Abstract

We report a numerical study showing how the random intensity and phase fluctuations across the bandwidth of a broadband optical super-continuum can be interpreted in terms of the random processes of random walks and Lévy flights. We also describe how the intensity fluctuations can be applied to physical random number generation. We conclude that the optical supercontinuum provides a highly versatile means of studying and generating a wide class of random processes at optical wavelengths.

© 2012 OSA

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  22. C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett.45, 217–218 (2009).
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2011

K. Hammani, A. Picozzi, and C. Finot, “Extreme statistics in Raman fiber amplifiers: From analytical description to experiments,” Opt. Commun.284, 2594–2603 (2011).
[CrossRef]

X. Li, A. B. Cohen, T. E. Murphy, and R. Roy, “Scalable parallel physical random number generator based on a superluminescent LED,” Opt. Lett.36, 1020–1022 (2011).
[CrossRef] [PubMed]

2010

M. H. Frosz, “Validation of input-noise model for simulations of supercontinuum generation and rogue waves,” Opt. Express18, 14778–14787 (2010).
[CrossRef] [PubMed]

P. Li, Y.-C. Wang, and J.-Z. Zhang, “All-optical fast random number generator,” Opt. Express18, 20360–20369 (2010).
[CrossRef] [PubMed]

C. R. S. Williams, J. C. Salevan, X. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express18, 23584–23597 (2010).
[CrossRef] [PubMed]

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top.185, 135–144 (2010).
[CrossRef]

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4, 58–61 (2010).
[CrossRef]

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

N. Akhmediev and E. Pelinovsky, “Editorial - Introductory remarks on Discussion & Debate: Rogue waves -Towards a unifying concept?,” Eur. Phys. J. Spec. Top.185, 1–4 (2010).
[CrossRef]

2009

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics3, 85–90 (2009).
[CrossRef]

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett.45, 217–218 (2009).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express17, 21497–21508 (2009).
[CrossRef] [PubMed]

N. Mercadier, W. Guerin, M. Chevrollier, and R. Kaiser, “Lévy flights of photons in hot atomic vapours,” Nat. Phys.5, 602–605 (2009).
[CrossRef]

2008

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express16, 3644–3651 (2008).
[CrossRef] [PubMed]

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature453, 495–498 (2008).
[CrossRef] [PubMed]

E. A. Codling, M. J. Plank, and S. Benhamou, “Random walk models in biology,” J. R. Soc. Interface5, 813–834 (2008).
[CrossRef] [PubMed]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

2007

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature450, 1054–1058 (2007).
[CrossRef] [PubMed]

2006

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys78, 1135–1184 (2006).
[CrossRef]

2002

M. N. Islam and O. Boyraz, “Fiber parametric amplifiers for wavelength band conversion,” IEEE J. Sel. Top. Quantum Electron.8, 527–537 (2002).
[CrossRef]

J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–1182 (2002).
[CrossRef]

1999

M. F. Shlesinger, J. Klafter, and G. Zumofen, “Above, below and beyond Brownian motion,” Am. J. Phys.67, 1253–1259 (1999).
[CrossRef]

1996

G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley, “Levy flight search patterns of wandering albatrosses,” Nature381, 413–415 (1996).
[CrossRef]

1991

1989

B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrödinger equation,” Phys. Rev. Lett.62, 2065–2068 (1989).
[CrossRef] [PubMed]

1983

G. H. Weiss and R. J. Rubin, “Random-walks - Theory and selected applications,” Adv. Chem. Phys.52, 363–505 (1983).
[CrossRef]

1905

K. Pearson, “The problem of the random walk,” Nature72, 294 (1905).
[CrossRef]

Ablowitz, M. J.

B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrödinger equation,” Phys. Rev. Lett.62, 2065–2068 (1989).
[CrossRef] [PubMed]

Afanasyev, V.

G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley, “Levy flight search patterns of wandering albatrosses,” Nature381, 413–415 (1996).
[CrossRef]

Akhmediev, N.

