Abstract

We show results obtained from a semiconductor saturable-absorber mirror mode-locked Ti:sapphire soliton laser that was operated in the multiple-pulse regime. Double, triple, and quadruple pulses were observed when the dispersion was decreased below a critical value. The pulse pairs and triplets were either widely separated or closely coupled, and spectra that resembled those of constant as well as rotating phase differences between pulses were observed. We explain our observations in the framework of the generalized complex Ginzburg–Landau equation as the master equation of the laser.

© 1999 Optical Society of America

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  1. F. Salin, P. Grangier, G. Roger, and A. Brun, “Experimental observation of nonsymmetrical N=2 solitons in a femtosecond laser,” Phys. Rev. Lett. 60, 569–572 (1988).
    [CrossRef] [PubMed]
  2. A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
    [CrossRef]
  3. C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
    [CrossRef]
  4. J. Aus der Au, D. Kopf, F. Morier-Genoud, M. Moser, and U. Keller, “60-fs pulses from a diode-pumped Nd:glass laser,” Opt. Lett. 22, 307–309 (1997).
    [CrossRef]
  5. A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Passive harmonic modelocking of a fibre soliton ring laser,” Electron. Lett. 29, 1860–1861 (1993).
    [CrossRef]
  6. M. E. Fermann, and J. D. Minelly, “Cladding-pumped passive harmonically mode-locked fiber laser,” Opt. Lett. 22, 970–972 (1996).
    [CrossRef]
  7. A. B. Grudinin and S. Gray, “Passive harmonic mode locking in soliton fiber lasers,” J. Opt. Soc. Am. B 14, 144–154 (1997).
    [CrossRef]
  8. B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
    [CrossRef]
  9. B. C. Collings, K. Bergmann, and W. H. Knox, “Stable multigigahertz pulse-train formation in a short-cavity passively harmonic mode-locked erbium/ytterbium fiber laser,” Opt. Lett. 23, 123–125 (1998).
    [CrossRef]
  10. B. C. Collings, K. Bergmann, and W. H. Knox, “True fundamental solitons in a passively mode-locked short-cavity Cr4+:YAG laser,” Opt. Lett. 22, 1098–1100 (1997).
    [CrossRef] [PubMed]
  11. A. N. Pilipetskii, E. A. Golovchenko, and C. R. Menyuk, “Acoustic effect in passively mode-locked fiber ring lasers,” Opt. Lett. 20, 907–909 (1994).
    [CrossRef]
  12. J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
    [CrossRef]
  13. F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers—what’s the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
    [CrossRef]
  14. M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “GaAs based antiresonant Fabry–Perot saturable absorber fabricated by metal organic vapor phase epitaxy and ion implantation,” Appl. Phys. Lett. 70, 3428–3430 (1997).
    [CrossRef]
  15. M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “An anti-resonant Fabry–Perot saturable absorber for passive mode-locking fabricated by metal organic vapor phase epitaxy and ion-implantation—design, characterization and mode-locking,” IEEE J. Quantum Electron. 34, 2150–2162 (1998).
    [CrossRef]
  16. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
  17. H. A. Haus, “Theory of modelocking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
    [CrossRef]
  18. F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
    [CrossRef]
  19. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).
  20. F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
    [CrossRef]

1998 (4)

B. C. Collings, K. Bergmann, and W. H. Knox, “Stable multigigahertz pulse-train formation in a short-cavity passively harmonic mode-locked erbium/ytterbium fiber laser,” Opt. Lett. 23, 123–125 (1998).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[CrossRef]

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers—what’s the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “An anti-resonant Fabry–Perot saturable absorber for passive mode-locking fabricated by metal organic vapor phase epitaxy and ion-implantation—design, characterization and mode-locking,” IEEE J. Quantum Electron. 34, 2150–2162 (1998).
[CrossRef]

1997 (5)

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “GaAs based antiresonant Fabry–Perot saturable absorber fabricated by metal organic vapor phase epitaxy and ion implantation,” Appl. Phys. Lett. 70, 3428–3430 (1997).
[CrossRef]

B. C. Collings, K. Bergmann, and W. H. Knox, “True fundamental solitons in a passively mode-locked short-cavity Cr4+:YAG laser,” Opt. Lett. 22, 1098–1100 (1997).
[CrossRef] [PubMed]

J. Aus der Au, D. Kopf, F. Morier-Genoud, M. Moser, and U. Keller, “60-fs pulses from a diode-pumped Nd:glass laser,” Opt. Lett. 22, 307–309 (1997).
[CrossRef]

