Abstract

We propose a model that is valid for ultrafast pulse propagation in a mode-locked laser cavity in the few- femtosecond pulse regime, thus deriving the equivalent of the master mode-locking equation for ultrashort pulses that has dominated mode-locking theory for two decades. The short-pulse equation with dissipative gain and loss terms allows for the generation of stable ultrashort optical pulses from initial white noise, thus providing the first theoretical framework for quantifying the pulse dynamics and stability as pulses widths approach the attosecond regime.

© 2010 Optical Society of America

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  1. R. Ell, G. Angelow, W. Seitz, M. Lederer, H. Huber, D. Kopf, J. Birge, and F. Kärtner, Opt. Express 13, 9292 (2005).
    [CrossRef] [PubMed]
  2. F. Kärtner, U. Morgner, R. Ell, T. Schibli, J. Fujimoto, E. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, J. Opt. Soc. Am. B 18, 882 (2001).
    [CrossRef]
  3. U. Keller, in Progress in Optics (Elsevier, 2004), Vol. 46, pp. 1–115.
    [CrossRef]
  4. M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
    [CrossRef] [PubMed]
  5. A. Heinrich, J. Biegert, and U. Keller, Phys. Rev. Lett. 92, 023003 (2004).
    [CrossRef] [PubMed]
  6. Y. Silberberg, Nature 414, 494 (2001).
    [CrossRef] [PubMed]
  7. A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, J. Phys. B 39, R1 (2006).
    [CrossRef]
  8. S. Cundiff, Nature Phys. 3, 16 (2007).
    [CrossRef]
  9. S. Cundiff, J. Korean Phys. Soc. 48, 1181 (2006).
  10. S. Cundiff, J. Ye, and J. Hall, Sci. Am. 298, 74 (2008).
    [CrossRef] [PubMed]
  11. H. Haus, IEEE J. Sel. Top. Quantum Electron. 6, 1173(2000).
    [CrossRef]
  12. H. Haus, J. Fujimoto, and E. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).
    [CrossRef]
  13. J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
    [CrossRef]
  14. T. Schafer and C. Wayne, Phys. D 196, 90 (2004).
    [CrossRef]
  15. Y. Chung, C. K. R. Jones, and T. Schafer, Nonlinearity 18, 1351 (2005).
    [CrossRef]
  16. Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, Phys. Rev. A 77, 063821 (2008).
    [CrossRef]
  17. M. Pietrzyk, I. Kanattsikov, and U. Bandelow, J. Nonlin. Math. Phys. 15, 262 (2008).
    [CrossRef]
  18. A. Sakovich and S. Sakovich, J. Phys. Soc. Jpn. 74, 239 (2005).
    [CrossRef]
  19. A. Sakovich and S. Sakovich, J. Phys. A 39, 361 (2006).
    [CrossRef]
  20. J. N. Kutz, SIAM Rev. 48, 629 (2006).
    [CrossRef]
  21. J. N. Kutz and B. Sandstede, Opt. Express 16, 636 (2008).
    [CrossRef] [PubMed]

2008

S. Cundiff, J. Ye, and J. Hall, Sci. Am. 298, 74 (2008).
[CrossRef] [PubMed]

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, Phys. Rev. A 77, 063821 (2008).
[CrossRef]

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, J. Nonlin. Math. Phys. 15, 262 (2008).
[CrossRef]

J. N. Kutz and B. Sandstede, Opt. Express 16, 636 (2008).
[CrossRef] [PubMed]

2007

S. Cundiff, Nature Phys. 3, 16 (2007).
[CrossRef]

2006

S. Cundiff, J. Korean Phys. Soc. 48, 1181 (2006).

A. Sakovich and S. Sakovich, J. Phys. A 39, 361 (2006).
[CrossRef]

J. N. Kutz, SIAM Rev. 48, 629 (2006).
[CrossRef]

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, J. Phys. B 39, R1 (2006).
[CrossRef]

