Abstract

A theoretical model that characterizes the physical process responsible for generating ultrashort, high-energy, mode-locked pulses in a normal-dispersion laser cavity with strong spectral filtering is developed. According to this model, two of the critical physical parameters used to achieve optimal performance are the ratio of the filter bandwidth to the gain bandwidth and the placement of the output coupler in the laser cavity. The spectral filtering plays a crucial role in maintaining a short pulse duration with high energy. This phenomenon is generic to mode locking with normal dispersion.

© 2008 Optical Society of America

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References

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  1. H. Haus, IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000).
    [CrossRef]
  2. I. N. Duling III and M. L. Dennis, Compact Sources of Ultrashort Pulses (Cambridge, 1995).
    [CrossRef]
  3. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys. Rev. Lett. 92, 213902 (2004).
    [CrossRef] [PubMed]
  4. A. Fernandez, T. Fuji, A. Poppe, A. Fürbach, F. Krausz, and A. Apolonski, Opt. Lett. 29, 1366 (2004).
    [CrossRef] [PubMed]
  5. A. Chong, J. Buckley, W. Renninger, and F. Wise, Opt. Express 14, 10095 (2006).
    [CrossRef] [PubMed]
  6. V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, Appl. Phys. B 83, 503 (2006).
    [CrossRef]
  7. P. A. Belanger, Opt. Express 14, 12174 (2006).
    [CrossRef] [PubMed]
  8. P. A. Belanger, Opt. Express 15, 11033 (2007).
    [CrossRef] [PubMed]
  9. J. N. Kutz and B. Sandstede, Opt. Express 16, 636 (2008).
    [CrossRef] [PubMed]
  10. J. Proctor and J. N. Kutz, J. Math. Comp. Sim. 74, 333 (2007).
    [CrossRef]
  11. D. Anderson, M. Lisak, and A. Berntson, Pramana, J. Phys. 57, 917 (2001).
    [CrossRef]
  12. B. Bale and J. N. Kutz, J. Opt. Soc. Am. B , submitted for publication.

2008 (1)

2007 (2)

P. A. Belanger, Opt. Express 15, 11033 (2007).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, J. Math. Comp. Sim. 74, 333 (2007).
[CrossRef]

2006 (3)

2004 (2)

A. Fernandez, T. Fuji, A. Poppe, A. Fürbach, F. Krausz, and A. Apolonski, Opt. Lett. 29, 1366 (2004).
[CrossRef] [PubMed]

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

2001 (1)

D. Anderson, M. Lisak, and A. Berntson, Pramana, J. Phys. 57, 917 (2001).
[CrossRef]

2000 (1)

H. Haus, IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000).
[CrossRef]

Anderson, D.

D. Anderson, M. Lisak, and A. Berntson, Pramana, J. Phys. 57, 917 (2001).
[CrossRef]

Apolonski, A.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, Appl. Phys. B 83, 503 (2006).
[CrossRef]

A. Fernandez, T. Fuji, A. Poppe, A. Fürbach, F. Krausz, and A. Apolonski, Opt. Lett. 29, 1366 (2004).
[CrossRef] [PubMed]

Bale, B.

B. Bale and J. N. Kutz, J. Opt. Soc. Am. B , submitted for publication.

Belanger, P. A.

Berntson, A.

D. Anderson, M. Lisak, and A. Berntson, Pramana, J. Phys. 57, 917 (2001).
[CrossRef]

Buckley, J.

Buckley, J. R.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Chernykh, A.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, Appl. Phys. B 83, 503 (2006).
[CrossRef]

Chong, A.

Clark, W. G.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Dennis, M. L.

I. N. Duling III and M. L. Dennis, Compact Sources of Ultrashort Pulses (Cambridge, 1995).
[CrossRef]

Duling, I. N.

I. N. Duling III and M. L. Dennis, Compact Sources of Ultrashort Pulses (Cambridge, 1995).
[CrossRef]

Fernandez, A.

Fuji, T.

Fürbach, A.

Haus, H.

