Abstract

An integrable nonlinear Schrödinger equation incorporating time-varying phase modulation is presented. A family of solutions is identified, including solitons that oscillate in position and frequency. The equation is used to model steady-state asynchronous laser mode locking. Numerical simulations are used to verify the model and to explore the breakdown of the model as the product of pulse width and modulation frequency is increased.

© 2001 Optical Society of America

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References

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  1. C. R. Doerr, H. A. Haus, and E. P. Ippen, Opt. Lett. 19, 1958 (1994).
    [CrossRef] [PubMed]
  2. H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, J. Lightwave Technol. 14, 622 (1996).
    [CrossRef]
  3. H. A. Haus and Y. Lai, J. Opt. Soc. Am. B 7, 386 (1990).
    [CrossRef]
  4. A. Hasegawa and Y. Kodama, Opt. Lett. 15, 1443 (1990).
    [CrossRef] [PubMed]
  5. D. J. Kaup, Phys. Rev. A 44, 4582 (1991).
    [CrossRef] [PubMed]
  6. J. N. Elgin, Phys. Rev. A 47, 4331 (1993).
    [CrossRef] [PubMed]

1996 (1)

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, J. Lightwave Technol. 14, 622 (1996).
[CrossRef]

1994 (1)

1993 (1)

J. N. Elgin, Phys. Rev. A 47, 4331 (1993).
[CrossRef] [PubMed]

1991 (1)

D. J. Kaup, Phys. Rev. A 44, 4582 (1991).
[CrossRef] [PubMed]

1990 (2)

Doerr, C. R.

Elgin, J. N.

J. N. Elgin, Phys. Rev. A 47, 4331 (1993).
[CrossRef] [PubMed]

Hasegawa, A.

Haus, H. A.

Ippen, E. P.

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, J. Lightwave Technol. 14, 622 (1996).
[CrossRef]

C. R. Doerr, H. A. Haus, and E. P. Ippen, Opt. Lett. 19, 1958 (1994).
[CrossRef] [PubMed]

Jones, D. J.

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, J. Lightwave Technol. 14, 622 (1996).
[CrossRef]

Kaup, D. J.

D. J. Kaup, Phys. Rev. A 44, 4582 (1991).
[CrossRef] [PubMed]

Kodama, Y.

Lai, Y.

Wong, W. S.

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, J. Lightwave Technol. 14, 622 (1996).
[CrossRef]

J. Lightwave Technol. (1)

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, J. Lightwave Technol. 14, 622 (1996).
[CrossRef]

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

Opt. Lett. (2)

Phys. Rev. A (2)

D. J. Kaup, Phys. Rev. A 44, 4582 (1991).
[CrossRef] [PubMed]

J. N. Elgin, Phys. Rev. A 47, 4331 (1993).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Narrow pulse width of 316  fs with 10-GHz phase modulation: (a) pulse mean frequency detuning versus distance, (b) pulse delay as a function of distance. Dotted curves, simulation; solid curves, theory. The curves overlap.

Fig. 2
Fig. 2

Pulse power versus delay on a logarithmic power scale after a propagation distance of 50  km, or 1978 soliton periods, for 316-fs pulses.

Fig. 3
Fig. 3

Broad pulse width of 1  ps with 10-GHz phase modulation. (a) Pulse mean frequency detuning versus distance, (b) pulse delay versus distance. Dashed curves, simulation; solid curves, theory.

Fig. 4
Fig. 4

Pulse power on a logarithmic scale after propagation over 100  km, or 396 soliton periods, for 1-ps pulses.

Equations (9)

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

iuz-k2utt+κu2u+Pz,tu=0,
Pz,t=h+K2T2ktsinKz-K2T28kcos2Kz,
FcosΩt-KzFcosKz+ΩtFsinKz=Pz,t.
hz=FKsinKz+kΩ2F24K2z+kΩ2F28K3sin2Kz+h0
T=2kΩ2FK2.
uz,t=vz,τexpiφz,t,
ivz-k2vττ+κv2v=0,
φz,t=hz-KT2ktcosKz,
τ=t-T2sinKz.

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