Abstract

We analyze the main methods for high-repetition-rate soliton generation in fiber lasers in view of the applications to fiber transmission. Particular emphasis is given to the problem of intracavity soliton interactions and stabilization methods. Because of the interactions we find that the maximum duty cycle achievable in a soliton source is 0.22. We also find that the relative phase difference between adjacent solitons may be controlled by means of an intracavity étalon, whereas without the étalon the solitons are stable only when they are out of phase. Moreover the introduction of the intracavity étalon has led to a strong reduction in the fluctuations of the soliton parameters induced by quantum noise.

© 1995 Optical Society of America

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References

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  1. E. M. Garmire and A. Yariv, "Laser mode-locking with saturable absorbers," IEEE J. Quantum Electron. QE-3, 222–225 (1967).
    [CrossRef]
  2. A. E. Siegman and D. J. Kuizenga, "Simple analytic expression for AM and FM mode-locked pulses in homogeneous lasers," Appl. Phys. Lett. 14, 181–182 (1969).
    [CrossRef]
  3. J. D. Kafka, T. Baer, and D. W. Hall, "Mode-locked erbium fiber laser," Opt. Lett. 14, 1269–1271 (1989).
    [CrossRef] [PubMed]
  4. A. Takada and H. Miyazawa, "30 GHz picosecond pulse generation from actively mode-locked erbium-doped fibre laser," Electron. Lett. 26, 216–217 (1990).
    [CrossRef]
  5. H. Takara, S. Kawanishi, and M. Saruwatari, "20 GHz transform-limited optical pulse generation and bit-error-free operation using a tunable, actively mode-locked Er-doped fibre ring laser," Electron. Lett. 29, 1149–1150 (1993).
    [CrossRef]
  6. Th. Pfeiffer and G. Veith, "40 GHz pulse generation using a widely tunable all-polarisation preserving erbium fibre ring laser," Electron. Lett. 29, 1849–1850 (1993).
    [CrossRef]
  7. F. Fontana, G. Bordogna, P. Franco, M. Midrio, and M. Romagnoli, "Conditions for soliton generation in harmonically modelocked erbium-doped fibre lasers," Electron. Lett. 29, 1652–1653 (1993).
    [CrossRef]
  8. I. N. Duling III, "Subpicosecond all-fiber erbium laser," Electron. Lett. 27, 544–545 (1991).
    [CrossRef]
  9. D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Selfstarting, passively modelocked erbium fibre ring laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542–543 (1991).
    [CrossRef]
  10. A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in the figure eight fiber laser," Electron. Lett. 28, 67–69 (1992).
    [CrossRef]
  11. E. Yoshida, Y. Kimura, and M. Nakazawa, "Laser diodepumped femtosecond erbium-doped fiber laser with a subring cavity for repetition rate control," Appl. Phys. Lett. 60, 932–934 (1992).
    [CrossRef]
  12. M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894–1896 (1992).
    [CrossRef]
  13. M. J. Guy, D. U. Noske, and J. R. Taylor, "Generation of femtosecond soliton pulses by passive mode locking of an ytterbium-erbium figure-eight fiber laser," Opt. Lett. 18, 1447–1449 (1993).
    [CrossRef] [PubMed]
  14. G. Eisenstein, Department of Electrical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel (personal communication, 1994).
  15. A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Passive harmonic mode-locking of a fibre soliton ring laser," Electron. Lett. 29, 1860–1861 (1993).
    [CrossRef]
  16. S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, and M. Saruwatari, "100 Gbit/s, 200 km optical transmission experiment using extremely low jitter PLL timing extraction and all-optical demultiplexing based on polarization insensitive four-wave mixing," Electron. Lett. 30, 800–801 (1994).
    [CrossRef]
  17. F. Matera and M. Settembre, "Exploitation of optical fibre capacity in long links," Electron. Lett. 30, 803–805 (1994).
    [CrossRef]
  18. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, "Sliding-frequency guiding filter: an improved form of soliton jitter control," Opt. Lett. 17, 1575–1577 (1992).
    [CrossRef] [PubMed]
  19. G. T. Harvey and L. F. Mollenauer, "Harmonically modelocked fiber ring laser with at internal Fabry–Perot stabilizer for soliton transmission," Opt. Lett. 18, 107–109 (1993).
    [CrossRef] [PubMed]
  20. J. P. Gordon and H. A. Haus, "Random walk of coherently amplified solitons in optical fiber transmission," Opt. Lett. 11, 665–667 (1986).
    [CrossRef] [PubMed]
  21. V. I. Karpman and V. V. Solov'ev, "A perturbational approach to the two-soliton systems," Physical D 3, 487–501 (1981).
    [CrossRef]
  22. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).
  23. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, "Modulation and filtering control of soliton transmission," J. Opt. Soc. Am. B 9, 1350–1357 (1992).
    [CrossRef]
  24. Y. Kodama, M. Romagnoli, and S. Wabnitz, "Soliton stability and interactions in fibre lasers," Electron. Lett. 28, 1981–1982 (1992).
    [CrossRef]
  25. H. A. Haus and A. Mecozzi, "Long-term storage of a bit stream of solitons," Opt. Lett. 17, 1500–1502 (1992).
    [CrossRef] [PubMed]
  26. H. A. Haus and A. Mecozzi, "Noise of mode-locked lasers," IEEE J. Quantum Electron. 29, 983–996 (1993).
    [CrossRef]
  27. V. Tzelpis, Th. Sphicopoulos, and C. Caroubalos, "Passive harmonic mode-locking in all-fiber soliton laser: energy states and stability aspects," IEEE Photon. Technol. Lett. 6, 47–49 (1994).
    [CrossRef]
  28. B. D. O. Anderson and J. B. Moore, Optimal Filtering (Prentice-Hall, Englewood Cliffs, N.J., 1979).

