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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  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,” (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

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

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,” (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,” (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,” (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]

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,” (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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