Abstract

A perturbation theory for the short-pulse equation is developed to investigate the effects of various perturbations on optical solitons propagating in nonlinear media in the few femtosecond to subfemtosecond regime. The theory is formulated using a variational approach since linearization of the exact solution is not tractable. A variety of physically realizable perturbations are considered, culminating in perturbations that result from considering short-pulse mode locking. In each case, the analytic results presented are in agreement with full numerical simulations of the short-pulse theory. Given the tremendous success of soliton perturbation theory in the theoretical realm of optical solitons, the short-pulse perturbation theory attempts to provide the same theoretical framework for understanding physically realizable mechanisms that affect pulse evolution and stability when nearing the attosecond regime.

© 2013 Optical Society of America

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  1. S. Backus, C. G. Durfee, M. M. Murname, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1233 (1998).
    [CrossRef]
  2. C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
    [CrossRef]
  3. D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
    [CrossRef]
  4. R. Ell, G. Angelow, W. Seitz, M. J. Lederer, H. Huber, D. Kopf, J. R. Birge, and F. X. Kärtner, “Quasi-synchronous pumping of modelocked few-cycle titanium sapphire lasers,” Opt. Express 13, 9292–9298 (2005).
    [CrossRef]
  5. F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
    [CrossRef]
  6. U. Keller, “Ultrafast solid-state lasers,” Prog. Opt. 46, 1–115 (2004).
    [CrossRef]
  7. M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
    [CrossRef]
  8. A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
    [CrossRef]
  9. Y. Silberberg, “Physics at the attosecond frontier,” Nature 414, 494–495 (2001).
    [CrossRef]
  10. A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, “Attosecond physics,” J. Phys. B 39, R1–R37 (2006).
    [CrossRef]
  11. S. T. Cundiff, “Attosecond physics: better by half,” Nat. Phys. 3, 16–18 (2007).
    [CrossRef]
  12. S. T. Cundiff, “Femtosecond comb technology,” J. Korean Phys. Soc. 48, 1181–1187 (2006).
  13. S. T. Cundiff, S. J. Ye, and J. Hall, “Rulers of light,” Sci. Am. 298, 74–81 (2008).
    [CrossRef]
  14. H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
    [CrossRef]
  15. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
    [CrossRef]
  16. Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
    [CrossRef]
  17. M. Pietrzyk, I. Kanattsikov, and U. Bandelow, “On the propagation of vector ultra-short pulses,” J. Nonlin. Math. Phys. 15, 170–262 (2008).
    [CrossRef]
  18. H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
    [CrossRef]
  19. T. Schafer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
    [CrossRef]
  20. Y. Chung, C. K. R. Jones, and T. Schafer, “Ultra-short pulses in linear and nonlinear media,” Nonlinearity 18, 1351–1374 (2005).
    [CrossRef]
  21. E. V. Kazantseva, A. I. Maimistov, and J. -G. Caputo, “Reduced Maxwell-Duffing description of extremely short pulses in nonresonant media,” Phys. Rev. E 71, 056622 (2005).
    [CrossRef]
  22. S. V. Sazonov, “Nonlinear theory of transverse perturbations of quasi-one-dimensional solitons,” J. Exp. Theor. Phys. 92, 361 (2001).
    [CrossRef]
  23. H. Leblond and D. Mihalache, “Few-optical cycle solitons: modified Korteweg-de Vries sine-Gordon equation versus other non-slowly varying envelope approximation models,” Phys. Rev. A 79, 063835 (2009).
    [CrossRef]
  24. E. Farnum and J. N. Kutz, “Master mode-locking theory for few-femtosecond pulses,” Opt. Lett. 35, 3033–3035 (2010).
    [CrossRef]
  25. E. Farnum and J. N. Kutz, “Mode locking in the few-femtosecond regime using waveguide arrays and the coupled short-pulse equations,” IEEE J. Sel. Top. Quantum Electron. 18, 113–118 (2012).
    [CrossRef]
  26. H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
    [CrossRef]
  27. J. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
    [CrossRef]
  28. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
    [CrossRef]
  29. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [CrossRef]
  30. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef]
  31. T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master-modelocking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002).
    [CrossRef]
  32. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  33. P. V. Mamyshev and L. F. Mollenauer, “Soliton collisions in wavelength-division-multiplexed dispersion-managed systems,” Opt. Lett. 24, 448–450 (1999).
    [CrossRef]
  34. P. V. Mamyshev and L. F. Mollenauer, “Pseudo-phase-matched four-wave mixing in soliton wavelength-division multiplexing transmission,” Opt. Lett. 21, 396–398 (1996).
    [CrossRef]
  35. A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
    [CrossRef]
  36. B. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008).
    [CrossRef]
  37. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
    [CrossRef]
  38. A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
    [CrossRef]
  39. A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
    [CrossRef]
  40. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
    [CrossRef]
  41. J. C. Brunelli, “The bi-Hamiltonian structure of the short pulse equation,” Phys. Lett. A 353, 475–478 (2006).
    [CrossRef]