1994

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, and M. Saruwatari, "100 Gbit/s, 200 km optical transmission experiment using extremely low jitter PLL timing extraction and all-optical demultiplexing based on polarization insensitive four-wave mixing," Electron. Lett. 30, 800–801 (1994).
[CrossRef]

F. Matera and M. Settembre, "Exploitation of optical fibre capacity in long links," Electron. Lett. 30, 803–805 (1994).
[CrossRef]

V. Tzelpis, Th. Sphicopoulos, and C. Caroubalos, "Passive harmonic mode-locking in all-fiber soliton laser: energy states and stability aspects," IEEE Photon. Technol. Lett. 6, 47–49 (1994).
[CrossRef]

1993

H. A. Haus and A. Mecozzi, "Noise of mode-locked lasers," IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

G. T. Harvey and L. F. Mollenauer, "Harmonically modelocked fiber ring laser with at internal Fabry–Perot stabilizer for soliton transmission," Opt. Lett. 18, 107–109 (1993).
[CrossRef] [PubMed]

M. J. Guy, D. U. Noske, and J. R. Taylor, "Generation of femtosecond soliton pulses by passive mode locking of an ytterbium-erbium figure-eight fiber laser," Opt. Lett. 18, 1447–1449 (1993).
[CrossRef] [PubMed]

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Passive harmonic mode-locking of a fibre soliton ring laser," Electron. Lett. 29, 1860–1861 (1993).
[CrossRef]

H. Takara, S. Kawanishi, and M. Saruwatari, "20 GHz transform-limited optical pulse generation and bit-error-free operation using a tunable, actively mode-locked Er-doped fibre ring laser," Electron. Lett. 29, 1149–1150 (1993).
[CrossRef]

Th. Pfeiffer and G. Veith, "40 GHz pulse generation using a widely tunable all-polarisation preserving erbium fibre ring laser," Electron. Lett. 29, 1849–1850 (1993).
[CrossRef]

F. Fontana, G. Bordogna, P. Franco, M. Midrio, and M. Romagnoli, "Conditions for soliton generation in harmonically modelocked erbium-doped fibre lasers," Electron. Lett. 29, 1652–1653 (1993).
[CrossRef]

1992

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in the figure eight fiber laser," Electron. Lett. 28, 67–69 (1992).
[CrossRef]

E. Yoshida, Y. Kimura, and M. Nakazawa, "Laser diodepumped femtosecond erbium-doped fiber laser with a subring cavity for repetition rate control," Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894–1896 (1992).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, "Sliding-frequency guiding filter: an improved form of soliton jitter control," Opt. Lett. 17, 1575–1577 (1992).
[CrossRef] [PubMed]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, "Modulation and filtering control of soliton transmission," J. Opt. Soc. Am. B 9, 1350–1357 (1992).
[CrossRef]

Y. Kodama, M. Romagnoli, and S. Wabnitz, "Soliton stability and interactions in fibre lasers," Electron. Lett. 28, 1981–1982 (1992).
[CrossRef]

H. A. Haus and A. Mecozzi, "Long-term storage of a bit stream of solitons," Opt. Lett. 17, 1500–1502 (1992).
[CrossRef] [PubMed]

