Abstract

Closed-form analytical solutions are obtained for a passively mode-locked laser for the case in which self-phase modulation and group-velocity dispersion, in addition to the more conventional mechanisms of saturable absorption and gain, shape the laser pulses. Provided that the self-phase modulation and group-velocity dispersion are related in a manner similar to that which causes soliton formation in optical fibers, this additional pulse shaping can reduce the pulse duration below the limit otherwise set by the laser bandwidth.

© 1984 Optical Society of America

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  1. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
    [CrossRef]
  2. H. A. Haus, IEEE J. Quantum Electron. QE-11, 736 (1975).
    [CrossRef]
  3. W. Dietel, E. Dopel, D. Kuhlke, B. Wilhelmi, Opt. Commun. 43, 433 (1982).
    [CrossRef]
  4. W. Dietel, J. J. Fontaine, J. C. Diels, Opt. Lett. 8, 4 (1983).
    [CrossRef] [PubMed]
  5. J. P. Gordon, R. L. Fork, Opt. Lett. 9, 153 (1984).
    [CrossRef] [PubMed]
  6. R. L. Fork, O. E. Martinez, J. P. Gordon, Opt. Lett. 9, 150 (1984).
    [CrossRef] [PubMed]
  7. L. F. Mollenauer, R. H. Stolen, Opt. Lett. 9, 13 (1984).
    [CrossRef] [PubMed]
  8. R. Cubeddu, O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
    [CrossRef]
  9. R. C. Eckardt, C. H. Lee, J. N. Bradford, Opto-Electron. 6, 67 (1974).
    [CrossRef]
  10. G. H. C. New, IEEE J. Quantum Electron. QE-10, 115 (1974).
    [CrossRef]
  11. R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
    [CrossRef]
  12. M. S. Stix, E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).

1984

1983

W. Dietel, J. J. Fontaine, J. C. Diels, Opt. Lett. 8, 4 (1983).
[CrossRef] [PubMed]

M. S. Stix, E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).

1982

W. Dietel, E. Dopel, D. Kuhlke, B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

1981

R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

1980

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

1975

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

1974

R. C. Eckardt, C. H. Lee, J. N. Bradford, Opto-Electron. 6, 67 (1974).
[CrossRef]

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

1969

R. Cubeddu, O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
[CrossRef]

Bradford, J. N.

R. C. Eckardt, C. H. Lee, J. N. Bradford, Opto-Electron. 6, 67 (1974).
[CrossRef]

Cubeddu, R.

R. Cubeddu, O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
[CrossRef]

Diels, J. C.

Dietel, W.

W. Dietel, J. J. Fontaine, J. C. Diels, Opt. Lett. 8, 4 (1983).
[CrossRef] [PubMed]

W. Dietel, E. Dopel, D. Kuhlke, B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Dopel, E.

W. Dietel, E. Dopel, D. Kuhlke, B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Eckardt, R. C.

R. C. Eckardt, C. H. Lee, J. N. Bradford, Opto-Electron. 6, 67 (1974).
[CrossRef]

Fontaine, J. J.

Fork, R. L.

Gordon, J. P.

Greene, B. I.

R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Haus, H. A.

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

Ippen, E. P.

M. S. Stix, E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).

Kuhlke, D.

W. Dietel, E. Dopel, D. Kuhlke, B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Lee, C. H.

R. C. Eckardt, C. H. Lee, J. N. Bradford, Opto-Electron. 6, 67 (1974).
[CrossRef]

Martinez, O. E.

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, Opt. Lett. 9, 13 (1984).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

New, G. H. C.

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

Shank, C. V.

R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

Stix, M. S.

M. S. Stix, E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, Opt. Lett. 9, 13 (1984).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Svelto, O.

R. Cubeddu, O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
[CrossRef]

Wilhelmi, B.

W. Dietel, E. Dopel, D. Kuhlke, B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Appl. Phys. Lett.

R. L. Fork, B. I. Greene, C. V. Shank, Appl. Phys. Lett. 38, 671 (1981).
[CrossRef]

IEEE J. Quantum Electron.

M. S. Stix, E. P. Ippen, IEEE J. Quantum Electron. QE-19, 521 (1983).

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

R. Cubeddu, O. Svelto, IEEE J. Quantum Electron. QE-5, 495 (1969).
[CrossRef]

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

Opt. Commun.

W. Dietel, E. Dopel, D. Kuhlke, B. Wilhelmi, Opt. Commun. 43, 433 (1982).
[CrossRef]

Opt. Lett.

Opto-Electron.

R. C. Eckardt, C. H. Lee, J. N. Bradford, Opto-Electron. 6, 67 (1974).
[CrossRef]

Phys. Rev. Lett.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Pulse width and chirp versus group-velocity dispersion for small self-phase modulation. In (a) we plot τ versus (−a2), and in (b) we plot x/τ2 versus (−a2). Note that positive a2 corresponds to negative group-velocity dispersion. For both examples the self-phase modulation is given by bUa = 0.2 fsec; αg = 0.05, αa = 0.035, γ = 0.02, s = 3, and a1 = 25 fsec.

Fig. 2
Fig. 2

Pulse width versus group-velocity dispersion for moderate self-phase modulation. Here τ is plotted versus −a2 for bUa =2.5 fsec. Curves are given for a range of pump rates, with oscillation threshold indicated by the dashed line. The asymmetry is typical of the case of moderate self-phase modulation. The gain parameter αg is 0.050, 0.0505, 0.0510, 0.0520, 0.0530, and 0.0540, with the largest gain yielding the smallest value of τ.

Equations (16)

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E ( t ) = υ ( t ) exp { i [ ω 0 t + ϕ ( t ) ] } ,
U = t υ ( t ) 2 d t .
υ ( t + T cav ) exp [ i ϕ ( t + T cav ) ] = exp ( O i O ) υ ( t ) exp [ i ϕ ( t ) ] ( 1 + O i O ) υ ( t ) exp [ i ϕ ( t ) ] ,
O = g 0 + g U + g U 2 + c 1 d d t + a 1 d 2 d t 2 ,
O = ϕ 0 + c 2 d d t + a 2 d 2 d t 2 + b υ 2 .
d 2 ϕ / d t 2 = ξ υ 2 .
V ( t ) = ( U 0 2 τ ) 1 / 2 sech ( t / τ )
ϕ ( t ) = x ln [ υ ( t ) ] ,
x = ξ ( U 0 τ / 2 ) = ξ y .
ϕ 0 τ 2 a 2 x 2 + 2 a 1 x + a 2 = 0 ,
c 1 x = c 2 ,
g y + c 2 x = c 1 ,
g 0 τ 2 + g y 2 a 1 x 2 2 a 2 x + a 1 = 0 ,
g y 2 a 1 x 2 3 a 2 x + 2 a 1 = 0 ,
a 2 x 2 b y + 3 a 1 x + 2 a 2 = 0 .
g = α g e ( u / s ) α a e u γ ,

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