Abstract

We experimentally observed optical rogue wave statistics during high power femtosecond pulse filamentation in air. We characterized wavelength-dependent intensity fluctuations across 300 nm broadband filament spectra generated by pulses with several times the critical power for filamentation. We show how the statistics vary from a near-Gaussian distribution in the vicinity of the pump to a long tailed “L-shaped” distribution at the short wavelength and long wavelength edges. The results are interpreted in terms of pump noise transfer via self-phase modulation.

© 2009 Optical Society of America

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References

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  1. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves," Nature 450, 1054 (2007).
    [CrossRef] [PubMed]
  2. A. I. Dyachenko and V. E. Zakharov, "Modulation instability of Stokes wave implies a freak wave," JETP Lett.,  81, 255-259 (2005).
    [CrossRef]
  3. J. M. Dudley, G. Genty, and B. J. Eggleton, "Harnessing and control of optical rogue waves in supercontinuum generation," Opt. Express 16, 3644-3651 (2008).
    [CrossRef] [PubMed]
  4. D. R. Solli, C. Ropers, and B. Jalali, "Active control of optical rogue waves for stimulated supercontinuum generation," Phys. Rev. Lett. 101, 233902 (2008).
    [CrossRef] [PubMed]
  5. G. Genty, J. M. Dudley, and B. J. Eggleton, "Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime," Appl. Phys. B 94, 187-194 (2009).
    [CrossRef]
  6. C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, "Direct detection of optical rogue waves energy statistics in supercontinuum generation," Electron. Lett. 45, 217-219 (2009).
    [CrossRef]
  7. N. Akhmediev, A. Ankiewicz, and M. Taki, "Waves that appear from nowhere and disappear without a trace," Phys. Lett. A 373, 675-678 (2009).
    [CrossRef]
  8. N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, "Extreme waves that appear from nowhere: on the nature of rogue waves," Phys. Lett. A (2009).
  9. D. Borlaug, S. Fathpour, and B. Jalali, "Extreme value statistics and the Pareto distribution in silicon photonics," arXiv:0809.2565v1 [Phys. Optics] (2008).
  10. K. Hammani, C. Finot, J. M. Dudley, and G. Millot, "Optical rogue-wave fluctuations in fiber Raman amplifiers," Opt. Express 16, 16467-16474 (2008).
    [CrossRef] [PubMed]
  11. S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
    [CrossRef]
  12. A. Couairon and A. Mysyrowicz. "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47-189, (2007).
    [CrossRef]
  13. L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf. "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70, 1633-1713 (2007).
    [CrossRef]
  14. J. Kasparian and J.-P. Wolf. Physics and applications of atmospheric nonlinear optics and filamentation.Opt. Express 16, 466-493, (2008).
    [CrossRef] [PubMed]
  15. P. Béjot, J. Kasparian, E. Salmon, R. Ackermann, and J.-P. Wolf, "Spectral correlation and noise reduction in laser filaments," Appl. Phys. B 87, 1-4 (2007).
    [CrossRef]
  16. S. Coles, An Introduction to Statistical Modeling of Extreme Values (Springer-Verlag, London, 2001).
  17. V. Pareto, Manuale di economia politica con una introduzione alla scienza sociale (Societa Editrice Libraria, Milano 1906)
  18. S. Bendersky, N. S. Komeika, and N. Blaunstein, "Atmospheric Optical Turbulence Over Land in Middle East Coastal Environments: Prediction Modeling and Measurements," Appl. Opt. 43, 4070-4079 (2004)
    [CrossRef] [PubMed]
  19. R. Salamé, N. Lascoux, E. Salmon, J. Kasparian, and J. P. Wolf, "Propagation of laser filaments through an extended turbulent region," Appl. Phys. Lett. 91, 171106 (2007).
    [CrossRef]

2009 (3)

G. Genty, J. M. Dudley, and B. J. Eggleton, "Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime," Appl. Phys. B 94, 187-194 (2009).
[CrossRef]

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, "Direct detection of optical rogue waves energy statistics in supercontinuum generation," Electron. Lett. 45, 217-219 (2009).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and M. Taki, "Waves that appear from nowhere and disappear without a trace," Phys. Lett. A 373, 675-678 (2009).
[CrossRef]