1991

I. N. Duling III, "Subpicosecond all-fiber erbium laser," Electron. Lett. 27, 544–545 (1991).
[CrossRef]

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Selfstarting, passively modelocked erbium fibre ring laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542–543 (1991).
[CrossRef]

1990

A. Takada and H. Miyazawa, "30 GHz picosecond pulse generation from actively mode-locked erbium-doped fibre laser," Electron. Lett. 26, 216–217 (1990).
[CrossRef]

1989

1986

1981

V. I. Karpman and V. V. Solov'ev, "A perturbational approach to the two-soliton systems," Physical D 3, 487–501 (1981).
[CrossRef]

1969

A. E. Siegman and D. J. Kuizenga, "Simple analytic expression for AM and FM mode-locked pulses in homogeneous lasers," Appl. Phys. Lett. 14, 181–182 (1969).
[CrossRef]

1967

E. M. Garmire and A. Yariv, "Laser mode-locking with saturable absorbers," IEEE J. Quantum Electron. QE-3, 222–225 (1967).
[CrossRef]

Anderson, B. D. O.

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Prentice-Hall, Englewood Cliffs, N.J., 1979).

Baer, T.

Bordogna, G.

F. Fontana, G. Bordogna, P. Franco, M. Midrio, and M. Romagnoli, "Conditions for soliton generation in harmonically modelocked erbium-doped fibre lasers," Electron. Lett. 29, 1652–1653 (1993).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).

Caroubalos, C.

V. Tzelpis, Th. Sphicopoulos, and C. Caroubalos, "Passive harmonic mode-locking in all-fiber soliton laser: energy states and stability aspects," IEEE Photon. Technol. Lett. 6, 47–49 (1994).
[CrossRef]

Dennis, M. L.

M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894–1896 (1992).
[CrossRef]

Duling, I. N.

M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894–1896 (1992).
[CrossRef]

I. N. Duling III, "Subpicosecond all-fiber erbium laser," Electron. Lett. 27, 544–545 (1991).
[CrossRef]

Eisenstein, G.

G. Eisenstein, Department of Electrical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel (personal communication, 1994).

Evangelides, S. G.

Fontana, F.

F. Fontana, G. Bordogna, P. Franco, M. Midrio, and M. Romagnoli, "Conditions for soliton generation in harmonically modelocked erbium-doped fibre lasers," Electron. Lett. 29, 1652–1653 (1993).
[CrossRef]

Franco, P.

F. Fontana, G. Bordogna, P. Franco, M. Midrio, and M. Romagnoli, "Conditions for soliton generation in harmonically modelocked erbium-doped fibre lasers," Electron. Lett. 29, 1652–1653 (1993).
[CrossRef]

Garmire, E. M.

E. M. Garmire and A. Yariv, "Laser mode-locking with saturable absorbers," IEEE J. Quantum Electron. QE-3, 222–225 (1967).
[CrossRef]

Gordon, J. P.

Grudinin, A. B.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Passive harmonic mode-locking of a fibre soliton ring laser," Electron. Lett. 29, 1860–1861 (1993).
[CrossRef]

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in the figure eight fiber laser," Electron. Lett. 28, 67–69 (1992).
[CrossRef]

Guy, M. J.

Hall, D. W.

Harvey, G. T.

Haus, H. A.

Kafka, J. D.

Kamatani, O.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, and M. Saruwatari, "100 Gbit/s, 200 km optical transmission experiment using extremely low jitter PLL timing extraction and all-optical demultiplexing based on polarization insensitive four-wave mixing," Electron. Lett. 30, 800–801 (1994).
[CrossRef]

Karpman, V. I.

V. I. Karpman and V. V. Solov'ev, "A perturbational approach to the two-soliton systems," Physical D 3, 487–501 (1981).
[CrossRef]

Kawanishi, S.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, and M. Saruwatari, "100 Gbit/s, 200 km optical transmission experiment using extremely low jitter PLL timing extraction and all-optical demultiplexing based on polarization insensitive four-wave mixing," Electron. Lett. 30, 800–801 (1994).
[CrossRef]

H. Takara, S. Kawanishi, and M. Saruwatari, "20 GHz transform-limited optical pulse generation and bit-error-free operation using a tunable, actively mode-locked Er-doped fibre ring laser," Electron. Lett. 29, 1149–1150 (1993).
[CrossRef]

Kimura, Y.

E. Yoshida, Y. Kimura, and M. Nakazawa, "Laser diodepumped femtosecond erbium-doped fiber laser with a subring cavity for repetition rate control," Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

Kodama, Y.

