Abstract

We show that a soliton of the nonlinear Schrödinger equation perturbed by filter losses and/or the finite gain bandwidth of amplifiers can be kept stable by saturable absorbers with a relaxation time much longer than the width of the soliton. This provides for ultrashort pulse generation with a slow saturable absorber only and may have possible applications in the stabilization of soliton storage rings.

© 1995 Optical Society of America

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References

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  1. H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
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  2. H. A. Haus, J. G. Fujimoto, E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).
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  6. K. Smith, E. J. Greer, R. Wyatt, P. Wheatley, N. J. Doran, Electron. Lett. 27, 244 (1990).
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  7. F. X. Kärtner, D. Kopf, U. Keller, “Solitary pulse stabilization and shortening in actively mode-locked lasers,” J. Opt. Soc. Am. B (to be published).
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    [CrossRef]
  9. J. P. Gordon, J. Opt. Soc. Am. B 9, 91 (1992).
    [CrossRef]
  10. S. Flügge, Practical Quantum Mechanics I, II (Springer-Verlag, Berlin, 1971), Vol. 1.
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1994

1993

1992

1991

1990

K. Smith, E. J. Greer, R. Wyatt, P. Wheatley, N. J. Doran, Electron. Lett. 27, 244 (1990).
[CrossRef]

1989

1985

1976

H. A. Haus, IEEE J. Quantum Electron. QE-12, 169 (1976).
[CrossRef]

1975

H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
[CrossRef]

1974

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

Baer, T.

Brovelli, L.

Chiu, T. H.

Doran, N. J.

K. Smith, E. J. Greer, R. Wyatt, P. Wheatley, N. J. Doran, Electron. Lett. 27, 244 (1990).
[CrossRef]

Ferguson, J. F.

Flügge, S.

S. Flügge, Practical Quantum Mechanics I, II (Springer-Verlag, Berlin, 1971), Vol. 1.

Fork, R. L.

Fujimoto, J. G.

Gordon, J. P.

Greer, E. J.

K. Smith, E. J. Greer, R. Wyatt, P. Wheatley, N. J. Doran, Electron. Lett. 27, 244 (1990).
[CrossRef]

Hall, D. W.

Haus, H. A.

Ippen, E. P.

Kafka, J. D.

Kamp, M.

Kärtner, F. X.

F. X. Kärtner, D. Kopf, U. Keller, in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper MA5.

F. X. Kärtner, D. Kopf, U. Keller, “Solitary pulse stabilization and shortening in actively mode-locked lasers,” J. Opt. Soc. Am. B (to be published).

Keller, U.

D. Kopf, K. J. Weingarten, L. Brovelli, M. Kamp, U. Keller, Opt. Lett. 19, 2143 (1994).
[CrossRef] [PubMed]

U. Keller, T. H. Chiu, J. F. Ferguson, Opt. Lett. 18, 217 (1993).
[CrossRef] [PubMed]

F. X. Kärtner, D. Kopf, U. Keller, in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper MA5.

F. X. Kärtner, D. Kopf, U. Keller, “Solitary pulse stabilization and shortening in actively mode-locked lasers,” J. Opt. Soc. Am. B (to be published).

Kopf, D.

D. Kopf, K. J. Weingarten, L. Brovelli, M. Kamp, U. Keller, Opt. Lett. 19, 2143 (1994).
[CrossRef] [PubMed]

F. X. Kärtner, D. Kopf, U. Keller, in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper MA5.

F. X. Kärtner, D. Kopf, U. Keller, “Solitary pulse stabilization and shortening in actively mode-locked lasers,” J. Opt. Soc. Am. B (to be published).

Martinez, O. E.

Mecozzi, A.

New, G. H. C.

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

Smith, K.

K. Smith, E. J. Greer, R. Wyatt, P. Wheatley, N. J. Doran, Electron. Lett. 27, 244 (1990).
[CrossRef]

Weingarten, K. J.

Wheatley, P.

K. Smith, E. J. Greer, R. Wyatt, P. Wheatley, N. J. Doran, Electron. Lett. 27, 244 (1990).
[CrossRef]

Wyatt, R.

K. Smith, E. J. Greer, R. Wyatt, P. Wheatley, N. J. Doran, Electron. Lett. 27, 244 (1990).
[CrossRef]

Electron. Lett.

K. Smith, E. J. Greer, R. Wyatt, P. Wheatley, N. J. Doran, Electron. Lett. 27, 244 (1990).
[CrossRef]

IEEE J. Quantum Electron.

H. A. Haus, IEEE J. Quantum Electron. QE-12, 169 (1976).
[CrossRef]

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

J. Appl. Phys.

H. A. Haus, J. Appl. Phys. 46, 3049 (1975).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Other

F. X. Kärtner, D. Kopf, U. Keller, in Ultrafast Phenomena, Vol. 7 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper MA5.

S. Flügge, Practical Quantum Mechanics I, II (Springer-Verlag, Berlin, 1971), Vol. 1.

F. X. Kärtner, D. Kopf, U. Keller, “Solitary pulse stabilization and shortening in actively mode-locked lasers,” J. Opt. Soc. Am. B (to be published).

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Figures (3)

Fig. 1
Fig. 1

Pulse-shaping mechanism owing to gain and loss dynamics in a mode-locked laser using (a) a fast saturable absorber only, (b) a slow saturable absorber plus slow gain saturation, (c) a slow saturable absorber plus soliton formation.

Fig. 2
Fig. 2

(a) Potential q(t) if the saturable absorber is immediately bleached after passage of the short intense soliton. The absorber recovers on a time scale much longer than the width of the soliton. (b) Real part of the ground state of the eigenvalue problem equations (5), assuming large negative GVD for the exponential potential and the V potential.

Fig. 3
Fig. 3

Numerical simulation of the pulse evolution in a laser in the time domain modeled by the master equation (1); however, the simulation is performed with the discrete action of SPM and GVD per round trip by use of the split-step Fourier-transform method. We start with a 2-ps-long pulse and end up with a 300-fs-long transform-limited sech pulse after 25,000 round trips.

Equations (6)

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T R T A ( T , t ) = [ - i D 2 t 2 + i δ A ( T , t ) 2 ] A ( T , t ) + [ g - l + D g 2 t 2 - q ( T , t ) ] A ( T , t ) ,
q ( T , t ) t = - q - q 0 T A - A ( T , t ) 2 E A ,
A s ( T , t ) = ( W 2 τ ) 1 / 2 sech ( t τ ) exp ( i Φ 0 T T R ) , Φ 0 = δ W 4 τ = D τ 2 ,
T R T G ( T , t ) = [ g - l + ( D g - i D ) 2 t 2 - q ( t ) ] × G ( T , t ) .
λ n = D g 3 τ 2 - E n , [ - D ¯ 2 t 2 + q ( t ) ] G n ( t ) = E n G n ( t ) ,
τ = ( 1 6 Ω g ) 3 / 4 ( T A g 3 / 2 q 0 ) 1 / 4 ( Φ 0 ) - 1 / 8 .

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