2013 (1)

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[CrossRef]

2012 (1)

E. Farnum and J. N. Kutz, “Mode locking in the few-femtosecond regime using waveguide arrays and the coupled short-pulse equations,” IEEE J. Sel. Top. Quantum Electron. 18, 113–118 (2012).
[CrossRef]

2010 (2)

H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
[CrossRef]

E. Farnum and J. N. Kutz, “Master mode-locking theory for few-femtosecond pulses,” Opt. Lett. 35, 3033–3035 (2010).
[CrossRef]

2009 (1)

H. Leblond and D. Mihalache, “Few-optical cycle solitons: modified Korteweg-de Vries sine-Gordon equation versus other non-slowly varying envelope approximation models,” Phys. Rev. A 79, 063835 (2009).
[CrossRef]

2008 (4)

B. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193–1202 (2008).
[CrossRef]

S. T. Cundiff, S. J. Ye, and J. Hall, “Rulers of light,” Sci. Am. 298, 74–81 (2008).
[CrossRef]

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[CrossRef]

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, “On the propagation of vector ultra-short pulses,” J. Nonlin. Math. Phys. 15, 170–262 (2008).
[CrossRef]

2007 (1)

S. T. Cundiff, “Attosecond physics: better by half,” Nat. Phys. 3, 16–18 (2007).
[CrossRef]

2006 (5)

S. T. Cundiff, “Femtosecond comb technology,” J. Korean Phys. Soc. 48, 1181–1187 (2006).

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, “Attosecond physics,” J. Phys. B 39, R1–R37 (2006).
[CrossRef]

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
[CrossRef]

J. C. Brunelli, “The bi-Hamiltonian structure of the short pulse equation,” Phys. Lett. A 353, 475–478 (2006).
[CrossRef]

2005 (4)

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[CrossRef]

Y. Chung, C. K. R. Jones, and T. Schafer, “Ultra-short pulses in linear and nonlinear media,” Nonlinearity 18, 1351–1374 (2005).
[CrossRef]

E. V. Kazantseva, A. I. Maimistov, and J. -G. Caputo, “Reduced Maxwell-Duffing description of extremely short pulses in nonresonant media,” Phys. Rev. E 71, 056622 (2005).
[CrossRef]

R. Ell, G. Angelow, W. Seitz, M. J. Lederer, H. Huber, D. Kopf, J. R. Birge, and F. X. Kärtner, “Quasi-synchronous pumping of modelocked few-cycle titanium sapphire lasers,” Opt. Express 13, 9292–9298 (2005).
[CrossRef]

2004 (4)

U. Keller, “Ultrafast solid-state lasers,” Prog. Opt. 46, 1–115 (2004).
[CrossRef]

A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef]

T. Schafer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[CrossRef]

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

2002 (1)

2001 (4)

S. V. Sazonov, “Nonlinear theory of transverse perturbations of quasi-one-dimensional solitons,” J. Exp. Theor. Phys. 92, 361 (2001).
[CrossRef]

Y. Silberberg, “Physics at the attosecond frontier,” Nature 414, 494–495 (2001).
[CrossRef]

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

2000 (1)

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[CrossRef]

1999 (1)

1998 (2)

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

S. Backus, C. G. Durfee, M. M. Murname, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1233 (1998).
[CrossRef]

1996 (1)

1994 (1)

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

1993 (1)

J. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
[CrossRef]

1992 (1)

1991 (1)

1990 (1)

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef]

1986 (2)

1979 (1)

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Amiranashvili, Sh.

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[CrossRef]

Anderson, D.