Y. Kodama, M. Romagnoli, and S. Wabnitz, "Soliton stability and interactions in fibre lasers," Electron. Lett. 28, 1981–1982 (1992).
[CrossRef]

Kuizenga, D. J.

A. E. Siegman and D. J. Kuizenga, "Simple analytic expression for AM and FM mode-locked pulses in homogeneous lasers," Appl. Phys. Lett. 14, 181–182 (1969).
[CrossRef]

Lai, Y.

Laming, R. I.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Selfstarting, passively modelocked erbium fibre ring laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542–543 (1991).
[CrossRef]

Matera, F.

F. Matera and M. Settembre, "Exploitation of optical fibre capacity in long links," Electron. Lett. 30, 803–805 (1994).
[CrossRef]

Matsas, V. J.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Selfstarting, passively modelocked erbium fibre ring laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542–543 (1991).
[CrossRef]

Mecozzi, A.

Midrio, M.

F. Fontana, G. Bordogna, P. Franco, M. Midrio, and M. Romagnoli, "Conditions for soliton generation in harmonically modelocked erbium-doped fibre lasers," Electron. Lett. 29, 1652–1653 (1993).
[CrossRef]

Miyazawa, H.

A. Takada and H. Miyazawa, "30 GHz picosecond pulse generation from actively mode-locked erbium-doped fibre laser," Electron. Lett. 26, 216–217 (1990).
[CrossRef]

Mollenauer, L. F.

Moore, J. B.

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Prentice-Hall, Englewood Cliffs, N.J., 1979).

Moores, J. D.

Morioka, T.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, and M. Saruwatari, "100 Gbit/s, 200 km optical transmission experiment using extremely low jitter PLL timing extraction and all-optical demultiplexing based on polarization insensitive four-wave mixing," Electron. Lett. 30, 800–801 (1994).
[CrossRef]

Nakazawa, M.

E. Yoshida, Y. Kimura, and M. Nakazawa, "Laser diodepumped femtosecond erbium-doped fiber laser with a subring cavity for repetition rate control," Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

Noske, D. U.

Payne, D. N.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Passive harmonic mode-locking of a fibre soliton ring laser," Electron. Lett. 29, 1860–1861 (1993).
[CrossRef]

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in the figure eight fiber laser," Electron. Lett. 28, 67–69 (1992).
[CrossRef]

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Selfstarting, passively modelocked erbium fibre ring laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542–543 (1991).
[CrossRef]

Pfeiffer, Th.

Th. Pfeiffer and G. Veith, "40 GHz pulse generation using a widely tunable all-polarisation preserving erbium fibre ring laser," Electron. Lett. 29, 1849–1850 (1993).
[CrossRef]

Phillips, M. W.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Selfstarting, passively modelocked erbium fibre ring laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542–543 (1991).
[CrossRef]

Richardson, D. J.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Passive harmonic mode-locking of a fibre soliton ring laser," Electron. Lett. 29, 1860–1861 (1993).
[CrossRef]

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in the figure eight fiber laser," Electron. Lett. 28, 67–69 (1992).
[CrossRef]

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Selfstarting, passively modelocked erbium fibre ring laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542–543 (1991).
[CrossRef]

Romagnoli, M.

F. Fontana, G. Bordogna, P. Franco, M. Midrio, and M. Romagnoli, "Conditions for soliton generation in harmonically modelocked erbium-doped fibre lasers," Electron. Lett. 29, 1652–1653 (1993).
[CrossRef]

Y. Kodama, M. Romagnoli, and S. Wabnitz, "Soliton stability and interactions in fibre lasers," Electron. Lett. 28, 1981–1982 (1992).
[CrossRef]

Saruwatari, M.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, and M. Saruwatari, "100 Gbit/s, 200 km optical transmission experiment using extremely low jitter PLL timing extraction and all-optical demultiplexing based on polarization insensitive four-wave mixing," Electron. Lett. 30, 800–801 (1994).
[CrossRef]

H. Takara, S. Kawanishi, and M. Saruwatari, "20 GHz transform-limited optical pulse generation and bit-error-free operation using a tunable, actively mode-locked Er-doped fibre ring laser," Electron. Lett. 29, 1149–1150 (1993).
[CrossRef]

Settembre, M.

F. Matera and M. Settembre, "Exploitation of optical fibre capacity in long links," Electron. Lett. 30, 803–805 (1994).
[CrossRef]

Siegman, A. E.

A. E. Siegman and D. J. Kuizenga, "Simple analytic expression for AM and FM mode-locked pulses in homogeneous lasers," Appl. Phys. Lett. 14, 181–182 (1969).
[CrossRef]

Solov'ev, V. V.

