Abstract

Passive mode locking of inhomogeneously broadened lasers is studied in three typical regimes: pure self-amplitude modulation (SAM) mode locking, soliton mode locking, and mode locking in the presence of self-phase modulation (SPM) and positive group-delay dispersion (GDD). When the mode-locking strength becomes weak and/or the GDD is large, the lockable spectral width reduces and mode locking becomes unstable, which is a common and unique feature for broadband inhomogeneously broadened lasers. Mode locking by the pure SAM is essentially a phase-locking process. The soliton mode locking best resists the impact of insufficient gain filtering, while mode locking in the presence of strong SPM and positive GDD is the weakest, and a certain amount of gain filtering (narrowing) is necessary for stable mode locking. The scaling relations of mode-locking characteristics with the gain linewidth, GDD, and nonlinearities, and the influence of the degree of inhomogeneity of the gain medium are examined. A linearized saturable absorber may not be able to mode lock a strongly inhomogeneously broadened laser, and a minimum unsaturated absorber loss is required for stable mode locking in most situations.

© 2007 Optical Society of America

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  1. J. Zhou, G. Taft, C. P. Huang, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, "Pulse evolution in a broad-bandwidth Ti:sapphire laser," Opt. Lett. 19, 1149-1151 (1994).
    [PubMed]
  2. A. Stingl, M. Lenzner, C. Spielmann, F. Krausz, and R. Szipöcs, "Sub-10-fs mirror-dispersion-controlled Ti:sapphire laser," Opt. Lett. 20, 602-604 (1995).
    [CrossRef] [PubMed]
  3. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, "Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser," Opt. Lett. 24, 411-413 (1999).
    [CrossRef]
  4. D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, "Semiconductor saturable-absorber mirror-assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime," Opt. Lett. 24, 631-633 (1999).
    [CrossRef]
  5. I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, H. P. Jenssen, and R. Szipöcs, "14-fs pulse generation in Kerr-lens mode-locked prismless Cr:LiSGaF and Cr:LiSAF lasers: observation of pulse frequency shift," Opt. Lett. 22, 1716-1718 (1997).
    [CrossRef]
  6. Z. Zhang, K. Torizuka, T. Itatani, K. Kobayashi, T. Sugaya, and T. Nakagawa, "Femtosecond Cr:forsterite laser with modelocking initiated by a quantum well saturable absorber," IEEE J. Quantum Electron. 33, 1975-1981 (1997).
    [CrossRef]
  7. A. Sennaroglu, C. R. Pollock, and H. Nathel, "Continuous-wave self-mode-locked operation of a femtosecond Cr4+:YAG laser," Opt. Lett. 19, 390-392 (1994).
    [PubMed]
  8. D. J. Jones, L. E. Nelson, H. A. Haus, and E. P. Ippen, "Diode-pumped environmentally stable stretched-pulse fiber laser," IEEE J. Sel. Top. Quantum Electron. 3, 1076-1079 (1997).
    [CrossRef]
  9. M. E. Fermann, L. M. Yang, M. L. Stock, and M. J. Andrejco, "Environmentally stable Kerr-type mode-locked erbium fiber laser producing 360-fs pulses," Opt. Lett. 19, 43-45 (1994).
    [CrossRef] [PubMed]
  10. D. Kopf, F. X. Kärtner, and U. Keller, "Diode-pumped mode-locked Nd:glass lasers with an antiresonant Fabry-Perot saturable absorber," Opt. Lett. 20, 1169-1171 (1995).
    [CrossRef] [PubMed]
  11. W. Lu, L. Yan, and C. R. Menyuk, "Kerr-lens mode locking of Nd:glass laser," Opt. Commun. 200, 159-163 (2001).
    [CrossRef]
  12. E. P. Ippen, "Principles of passive mode locking," Appl. Phys. B 58, 159-170 (1994).
    [CrossRef]
  13. D. E. Spence, P. N. Kean, and W. Sibbett, "60-fsec pulse generation from a self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 42-44 (1991).
