Abstract

The stability of passively mode-locked fiber lasers with fast saturable absorption is examined by a direct numerical approach. The laser operation is described by use of a standard model, closely related to the Ginzburg–Landau equation, which is valid when the change of the laser pulse during one round trip through the laser is small. This equation is then linearized around an equilibrium solution, and the eigenmodes and eigenvalues of the linearized equation are determined numerically. This approach retains the advantage of previous analytical research of permitting one to explore a wide range of parameter space while also permitting one to avoid approximations in the stability calculation. The stability of a figure-eight loop laser with the amplifier in the external loop is then studied by use of this approach.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See, for example, A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 28, and references therein.
  2. E. P. Ippen, H. A. Haus, L. Y. Liu, J. Opt. Soc. Am. B 6,1736 (1989); H. A. Haus, J. G. Fujimoto, E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).
    [Crossref]
  3. I. N. Duling, Electron. Lett. 27, 544 (1991); D. J. Richardson, R. I. Laming, D. N. Payne, V. Matsas, M. W. Phillips, Electron. Lett. 27, 542 (1991); N. Pandit, D. V. Noske, S. M. J. Kelley, J. R. Taylor, Electron. Lett. 28, 455 (1992); M. Nakazawa, E. Yoshida, Y. Kimura, Appl. Phys. Lett. 59, 2073 (1991).
    [Crossref]
  4. S. Wu, J. Strait, R. L. Fork, T. F. Morse, Opt. Lett. 18, 1444 (1993).
    [Crossref] [PubMed]
  5. C.-J. Chen, P. K A. Wai, C. R. Menyuk, Opt. Lett. 17, 417 (1992); V. J. Matsas, T. P. Newton, D. J. Richardson, D. N. Payne, Electron. Lett. 28, 1391 (1992); D. V. Noske, N. Pandit, J. R. Taylor, Electron. Lett. 28, 2185 (1992); K. Tamura, H. A. Haus, E. P. Ippen, Electron. Lett. 28, 2226 (1992); M. Nakazawa, E. Yashida, T. Sugawa, Y. Kimura, Electron. Lett. 29, 1327 (1993).
    [Crossref] [PubMed]
  6. H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
    [Crossref]
  7. W. H. Press, B. P. Flannery, S. A. Teukosky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chaps. 2 and 11.
  8. P. A. Bélanger, J. Opt. Soc. Am. B 8, 2077 (1991); O. E. Martinez, R. L. Fork, J. P. Gordon, J. Opt. Soc. Am. B 2, 753 (1985);N. R. Pereira, L. Stenflo, Phys. Fluids 20, 1733 (1977).
    [Crossref]
  9. See, for example, D. Zwillinger, Handbook of Differential Equations (Academic, San Diego, Calif., 1989), p. 133, and references therein.
  10. S. M. J. Kelley, Electron. Lett. 28, 806 (1992); M. L. Dennis, I. N. Duling, Appl. Phys. Lett. 62, 2911 (1993). The mechanism is that dispersive waves grow at the expense of the mode-locked pulse when they are phase matched. These waves are then absorbed by the saturable absorber. The rate at which these waves grow increases rapidly with the pulse’s bandwidth; thus the pulse’s loss is bandwidth dependent.
    [Crossref]

1993 (1)

1992 (2)

C.-J. Chen, P. K A. Wai, C. R. Menyuk, Opt. Lett. 17, 417 (1992); V. J. Matsas, T. P. Newton, D. J. Richardson, D. N. Payne, Electron. Lett. 28, 1391 (1992); D. V. Noske, N. Pandit, J. R. Taylor, Electron. Lett. 28, 2185 (1992); K. Tamura, H. A. Haus, E. P. Ippen, Electron. Lett. 28, 2226 (1992); M. Nakazawa, E. Yashida, T. Sugawa, Y. Kimura, Electron. Lett. 29, 1327 (1993).
[Crossref] [PubMed]

S. M. J. Kelley, Electron. Lett. 28, 806 (1992); M. L. Dennis, I. N. Duling, Appl. Phys. Lett. 62, 2911 (1993). The mechanism is that dispersive waves grow at the expense of the mode-locked pulse when they are phase matched. These waves are then absorbed by the saturable absorber. The rate at which these waves grow increases rapidly with the pulse’s bandwidth; thus the pulse’s loss is bandwidth dependent.
[Crossref]

1991 (2)

P. A. Bélanger, J. Opt. Soc. Am. B 8, 2077 (1991); O. E. Martinez, R. L. Fork, J. P. Gordon, J. Opt. Soc. Am. B 2, 753 (1985);N. R. Pereira, L. Stenflo, Phys. Fluids 20, 1733 (1977).
[Crossref]

I. N. Duling, Electron. Lett. 27, 544 (1991); D. J. Richardson, R. I. Laming, D. N. Payne, V. Matsas, M. W. Phillips, Electron. Lett. 27, 542 (1991); N. Pandit, D. V. Noske, S. M. J. Kelley, J. R. Taylor, Electron. Lett. 28, 455 (1992); M. Nakazawa, E. Yoshida, Y. Kimura, Appl. Phys. Lett. 59, 2073 (1991).
[Crossref]

1989 (1)

1975 (1)

H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
[Crossref]

Bélanger, P. A.

