Abstract

We propose a new model for passive mode locking that is a set of ordinary delay differential equations. We assume a ring-cavity geometry and Lorentzian spectral filtering of the pulses but do not use small gain and loss and weak saturation approximations. By means of a continuation method, we study mode-locking solutions and their stability. We find that stable mode locking can exist even when the nonlasing state between pulses becomes unstable.

© 2004 Optical Society of America

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References

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  1. P. Vasil'ev, Ultrafast Diode Lasers: Fundamentals and Applications (Artech House, Norwood, Mass., 1995).
  2. G. H. C. New, IEEE J. Quantum Electron. QE-10, 115 (1974).
    [CrossRef]
  3. H. Haus, IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000).
    [CrossRef]
  4. E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, IEE Proc. Optoelectron. 147, 251 (2000).
    [CrossRef]
  5. H. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
    [CrossRef]
  6. H. Haus, Jpn. J. Appl. Phys. 20, 1007 (1981).
    [CrossRef]
  7. H. A. Haus, C. V. Shank, and E. P. Ippen, Opt. Commun. 15, 29 (1975).
    [CrossRef]
  8. J. C. Chen, H. A. Haus, and E. P. Ippen, IEEE J. Quantum Electron. 29, 1228 (1993).
    [CrossRef]
  9. V. B. Khalfin, J. M. Arnold, and J. H. Marsh, IEEE J. Sel. Top. Quantum Electron. 1, 523 (1995).
    [CrossRef]
  10. S. L. McCall and P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
    [CrossRef]
  11. K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL v.2.00:a Matlab package for bifurcation analysis of delay differential equations,” Tech. Rep. TW-330 (Department of Computer Science, Katholicke Universitet Leuven, Leuven, Belgium, 2001).
  12. J. Palaski and K. Y. Lau, Appl. Phys. Lett. 59, 7 (1991).
    [CrossRef]
  13. T. Erneux, J. Opt. Soc. Am. B 5, 1063 (1988).
    [CrossRef]
  14. R. Paschotta and U. Keller, Appl. Phys. B 73, 653 (2001).
    [CrossRef]

2001

K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL v.2.00:a Matlab package for bifurcation analysis of delay differential equations,” Tech. Rep. TW-330 (Department of Computer Science, Katholicke Universitet Leuven, Leuven, Belgium, 2001).

R. Paschotta and U. Keller, Appl. Phys. B 73, 653 (2001).
[CrossRef]

2000

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

E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, IEE Proc. Optoelectron. 147, 251 (2000).
[CrossRef]

1995

P. Vasil'ev, Ultrafast Diode Lasers: Fundamentals and Applications (Artech House, Norwood, Mass., 1995).

V. B. Khalfin, J. M. Arnold, and J. H. Marsh, IEEE J. Sel. Top. Quantum Electron. 1, 523 (1995).
[CrossRef]

1993

J. C. Chen, H. A. Haus, and E. P. Ippen, IEEE J. Quantum Electron. 29, 1228 (1993).
[CrossRef]

1991

J. Palaski and K. Y. Lau, Appl. Phys. Lett. 59, 7 (1991).
[CrossRef]

1988

1985

S. L. McCall and P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
[CrossRef]

1981

H. Haus, Jpn. J. Appl. Phys. 20, 1007 (1981).
[CrossRef]

1975

H. A. Haus, C. V. Shank, and E. P. Ippen, Opt. Commun. 15, 29 (1975).
[CrossRef]

H. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
[CrossRef]

1974

G. H. C. New, IEEE J. Quantum Electron. QE-10, 115 (1974).
[CrossRef]

Arnold, J. M.

V. B. Khalfin, J. M. Arnold, and J. H. Marsh, IEEE J. Sel. Top. Quantum Electron. 1, 523 (1995).
[CrossRef]

Avrutin, E. A.

E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, IEE Proc. Optoelectron. 147, 251 (2000).
[CrossRef]

Chen, J. C.

J. C. Chen, H. A. Haus, and E. P. Ippen, IEEE J. Quantum Electron. 29, 1228 (1993).
[CrossRef]

Engelborghs, K.

K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL v.2.00:a Matlab package for bifurcation analysis of delay differential equations,” Tech. Rep. TW-330 (Department of Computer Science, Katholicke Universitet Leuven, Leuven, Belgium, 2001).

Erneux, T.

Haus, H.

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

H. Haus, Jpn. J. Appl. Phys. 20, 1007 (1981).
[CrossRef]

H. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
[CrossRef]

Haus, H. A.

J. C. Chen, H. A. Haus, and E. P. Ippen, IEEE J. Quantum Electron. 29, 1228 (1993).
[CrossRef]

H. A. Haus, C. V. Shank, and E. P. Ippen, Opt. Commun. 15, 29 (1975).
[CrossRef]

Ippen, E. P.

J. C. Chen, H. A. Haus, and E. P. Ippen, IEEE J. Quantum Electron. 29, 1228 (1993).
[CrossRef]

H. A. Haus, C. V. Shank, and E. P. Ippen, Opt. Commun. 15, 29 (1975).
[CrossRef]

Keller, U.