2008 (4)

2007 (5)

P. Béjot, J. Kasparian, E. Salmon, R. Ackermann, and J.-P. Wolf, "Spectral correlation and noise reduction in laser filaments," Appl. Phys. B 87, 1-4 (2007).
[CrossRef]

R. Salamé, N. Lascoux, E. Salmon, J. Kasparian, and J. P. Wolf, "Propagation of laser filaments through an extended turbulent region," Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

A. Couairon and A. Mysyrowicz. "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47-189, (2007).
[CrossRef]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf. "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70, 1633-1713 (2007).
[CrossRef]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves," Nature 450, 1054 (2007).
[CrossRef] [PubMed]

2005 (2)

A. I. Dyachenko and V. E. Zakharov, "Modulation instability of Stokes wave implies a freak wave," JETP Lett.,  81, 255-259 (2005).
[CrossRef]

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

2004 (1)

Ackermann, R.

P. Béjot, J. Kasparian, E. Salmon, R. Ackermann, and J.-P. Wolf, "Spectral correlation and noise reduction in laser filaments," Appl. Phys. B 87, 1-4 (2007).
[CrossRef]

Akhmediev, N.

N. Akhmediev, A. Ankiewicz, and M. Taki, "Waves that appear from nowhere and disappear without a trace," Phys. Lett. A 373, 675-678 (2009).
[CrossRef]

Akozbek, N.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Ankiewicz, A.

N. Akhmediev, A. Ankiewicz, and M. Taki, "Waves that appear from nowhere and disappear without a trace," Phys. Lett. A 373, 675-678 (2009).
[CrossRef]

Becker, A.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Béjot, P.

P. Béjot, J. Kasparian, E. Salmon, R. Ackermann, and J.-P. Wolf, "Spectral correlation and noise reduction in laser filaments," Appl. Phys. B 87, 1-4 (2007).
[CrossRef]

Bendersky, S.

Bergé, L.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf. "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70, 1633-1713 (2007).
[CrossRef]

Blaunstein, N.

Bolger, J.

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, "Direct detection of optical rogue waves energy statistics in supercontinuum generation," Electron. Lett. 45, 217-219 (2009).
[CrossRef]

Chin, S. L.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Couairon, A.

A. Couairon and A. Mysyrowicz. "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47-189, (2007).
[CrossRef]

Dias, F.

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, "Direct detection of optical rogue waves energy statistics in supercontinuum generation," Electron. Lett. 45, 217-219 (2009).
[CrossRef]

Dudley, J. M.

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, "Direct detection of optical rogue waves energy statistics in supercontinuum generation," Electron. Lett. 45, 217-219 (2009).
[CrossRef]

G. Genty, J. M. Dudley, and B. J. Eggleton, "Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime," Appl. Phys. B 94, 187-194 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and B. J. Eggleton, "Harnessing and control of optical rogue waves in supercontinuum generation," Opt. Express 16, 3644-3651 (2008).
[CrossRef] [PubMed]

K. Hammani, C. Finot, J. M. Dudley, and G. Millot, "Optical rogue-wave fluctuations in fiber Raman amplifiers," Opt. Express 16, 16467-16474 (2008).
[CrossRef] [PubMed]

Dyachenko, A. I.

A. I. Dyachenko and V. E. Zakharov, "Modulation instability of Stokes wave implies a freak wave," JETP Lett.,  81, 255-259 (2005).
[CrossRef]

Eggleton, B. J.

G. Genty, J. M. Dudley, and B. J. Eggleton, "Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime," Appl. Phys. B 94, 187-194 (2009).
[CrossRef]

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, "Direct detection of optical rogue waves energy statistics in supercontinuum generation," Electron. Lett. 45, 217-219 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and B. J. Eggleton, "Harnessing and control of optical rogue waves in supercontinuum generation," Opt. Express 16, 3644-3651 (2008).
[CrossRef] [PubMed]

Finot, C.

Genty, G.

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, "Direct detection of optical rogue waves energy statistics in supercontinuum generation," Electron. Lett. 45, 217-219 (2009).
[CrossRef]

G. Genty, J. M. Dudley, and B. J. Eggleton, "Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime," Appl. Phys. B 94, 187-194 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and B. J. Eggleton, "Harnessing and control of optical rogue waves in supercontinuum generation," Opt. Express 16, 3644-3651 (2008).
[CrossRef] [PubMed]

Hammani, K.