A. B. Grudinin and S. Gray, “Passive harmonic mode locking in soliton fiber lasers,” J. Opt. Soc. Am. B 14, 144–154 (1997).
[CrossRef]

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

1996 (2)

M. E. Fermann, and J. D. Minelly, “Cladding-pumped passive harmonically mode-locked fiber laser,” Opt. Lett. 22, 970–972 (1996).
[CrossRef]

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

1994 (2)

A. N. Pilipetskii, E. A. Golovchenko, and C. R. Menyuk, “Acoustic effect in passively mode-locked fiber ring lasers,” Opt. Lett. 20, 907–909 (1994).
[CrossRef]

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

1993 (1)

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

1992 (2)

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

1988 (1)

F. Salin, P. Grangier, G. Roger, and A. Brun, “Experimental observation of nonsymmetrical N=2 solitons in a femtosecond laser,” Phys. Rev. Lett. 60, 569–572 (1988).
[CrossRef] [PubMed]

1975 (1)

H. A. Haus, “Theory of modelocking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

Aus der Au, J.

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers—what’s the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

J. Aus der Au, D. Kopf, F. Morier-Genoud, M. Moser, and U. Keller, “60-fs pulses from a diode-pumped Nd:glass laser,” Opt. Lett. 22, 307–309 (1997).
[CrossRef]

Bergman, K.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[CrossRef]

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

Bergmann, K.

Brabec, T.

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Brun, A.

F. Salin, P. Grangier, G. Roger, and A. Brun, “Experimental observation of nonsymmetrical N=2 solitons in a femtosecond laser,” Phys. Rev. Lett. 60, 569–572 (1988).
[CrossRef] [PubMed]

Collings, B. C.

B. C. Collings, K. Bergmann, and W. H. Knox, “Stable multigigahertz pulse-train formation in a short-cavity passively harmonic mode-locked erbium/ytterbium fiber laser,” Opt. Lett. 23, 123–125 (1998).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[CrossRef]

B. C. Collings, K. Bergmann, and W. H. Knox, “True fundamental solitons in a passively mode-locked short-cavity Cr4+:YAG laser,” Opt. Lett. 22, 1098–1100 (1997).
[CrossRef] [PubMed]

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

Cundiff, S. T.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

Cunningham, J. E.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

Curley, P.

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Curley, P. F.

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

Fermann, M.

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Fermann, M. E.

Golovchenko, E. A.

Grangier, P.

F. Salin, P. Grangier, G. Roger, and A. Brun, “Experimental observation of nonsymmetrical N=2 solitons in a femtosecond laser,” Phys. Rev. Lett. 60, 569–572 (1988).
[CrossRef] [PubMed]

Gray, S.

Grudinin, A. B.

A. B. Grudinin and S. Gray, “Passive harmonic mode locking in soliton fiber lasers,” J. Opt. Soc. Am. B 14, 144–154 (1997).
[CrossRef]

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

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

Haus, H. A.

H. A. Haus, “Theory of modelocking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

Hofer, M.

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Jagadish, C.

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “An anti-resonant Fabry–Perot saturable absorber for passive mode-locking fabricated by metal organic vapor phase epitaxy and ion-implantation—design, characterization and mode-locking,” IEEE J. Quantum Electron. 34, 2150–2162 (1998).
[CrossRef]

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “GaAs based antiresonant Fabry–Perot saturable absorber fabricated by metal organic vapor phase epitaxy and ion implantation,” Appl. Phys. Lett. 70, 3428–3430 (1997).
[CrossRef]

Jan, W. Y.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

Jung, I. D.

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

Kärtner, F. X.

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers—what’s the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

Keller, U.

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers—what’s the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

J. Aus der Au, D. Kopf, F. Morier-Genoud, M. Moser, and U. Keller, “60-fs pulses from a diode-pumped Nd:glass laser,” Opt. Lett. 22, 307–309 (1997).
[CrossRef]

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

Knox, W. H.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[CrossRef]

B. C. Collings, K. Bergmann, and W. H. Knox, “Stable multigigahertz pulse-train formation in a short-cavity passively harmonic mode-locked erbium/ytterbium fiber laser,” Opt. Lett. 23, 123–125 (1998).
[CrossRef]

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

B. C. Collings, K. Bergmann, and W. H. Knox, “True fundamental solitons in a passively mode-locked short-cavity Cr4+:YAG laser,” Opt. Lett. 22, 1098–1100 (1997).
[CrossRef] [PubMed]

Koch, M.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

Kopf, D.