N. Akhmediev and E. Pelinovsky, “Editorial - Introductory remarks on Discussion & Debate: Rogue waves -Towards a unifying concept?,” Eur. Phys. J. Spec. Top.185, 1–4 (2010).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express17, 21497–21508 (2009).
[CrossRef] [PubMed]

Amano, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Andersen, U. L.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

Aviad, Y.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4, 58–61 (2010).
[CrossRef]

Barthelemy, P.

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature453, 495–498 (2008).
[CrossRef] [PubMed]

Benhamou, S.

E. A. Codling, M. J. Plank, and S. Benhamou, “Random walk models in biology,” J. R. Soc. Interface5, 813–834 (2008).
[CrossRef] [PubMed]

Bertolotti, J.

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature453, 495–498 (2008).
[CrossRef] [PubMed]

Bolger, J.

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett.45, 217–218 (2009).
[CrossRef]

Boyraz, O.

M. N. Islam and O. Boyraz, “Fiber parametric amplifiers for wavelength band conversion,” IEEE J. Sel. Top. Quantum Electron.8, 527–537 (2002).
[CrossRef]

Buldyrev, S. V.

G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley, “Levy flight search patterns of wandering albatrosses,” Nature381, 413–415 (1996).
[CrossRef]

Chevrollier, M.

N. Mercadier, W. Guerin, M. Chevrollier, and R. Kaiser, “Lévy flights of photons in hot atomic vapours,” Nat. Phys.5, 602–605 (2009).
[CrossRef]

Codling, E. A.

E. A. Codling, M. J. Plank, and S. Benhamou, “Random walk models in biology,” J. R. Soc. Interface5, 813–834 (2008).
[CrossRef] [PubMed]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys78, 1135–1184 (2006).
[CrossRef]

J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–1182 (2002).
[CrossRef]

Cohen, A. B.

Cohen, E.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4, 58–61 (2010).
[CrossRef]

Cover, T.

T. Cover and J. Thomas, Elements of Information Theory (Wiley & Sons, New York, 1991).
[CrossRef]

Davis, P.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Dias, F.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett.45, 217–218 (2009).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express17, 21497–21508 (2009).
[CrossRef] [PubMed]

Dong, R.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

Dudley, J. M.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top.185, 135–144 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express17, 21497–21508 (2009).
[CrossRef] [PubMed]

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett.45, 217–218 (2009).
[CrossRef]

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics3, 85–90 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express16, 3644–3651 (2008).
[CrossRef] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys78, 1135–1184 (2006).
[CrossRef]

J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–1182 (2002).
[CrossRef]

J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers (Cambridge University Press, 2010).
[CrossRef]

Eggleton, B. J.

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett.45, 217–218 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express16, 3644–3651 (2008).
[CrossRef] [PubMed]

Erkintalo, M.

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top.185, 135–144 (2010).
[CrossRef]

Fatome, J.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

Finot, C.

K. Hammani, A. Picozzi, and C. Finot, “Extreme statistics in Raman fiber amplifiers: From analytical description to experiments,” Opt. Commun.284, 2594–2603 (2011).
[CrossRef]

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

Frosz, M. H.

Gabriel, C.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

Genty, G.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top.185, 135–144 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express17, 21497–21508 (2009).
[CrossRef] [PubMed]

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett.45, 217–218 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express16, 3644–3651 (2008).
[CrossRef] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys78, 1135–1184 (2006).
[CrossRef]

Guerin, W.

N. Mercadier, W. Guerin, M. Chevrollier, and R. Kaiser, “Lévy flights of photons in hot atomic vapours,” Nat. Phys.5, 602–605 (2009).
[CrossRef]

Hammani, K.

K. Hammani, A. Picozzi, and C. Finot, “Extreme statistics in Raman fiber amplifiers: From analytical description to experiments,” Opt. Commun.284, 2594–2603 (2011).
[CrossRef]

Herbst, B. M.

