Wood, D.

Yamabayashi, Y.

J. B. Schlager, Y. Yamabayashi, D. L. Franzen, and R. I. Juneau, “Mode-locked, long-cavity, erbium fiber lasers with subsequent soliton-like compression,” IEEE Photon. Technol. Lett. 1, 264–266 (1989).
[CrossRef]

Yamada, E.

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]

M. Nakazawa, K. Kurokawa, H. Kubota, K. Suzuki, and Y. Kimura, “Femtosecond erbium-doped optical fiber amplifier,” Appl. Phys. Lett. 57, 653–655 (1990).
[CrossRef]

Electron. Lett. (2)

I. N. Duling, “Subpicosecond all-fibre erbium laser,” Electron. Lett. 27, 544–545 (1991).
[CrossRef]

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fibre ring laser,” Electron. Lett. 28, 2226–2228 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. B. Schlager, Y. Yamabayashi, D. L. Franzen, and R. I. Juneau, “Mode-locked, long-cavity, erbium fiber lasers with subsequent soliton-like compression,” IEEE Photon. Technol. Lett. 1, 264–266 (1989).
[CrossRef]

G. P. Agrawal, “Amplification of ultrashort solitons in erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 2, 875–877 (1990).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (2)

V. Petrov and W. Rudolph, “Femtosecond solitary waves in the presence of resonant absorption,” Opt. Commun. 76, 53–55 (1990).
[CrossRef]

M. P. Soerensen, K. A. Shore, T. Geisler, P. L. Christiansen, J. Mørk, and J. Mark, “Dynamics of additive-pulse mode-locked fibre lasers,” Opt. Commun. 90, 65–69 (1992).
[CrossRef]

Opt. Lett. (7)

Phys. Rev. A (3)

O. Legrand, “Kink–antikink dissociation and annihilation: a collective-coordinate description,” Phys. Rev. A 36, 5068–5073 (1987).
[CrossRef] [PubMed]

L. Gagnon and P. A. Bélanger, “Adiabatic amplification of optical solitons,” Phys. Rev. A 43, 6187–6193 (1991).
[CrossRef] [PubMed]

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 24, 5689–5694 (1990).
[CrossRef]

Phys. Scr. (1)

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Rev. Mod. Phys. (1)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[CrossRef]

Other (2)

D. J. Kaup, “Second-order perturbations for solitons,” in Nonlinearity with Disorder, F. Abdullaev, A. R. Bishop, and S. Pnevmatikos, eds., Vol. 67 of Springer Proceedings in Physics (Springer-Verlag, Berlin, 1990), pp. 14–22.
[CrossRef]

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

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Figures (6)

Fig. 1
Fig. 1

Fiber laser ring configuration.

Fig. 2
Fig. 2

η versus n for the ring laser with Eqs. (14) used for κ2 = 0 (dashed curve) and for κ2 = 0.01 (solid curve). The parameter values are from Table 1, and w = 1.13 × 10−2, g0 = 15, la = 0.1, and lp = 0.1. The initial conditions are η = 10−5, ξ = 9, t0 = 0, and σ0 = 0.

Fig. 3
Fig. 3

(a) max|u| = ηn versus n for the map [Eq. (16)] (solid curve) and for direct numerical simulation (squares) of the fiber ring laser. The parameter values are from Table 1, except that Γ = 2.08 × 10−2. g0 = 2.0, la = 0.01, lp = 0.34, w = 0, b3 = 0, κ1 = 0, κ2 = 0. (b) |u(zn, t)|, zn = n(la + lp) versus (n, t) from the direct numerical simulation of the fiber ring laser.

Fig. 4
Fig. 4

(a) ηn versus n for the map [Eq. (16)] (solid curve) with w = 0 and for the map [Eqs. (21)] (dashed curve) with w = 0.0113. Parameter values as from Table 1, and g0 = 0.2, la = 0.01, and lp = 0.34. (b) η (stationary state) versus g0 for the maps [Eq. (16)] (solid curve) and [Eqs. (21)] (dashed curve). (c) The associated FWHM (fs) versus the physical gain G0 (m−1). The long-dashed curve represents the asymptotic value from relation (22) for G0 → ∞.

Fig. 5
Fig. 5

max|u| = ηn versus n for the map [Eq. (16)] (dashed curve) and for direct numerical simulation (solid curve) of the fiber ring laser with a bandwidth-limited gain and output coupler. Parameter values are from Table 1. (a) g0 = 0.4, la = 0.01, lp = 0.1, Γc = 0.1, lc = 0.01, zc = lp/2, and w = 1.13 × 10−2. (b) g0 = 1.5, la = 0.1, lp = 0.1, Γc = 5, lc = 0.01, zc = lp/2, and w = 1.13 × 10−2.

Fig. 6
Fig. 6

|u(zn, t)|2, zn = n(la + lp) versus (n, t) for an initial condition consisting of random noise of maximum amplitude 5 × 10−4 in a time window of extension [−0.5; 0.5] and zero outside. The parameters are as in Fig. 5(b).