Chen, C.-J.

Duling, I. N.

I. N. Duling, Electron. Lett. 27, 544 (1991); D. J. Richardson, R. I. Laming, D. N. Payne, V. Matsas, M. W. Phillips, Electron. Lett. 27, 542 (1991); N. Pandit, D. V. Noske, S. M. J. Kelley, J. R. Taylor, Electron. Lett. 28, 455 (1992); M. Nakazawa, E. Yoshida, Y. Kimura, Appl. Phys. Lett. 59, 2073 (1991).
[Crossref]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukosky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chaps. 2 and 11.

Fork, R. L.

Haus, H. A.

Ippen, E. P.

Kelley, S. M. J.

S. M. J. Kelley, Electron. Lett. 28, 806 (1992); M. L. Dennis, I. N. Duling, Appl. Phys. Lett. 62, 2911 (1993). The mechanism is that dispersive waves grow at the expense of the mode-locked pulse when they are phase matched. These waves are then absorbed by the saturable absorber. The rate at which these waves grow increases rapidly with the pulse’s bandwidth; thus the pulse’s loss is bandwidth dependent.
[Crossref]

Liu, L. Y.

Menyuk, C. R.

Morse, T. F.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukosky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chaps. 2 and 11.

Siegman, A. E.

See, for example, A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 28, and references therein.

Strait, J.

Teukosky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukosky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chaps. 2 and 11.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukosky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chaps. 2 and 11.

Wai, P. K A.

Wu, S.

Zwillinger, D.

See, for example, D. Zwillinger, Handbook of Differential Equations (Academic, San Diego, Calif., 1989), p. 133, and references therein.

Electron. Lett. (2)

I. N. Duling, Electron. Lett. 27, 544 (1991); D. J. Richardson, R. I. Laming, D. N. Payne, V. Matsas, M. W. Phillips, Electron. Lett. 27, 542 (1991); N. Pandit, D. V. Noske, S. M. J. Kelley, J. R. Taylor, Electron. Lett. 28, 455 (1992); M. Nakazawa, E. Yoshida, Y. Kimura, Appl. Phys. Lett. 59, 2073 (1991).
[Crossref]

S. M. J. Kelley, Electron. Lett. 28, 806 (1992); M. L. Dennis, I. N. Duling, Appl. Phys. Lett. 62, 2911 (1993). The mechanism is that dispersive waves grow at the expense of the mode-locked pulse when they are phase matched. These waves are then absorbed by the saturable absorber. The rate at which these waves grow increases rapidly with the pulse’s bandwidth; thus the pulse’s loss is bandwidth dependent.
[Crossref]

J. Appl. Phys. (1)

H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
[Crossref]

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

Opt. Lett. (2)

Other (3)

W. H. Press, B. P. Flannery, S. A. Teukosky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chaps. 2 and 11.

See, for example, D. Zwillinger, Handbook of Differential Equations (Academic, San Diego, Calif., 1989), p. 133, and references therein.

See, for example, A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 28, and references therein.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Eigenvalues of the linearized equation are shown in the complex λ plane when β = 0.4 and a = 0.52. The two solid lines correspond to radiation modes, and the circles correspond to discrete modes. The filled circles correspond to the case without gain saturation (p = 0). With gain saturation (p = 0.1) the unstable mode becomes stable, as shown by the arrow.

Fig. 2
Fig. 2

Stability diagram of the minimum p value that can stabilize the laser with anomalous intracavity dispersion. Normal intracavity dispersion corresponds to setting β → −β. The quantity β is the equilibrium solution’s chirp parameter, and the quantity a = B/2|D| measures the relative importance of the frequency limiter and dispersion. As a increases, the amount of chirp required for stabilizing the pulse decreases, but pump power (which determines p) and Γ/K (which determines the chirp) must increase. Note that determining the stability requires only the normalized parameters β, a, and ρ—a key advantage of the approach that we are using.

Equations (3)

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

U z = ( g l + i θ ) U + ( B + i D ) 2 U t 2 + ( Γ + i K ) | U | 2 U = 0
U 0 = A sech 1 + i β ( t / τ ) exp ( i ψ z ) ,
u ˜ ξ = ( a ± i 2 ) { b 1 u ˜ + 2 u ˜ s 2 b 2 [ 2 sech 2 ( s ) u ˜ + sech 2 + 2 i β ( s ) υ ˜ ] } p sech 1 + i β ( s ) [ sech 1 i β ( s ) u ˜ + sech 1 + i β ( s ) υ ˜ ] d s , υ ˜ ξ = ( a i 2 ) { b 1 * υ ˜ + 2 υ ˜ s 2 b 2 * [ sech 2 2 i β ( s ) u ˜ + 2 sech 2 ( s ) υ ˜ ] } p sech 1 i β ( s ) [ sech 1 i β ( s ) u ˜ + sech 1 + i β ( s ) υ ˜ ] d s ,

Metrics