V. I. Karpman and V. V. Solov'ev, "A perturbational approach to the two-soliton systems," Physical D 3, 487–501 (1981).
[CrossRef]

Sphicopoulos, Th.

V. Tzelpis, Th. Sphicopoulos, and C. Caroubalos, "Passive harmonic mode-locking in all-fiber soliton laser: energy states and stability aspects," IEEE Photon. Technol. Lett. 6, 47–49 (1994).
[CrossRef]

Takada, A.

A. Takada and H. Miyazawa, "30 GHz picosecond pulse generation from actively mode-locked erbium-doped fibre laser," Electron. Lett. 26, 216–217 (1990).
[CrossRef]

Takara, H.

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, and M. Saruwatari, "100 Gbit/s, 200 km optical transmission experiment using extremely low jitter PLL timing extraction and all-optical demultiplexing based on polarization insensitive four-wave mixing," Electron. Lett. 30, 800–801 (1994).
[CrossRef]

H. Takara, S. Kawanishi, and M. Saruwatari, "20 GHz transform-limited optical pulse generation and bit-error-free operation using a tunable, actively mode-locked Er-doped fibre ring laser," Electron. Lett. 29, 1149–1150 (1993).
[CrossRef]

Taylor, J. R.

Tzelpis, V.

V. Tzelpis, Th. Sphicopoulos, and C. Caroubalos, "Passive harmonic mode-locking in all-fiber soliton laser: energy states and stability aspects," IEEE Photon. Technol. Lett. 6, 47–49 (1994).
[CrossRef]

Veith, G.

Th. Pfeiffer and G. Veith, "40 GHz pulse generation using a widely tunable all-polarisation preserving erbium fibre ring laser," Electron. Lett. 29, 1849–1850 (1993).
[CrossRef]

Wabnitz, S.

Y. Kodama, M. Romagnoli, and S. Wabnitz, "Soliton stability and interactions in fibre lasers," Electron. Lett. 28, 1981–1982 (1992).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).

Yariv, A.

E. M. Garmire and A. Yariv, "Laser mode-locking with saturable absorbers," IEEE J. Quantum Electron. QE-3, 222–225 (1967).
[CrossRef]

Yoshida, E.

E. Yoshida, Y. Kimura, and M. Nakazawa, "Laser diodepumped femtosecond erbium-doped fiber laser with a subring cavity for repetition rate control," Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

Appl. Phys. Lett.

E. Yoshida, Y. Kimura, and M. Nakazawa, "Laser diodepumped femtosecond erbium-doped fiber laser with a subring cavity for repetition rate control," Appl. Phys. Lett. 60, 932–934 (1992).
[CrossRef]

A. E. Siegman and D. J. Kuizenga, "Simple analytic expression for AM and FM mode-locked pulses in homogeneous lasers," Appl. Phys. Lett. 14, 181–182 (1969).
[CrossRef]

Electron. Lett.

Y. Kodama, M. Romagnoli, and S. Wabnitz, "Soliton stability and interactions in fibre lasers," Electron. Lett. 28, 1981–1982 (1992).
[CrossRef]

M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894–1896 (1992).
[CrossRef]

A. Takada and H. Miyazawa, "30 GHz picosecond pulse generation from actively mode-locked erbium-doped fibre laser," Electron. Lett. 26, 216–217 (1990).
[CrossRef]

H. Takara, S. Kawanishi, and M. Saruwatari, "20 GHz transform-limited optical pulse generation and bit-error-free operation using a tunable, actively mode-locked Er-doped fibre ring laser," Electron. Lett. 29, 1149–1150 (1993).
[CrossRef]

Th. Pfeiffer and G. Veith, "40 GHz pulse generation using a widely tunable all-polarisation preserving erbium fibre ring laser," Electron. Lett. 29, 1849–1850 (1993).
[CrossRef]

F. Fontana, G. Bordogna, P. Franco, M. Midrio, and M. Romagnoli, "Conditions for soliton generation in harmonically modelocked erbium-doped fibre lasers," Electron. Lett. 29, 1652–1653 (1993).
[CrossRef]

I. N. Duling III, "Subpicosecond all-fiber erbium laser," Electron. Lett. 27, 544–545 (1991).
[CrossRef]

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Selfstarting, passively modelocked erbium fibre ring laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542–543 (1991).
[CrossRef]

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in the figure eight fiber laser," Electron. Lett. 28, 67–69 (1992).
[CrossRef]

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Passive harmonic mode-locking of a fibre soliton ring laser," Electron. Lett. 29, 1860–1861 (1993).
[CrossRef]

S. Kawanishi, T. Morioka, O. Kamatani, H. Takara, and M. Saruwatari, "100 Gbit/s, 200 km optical transmission experiment using extremely low jitter PLL timing extraction and all-optical demultiplexing based on polarization insensitive four-wave mixing," Electron. Lett. 30, 800–801 (1994).
[CrossRef]

F. Matera and M. Settembre, "Exploitation of optical fibre capacity in long links," Electron. Lett. 30, 803–805 (1994).
[CrossRef]

IEEE J. Quantum Electron.