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Angelow, G.

R. Ell, G. Angelow, W. Seitz, M. J. Lederer, H. Huber, D. Kopf, J. R. Birge, and F. X. Kärtner, “Quasi-synchronous pumping of modelocked few-cycle titanium sapphire lasers,” Opt. Express 13, 9292–9298 (2005).
[CrossRef]

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

Backus, S.

S. Backus, C. G. Durfee, M. M. Murname, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1233 (1998).
[CrossRef]

Bale, B.

Bandelow, U.

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[CrossRef]

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, “On the propagation of vector ultra-short pulses,” J. Nonlin. Math. Phys. 15, 170–262 (2008).
[CrossRef]

Biegert, J.

A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef]

Birge, J. R.

Bondeson, A.

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Brabec, T.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

Brunelli, J. C.

J. C. Brunelli, “The bi-Hamiltonian structure of the short pulse equation,” Phys. Lett. A 353, 475–478 (2006).
[CrossRef]

Caputo, J. -G.

E. V. Kazantseva, A. I. Maimistov, and J. -G. Caputo, “Reduced Maxwell-Duffing description of extremely short pulses in nonresonant media,” Phys. Rev. E 71, 056622 (2005).
[CrossRef]

Chung, Y.

Y. Chung, C. K. R. Jones, and T. Schafer, “Ultra-short pulses in linear and nonlinear media,” Nonlinearity 18, 1351–1374 (2005).
[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, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

Cundiff, S. T.

S. T. Cundiff, S. J. Ye, and J. Hall, “Rulers of light,” Sci. Am. 298, 74–81 (2008).
[CrossRef]

S. T. Cundiff, “Attosecond physics: better by half,” Nat. Phys. 3, 16–18 (2007).
[CrossRef]

S. T. Cundiff, “Femtosecond comb technology,” J. Korean Phys. Soc. 48, 1181–1187 (2006).

Curley, P. F.

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

Drescher, M.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

Durfee, C. G.

S. Backus, C. G. Durfee, M. M. Murname, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1233 (1998).
[CrossRef]

Elgin, J.

J. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
[CrossRef]

Ell, R.

R. Ell, G. Angelow, W. Seitz, M. J. Lederer, H. Huber, D. Kopf, J. R. Birge, and F. X. Kärtner, “Quasi-synchronous pumping of modelocked few-cycle titanium sapphire lasers,” Opt. Express 13, 9292–9298 (2005).
[CrossRef]

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

Farnum, E.

E. Farnum and J. N. Kutz, “Mode locking in the few-femtosecond regime using waveguide arrays and the coupled short-pulse equations,” IEEE J. Sel. Top. Quantum Electron. 18, 113–118 (2012).
[CrossRef]

E. Farnum and J. N. Kutz, “Master mode-locking theory for few-femtosecond pulses,” Opt. Lett. 35, 3033–3035 (2010).
[CrossRef]

Fujimoto, J. G.

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

Gordon, J. P.

Hall, J.

S. T. Cundiff, S. J. Ye, and J. Hall, “Rulers of light,” Sci. Am. 298, 74–81 (2008).
[CrossRef]

Haus, H. A.

Heinrich, A.

A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef]

Heinzmann, U.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

Hentschel, M.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

Huber, H.

Ippen, E. P.

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

Ivanov, M. Yu.

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, “Attosecond physics,” J. Phys. B 39, R1–R37 (2006).
[CrossRef]

Jones, C. K. R.

Y. Chung, C. K. R. Jones, and T. Schafer, “Ultra-short pulses in linear and nonlinear media,” Nonlinearity 18, 1351–1374 (2005).
[CrossRef]

Jung, I. D.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Kanattsikov, I.

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, “On the propagation of vector ultra-short pulses,” J. Nonlin. Math. Phys. 15, 170–262 (2008).
[CrossRef]

Kapitula, T.

Kapteyn, H. C.

S. Backus, C. G. Durfee, M. M. Murname, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1233 (1998).
[CrossRef]

Kärtner, F. X.

R. Ell, G. Angelow, W. Seitz, M. J. Lederer, H. Huber, D. Kopf, J. R. Birge, and F. X. Kärtner, “Quasi-synchronous pumping of modelocked few-cycle titanium sapphire lasers,” Opt. Express 13, 9292–9298 (2005).
[CrossRef]

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Kaup, D. J.