2005

Y. Chung, C. K. R. Jones, and T. Schafer, Nonlinearity 18, 1351 (2005).
[CrossRef]

A. Sakovich and S. Sakovich, J. Phys. Soc. Jpn. 74, 239 (2005).
[CrossRef]

R. Ell, G. Angelow, W. Seitz, M. Lederer, H. Huber, D. Kopf, J. Birge, and F. Kärtner, Opt. Express 13, 9292 (2005).
[CrossRef] [PubMed]

2004

T. Schafer and C. Wayne, Phys. D 196, 90 (2004).
[CrossRef]

U. Keller, in Progress in Optics (Elsevier, 2004), Vol. 46, pp. 1–115.
[CrossRef]

A. Heinrich, J. Biegert, and U. Keller, Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

2001

Y. Silberberg, Nature 414, 494 (2001).
[CrossRef] [PubMed]

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

F. Kärtner, U. Morgner, R. Ell, T. Schibli, J. Fujimoto, E. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, J. Opt. Soc. Am. B 18, 882 (2001).
[CrossRef]

2000

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

1991

Amiranashvili, Sh.

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, Phys. Rev. A 77, 063821 (2008).
[CrossRef]

Angelow, G.

Bandelow, U.

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, Phys. Rev. A 77, 063821 (2008).
[CrossRef]

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, J. Nonlin. Math. Phys. 15, 262 (2008).
[CrossRef]

Biegert, J.

A. Heinrich, J. Biegert, and U. Keller, Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

Birge, J.

Brabec, T.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Chung, Y.

Y. Chung, C. K. R. Jones, and T. Schafer, Nonlinearity 18, 1351 (2005).
[CrossRef]

Coen, S.

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Corkum, P.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Cundiff, S.

S. Cundiff, J. Ye, and J. Hall, Sci. Am. 298, 74 (2008).
[CrossRef] [PubMed]

S. Cundiff, Nature Phys. 3, 16 (2007).
[CrossRef]

S. Cundiff, J. Korean Phys. Soc. 48, 1181 (2006).

Drescher, M.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Dudley, J.

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Ell, R.

Fujimoto, J.

Genty, G.

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Hall, J.

S. Cundiff, J. Ye, and J. Hall, Sci. Am. 298, 74 (2008).
[CrossRef] [PubMed]

Haus, H.

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

H. Haus, J. Fujimoto, and E. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).
[CrossRef]

Heinrich, A.

A. Heinrich, J. Biegert, and U. Keller, Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

Heinzmann, U.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Hentschel, M.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Huber, H.

Ippen, E.

Ivanov, M. Yu.

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, J. Phys. B 39, R1 (2006).
[CrossRef]

Jones, C. K. R.

Y. Chung, C. K. R. Jones, and T. Schafer, Nonlinearity 18, 1351 (2005).
[CrossRef]

Kanattsikov, I.

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, J. Nonlin. Math. Phys. 15, 262 (2008).
[CrossRef]

Kärtner, F.

Keller, U.

U. Keller, in Progress in Optics (Elsevier, 2004), Vol. 46, pp. 1–115.
[CrossRef]

A. Heinrich, J. Biegert, and U. Keller, Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

Kienberger, R.

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, J. Phys. B 39, R1 (2006).
[CrossRef]

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Kopf, D.

Krausz, F.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Kutz, J. N.

Lederer, M.

Milosevic, N.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Morgner, U.

Pietrzyk, M.

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, J. Nonlin. Math. Phys. 15, 262 (2008).
[CrossRef]

Reider, G. A.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Sakovich, A.

A. Sakovich and S. Sakovich, J. Phys. A 39, 361 (2006).
[CrossRef]

A. Sakovich and S. Sakovich, J. Phys. Soc. Jpn. 74, 239 (2005).
[CrossRef]

Sakovich, S.

A. Sakovich and S. Sakovich, J. Phys. A 39, 361 (2006).
[CrossRef]

A. Sakovich and S. Sakovich, J. Phys. Soc. Jpn. 74, 239 (2005).
[CrossRef]

Sandstede, B.

Schafer, T.

Y. Chung, C. K. R. Jones, and T. Schafer, Nonlinearity 18, 1351 (2005).
[CrossRef]

T. Schafer and C. Wayne, Phys. D 196, 90 (2004).
[CrossRef]

Scheuer, V.