H. A. Haus and A. Mecozzi, "Noise of mode-locked lasers," IEEE J. Quantum Electron. 29, 983–996 (1993).
[CrossRef]

E. M. Garmire and A. Yariv, "Laser mode-locking with saturable absorbers," IEEE J. Quantum Electron. QE-3, 222–225 (1967).
[CrossRef]

IEEE Photon. Technol. Lett.

V. Tzelpis, Th. Sphicopoulos, and C. Caroubalos, "Passive harmonic mode-locking in all-fiber soliton laser: energy states and stability aspects," IEEE Photon. Technol. Lett. 6, 47–49 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Physical D

V. I. Karpman and V. V. Solov'ev, "A perturbational approach to the two-soliton systems," Physical D 3, 487–501 (1981).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).

G. Eisenstein, Department of Electrical Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel (personal communication, 1994).

B. D. O. Anderson and J. B. Moore, Optimal Filtering (Prentice-Hall, Englewood Cliffs, N.J., 1979).

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Figures (3)

Fig. 1
Fig. 1

Fiber loop including a pair of solitons labeled A and B: (a) closed loop, (b) the equivalent open loop. The length of the loop is L, and the soliton spacings are ξBξA = ξBA and LξBA.

Fig. 2
Fig. 2

Spectra and temporal forms of the train obtained in the active mode-locking case. The propagation distance is Z = 600. The initial conditions are β = 0.15, αM = 0.36, Δ = 6, ΩM = 2π/Δ, γ = 0. Moreover, (a) Ψ = σ = π, R = 0; (b) Ψ = σ = 0, R = 0; (c) Ψ = σ = 0, R = 0.2. The equilibrium value of the soliton spacing is Δ = 8 (DC = 0.22).

Fig. 3
Fig. 3

Spectra and temporal forms of the train obtained in the passive mode-locking case. The propagation distance is Z = 600. The initial conditions are β = 0.15, αM = 0, Δ = 6, ΩM = 0, γ = 0.06. Moreover, (a) Ψ = σ = π, R = 0; (b) Ψ = σ = 0, R = 0; (c) Ψ = σ = 0, R = 0.2. The final spacing, as in Fig. 2, is Δ = 8 (DC = 0.22).

Equations (69)