R. Paschotta and U. Keller, Appl. Phys. B 73, 653 (2001).
[CrossRef]

Khalfin, V. B.

V. B. Khalfin, J. M. Arnold, and J. H. Marsh, IEEE J. Sel. Top. Quantum Electron. 1, 523 (1995).
[CrossRef]

Lau, K. Y.

J. Palaski and K. Y. Lau, Appl. Phys. Lett. 59, 7 (1991).
[CrossRef]

Luzyanina, T.

K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL v.2.00:a Matlab package for bifurcation analysis of delay differential equations,” Tech. Rep. TW-330 (Department of Computer Science, Katholicke Universitet Leuven, Leuven, Belgium, 2001).

Marsh, J. H.

E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, IEE Proc. Optoelectron. 147, 251 (2000).
[CrossRef]

V. B. Khalfin, J. M. Arnold, and J. H. Marsh, IEEE J. Sel. Top. Quantum Electron. 1, 523 (1995).
[CrossRef]

McCall, S. L.

S. L. McCall and P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
[CrossRef]

New, G. H. C.

G. H. C. New, IEEE J. Quantum Electron. QE-10, 115 (1974).
[CrossRef]

Palaski, J.

J. Palaski and K. Y. Lau, Appl. Phys. Lett. 59, 7 (1991).
[CrossRef]

Paschotta, R.

R. Paschotta and U. Keller, Appl. Phys. B 73, 653 (2001).
[CrossRef]

Platzman, P. M.

S. L. McCall and P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
[CrossRef]

Portnoi, E. L.

E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, IEE Proc. Optoelectron. 147, 251 (2000).
[CrossRef]

Samaey, G.

K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL v.2.00:a Matlab package for bifurcation analysis of delay differential equations,” Tech. Rep. TW-330 (Department of Computer Science, Katholicke Universitet Leuven, Leuven, Belgium, 2001).

Shank, C. V.

H. A. Haus, C. V. Shank, and E. P. Ippen, Opt. Commun. 15, 29 (1975).
[CrossRef]

Vasil'ev, P.

P. Vasil'ev, Ultrafast Diode Lasers: Fundamentals and Applications (Artech House, Norwood, Mass., 1995).

Appl. Phys. B

R. Paschotta and U. Keller, Appl. Phys. B 73, 653 (2001).
[CrossRef]

Appl. Phys. Lett.

J. Palaski and K. Y. Lau, Appl. Phys. Lett. 59, 7 (1991).
[CrossRef]

IEE Proc. Optoelectron.

E. A. Avrutin, J. H. Marsh, and E. L. Portnoi, IEE Proc. Optoelectron. 147, 251 (2000).
[CrossRef]

IEEE J. Quantum Electron.

H. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
[CrossRef]

G. H. C. New, IEEE J. Quantum Electron. QE-10, 115 (1974).
[CrossRef]

J. C. Chen, H. A. Haus, and E. P. Ippen, IEEE J. Quantum Electron. 29, 1228 (1993).
[CrossRef]

S. L. McCall and P. M. Platzman, IEEE J. Quantum Electron. QE-21, 1899 (1985).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

V. B. Khalfin, J. M. Arnold, and J. H. Marsh, IEEE J. Sel. Top. Quantum Electron. 1, 523 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys.

H. Haus, Jpn. J. Appl. Phys. 20, 1007 (1981).
[CrossRef]

Opt. Commun.

H. A. Haus, C. V. Shank, and E. P. Ippen, Opt. Commun. 15, 29 (1975).
[CrossRef]

Other

K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL v.2.00:a Matlab package for bifurcation analysis of delay differential equations,” Tech. Rep. TW-330 (Department of Computer Science, Katholicke Universitet Leuven, Leuven, Belgium, 2001).

P. Vasil'ev, Ultrafast Diode Lasers: Fundamentals and Applications (Artech House, Norwood, Mass., 1995).

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Figures (3)

Fig. 1
Fig. 1

Andronov–Hopf bifurcations of the cw solution of Eqs. (2)–(4). The parameters are T=25 ps, γ-1=0.4 ps, αg,q=0, s=5, γg-1=1 ns, γq-1=10 ps, and κ=0.5.

Fig. 2
Fig. 2

Branches of ML solutions bifurcating from the Andronov–Hopf bifurcation curves shown in Fig. 1. The solid (dashed) curves indicate stable (unstable) solutions. The branch of constant-intensity solutions is labeled cw. q0=2γq. The other parameters are the same as in Fig. 1.

Fig. 3
Fig. 3

Time dependence of the amplitude (solid curves) of a fundamental ML pulse and the net gain parameter (dashed curves). (a) g0=0.4γq, (b) g0=0.8γq, (c) g0=1.32γq. The other parameters are the same as in Fig. 2.

Equations (4)

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

at+T=κ-tft-θexp1-iαggθ/2-1-iαqqθ/2aθdθ.
g·t=g0-γggt-exp-qtexpgt-1at2/Eg,
q·t=q0-γqqt-1-exp-qtat2/Eq.
γ-1a·t+at=κexp1-iαggt-T/2-1-iαqqt-T/2at-T.

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