    [CrossRef] [PubMed]
  14. T. Brabec, Ch. Spielmann, and F. Krausz, "Modelocking in solitary lasers," Opt. Lett. 16, 1961-1963 (1991).
    [CrossRef] [PubMed]
  15. U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, "Semiconductor saturable absorber mirrors (SASEMs) for femtosecond to nanosecond pulse generation in solid-state lasers," IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
    [CrossRef]
  16. M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, "Characterization of ultrashort pulse formation in passively modelocked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
    [CrossRef]
  17. H. A. Haus, "Theory of mode-locking with a slow saturable absorber," J. Appl. Phys. 46, 3049-3058 (1975).
    [CrossRef]
  18. H. A. Haus, "Theory of mode-locking with a fast saturable absorber," IEEE J. Quantum Electron. 11, 736-746 (1975).
    [CrossRef]
  19. O. E. Martinez, R. L. Fork, and J. P. Gordon, "Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion," Opt. Lett. 9, 156-158 (1984).
    [CrossRef] [PubMed]
  20. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Structures for additive pulse mode locking," J. Opt. Soc. Am. B 8, 2068-2076 (1991).
    [CrossRef]
  21. F. Krausz, T. Brabec, and Ch. Spielmann, "Self-starting passive mode locking," Opt. Lett. 16, 235-237 (1991).
    [CrossRef] [PubMed]
  22. B. Braun, K. J. Weingarten, F. X. Kärtner, and U. Keller, "Continuous-wave mode-locked solid-state lasers with enhanced spatial hole burning, part I: experiments," Appl. Phys. B 61, 429-437 (1995).
    [CrossRef]
  23. F. X. Kärtner, B. Braun, and U. Keller, "Continuous-wave mode-locked solid-state lasers with enhanced spatial hole burning, part II: theory," Appl. Phys. B 61, 569-579 (1995).
    [CrossRef]
  24. R. Paschotta, J. Aus der Au, G. J. Spühler, S. Erhard, A. Giesen, and U. Keller, "Passive mode locking of thin-disk lasers: effects of spatial hole burning," Appl. Phys. B 72, 267-278 (2001).
    [CrossRef]
  25. L. Yan, "Pulse coherence of actively mode-locked inhomogeneously broadened lasers," Opt. Commun. 162, 75-78 (1999).
    [CrossRef]
  26. W. Lu, L. Yan, and C. R. Menyuk, "Dispersion effects in an actively mode-locked inhomogeneously broadened laser," IEEE J. Quantum Electron. 38, 1317-1324 (2002).
    [CrossRef]
  27. L. Yan, W. Lu, and S. Han, "Soliton stability in inhomogeneously broadened lasers," in Proceedings of the Conference on Lasers and Electro-Optics Pacific Rim (Optical Society of America, 2004), paper ThP0899.
  28. W. Lu and L. Yan, "Resonant dispersion effect on soliton stability in actively mode-locked inhomogeneously broadened lasers," in Proceedings of IEEE Lasers and Electro-Optics Society Annual Meeting (IEEE, 2004), paper WR5.
  29. H. A. Haus, "Parameter ranges for CW passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
    [CrossRef]
  30. C. Hönninger, R. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, "Q-switching stability limits of continuous-wave passive mode locking," J. Opt. Soc. Am. B 16, 46-56 (1999).
    [CrossRef]
  31. T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, "Suppression of Q-switched mode locking and break-up into multiple pulses by inverse saturable absorption," Appl. Phys. B 70, s41-s49 (2000).
    [CrossRef]
  32. L. Yan, "Continuous-wave lasing of hybrid lasers," IEEE J. Quantum Electron. 33, 1075-1083 (1997).
    [CrossRef]
  33. D. W. Hall and W. J. Weber, "Fluorescence line narrowing in neodymium laser glasses," J. Appl. Phys. 55, 2642-2647 (1984).
    [CrossRef]
  34. C. R. Menyuk, "Why are solitons robust in optical fibers?" in Guided Wave Nonlinear Optics, D.Ostrowsky and R.Reinisch, eds. (Kluwer, 1992), pp. 457-488.