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i u Z + 1 2 2 u T 2 + u 2 u = i Q ( u ) + i I ( u ) .
u A , B ( Z , T ) = η A , B sech [ η A , B ( T - ξ A , B ) ] × exp { i [ κ A , B ( T - ξ A , B ) + ½ ( η A , B 2 - κ A , B 2 ) Z + θ A , B ] } = η A , B sech [ η A , B ( T - ξ A , B ) ] exp ( i ϕ A , B ) ,
i u A , B Z + 1 2 2 u A , B T 2 + u A , B 2 u A , B = + i I A , B ( u A , B ) ,
I A = 2 i u A 2 u B + i u A 2 u B * ,             I B = 2 i u B 2 u A + i u B 2 u A * .
( d η d Z ) int = 0 ,
( d p d Z ) int = 4 η 3 { exp [ - η ( L - ξ B A ) ] sin [ κ ( L - ξ B A ) + ϕ B A ] - exp ( - η ξ B A ) sin ( κ ξ B A - ϕ B A ) } ,
( d κ d Z ) int = 0 ,
( d q d Z ) int = 4 η { exp [ - η ( L - ξ B A ) ] cos [ κ ( L - ξ B A ) + ϕ B A ] - exp ( - η ξ B A ) cos ( κ ξ B A - ϕ B A ) } ,
( d ξ B A d Z ) int = 2 q ,
( d ϕ B A d Z ) int = 2 η p .
H E ( Ω ) = 1 - r ( 1 - 2 r cos μ + r 2 ) 1 / 2 × exp [ i arctan ( r sin μ 1 - r cos μ ) ] ,
h E ( Ω ) = z 0 z A log H E ( Ω ) ,
h E ( Ω ) = z 0 z A [ log ( 1 - r ) + n = 1 r n n exp ( i n μ ) ] R [ exp ( i μ ) - 1 ] ,
Q ( u ) = F - 1 [ h E ( Ω ) u ^ ( Ω ) ] = 1 2 π - + h E ( Ω ) u ^ ( Ω ) exp ( i Ω t ) d Ω = R [ - u ( T ) + exp ( - i σ ) u ( T + Ξ 0 ) ] .
( d η d Z ) et = - 2 R η + 2 R η cos [ κ ( Ξ 0 - ξ B A ) + ϕ B A ] cos ( σ ) ,
( d p d Z ) et = - 2 R p - 2 R η sin [ κ ( Ξ 0 - ξ B A ) + ϕ B A ] sin ( σ ) + - 2 R p cos [ κ ( Ξ 0 - ξ B A ) + ϕ B A ] cos ( σ ) ,
( d κ d Z ) et = - R η 2 ( Ξ 0 - ξ B A ) sin [ κ ( Ξ 0 - ξ B A ) + ϕ B A ] × cos ( σ ) ,
( d q d Z ) et = - R η 2 ( Ξ 0 - ξ B A ) cos [ κ ( Ξ 0 - ξ B A ) + ϕ B A ] × sin ( σ ) ,
( d ξ B A d Z ) et = 2 R ( Ξ 0 - ξ B A ) cos [ κ ( Ξ 0 - ξ B A ) + ϕ B A ] cos ( σ ) ,
( d ϕ B A d Z ) e t = κ d ξ B A d Z - 2 R sin [ κ ( Ξ 0 - ξ B A ) + ϕ B A ] cos ( σ ) .
d ξ B A d Z = 2 R ( Ξ 0 - ξ B A ) cos ( ϕ B A ) cos ( σ ) .
Q ( u ) = F - 1 [ h F ( Ω ) u ^ ( Ω ) = F - 1 [ z 0 z A log 1 1 + i 2 Ω B u ^ ( Ω ) ] z 0 z A 2 B u T + β 2 u T 2 ,
β = z 0 z A 2 B 2 .
Q ( u ) = [ ½ α m Ω m 2 T 2 + / γ u 2 ] u ,
( d η d Z ) ope = 2 δ η + 4 3 γ η 3 - 2 β η ( η 2 3 + κ 2 ) - π 2 12 α M Ω M 2 1 η ,
( d p d Z ) ope = 2 δ p + 4 γ η 2 p - 2 β p ( η 2 + κ 2 ) - 4 β η κ q + π 2 12 α M Ω M 2 p η ,
( d κ d Z ) ope = - / β η 2 κ ,
( d q d Z ) ope = - / β η 2 q - / β η κ p ,
( d ξ B A d Z ) ope = - π 2 6 α M Ω M 2 1 η 2 ( ξ B A = Ξ M ) ,
( d ϕ B A d Z ) ope = - π 2 6 α M Ω M 2 κ 1 η 2 ( ξ B A - Ξ M ) .
d Y ¯ d z = ( d Y ¯ d z ) int + ( d Y ¯ d z ) et + ( d Y ¯ d z ) ope ,
P ( λ ) = ( λ + Λ 1 ) ( λ + Λ 2 ) [ λ 2 + λ ( Λ 1 + 6 ζ 1 ) + 2 ζ 1 Λ 1 + 8 ζ 1 2 - 2 ζ 3 + 4 ζ 2 ] [ λ 2 + λ ( Λ 2 + Λ 3 ) + Λ 2 Λ 3 - 2 ζ 3 - / ζ 2 ] ,