Tables (1)

Tables Icon

Table 1 Fixed Parameter Values Used throughout in the Numerical Computations

Equations (43)

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

A Z + β 1 A T + i 2 β 2 2 A T 2 i γ | A | 2 A + α 2 A 1 6 β 3 3 A T 3 + K 1 T ( | A | 2 A ) + K 2 A T ( | A | 2 ) = G 0 1 + ( P / P sat ) ( A + 1 ω b 2 2 A T 2 ) ,
P = F rep | A ( Z , T ) | 2 d T .
t = k t ( T β 1 Z ) , z = k z Z , A ( Z , T ) = k u ( z , t ) ,
k z = ½ β 2 k t 2 , k 2 = 2 k z / γ ,
b 3 = k t 2 β 3 / ( 6 k z ) , κ 1 = k 2 k t K 1 / k z , κ 2 = i k 2 k t K 2 / k z ,
Γ = α / ( 2 k z ) , g 0 = G 0 / k z ,
p sat = k t P sat / ( F rep k 2 ) , w = k t 2 / ω b 2 .
i u z + 2 u t 2 + 2 | u | 2 u = i Γ u + i g 0 1 + ( P / p sat ) ( u + w 2 u t 2 ) + i b 3 3 u t 3 i κ 1 t ( | u | 2 u ) + κ 2 u t ( | u | 2 ) = R .
p = | u ( z , t ) | 2 d t .
u ( z , t ) = η sech ( η θ ) exp ( i ξ θ + i σ ) ,
θ = t 2 ξ z t 0 ,
σ = ( η 2 + ξ 2 ) z σ 0 .
L = { i 2 ( u * u z u z * u ) u t u t * + u 2 u * 2 } d t = d t .
L y = { u u y + u z u z y + u t u t y } d t + c.c . ,
L y z = u z u z y z d t + c.c . ,
L y d d z ( L y z ) = + d t [ u t ( u t ) z ( u z ) ] u y + c.c . ,
L y d d z ( L y z ) = R u * y d t + c.c .
L = 2 3 η 3 2 η ξ 2 2 η ξ d θ d z 2 η d σ d z .
d η d z = 2 Γ η + 2 g 0 η 1 + ( p / p sat ) [ 1 w ( η 2 3 + ξ 2 ) ] ,
d σ d z = η 2 + ξ 2 + 2 b 3 ξ ( ξ 2 η 2 ) ,
d θ d z = 2 ξ b 3 ( η 2 + 3 ξ 2 ) κ 1 η 2 ,
d ξ d z = 4 3 g 0 w 1 + ( p / p sat ) ξ η 2 + 8 15 κ 2 η 4 .
d η d z = 2 Γ η + 2 g 0 η 1 + ( p / p sat ) [ 1 w ( η 2 3 + ξ 2 ) ] ,
d ξ d z = 4 3 g 0 w 1 + ( p / p sat ) ξ η 2 + 8 15 κ 2 η 4 ,
d t 0 d z = 2 z d ξ d z + b 3 ( η 2 + 3 ξ 2 ) + κ 1 η 2 ,
d σ 0 d z = 2 z ( η d η d z + ξ d ξ d z ) 2 b 3 ξ ( ξ 2 η 2 ) .
d η d z = 2 Γ η + 2 g 0 η 1 + ( 2 η / p sat ) ,
σ 0 = η 2 z 0 z η 2 d z + c 0 ,
η n + 1 = η n exp [ 2 g 0 l a 1 + ( 2 η n / p sat ) 2 Γ ( l a + l p ) ] .
η = p sat 2 [ g 0 l a Γ ( l a + l p ) 1 ] .
g 0 l a > 2 Γ 2 ( l a + l p ) 2 p sat .
FWHM = 2 arccosh ( 2 ) η .
d η d z = 2 Γ η + 2 g 0 η 1 + ( p / p sat ) [ 1 w ( η 2 3 + ξ 2 ) ] ,
d σ 0 d z = 2 z ( η d η d z + ξ d ξ d z ) ,
d ξ d z = 4 3 [ g 0 ξ w 1 + ( p / p sat ) ] η 2 ,
d t 0 d z = 2 z d ξ d z .
η n + 1 = η n exp [ a n l a 2 Γ ( l a + l p ) ] ( 1 + ( η n 2 / A n ) { exp [ 2 ( a n 2 Γ ) l a ] 1 } ) 1 / 2 ,
a n = 2 g 0 1 + ( 2 η n / p sat ) ,
A n = 3 w ( 1 2 Γ a n ) .
FWHM 2 arccosh ( 2 ) w / 3 exp ( 2 Γ l p ) for g 0 ,
Γ ( z ) = Γ + Γ c sech 2 ( z z c l c ) ,
η ( l p ) = η ( 0 ) exp { 2 Γ l p 2 Γ c l c × [ tanh ( l p z c l c ) tanh ( z c l c ) ] } .
η = p sat 2 [ g 0 l a Γ ( l a + l p ) + Γ c l c 1 ] .

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