Krausz, F.

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Kutz, J. N.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[CrossRef]

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

Lederer, M. J.

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “An anti-resonant Fabry–Perot saturable absorber for passive mode-locking fabricated by metal organic vapor phase epitaxy and ion-implantation—design, characterization and mode-locking,” IEEE J. Quantum Electron. 34, 2150–2162 (1998).
[CrossRef]

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “GaAs based antiresonant Fabry–Perot saturable absorber fabricated by metal organic vapor phase epitaxy and ion implantation,” Appl. Phys. Lett. 70, 3428–3430 (1997).
[CrossRef]

Luther-Davies, B.

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “An anti-resonant Fabry–Perot saturable absorber for passive mode-locking fabricated by metal organic vapor phase epitaxy and ion-implantation—design, characterization and mode-locking,” IEEE J. Quantum Electron. 34, 2150–2162 (1998).
[CrossRef]

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “GaAs based antiresonant Fabry–Perot saturable absorber fabricated by metal organic vapor phase epitaxy and ion implantation,” Appl. Phys. Lett. 70, 3428–3430 (1997).
[CrossRef]

Menyuk, C. R.

Minelly, J. D.

Morier-Genoud, F.

Moser, M.

Ober, M.

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Payne, D. N.

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

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

Pilipetskii, A. N.

Richardson, D. J.

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

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

Roger, G.

F. Salin, P. Grangier, G. Roger, and A. Brun, “Experimental observation of nonsymmetrical N=2 solitons in a femtosecond laser,” Phys. Rev. Lett. 60, 569–572 (1988).
[CrossRef] [PubMed]

Salin, F.

F. Salin, P. Grangier, G. Roger, and A. Brun, “Experimental observation of nonsymmetrical N=2 solitons in a femtosecond laser,” Phys. Rev. Lett. 60, 569–572 (1988).
[CrossRef] [PubMed]

Schmidt, A.

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Spielmann, C.

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Tan, H. H.

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “An anti-resonant Fabry–Perot saturable absorber for passive mode-locking fabricated by metal organic vapor phase epitaxy and ion-implantation—design, characterization and mode-locking,” IEEE J. Quantum Electron. 34, 2150–2162 (1998).
[CrossRef]

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “GaAs based antiresonant Fabry–Perot saturable absorber fabricated by metal organic vapor phase epitaxy and ion implantation,” Appl. Phys. Lett. 70, 3428–3430 (1997).
[CrossRef]

Tsuda, S.

B. C. Collings, K. Bergman, S. T. Cundiff, S. Tsuda, J. N. Kutz, J. E. Cunningham, W. Y. Jan, M. Koch, and W. H. Knox, “Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector,” IEEE J. Sel. Top. Quantum Electron. 2, 1065–1074 (1997).
[CrossRef]

Wintner, E.

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

Appl. Phys. Lett. (1)

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “GaAs based antiresonant Fabry–Perot saturable absorber fabricated by metal organic vapor phase epitaxy and ion implantation,” Appl. Phys. Lett. 70, 3428–3430 (1997).
[CrossRef]

Electron. Lett. (2)

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantisation in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

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

IEEE J. Quantum Electron. (4)

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

M. J. Lederer, B. Luther-Davies, H. H. Tan, and C. Jagadish, “An anti-resonant Fabry–Perot saturable absorber for passive mode-locking fabricated by metal organic vapor phase epitaxy and ion-implantation—design, characterization and mode-locking,” IEEE J. Quantum Electron. 34, 2150–2162 (1998).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[CrossRef]

F. Krausz, M. Fermann, T. Brabec, P. Curley, M. Hofer, M. Ober, C. Spielmann, E. Wintner, and A. Schmidt, “Femtosecond solid-state lasers,” IEEE J. Quantum Electron. 28, 2097–2122 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (3)

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers—what’s the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Schematic of the Ti:sapphire laser cavity used in the mode-locking experiment: M’s, mirrors; P’s, prisms; BRF’s, birefringent filters. ROC, radius of curvature.

Fig. 2
Fig. 2

Measured autocorrelation width versus total intracavity dispersion β2 in the range of the single-to-double soliton transition with soliton number N annotated.