Schibli, T.

Scrinzi, A.

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, J. Phys. B 39, R1 (2006).
[CrossRef]

Seitz, W.

Silberberg, Y.

Y. Silberberg, Nature 414, 494 (2001).
[CrossRef] [PubMed]

Spielmann, Ch.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Tschudi, T.

Villeneuve, D. M.

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, J. Phys. B 39, R1 (2006).
[CrossRef]

Vladimirov, A. G.

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, Phys. Rev. A 77, 063821 (2008).
[CrossRef]

Wayne, C.

T. Schafer and C. Wayne, Phys. D 196, 90 (2004).
[CrossRef]

Ye, J.

S. Cundiff, J. Ye, and J. Hall, Sci. Am. 298, 74 (2008).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron.

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

J. Korean Phys. Soc.

S. Cundiff, J. Korean Phys. Soc. 48, 1181 (2006).

J. Nonlin. Math. Phys.

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, J. Nonlin. Math. Phys. 15, 262 (2008).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. A

A. Sakovich and S. Sakovich, J. Phys. A 39, 361 (2006).
[CrossRef]

J. Phys. B

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, J. Phys. B 39, R1 (2006).
[CrossRef]

J. Phys. Soc. Jpn.

A. Sakovich and S. Sakovich, J. Phys. Soc. Jpn. 74, 239 (2005).
[CrossRef]

Nature

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, Nature 414, 509 (2001).
[CrossRef] [PubMed]

Y. Silberberg, Nature 414, 494 (2001).
[CrossRef] [PubMed]

Nature Phys.

S. Cundiff, Nature Phys. 3, 16 (2007).
[CrossRef]

Nonlinearity

Y. Chung, C. K. R. Jones, and T. Schafer, Nonlinearity 18, 1351 (2005).
[CrossRef]

Opt. Express

Phys. D

T. Schafer and C. Wayne, Phys. D 196, 90 (2004).
[CrossRef]

Phys. Rev. A

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, Phys. Rev. A 77, 063821 (2008).
[CrossRef]

Phys. Rev. Lett.

A. Heinrich, J. Biegert, and U. Keller, Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

Rev. Mod. Phys.

J. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).
[CrossRef]

Sci. Am.

S. Cundiff, J. Ye, and J. Hall, Sci. Am. 298, 74 (2008).
[CrossRef] [PubMed]

SIAM Rev.

J. N. Kutz, SIAM Rev. 48, 629 (2006).
[CrossRef]

Other

U. Keller, in Progress in Optics (Elsevier, 2004), Vol. 46, pp. 1–115.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Spectra of SPE solutions are localized around the carrier frequency, motivating symmetric gain windows. Dashed lines show vertically rescaled gain windows for P ( k ) = 1 2 ( | k | 1 ) 2 (top) and P ( k ) = 1 0.75 ( | k | 1.6 ) 2 , with SPE solutions for m = 0.10 and 0.35.

Fig. 2
Fig. 2

Mode-locking formation from initial white noise with saturating gain g ( t ) ( 1 0.75 ( | k | 1.6 ) 2 ) . The gain window is shown in the bottom panel of Fig. 1.

Fig. 3
Fig. 3

Increased cavity gain leads to multipulse instability. Shown are the longtime mode-locked pulse profiles and amplitudes for gain levels of g 0 = 0.2 , 0.34, and 0.4125. Consistent with experimental observations, one, two, and three pulses per round trip can be stabilized.

Equations (5)

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

u x t u 1 6 ( u 3 ) x x = ( g ( t ) F 1 [ P ( k ) U ^ ] γ u + β u 3 ) x ,
g ( t ) = 2 g 0 1 + u 2 / E 0 ,
u = 4 m n sin ψ sinh ϕ + n cos ψ cosh ϕ m 2 sin 2 ψ + n 2 cosh 2 ϕ ,
x = y + 2 m n m sin 2 ψ n sinh 2 ϕ m 2 sin 2 ψ + n 2 cosh 2 ϕ ,
P ( k ) = ( 1 a ( | k | b ) 2 ) ,

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