Abstract

Actively mode-locked lasers with noise are studied employing statistical mechanics. A mapping of the system to the spherical model (related to the Ising model) of ferromagnets in one dimension that has an exact solution is established. It gives basic features, such as analytical expressions for the correlation function between modes, and the widths and shapes of the pulses [different from the Kuizenga–Siegman expression; IEEE J. Quantum Electron. QE-6, 803 (1970)] and reveals the susceptibility to noise of mode ordering compared with passive mode locking.

© 2004 Optical Society of America

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References

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  1. D. I. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 803 (1970).
  2. A. Gordon and B. Fischer, Phys. Rev. Lett. 89, 103901 (2002).
    [CrossRef]
  3. A. Gordon and B. Fischer, Opt. Commun. 223, 151 (2003).
    [CrossRef]
  4. A. Gordon and B. Fischer, Opt. Lett. 18, 1326 (2003).
    [CrossRef]
  5. T. H. Berlin and M. Kac, Phys. Rev. 86, 821 (1952).
    [CrossRef]
  6. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford U. Press, Oxford, UK, 1971).
  7. H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
    [CrossRef]
  8. H. A. Haus, IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000).
    [CrossRef]
  9. H. Risken, The Fokker–Planck Equation, 2nd ed. (Springer-Verlag, Berlin, 1989).
  10. H. E. Stanley, Phys. Rev. 179, 570 (1969).
    [CrossRef]

2003 (2)

A. Gordon and B. Fischer, Opt. Commun. 223, 151 (2003).
[CrossRef]

A. Gordon and B. Fischer, Opt. Lett. 18, 1326 (2003).
[CrossRef]

2002 (1)

A. Gordon and B. Fischer, Phys. Rev. Lett. 89, 103901 (2002).
[CrossRef]

2000 (1)

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

1993 (1)

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

1970 (1)

D. I. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 803 (1970).

1969 (1)

H. E. Stanley, Phys. Rev. 179, 570 (1969).
[CrossRef]

1952 (1)

T. H. Berlin and M. Kac, Phys. Rev. 86, 821 (1952).
[CrossRef]

Berlin, T. H.

T. H. Berlin and M. Kac, Phys. Rev. 86, 821 (1952).
[CrossRef]

Fischer, B.

A. Gordon and B. Fischer, Opt. Commun. 223, 151 (2003).
[CrossRef]

A. Gordon and B. Fischer, Opt. Lett. 18, 1326 (2003).
[CrossRef]

A. Gordon and B. Fischer, Phys. Rev. Lett. 89, 103901 (2002).
[CrossRef]

Gordon, A.

A. Gordon and B. Fischer, Opt. Lett. 18, 1326 (2003).
[CrossRef]

A. Gordon and B. Fischer, Opt. Commun. 223, 151 (2003).
[CrossRef]

A. Gordon and B. Fischer, Phys. Rev. Lett. 89, 103901 (2002).
[CrossRef]

Haus, H. A.

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

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

Kac, M.

T. H. Berlin and M. Kac, Phys. Rev. 86, 821 (1952).
[CrossRef]

Kuizenga, D. I.

D. I. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 803 (1970).

Mecozzi, A.

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

Risken, H.

H. Risken, The Fokker–Planck Equation, 2nd ed. (Springer-Verlag, Berlin, 1989).

Siegman, A. E.

D. I. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 803 (1970).

Stanley, H. E.

H. E. Stanley, Phys. Rev. 179, 570 (1969).
[CrossRef]

H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford U. Press, Oxford, UK, 1971).

IEEE J. Quantum Electron. (2)

D. I. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 803 (1970).

H. A. Haus and A. Mecozzi, IEEE J. Quantum Electron. 29, 983 (1993).
[CrossRef]

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

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

Opt. Commun. (1)

A. Gordon and B. Fischer, Opt. Commun. 223, 151 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (2)

H. E. Stanley, Phys. Rev. 179, 570 (1969).
[CrossRef]

T. H. Berlin and M. Kac, Phys. Rev. 86, 821 (1952).
[CrossRef]

Phys. Rev. Lett. (1)

A. Gordon and B. Fischer, Phys. Rev. Lett. 89, 103901 (2002).
[CrossRef]

Other (2)

H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford U. Press, Oxford, UK, 1971).

H. Risken, The Fokker–Planck Equation, 2nd ed. (Springer-Verlag, Berlin, 1989).

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Figures (2)

Fig. 1
Fig. 1

Axial mode system in the frequency (ν) domain, where the arrows describe the mode amplitudes and phases (phasors) and the corresponding one-period light-intensity profiles in the time domain (τR is the cavity round-trip time). Note the resemblance to the magnetic spin system, where the arrows describe spins. The traces at the right are given for various ordering levels, from (top to bottom) complete disorder, through partial order with a finite correlation length, to a highly ordered and then to a completely ordered state. The degree of order is determined by AP0/W.

Fig. 2
Fig. 2

Average pulse intensity profiles [Eq. (9)], showing one period, for various correlation lengths.

Equations (10)

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a˙m=A/2am-1+am+1+gP-lam+Γm,
HI=-A2J-g0Psat lnPsat+P+lP, J=mamam+1*+am*am+1,
a˙m=-HIam*+Γm,  a˙m*=-HIam+Γm*.
ρa1,aNexp-HIT=expg0Psat lnPsat+P-lPT×expAJ2T.
ρa1,aNδP-P0expANJW.
akak+n*=P0N2AP0/W1+1+2AP0/W21/2n.
akak+n*P0N1-W2AP0nP0Nexp-nW2AP0.
Ncor=2AP0W.
W>2AP0/N.
ψt2=m,naman*exp2πim-nt/τR=P01+2AP0/W21/2-2AP0 cos2πt/τR/W,

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