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef]

Kazantseva, E. V.

E. V. Kazantseva, A. I. Maimistov, and J. -G. Caputo, “Reduced Maxwell-Duffing description of extremely short pulses in nonresonant media,” Phys. Rev. E 71, 056622 (2005).
[CrossRef]

Keller, U.

A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef]

U. Keller, “Ultrafast solid-state lasers,” Prog. Opt. 46, 1–115 (2004).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Kienberger, R.

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, “Attosecond physics,” J. Phys. B 39, R1–R37 (2006).
[CrossRef]

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

Kolesik, M.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

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, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

Kutz, J. N.

Leblond, H.

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[CrossRef]

H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
[CrossRef]

H. Leblond and D. Mihalache, “Few-optical cycle solitons: modified Korteweg-de Vries sine-Gordon equation versus other non-slowly varying envelope approximation models,” Phys. Rev. A 79, 063835 (2009).
[CrossRef]

Lederer, M. J.

Lisak, M.

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Maimistov, A. I.

E. V. Kazantseva, A. I. Maimistov, and J. -G. Caputo, “Reduced Maxwell-Duffing description of extremely short pulses in nonresonant media,” Phys. Rev. E 71, 056622 (2005).
[CrossRef]

Mamyshev, P. V.

Matuschek, N.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Mihalache, D.

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[CrossRef]

H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
[CrossRef]

H. Leblond and D. Mihalache, “Few-optical cycle solitons: modified Korteweg-de Vries sine-Gordon equation versus other non-slowly varying envelope approximation models,” Phys. Rev. A 79, 063835 (2009).
[CrossRef]

Milosevic, N.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

Mollenauer, L. F.

Moloney, J. V.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

Morgner, U.

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

Morier-Genoud, F.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Murname, M. M.

S. Backus, C. G. Durfee, M. M. Murname, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1233 (1998).
[CrossRef]

Pietrzyk, M.

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, “On the propagation of vector ultra-short pulses,” J. Nonlin. Math. Phys. 15, 170–262 (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, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

Sakovich, A.

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
[CrossRef]

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[CrossRef]

Sakovich, S.

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
[CrossRef]

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[CrossRef]

Sandstede, B.

Sazonov, S. V.

S. V. Sazonov, “Nonlinear theory of transverse perturbations of quasi-one-dimensional solitons,” J. Exp. Theor. Phys. 92, 361 (2001).
[CrossRef]

Schafer, T.

Y. Chung, C. K. R. Jones, and T. Schafer, “Ultra-short pulses in linear and nonlinear media,” Nonlinearity 18, 1351–1374 (2005).
[CrossRef]

T. Schafer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[CrossRef]

Scheuer, V.

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Schibli, T.

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

Scrinzi, A.

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, “Attosecond physics,” J. Phys. B 39, R1–R37 (2006).
[CrossRef]

Seitz, W.

Silberberg, Y.

Y. Silberberg, “Physics at the attosecond frontier,” Nature 414, 494–495 (2001).
[CrossRef]

Spielmann, C.

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

Spielmann, Ch.

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

Sutter, D. H.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Tilsch, M.

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Tschudi, T.

F. X. Kärtner, U. Morgner, R. Ell, T. Schibli, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Ultrabroadband double-chirped mirror pairs for generation of octave spectra,” J. Opt. Soc. Am. B 19, 382–385 (2001).
[CrossRef]

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

Villeneuve, D. M.

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, “Attosecond physics,” J. Phys. B 39, R1–R37 (2006).
[CrossRef]

Vladimirov, A. G.

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[CrossRef]

Wayne, C. E.

T. Schafer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[CrossRef]

Ye, S. J.