2002 (1)

W. Lu, L. Yan, and C. R. Menyuk, "Dispersion effects in an actively mode-locked inhomogeneously broadened laser," IEEE J. Quantum Electron. 38, 1317-1324 (2002).
[CrossRef]

2001 (2)

R. Paschotta, J. Aus der Au, G. J. Spühler, S. Erhard, A. Giesen, and U. Keller, "Passive mode locking of thin-disk lasers: effects of spatial hole burning," Appl. Phys. B 72, 267-278 (2001).
[CrossRef]

W. Lu, L. Yan, and C. R. Menyuk, "Kerr-lens mode locking of Nd:glass laser," Opt. Commun. 200, 159-163 (2001).
[CrossRef]

2000 (1)

T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, "Suppression of Q-switched mode locking and break-up into multiple pulses by inverse saturable absorption," Appl. Phys. B 70, s41-s49 (2000).
[CrossRef]

1999 (4)

1997 (4)

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, H. P. Jenssen, and R. Szipöcs, "14-fs pulse generation in Kerr-lens mode-locked prismless Cr:LiSGaF and Cr:LiSAF lasers: observation of pulse frequency shift," Opt. Lett. 22, 1716-1718 (1997).
[CrossRef]

L. Yan, "Continuous-wave lasing of hybrid lasers," IEEE J. Quantum Electron. 33, 1075-1083 (1997).
[CrossRef]

Z. Zhang, K. Torizuka, T. Itatani, K. Kobayashi, T. Sugaya, and T. Nakagawa, "Femtosecond Cr:forsterite laser with modelocking initiated by a quantum well saturable absorber," IEEE J. Quantum Electron. 33, 1975-1981 (1997).
[CrossRef]

D. J. Jones, L. E. Nelson, H. A. Haus, and E. P. Ippen, "Diode-pumped environmentally stable stretched-pulse fiber laser," IEEE J. Sel. Top. Quantum Electron. 3, 1076-1079 (1997).
[CrossRef]

1996 (1)

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, "Semiconductor saturable absorber mirrors (SASEMs) for femtosecond to nanosecond pulse generation in solid-state lasers," IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

1995 (4)

B. Braun, K. J. Weingarten, F. X. Kärtner, and U. Keller, "Continuous-wave mode-locked solid-state lasers with enhanced spatial hole burning, part I: experiments," Appl. Phys. B 61, 429-437 (1995).
[CrossRef]

F. X. Kärtner, B. Braun, and U. Keller, "Continuous-wave mode-locked solid-state lasers with enhanced spatial hole burning, part II: theory," Appl. Phys. B 61, 569-579 (1995).
[CrossRef]

A. Stingl, M. Lenzner, C. Spielmann, F. Krausz, and R. Szipöcs, "Sub-10-fs mirror-dispersion-controlled Ti:sapphire laser," Opt. Lett. 20, 602-604 (1995).
[CrossRef] [PubMed]

D. Kopf, F. X. Kärtner, and U. Keller, "Diode-pumped mode-locked Nd:glass lasers with an antiresonant Fabry-Perot saturable absorber," Opt. Lett. 20, 1169-1171 (1995).
[CrossRef] [PubMed]

1994 (4)

1992 (1)

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, "Characterization of ultrashort pulse formation in passively modelocked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
[CrossRef]

1991 (4)

1984 (2)

1976 (1)

H. A. Haus, "Parameter ranges for CW passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

1975 (2)

H. A. Haus, "Theory of mode-locking with a slow saturable absorber," J. Appl. Phys. 46, 3049-3058 (1975).
[CrossRef]

H. A. Haus, "Theory of mode-locking with a fast saturable absorber," IEEE J. Quantum Electron. 11, 736-746 (1975).
[CrossRef]

