Abstract

A theoretical model is developed to quantify the experimental observations of self-similar parabolic pulses in a mode-locked laser cavity with net-zero dispersion. The averaging procedure used shows the pulse amplitude to be governed by the porous media equation that has the well-known Barenblatt similarity (parabolic) solution, suggesting that it is a viable theoretical description of temporal profiles observed in mode-locked cavities. To the best of our knowledge, this is the first analytic model proposing a mechanism responsible for generating temporal parabolic pulses.

© 2008 Optical Society of America

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References

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  1. G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge U. Press, 1996).
  2. C. Menyuk, D. Levi, and P. Winternitz, Phys. Rev. Lett. 69, 3048 (1992).
    [CrossRef] [PubMed]
  3. M. Soljacic, M. Segev, and C. Menyuk, Phys. Rev. E 61, R1048 (2000).
    [CrossRef]
  4. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. Quiroga-Teixiero, J. Opt. Soc. Am. B 10, 1185 (1993).
    [CrossRef]
  5. M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, and J. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
    [CrossRef] [PubMed]
  6. C. Finot, G. Millot, C. Billet, and J. Dudley, Opt. Express 11, 1547 (2003).
    [CrossRef] [PubMed]
  7. G. Chang, H. Winful, A. Galvanaukas, and T. Norris, Phys. Rev. E 72, 016609 (2005).
    [CrossRef]
  8. F. Ilday, J. Buckley, W. Clark, and F. Wise, Phys. Rev. Lett. 92, 213902 (2004).
    [CrossRef] [PubMed]
  9. P. Belanger, Opt. Express 14, 12174 (2006).
    [CrossRef] [PubMed]
  10. P. Belanger, Opt. Express 15, 11033 (2007).
    [CrossRef] [PubMed]
  11. H. Haus, IEEE J. Sel. Top. Quant. Electron. 6, 1173 (2000).
    [CrossRef]
  12. J. N. Kutz, SIAM Rev. 48, 629 (2006).
    [CrossRef]
  13. J. Bronski and J. N. Kutz, Physica D 108, 315 (1997).
    [CrossRef]
  14. C. Bender and S. Orzag, Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999).
  15. J. Bronski and J. N. Kutz, J. Opt. Soc. Am. B 14, 903 (1997).
    [CrossRef]
  16. A. Hasegawa and Y. Kodama, Opt. Lett. 15, 1443 (1990).
    [CrossRef] [PubMed]
  17. T. Witelski and A. Bernoff, Stud. Appl. Math. 100, 153 (1998).
    [CrossRef]

2007

2006

2005

G. Chang, H. Winful, A. Galvanaukas, and T. Norris, Phys. Rev. E 72, 016609 (2005).
[CrossRef]

2004

F. Ilday, J. Buckley, W. Clark, and F. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

2003

2000

M. Soljacic, M. Segev, and C. Menyuk, Phys. Rev. E 61, R1048 (2000).
[CrossRef]

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, and J. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

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

1998

T. Witelski and A. Bernoff, Stud. Appl. Math. 100, 153 (1998).
[CrossRef]

1997

J. Bronski and J. N. Kutz, Physica D 108, 315 (1997).
[CrossRef]

J. Bronski and J. N. Kutz, J. Opt. Soc. Am. B 14, 903 (1997).
[CrossRef]

1993

1992

C. Menyuk, D. Levi, and P. Winternitz, Phys. Rev. Lett. 69, 3048 (1992).
[CrossRef] [PubMed]

1990

Anderson, D.

Barenblatt, G. I.

G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge U. Press, 1996).

Belanger, P.

Bender, C.

C. Bender and S. Orzag, Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999).

Bernoff, A.

T. Witelski and A. Bernoff, Stud. Appl. Math. 100, 153 (1998).
[CrossRef]

Billet, C.

Bronski, J.

J. Bronski and J. N. Kutz, Physica D 108, 315 (1997).
[CrossRef]

J. Bronski and J. N. Kutz, J. Opt. Soc. Am. B 14, 903 (1997).
[CrossRef]

Buckley, J.

F. Ilday, J. Buckley, W. Clark, and F. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Chang, G.

G. Chang, H. Winful, A. Galvanaukas, and T. Norris, Phys. Rev. E 72, 016609 (2005).
[CrossRef]

Clark, W.

F. Ilday, J. Buckley, W. Clark, and F. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Desaix, M.

Dudley, J.

C. Finot, G. Millot, C. Billet, and J. Dudley, Opt. Express 11, 1547 (2003).
[CrossRef] [PubMed]

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, and J. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

Fermann, M.

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, and J. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

Finot, C.

Galvanaukas, A.

G. Chang, H. Winful, A. Galvanaukas, and T. Norris, Phys. Rev. E 72, 016609 (2005).
[CrossRef]

Harvey, J.

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, and J. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

Hasegawa, A.

Haus, H.

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

Ilday, F.