Hosseini, S. A.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Jalali, B.

D. R. Solli, C. Ropers, and B. Jalali, "Active control of optical rogue waves for stimulated supercontinuum generation," Phys. Rev. Lett. 101, 233902 (2008).
[CrossRef] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves," Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Kandidov, V. P.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Kasparian, J.

J. Kasparian and J.-P. Wolf. Physics and applications of atmospheric nonlinear optics and filamentation.Opt. Express 16, 466-493, (2008).
[CrossRef] [PubMed]

P. Béjot, J. Kasparian, E. Salmon, R. Ackermann, and J.-P. Wolf, "Spectral correlation and noise reduction in laser filaments," Appl. Phys. B 87, 1-4 (2007).
[CrossRef]

R. Salamé, N. Lascoux, E. Salmon, J. Kasparian, and J. P. Wolf, "Propagation of laser filaments through an extended turbulent region," Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf. "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70, 1633-1713 (2007).
[CrossRef]

Komeika, N. S.

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves," Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Kosareva, O. G.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Lafargue, C.

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, "Direct detection of optical rogue waves energy statistics in supercontinuum generation," Electron. Lett. 45, 217-219 (2009).
[CrossRef]

Lascoux, N.

R. Salamé, N. Lascoux, E. Salmon, J. Kasparian, and J. P. Wolf, "Propagation of laser filaments through an extended turbulent region," Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

Liu, W.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Luo, Q.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Millot, G.

Mysyrowicz, A.

A. Couairon and A. Mysyrowicz. "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47-189, (2007).
[CrossRef]

Nuter, R.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf. "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70, 1633-1713 (2007).
[CrossRef]

Ropers, C.

D. R. Solli, C. Ropers, and B. Jalali, "Active control of optical rogue waves for stimulated supercontinuum generation," Phys. Rev. Lett. 101, 233902 (2008).
[CrossRef] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves," Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Salamé, R.

R. Salamé, N. Lascoux, E. Salmon, J. Kasparian, and J. P. Wolf, "Propagation of laser filaments through an extended turbulent region," Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

Salmon, E.

R. Salamé, N. Lascoux, E. Salmon, J. Kasparian, and J. P. Wolf, "Propagation of laser filaments through an extended turbulent region," Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

P. Béjot, J. Kasparian, E. Salmon, R. Ackermann, and J.-P. Wolf, "Spectral correlation and noise reduction in laser filaments," Appl. Phys. B 87, 1-4 (2007).
[CrossRef]

Schroeder, H.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Skupin, S.

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf. "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70, 1633-1713 (2007).
[CrossRef]

Solli, D. R.

D. R. Solli, C. Ropers, and B. Jalali, "Active control of optical rogue waves for stimulated supercontinuum generation," Phys. Rev. Lett. 101, 233902 (2008).
[CrossRef] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves," Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Taki, M.

N. Akhmediev, A. Ankiewicz, and M. Taki, "Waves that appear from nowhere and disappear without a trace," Phys. Lett. A 373, 675-678 (2009).
[CrossRef]

Théberge, F.

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Wolf, J. P.

R. Salamé, N. Lascoux, E. Salmon, J. Kasparian, and J. P. Wolf, "Propagation of laser filaments through an extended turbulent region," Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

Wolf, J.-P.

J. Kasparian and J.-P. Wolf. Physics and applications of atmospheric nonlinear optics and filamentation.Opt. Express 16, 466-493, (2008).
[CrossRef] [PubMed]

P. Béjot, J. Kasparian, E. Salmon, R. Ackermann, and J.-P. Wolf, "Spectral correlation and noise reduction in laser filaments," Appl. Phys. B 87, 1-4 (2007).
[CrossRef]

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf. "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70, 1633-1713 (2007).
[CrossRef]

Zakharov, V. E.