Appl. Phys. B (5)

E. P. Ippen, "Principles of passive mode locking," Appl. Phys. B 58, 159-170 (1994).
[CrossRef]

B. Braun, K. J. Weingarten, F. X. Kärtner, and U. Keller, "Continuous-wave mode-locked solid-state lasers with enhanced spatial hole burning, part I: experiments," Appl. Phys. B 61, 429-437 (1995).
[CrossRef]

F. X. Kärtner, B. Braun, and U. Keller, "Continuous-wave mode-locked solid-state lasers with enhanced spatial hole burning, part II: theory," Appl. Phys. B 61, 569-579 (1995).
[CrossRef]

R. Paschotta, J. Aus der Au, G. J. Spühler, S. Erhard, A. Giesen, and U. Keller, "Passive mode locking of thin-disk lasers: effects of spatial hole burning," Appl. Phys. B 72, 267-278 (2001).
[CrossRef]

T. R. Schibli, E. R. Thoen, F. X. Kärtner, and E. P. Ippen, "Suppression of Q-switched mode locking and break-up into multiple pulses by inverse saturable absorption," Appl. Phys. B 70, s41-s49 (2000).
[CrossRef]

IEEE J. Quantum Electron. (6)

L. Yan, "Continuous-wave lasing of hybrid lasers," IEEE J. Quantum Electron. 33, 1075-1083 (1997).
[CrossRef]

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, "Characterization of ultrashort pulse formation in passively modelocked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
[CrossRef]

W. Lu, L. Yan, and C. R. Menyuk, "Dispersion effects in an actively mode-locked inhomogeneously broadened laser," IEEE J. Quantum Electron. 38, 1317-1324 (2002).
[CrossRef]

Z. Zhang, K. Torizuka, T. Itatani, K. Kobayashi, T. Sugaya, and T. Nakagawa, "Femtosecond Cr:forsterite laser with modelocking initiated by a quantum well saturable absorber," IEEE J. Quantum Electron. 33, 1975-1981 (1997).
[CrossRef]

H. A. Haus, "Theory of mode-locking with a fast saturable absorber," IEEE J. Quantum Electron. 11, 736-746 (1975).
[CrossRef]

H. A. Haus, "Parameter ranges for CW passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

D. J. Jones, L. E. Nelson, H. A. Haus, and E. P. Ippen, "Diode-pumped environmentally stable stretched-pulse fiber laser," IEEE J. Sel. Top. Quantum Electron. 3, 1076-1079 (1997).
[CrossRef]

U. Keller, K. J. Weingarten, F. X. Kärtner, D. Kopf, B. Braun, I. D. Jung, R. Fluck, C. Hönninger, N. Matuschek, and J. Aus der Au, "Semiconductor saturable absorber mirrors (SASEMs) for femtosecond to nanosecond pulse generation in solid-state lasers," IEEE J. Sel. Top. Quantum Electron. 2, 435-453 (1996).
[CrossRef]

J. Appl. Phys. (2)

H. A. Haus, "Theory of mode-locking with a slow saturable absorber," J. Appl. Phys. 46, 3049-3058 (1975).
[CrossRef]

D. W. Hall and W. J. Weber, "Fluorescence line narrowing in neodymium laser glasses," J. Appl. Phys. 55, 2642-2647 (1984).
[CrossRef]

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

Opt. Commun. (2)

W. Lu, L. Yan, and C. R. Menyuk, "Kerr-lens mode locking of Nd:glass laser," Opt. Commun. 200, 159-163 (2001).
[CrossRef]

L. Yan, "Pulse coherence of actively mode-locked inhomogeneously broadened lasers," Opt. Commun. 162, 75-78 (1999).
[CrossRef]

Opt. Lett. (12)

O. E. Martinez, R. L. Fork, and J. P. Gordon, "Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion," Opt. Lett. 9, 156-158 (1984).
[CrossRef] [PubMed]