F. Ilday, J. Buckley, W. Clark, and F. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Karlsson, M.

Kodama, Y.

Kruglov, V.

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, and J. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

Kutz, J. N.

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

J. Bronski and J. N. Kutz, Physica D 108, 315 (1997).
[CrossRef]

J. Bronski and J. N. Kutz, J. Opt. Soc. Am. B 14, 903 (1997).
[CrossRef]

Levi, D.

C. Menyuk, D. Levi, and P. Winternitz, Phys. Rev. Lett. 69, 3048 (1992).
[CrossRef] [PubMed]

Lisak, M.

Menyuk, C.

M. Soljacic, M. Segev, and C. Menyuk, Phys. Rev. E 61, R1048 (2000).
[CrossRef]

C. Menyuk, D. Levi, and P. Winternitz, Phys. Rev. Lett. 69, 3048 (1992).
[CrossRef] [PubMed]

Millot, G.

Norris, T.

G. Chang, H. Winful, A. Galvanaukas, and T. Norris, Phys. Rev. E 72, 016609 (2005).
[CrossRef]

Orzag, S.

C. Bender and S. Orzag, Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999).

Quiroga-Teixiero, M.

Segev, M.

M. Soljacic, M. Segev, and C. Menyuk, Phys. Rev. E 61, R1048 (2000).
[CrossRef]

Soljacic, M.

M. Soljacic, M. Segev, and C. Menyuk, Phys. Rev. E 61, R1048 (2000).
[CrossRef]

Thomsen, B.

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, and J. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

Winful, H.

G. Chang, H. Winful, A. Galvanaukas, and T. Norris, Phys. Rev. E 72, 016609 (2005).
[CrossRef]

Winternitz, P.

C. Menyuk, D. Levi, and P. Winternitz, Phys. Rev. Lett. 69, 3048 (1992).
[CrossRef] [PubMed]

Wise, F.

F. Ilday, J. Buckley, W. Clark, and F. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Witelski, T.

T. Witelski and A. Bernoff, Stud. Appl. Math. 100, 153 (1998).
[CrossRef]

IEEE J. Sel. Top. Quant. Electron.

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

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. E

M. Soljacic, M. Segev, and C. Menyuk, Phys. Rev. E 61, R1048 (2000).
[CrossRef]

G. Chang, H. Winful, A. Galvanaukas, and T. Norris, Phys. Rev. E 72, 016609 (2005).
[CrossRef]

Phys. Rev. Lett.

F. Ilday, J. Buckley, W. Clark, and F. Wise, Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

M. Fermann, V. Kruglov, B. Thomsen, J. Dudley, and J. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

C. Menyuk, D. Levi, and P. Winternitz, Phys. Rev. Lett. 69, 3048 (1992).
[CrossRef] [PubMed]

Physica D

J. Bronski and J. N. Kutz, Physica D 108, 315 (1997).
[CrossRef]

SIAM Rev.

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

Stud. Appl. Math.

T. Witelski and A. Bernoff, Stud. Appl. Math. 100, 153 (1998).
[CrossRef]

Other

C. Bender and S. Orzag, Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999).

G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics (Cambridge U. Press, 1996).

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Figures (1)

Fig. 1
Fig. 1

Typical evolution of the Barenblatt similarity solution (11) over three laser cavity round trips with D ( Z ) = 1 for 0 < Z < P 2 and D ( Z ) = 1 for P 2 < Z < P . The breathing dynamics are induced by the periodically varying diffusion coefficient μ ( Z ) O ( ϵ ) .

Equations (12)

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

i Q Z + D ( Z ) 2 2 Q T 2 + Q 2 Q + i γ Q i g ( Z ) ( 1 + τ 2 T 2 ) Q + F ( Q ) = 0 ,
g ( Z ) = 2 g 0 1 + Q 2 e 0
ϵ = P = Z cav Z 0 1 .
i Q Z + 1 2 D ( Z ϵ ) 2 Q 2 T + Q 2 Q = ϵ 2 G ( Q ) ,
i G Z + 1 2 D ( Z ϵ ) 2 G T 2 = 0 ,
G ( T , T , Z ) = exp ( i π 4 ) 2 π μ ( Z ) exp ( i ( T T ) 2 2 μ ( Z ) ) .
A ( ξ , Z ) = G ( ξ , T , Z ) Q ( T , Z ) d T .
i A Z = A 2 A + i μ [ A ¯ ( A T ) 2 + 2 A A T 2 + A 2 2 A ¯ T 2 ] + O ( μ 2 ) .
A ( T , Z ) = ρ ( T , Z ) exp ( i ϴ ( T , Z ) ) .
ρ Z = μ 2 2 T 2 ( ρ 2 ) ,
ϴ Z = μ ρ 2 ϴ T 2 ρ .
Q ( T , Z ) 2 ρ ( T , Z ) [ a * 2 ( ( T T * ) ( σ + Z * ) 1 3 ) 2 ] + 12 ( σ + Z * ) 1 3 ,

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