Λ 1 = 4 3 β - 8 3 γ - π 2 6 α M Ω M 2 ,             Λ 2 = 4 3 β , Λ 3 = 2 ζ 1 + π 2 6 α M Ω M 2 ,
ζ 1 = R cos ( ϕ B A ) cos ( σ ) ,             ζ 2 = R cos ( ϕ B A ) sin ( σ ) , ζ 3 = 8 exp ( - Ξ 0 ) cos ( ϕ B A ) .
Λ 1 = 4 3 β - 8 3 γ - π 2 6 α M Ω M 2 > 0 , Λ 2 = 4 3 β > 0 ,
Λ 1 > 0 Λ 2 > 0 R > max ( Λ 1 8 { [ 1 + 128 Λ 1 exp ( - Ξ 0 ) ] 1 / 2 - 1 } , 6 exp ( - Ξ 0 β - π 2 12 α M Ω M 2 ) .
u ( 0 , T ) = p = 1 16 sech [ ( T - ( p - 1 / 2 ) Δ ] exp ( i p Ψ ) ,
d d z Δ η = ( - 4 3 β + 8 3 γ + π 2 6 α M Ω M 2 ) Δ η + S η ( z ) ,
S η ( z ) S η * ( z ) = D η η δ ( z - z ) ,
D η η = 4 n sp n ph G .
Δ η 2 = 1 2 D η η 4 3 β - 8 3 γ - π 2 6 α M Ω M 2 .
d d z Δ κ = - 4 3 β Δ κ + S κ ( z ) ,
S κ ( z ) S κ * ( z ) = D κ κ δ ( z - z ) .
D κ κ = 2 3 n s p n p h G ,
Δ κ 2 = 1 2 D κ κ / β .
d d z Δ q = - 4 3 β Δ q + [ 8 exp ( - L / 2 ) cos ( ϕ B A ) + R cos ( ϕ B A ) sin ( ϕ ) ] δ ξ B A + S q ( z ) ,
d d z Δ ξ B A = 2 Δ q + [ - π 2 6 α M Ω M 2 - 2 R cos ( ϕ B A ) cos ( ϕ B A ) ] δ ξ B A + S ξ ( z ) ,
S q ( z ) S q * ( z ) = D q q δ ( z - z ) ,
S ξ ( z ) S ξ * ( z ) = D ξ ξ δ ( z - z )
D q q = 2 3 n sp n ph G ,
D ξ ξ = 4 3 π 2 n sp n ph G .
d d z Δ q = - / β Δ q + [ 8 exp ( - L / 2 ) cos ( ϕ B A ) ] Δ ξ B A + S q ( z ) ,
d d z Δ ξ B A = 2 Δ q + ( - π 2 6 α M Ω M 2 - 2 R ) Δ ξ B A + S ξ ( z ) .
F = [ - 4 3 β 8 exp ( - L / 2 ) cos ( ϕ B A ) 2 - π 2 6 α M Ω M 2 - 2 R ] .
F X + X F T + G G T = 0 ,
X = [ Δ q 2 Δ q Δ ξ B A Δ ξ B A Δ q Δ ξ B A 2 ] ,
G G T = [ D q q 0 0 D ξ ξ ] .
Δ q 2 = ν 1 ν 2 + ν 3 + ν 2 2 2 ( ν 1 + ν 2 ) ( ν 1 ν 2 + ν 3 ) D q q + ν 3 2 8 ( ν 1 + ν 2 ) ( ν 1 ν 2 + ν 3 ) D ξ ξ ,
Δ ξ B A 2 = 2 ( ν 1 + ν 2 ) ( ν 1 ν 2 + ν 3 ) D q q + ν 1 ν 2 + ν 3 + ν 1 2 2 ( ν 1 + ν 2 ) ( ν 1 ν 2 + ν 3 ) D ξ ξ ,
ν 1 = 4 3 β ,             ν 2 = π 2 6 α M Ω M 2 + 2 R , ν 3 = - 16 exp ( - L / 2 ) cos ( ϕ B A ) .
d d z Δ p = ( - 4 3 β + 8 3 γ + π 2 6 α M Ω M 2 - 4 R ) Δ p + [ 8 exp ( - L / 2 ) cos ( ϕ B A ) ] Δ ϕ B A + S p ,
d d z Δ ϕ B A = 2 Δ p - 2 R Δ ϕ B A + S ϕ ( r ) ,
S p ( z ) S p * ( z ) = D p p δ ( z - z ) ,
S ϕ ( z ) S ϕ * ( z ) = D ϕ ϕ δ ( z - z )
D p p = 4 n sp n ph G ,
D ϕ ϕ = 16 3 ( 1 + π 2 12 ) n sp n ph G .
Δ p 2 = μ 1 μ 2 + ν 3 + μ 2 2 2 ( μ 1 + μ 2 ) ( μ 1 μ 2 + ν 3 ) D p p + ν 3 2 8 ( μ 1 + μ 2 ) ( μ 1 μ 2 + ν 3 ) D ϕ ϕ ,
Δ ϕ B A 2 = 2 ( μ 1 + μ 2 ) ( μ 1 μ 2 + ν 3 ) D p p + μ 1 μ 2 + ν 3 + μ 1 2 2 ( μ 1 + μ 2 ) ( μ 1 μ 2 + ν 3 ) D ϕ ϕ ,
μ 1 = 4 3 β - 8 3 γ - π 2 6 α M Ω M 2 ,             μ 2 = - 2 R .

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