D. E. Spence, P. N. Kean, and W. Sibbett, "60-fsec pulse generation from a self-mode-locked Ti:sapphire laser," Opt. Lett. 16, 42-44 (1991).
[CrossRef] [PubMed]

F. Krausz, T. Brabec, and Ch. Spielmann, "Self-starting passive mode locking," Opt. Lett. 16, 235-237 (1991).
[CrossRef] [PubMed]

T. Brabec, Ch. Spielmann, and F. Krausz, "Modelocking in solitary lasers," Opt. Lett. 16, 1961-1963 (1991).
[CrossRef] [PubMed]

M. E. Fermann, L. M. Yang, M. L. Stock, and M. J. Andrejco, "Environmentally stable Kerr-type mode-locked erbium fiber laser producing 360-fs pulses," Opt. Lett. 19, 43-45 (1994).
[CrossRef] [PubMed]

A. Sennaroglu, C. R. Pollock, and H. Nathel, "Continuous-wave self-mode-locked operation of a femtosecond Cr4+:YAG laser," Opt. Lett. 19, 390-392 (1994).
[PubMed]

J. Zhou, G. Taft, C. P. Huang, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, "Pulse evolution in a broad-bandwidth Ti:sapphire laser," Opt. Lett. 19, 1149-1151 (1994).
[PubMed]

A. Stingl, M. Lenzner, C. Spielmann, F. Krausz, and R. Szipöcs, "Sub-10-fs mirror-dispersion-controlled Ti:sapphire laser," Opt. Lett. 20, 602-604 (1995).
[CrossRef] [PubMed]

D. Kopf, F. X. Kärtner, and U. Keller, "Diode-pumped mode-locked Nd:glass lasers with an antiresonant Fabry-Perot saturable absorber," Opt. Lett. 20, 1169-1171 (1995).
[CrossRef] [PubMed]

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, H. P. Jenssen, and R. Szipöcs, "14-fs pulse generation in Kerr-lens mode-locked prismless Cr:LiSGaF and Cr:LiSAF lasers: observation of pulse frequency shift," Opt. Lett. 22, 1716-1718 (1997).
[CrossRef]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, "Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser," Opt. Lett. 24, 411-413 (1999).
[CrossRef]

D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, "Semiconductor saturable-absorber mirror-assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime," Opt. Lett. 24, 631-633 (1999).
[CrossRef]

Other (3)

C. R. Menyuk, "Why are solitons robust in optical fibers?" in Guided Wave Nonlinear Optics, D.Ostrowsky and R.Reinisch, eds. (Kluwer, 1992), pp. 457-488.

L. Yan, W. Lu, and S. Han, "Soliton stability in inhomogeneously broadened lasers," in Proceedings of the Conference on Lasers and Electro-Optics Pacific Rim (Optical Society of America, 2004), paper ThP0899.

W. Lu and L. Yan, "Resonant dispersion effect on soliton stability in actively mode-locked inhomogeneously broadened lasers," in Proceedings of IEEE Lasers and Electro-Optics Society Annual Meeting (IEEE, 2004), paper WR5.

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Figures (10)

Fig. 1
Fig. 1

Parameters of stably mode-locked pulse from an inhomogeneously broadened laser ( M = 10 ) and a purely homogeneously broadened laser. (a) Pulse bandwidth ( Δ ν p Δ ν g 0 ) , (b) Pulse width ( τ p Δ ν g 0 ) , and (c) chirp parameter.

Fig. 2
Fig. 2

Spectra of the saturated gain of an inhomogeneously broadened laser ( M = 10 ) in different mode-locking regimes. (a) Pure SAM mode locking, (b) soliton mode locking, (c) mode locking with SPM and positive GDD.

Fig. 3
Fig. 3

Temporal shape and spectrum when the pure SAM mode locking fails, GDD = 120 fs 2 . (a) Temporal shape, (b) spectra of individual pulses; the wider spectrum is at GDD = 80 fs 2 , before mode locking fails. P ave P s , abs = 1 × 10 4 .