Fig. 3
Fig. 3

Autocorrelation traces and spectra of (a) a widely separated doublet, (b) a rotating ϕ doublet, (c) a ϕπ doublet, and (d) ϕ0 doublet.

Fig. 4
Fig. 4

Autocorrelation traces and spectra of (a) a doublet with a widely separated companion, (b) a ϕπ triplet, (c) a ϕ0 triplet, (d) irregular output for small |β2|, and (e) chirped picosecond pulses for positive β2.

Fig. 5
Fig. 5

(a) g-l0-q0 as a function of β2 around the single-to-double transition. Inset, diagram clarifying the energy balance in equilibrium. (b) Evolution of the single soliton into two solitons with ∼5-ps separation at β2-1200 fs2. g0=0.1687, q0=0.006, Ta=0.3 ps, Ea=10 nJ, Pg=4 W (see also Table 1).

Fig. 6
Fig. 6

Total soliton loss versus separation of two solitons of equal energy for Ep/Ea=2.35 and Ep/Ea=23.5 (inset). β2=-900 fs2, Ta=0.3 ps, Ea=10 nJ, q0=0.006, t0=77 fs (see also Table 1).

Fig. 7
Fig. 7

Simulation results after 10,000 round trips: (a) intensity autocorrelation, (b) time-integrated spectrum, (c) evolution in the time domain, (d) interaction plane of two closely spaced pulses with rotating phase difference. β2=-1100 fs2, Ta=0.3 ps, Ea=10 nJ, q0=0.006, g0=0.2441, Pg=2 W (see also Table 1).

Fig. 8
Fig. 8

Simulation results after 94,000 round trips: (a) intensity autocorrelation, (b) time-integrated spectrum, (c) evolution in the time domain, (d) interaction plane of two closely spaced pulses with mean phase difference ϕ between π/2 and π. β2=-688 fs2, Ta=0.1 ps, Ea=3 nJ, q0=0.01, g0=0.25, Pg=2 W (see also Table 1).

Fig. 9
Fig. 9

Simulation results after 25,000 round trips: (a) intensity autocorrelation, (b) time-integrated spectrum, (c) evolution in the time domain, (d) interaction plane of two closely spaced pulses with mean phase difference ϕ between 0 and π/2. β2=-688 fs2, Ta=0.2 ps, Ea=3 nJ, q0=0.01, g0=0.249, Pg=2 W (see also Table 1).

Fig. 10
Fig. 10

(a) (g0, |β2|) plane showing regions of existence for single and multiple pulses according to simulations in the framework of soliton perturbation theory. Insets, g-l0-q0, δ, and α for constant g0=0.16 and |β2|=600 fs2 respectively, as calculated from the steady-state energy balance [Eq. (7)] with α+δ>q0 as the switching condition. (b) (g0, |β2|) plane obtained from simulation of the complete GCGLE [Eqs. (1)–(3)]. Ta=0.3 ps, Ea=10 nJ, q0=0.006, g0=0.1687, Pg=4 W (see also Table 1).

Fig. 11
Fig. 11

(a) Simulation of the laser in nonequilibrium. The energy balance condition g-l0-α-δ=0 is permanently violated. Also shown are the intracavity |ψ|2 after various numbers of round trips; β2=-200 fs2. (b) Simulation of |ψ|2 and spectrum after 10,000 round trips for positive dispersion; β2=+200 fs2. Ta=0.3 ps, Ea=10 nJ, q0=0.006, g0=0.1687, Pg=4 W (see also Table 1).

Tables (1)

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Table 1 Relevant Modeling Parameters

Equations (14)

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iTRψT-β22ψtt+κ|ψ|2ψ=i(g-q-l0)ψ+iβψtt,
qt=q-q0Ta-|ψ|2qEa,
gT=-g-g0Tg-QgPgTgTR.
r=Q2κ22π|β2|=Qκπt0.
ψ=κQ24|β2|sechκQ2×|β2|texpiQ2κ28|β2|TRT.
ls=δ+α=β3t02+12t0- sech2tt0q(t)dt
g-l0-δ-α=0
(α1+δ1+)Q1q0,
(α1+δ1)Q1>(α2+δ2)Q2
TRdQadT=2[g(Qa+Qb)-l0-δa-αa]Qa,
TRdQbdT=2[g(Qa+Qb)-l0-δb-αb]Qb,
dgdT=-g-g0Tg-(Qa+Qb)gPgTgTR,
TRdΔQdT=2ΔQ(Δg-Δα-Δδ).
Δδ>-Δα.

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