S. T. Cundiff, S. J. Ye, and J. Hall, “Rulers of light,” Sci. Am. 298, 74–81 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. 30, 1100–1114 (1994).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (3)

D. H. Sutter, I. D. Jung, F. X. Kärtner, N. Matuschek, F. Morier-Genoud, V. Scheuer, M. Tilsch, T. Tschudi, and U. Keller, “Self-starting 6.5 fs pulses from a Ti:sapphire laser using a semiconductor saturable absorber and double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 169–178 (1998).
[CrossRef]

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[CrossRef]

E. Farnum and J. N. Kutz, “Mode locking in the few-femtosecond regime using waveguide arrays and the coupled short-pulse equations,” IEEE J. Sel. Top. Quantum Electron. 18, 113–118 (2012).
[CrossRef]

J. Exp. Theor. Phys. (1)

S. V. Sazonov, “Nonlinear theory of transverse perturbations of quasi-one-dimensional solitons,” J. Exp. Theor. Phys. 92, 361 (2001).
[CrossRef]

J. Korean Phys. Soc. (1)

S. T. Cundiff, “Femtosecond comb technology,” J. Korean Phys. Soc. 48, 1181–1187 (2006).

J. Nonlin. Math. Phys. (1)

M. Pietrzyk, I. Kanattsikov, and U. Bandelow, “On the propagation of vector ultra-short pulses,” J. Nonlin. Math. Phys. 15, 170–262 (2008).
[CrossRef]

J. Opt. Soc. Am. B (5)

J. Phys. A (2)

A. Sakovich and S. Sakovich, “Solitary wave solutions of the short pulse equation,” J. Phys. A 39, L361–L367 (2006).
[CrossRef]

H. Leblond and D. Mihalache, “Few-optical-cycle dissipative solitons,” J. Phys. A 43, 375205 (2010).
[CrossRef]

J. Phys. B (1)

A. Scrinzi, M. Yu. Ivanov, R. Kienberger, and D. M. Villeneuve, “Attosecond physics,” J. Phys. B 39, R1–R37 (2006).
[CrossRef]

J. Phys. Soc. Jpn. (1)

A. Sakovich and S. Sakovich, “The short pulse equation is integrable,” J. Phys. Soc. Jpn. 74, 239–241 (2005).
[CrossRef]

Nat. Phys. (1)

S. T. Cundiff, “Attosecond physics: better by half,” Nat. Phys. 3, 16–18 (2007).
[CrossRef]

Nature (2)

M. Hentschel, R. Kienberger, Ch. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001).
[CrossRef]

Y. Silberberg, “Physics at the attosecond frontier,” Nature 414, 494–495 (2001).
[CrossRef]

Nonlinearity (1)

Y. Chung, C. K. R. Jones, and T. Schafer, “Ultra-short pulses in linear and nonlinear media,” Nonlinearity 18, 1351–1374 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Lett. A (1)

J. C. Brunelli, “The bi-Hamiltonian structure of the short pulse equation,” Phys. Lett. A 353, 475–478 (2006).
[CrossRef]

Phys. Rep. (1)

H. Leblond and D. Mihalache, “Models of few optical cycle solitons beyond the slowly varying envelope approximation,” Phys. Rep. 523, 61–126 (2013).
[CrossRef]

Phys. Rev. A (4)

Sh. Amiranashvili, A. G. Vladimirov, and U. Bandelow, “Solitary-wave solutions for few-cycle optical pulses,” Phys. Rev. A 77, 063821 (2008).
[CrossRef]

H. Leblond and D. Mihalache, “Few-optical cycle solitons: modified Korteweg-de Vries sine-Gordon equation versus other non-slowly varying envelope approximation models,” Phys. Rev. A 79, 063835 (2009).
[CrossRef]

J. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
[CrossRef]

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef]

Phys. Rev. E (2)

E. V. Kazantseva, A. I. Maimistov, and J. -G. Caputo, “Reduced Maxwell-Duffing description of extremely short pulses in nonresonant media,” Phys. Rev. E 71, 056622 (2005).
[CrossRef]

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwells to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef]

Phys. Scr. (1)

A. Bondeson, M. Lisak, and D. Anderson, “Soliton perturbations: a variational principle for the soliton parameters,” Phys. Scr. 20, 479–485 (1979).
[CrossRef]

Physica D (1)

T. Schafer and C. E. Wayne, “Propagation of ultra-short optical pulses in cubic nonlinear media,” Physica D 196, 90–105 (2004).
[CrossRef]

Prog. Opt. (1)

U. Keller, “Ultrafast solid-state lasers,” Prog. Opt. 46, 1–115 (2004).
[CrossRef]

Rev. Sci. Instrum. (1)

S. Backus, C. G. Durfee, M. M. Murname, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207–1233 (1998).
[CrossRef]

Sci. Am. (1)

S. T. Cundiff, S. J. Ye, and J. Hall, “Rulers of light,” Sci. Am. 298, 74–81 (2008).
[CrossRef]

SIAM Rev. (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629–678 (2006).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Top panel shows evolution of pulse amplitude (η) and inverse width (k). For appropriate initial conditions, solutions maintain their shape and ODE agrees with parameter fit of full PDE. Bottom panel indicates that pulse position and phase (b and θ, respectively) grow linearly, as predicted by ODE.