A. I. Dyachenko and V. E. Zakharov, "Modulation instability of Stokes wave implies a freak wave," JETP Lett.,  81, 255-259 (2005).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (2)

G. Genty, J. M. Dudley, and B. J. Eggleton, "Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime," Appl. Phys. B 94, 187-194 (2009).
[CrossRef]

P. Béjot, J. Kasparian, E. Salmon, R. Ackermann, and J.-P. Wolf, "Spectral correlation and noise reduction in laser filaments," Appl. Phys. B 87, 1-4 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

R. Salamé, N. Lascoux, E. Salmon, J. Kasparian, and J. P. Wolf, "Propagation of laser filaments through an extended turbulent region," Appl. Phys. Lett. 91, 171106 (2007).
[CrossRef]

Can. J. Phys. (1)

S. L. Chin, S. A. Hosseini, W. Liu, Q. Luo, F. Théberge, N. Akozbek, A. Becker, V. P. Kandidov, O. G.  Kosareva, and H. Schroeder, "The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges," Can. J. Phys. 83, 863-905 (2005).
[CrossRef]

Electron. Lett. (1)

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, "Direct detection of optical rogue waves energy statistics in supercontinuum generation," Electron. Lett. 45, 217-219 (2009).
[CrossRef]

JETP Lett. (1)

A. I. Dyachenko and V. E. Zakharov, "Modulation instability of Stokes wave implies a freak wave," JETP Lett.,  81, 255-259 (2005).
[CrossRef]

Nature (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves," Nature 450, 1054 (2007).
[CrossRef] [PubMed]

Opt. Express (3)

Phys. Lett. A (1)

N. Akhmediev, A. Ankiewicz, and M. Taki, "Waves that appear from nowhere and disappear without a trace," Phys. Lett. A 373, 675-678 (2009).
[CrossRef]

Phys. Rep. (1)

A. Couairon and A. Mysyrowicz. "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47-189, (2007).
[CrossRef]

Phys. Rev. Lett. (1)

D. R. Solli, C. Ropers, and B. Jalali, "Active control of optical rogue waves for stimulated supercontinuum generation," Phys. Rev. Lett. 101, 233902 (2008).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf. "Ultrashort filaments of light in weakly ionized, optically transparent media," Rep. Prog. Phys. 70, 1633-1713 (2007).
[CrossRef]

Other (4)

S. Coles, An Introduction to Statistical Modeling of Extreme Values (Springer-Verlag, London, 2001).

V. Pareto, Manuale di economia politica con una introduzione alla scienza sociale (Societa Editrice Libraria, Milano 1906)

N. Akhmediev, J. M. Soto-Crespo, and A. Ankiewicz, "Extreme waves that appear from nowhere: on the nature of rogue waves," Phys. Lett. A (2009).

D. Borlaug, S. Fathpour, and B. Jalali, "Extreme value statistics and the Pareto distribution in silicon photonics," arXiv:0809.2565v1 [Phys. Optics] (2008).

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Figures (5)

Fig. 1.
Fig. 1.

Experimental setup

Fig. 2.
Fig. 2.

(a) Typical series of 500 individual spectra (gray traces), together with the calculated mean over the same 500-shot ensemble (bold line). These results were measured at the output of a filament generated by a 15 GW (i.e. 5 Pcr ) pump. (b) and (c) show time series of the intensity at (b) 815 nm in the spectrum center, and (c) 630 nm on the spectrum edge.

Fig. 3.
Fig. 3.

Histograms at (a) 630 nm (b) 815 nm and (c) 966 nm. Insets show the same data on a log-log representation to highlight the differences in the distributions – while the distribution in (b) is bell-shaped, the distributions on the edges exhibit L-shaped characteristics.

Fig. 4.
Fig. 4.

Filament spectra for peak powers as shown. (a) plots the wavelength-dependence of the metric M(λ) and (b) shows the corresponding spectra. Values of M>0.5 (dotted line) indicate the development of a long tail on the probability distribution.

Fig. 5.
Fig. 5.

(a) Mean spectrum (log scale, left axis) and Pareto metric M(λ) (right axis) for 1000 realizations of noise-induced SPM. Peak spectral intensity is 57 dB. (b) and (c) Histograms of spectral intensities at 650 nm and 1100 nm respectively. Insets use a log-log representation.

Equations (1)

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M(λ0)=Σi=1kNIi(λ0)Σi=1NIi(λ0)

Metrics