Fig. 4
Fig. 4

Temporal shape and spectrum when the soliton mode locking fails, GDD = 1220 fs 2 . (a) Temporal shape, (b) spectrum. P ave P s , abs = 1 × 10 2 .

Fig. 5
Fig. 5

Dependence of pulse bandwidth ( Δ ν p Δ ν g 0 ) on absorber’s saturation level for an inhomogeneously broadened laser ( M = 10 ) : (a) pure SAM mode locking with GDD = 30 fs 2 , (b) soliton mode locking with GDD = 900 fs 2 , (c) mode locking with presence of SPM at GDD = 310 fs 2 .

Fig. 6
Fig. 6

Pulse bandwidth ( Δ ν p Δ ν g 0 ) versus GDD for an inhomogeneously broadened laser ( M = 10 ) with different unsaturated absorber loss. (a) Pure SAM mode locking, P ave P s , abs = 1 × 10 4 , (b) mode locking with SPM, P ave P s , abs = 1 × 10 2 .

Fig. 7
Fig. 7

GDD range for stable mode locking of an inhomogeneously broadened laser ( M = 10 ) under different unsaturated absorber loss. P ave P s , abs = 1 × 10 2 .

Fig. 8
Fig. 8

Scaling of pulse bandwidth with the gain linewidth ( M = 10 fixed), GDD, and Kerr nonlinearity: (a) pure SAM mode locking, (b) mode locking with presence of SPM and positive GDD, (c) soliton mode locking. Corresponding values of Δ ν g 0 and κ spm are used in the normalizations. For (a) P ave P s , abs = 2 × 10 4 , 1 × 10 4 , and 0.5 × 10 4 for Δ ν g 0 = 1500 , 3000, and 6000 GHz , respectively. For (b) P ave P s , abs = 1 × 10 3 ; for (c) P ave P s , abs = 1 × 10 2 .

Fig. 9
Fig. 9

Normalized GDD range for stable mode locking of an inhomogeneously broadened laser ( M = 10 ) under different unsaturated absorber loss for two gain linewidths. For soliton mode locking, P ave P s , abs = 1 × 10 2 ; for mode locking with presence of SPM and positive GDD, P ave P s , abs = 1 × 10 2 (optimal) for Δ ν g 0 = 3000 GHz and P ave P s , abs = 1 × 10 3 (optimal) for Δ ν g 0 = 6000 GHz .

Fig. 10
Fig. 10

Stable mode-locking regions for inhomogeneously broadened lasers with different degree of inhomogeneity: M = (a) 10, (b) 4, (c) 2, (d) 1.

Equations (9)

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g ( ν n ) = ν ξ g ( ν ξ , ν n ) = ν ξ N ( ν ξ ) σ ( ν ξ , ν n ) L m ,
σ ( ν ξ , ν n ) = σ 0 g ¯ ( ν n ν ξ ) ,
g ¯ ( ν n ν ξ ) = 1 1 + 4 ( ν n ν ξ Δ ν h ) 2 ,
g ( ν ξ , ν n ) = g ( ν 0 ) η 0 p ( ν ξ ) d ν ξ g ¯ ( ν n ν ξ ) 1 + ν n P ( ν n ) P s g ¯ ( ν n ν ξ ) ,
E ( t ) = Re [ E ̃ ( t ) e j ω 0 t ] .
T R d E ̃ n d T = δ c 2 E ̃ n + g ( ν n ) 2 E ̃ n j 1 2 D ( ω n ω 0 ) 2 E ̃ n ,
Δ E ̃ ( T , t ) = [ exp [ δ A ( t ) 2 ] 1 ] E ̃ ( T , t ) + [ exp ( j κ spm E ̃ ( T , t ) 2 1 ) ] E ̃ ( T , t ) .
δ A ( t ) = δ a 0 1 + E ̃ ( t ) 2 P s , abs ,
exp [ j β ln E ̃ ( t ) E ̃ ( 0 ) ] .

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