Fig. 2.
Fig. 2.

For lossy system, amplitude (η) decays nearly exponentially with loss parameter γ, as predicted by ODE. Inverse pulse width (k) does not remain constant, but the agreement is fair. Parameter fit to PDE indicates that, even as pulses decay, the pulse speed (average slope of b) remains nearly constant, in contrast to ODE prediction.

Fig. 3.
Fig. 3.

When all mode-locking terms are included, stable pulses are possible. Here parameter values are chosen to be perturbatively small, gain parameter g0=0.015, loss γ=0.0035, cubic coefficient β=0.001. ODE shows reasonable agreement with the full PDE, but does not reflect the oscillations inherent in the PDE model at this low dissipation. This shows the evolution of all four parameters.

Fig. 4.
Fig. 4.

When the magnitude of dissipative perturbation is increased, the ODE agrees only qualitatively with PDE. However, stability of such solutions is greatly enhanced, as evidenced by the robust mode locking in the PDE model as well as the large negative eigenvalues associated with the ODE. Parameter values are gain parameter g0=0.25, loss γ=0.25, cubic coefficient β=0.05.

Equations (42)

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

uxtu16(u3)xx=εF(u,x,t),
u=4mnsinψsinhϕ+ncosψcoshϕm2sin2ψ+n2cosh2ϕ,
x=y+2mnmsin2ψnsinh2ϕm2sin2ψ+n2cosh2ϕ,
L=12ϕtϕx124ϕx4+12ϕ2.
ϕ(t,x)=η(t)sech(k(t)(xb(t)))sin(ω(xθ(t)).
L=ϕtu2dxu424dx+ϕ22dx=L1+L2+L3.
ϕt=ϕηη+ϕkk+ϕbb+ϕθθ=i=14ϕρiρi,
L=12i=14ρiIi+L2+L3,
Ii=ϕρiudx.
δLδρj=Lρjt(Lρj),
=(12i=14ρiIiρj+ρj(L2+L3))12Ijt,
Ijt=i=14Ijρiρi.
δLδρj=12i=14ρi(IiρjIjρi)+ρj(L2+L3).
12MdXdt+(L2+L3)=0,
Fj=Fϕρjdx,
dXdt=2M1((L2+L3)+F),
I1(η,k,b,θ)=0,
I2(η,k,b,θ)0,
I3(η,k,b,θ)13η2k,
I4(η,k,b,θ)=η2(1/k).
L20.02η4/k,
L30.5η2/k.
M1M01=[0034ηkk4η0032η2k22η234ηk32η200k4ηk22η200].
(L2+L3)=[(0.08η3η)/k(0.02η40.5η2)/k200].
η=F˜1,
k=F˜2,
b=0.06(η2/k2),
θ=10.06η2.
ddt[ηkbθ]=[000.06η2k210.06η2].
F^[ux(x,t)]=ikF^[u(x,t)]
Fγ=γuϕρjdx=γ[I1,I2,I3,I4]T.
ddt[ηγkγbγθγ]=[γη000].
F=Fγ+FG+Fβ,
η=η0+ηγ(t)+ηG(t)+ηβ(t),
k=k0+kγ(t)+kG(t)+kβ(t).
ddt[ηk]=[ηγkγ]+[ηGkG]+[ηβkβ].
FG=g(t)(τ2uxx+τ4uxxxx),
g(t)=2g01+u2.
g(t)=2g01+u22g01+η2(1k+k3),
FGj=g(t)(τ2uxx+τ4uxxxx)ϕρjdx.
ddt[ηGkG]=2g(t)[η((1.075τ21.51τ4)+(0.25τ24.56τ4)k2)(1.15τ22.02τ4)k(0.5τ2+5.125τ4)k3].
ddt[ηβkβ]=β[0.40η30.20η2k].

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