H. Haus, IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000).
[CrossRef]

Ilday, F. O.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Kalashnikov, V. L.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, Appl. Phys. B 83, 503 (2006).
[CrossRef]

Krausz, F.

Kutz, J. N.

J. N. Kutz and B. Sandstede, Opt. Express 16, 636 (2008).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, J. Math. Comp. Sim. 74, 333 (2007).
[CrossRef]

B. Bale and J. N. Kutz, J. Opt. Soc. Am. B , submitted for publication.

Lisak, M.

D. Anderson, M. Lisak, and A. Berntson, Pramana, J. Phys. 57, 917 (2001).
[CrossRef]

Podivilov, E.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, Appl. Phys. B 83, 503 (2006).
[CrossRef]

Poppe, A.

Proctor, J.

J. Proctor and J. N. Kutz, J. Math. Comp. Sim. 74, 333 (2007).
[CrossRef]

Renninger, W.

Sandstede, B.

Wise, F.

Wise, F. W.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Appl. Phys. B (1)

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, Appl. Phys. B 83, 503 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

H. Haus, IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000).
[CrossRef]

J. Math. Comp. Sim. (1)

J. Proctor and J. N. Kutz, J. Math. Comp. Sim. 74, 333 (2007).
[CrossRef]

J. Opt. Soc. Am. B (1)

B. Bale and J. N. Kutz, J. Opt. Soc. Am. B , submitted for publication.

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Pramana, J. Phys. (1)

D. Anderson, M. Lisak, and A. Berntson, Pramana, J. Phys. 57, 917 (2001).
[CrossRef]

Other (1)

I. N. Duling III and M. L. Dennis, Compact Sources of Ultrashort Pulses (Cambridge, 1995).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Values of η 2 , A, and ω versus R after filtering has been applied to the fixed-point solution. (b) Phase-plane projection in the ( η 2 , A ) and ( η 2 , ω ) planes depicting the jump condition from the fixed-point (solid circle) to the post-filtered pulse (gray square, circle, and triangle), followed by the evolution along the flow lines back to the fixed point. The gray square, circle, and triangle in (b) correspond to R = 0.06 , 0.26, and 0.66 in (a). Note that, for the specific parameters chosen, the gray triangles ( R = 0.26 ) allow for the highest intensities.

Fig. 2
Fig. 2

Stable and robust mode locking with spectral filtering starting from initial white noise. (a) Evolution of intensity to the optimal mode-locked solution. The inset demonstrates the increased mode-locked intensity with filter (solid curve) in comparison with the mode-locking without filter (dotted curve). (b) and (c) Intensity over three periods for the full model Eq. (1) and variational model Eq. (3), respectively, with filter (dotted line) and output coupler (gray area).

Fig. 3
Fig. 3

Geometric representation of the mode-locking operation. The jump condition associated with spectral filtering on the parameters η, A, and ω is demonstrated with the gray triangles (before/after, −/+, filtering). The output coupler (OC) is placed in the optimal location where the intensity ( η 2 ) is maximized. The flow lines determine the path from the post-filtered location to the fixed point (solid circle). The filter application after the OC returns the pulse to the +gray triangle. The gray curves correspond to the full model Eq. (1), illustrating good agreement with the geometric description of mode locking.

Equations (7)

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i u Z + D 2 2 u T 2 + α u 2 u + C v i g ( Z ) ( 1 + τ 2 T 2 ) u = 0 ,
i v Z + C ( u + w ) + i γ 1 v = 0 ,
i w Z + C v + i γ 2 w = 0 ,
u ( Z , T ) = η sech ( ω T ) 1 + i A e i θ Z ,
d η d Z = 1 3 D η ω 2 A 1 9 τ g η ω 2 ( A 2 + 7 ) + Γ l η ,
d A d Z = 2 3 α η 2 2 3 ω 2 ( 1 + A 2 ) ( D + 2 τ g A ) ,
d ω d Z = 2 3 ω 3 ( D A + 2 3 τ g ( A 2 2 ) ) ,

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