B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrödinger equation,” Phys. Rev. Lett.62, 2065–2068 (1989).
[CrossRef] [PubMed]

Hirano, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Inoue, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Islam, M. N.

M. N. Islam and O. Boyraz, “Fiber parametric amplifiers for wavelength band conversion,” IEEE J. Sel. Top. Quantum Electron.8, 527–537 (2002).
[CrossRef]

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature450, 1054–1058 (2007).
[CrossRef] [PubMed]

Kaiser, R.

N. Mercadier, W. Guerin, M. Chevrollier, and R. Kaiser, “Lévy flights of photons in hot atomic vapours,” Nat. Phys.5, 602–605 (2009).
[CrossRef]

Kanter, I.

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4, 58–61 (2010).
[CrossRef]

Kibler, B.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev breathers and continuous wave supercontinuum generation,” Opt. Express17, 21497–21508 (2009).
[CrossRef] [PubMed]

Klafter, J.

M. F. Shlesinger, J. Klafter, and G. Zumofen, “Above, below and beyond Brownian motion,” Am. J. Phys.67, 1253–1259 (1999).
[CrossRef]

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature450, 1054–1058 (2007).
[CrossRef] [PubMed]

Kurashige, T.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Lafargue, C.

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett.45, 217–218 (2009).
[CrossRef]

Leuchs, G.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

Li, P.

Li, X.

Marquardt, C.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

Martino, A.

Mauerer, W.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

Mercadier, N.

N. Mercadier, W. Guerin, M. Chevrollier, and R. Kaiser, “Lévy flights of photons in hot atomic vapours,” Nat. Phys.5, 602–605 (2009).
[CrossRef]

Millot, G.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

Morris, G.

Murphy, E. J.

G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley, “Levy flight search patterns of wandering albatrosses,” Nature381, 413–415 (1996).
[CrossRef]

Murphy, T. E.

Naito, S.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Oowada, I.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Pearson, K.

K. Pearson, “The problem of the random walk,” Nature72, 294 (1905).
[CrossRef]

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N. Akhmediev and E. Pelinovsky, “Editorial - Introductory remarks on Discussion & Debate: Rogue waves -Towards a unifying concept?,” Eur. Phys. J. Spec. Top.185, 1–4 (2010).
[CrossRef]

Picozzi, A.

K. Hammani, A. Picozzi, and C. Finot, “Extreme statistics in Raman fiber amplifiers: From analytical description to experiments,” Opt. Commun.284, 2594–2603 (2011).
[CrossRef]

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E. A. Codling, M. J. Plank, and S. Benhamou, “Random walk models in biology,” J. R. Soc. Interface5, 813–834 (2008).
[CrossRef] [PubMed]

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G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley, “Levy flight search patterns of wandering albatrosses,” Nature381, 413–415 (1996).
[CrossRef]

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I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4, 58–61 (2010).
[CrossRef]

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D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature450, 1054–1058 (2007).
[CrossRef] [PubMed]

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I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4, 58–61 (2010).
[CrossRef]

Roy, R.

Rubin, R. J.

G. H. Weiss and R. J. Rubin, “Random-walks - Theory and selected applications,” Adv. Chem. Phys.52, 363–505 (1983).
[CrossRef]

Salevan, J. C.

Shiki, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

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M. F. Shlesinger, J. Klafter, and G. Zumofen, “Above, below and beyond Brownian motion,” Am. J. Phys.67, 1253–1259 (1999).
[CrossRef]

Solli, D. R.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature450, 1054–1058 (2007).
[CrossRef] [PubMed]

Someya, H.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Stanley, H. E.

G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley, “Levy flight search patterns of wandering albatrosses,” Nature381, 413–415 (1996).
[CrossRef]

Sych, D.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

Taylor, J. R.

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics3, 85–90 (2009).
[CrossRef]

J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers (Cambridge University Press, 2010).
[CrossRef]

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T. Cover and J. Thomas, Elements of Information Theory (Wiley & Sons, New York, 1991).
[CrossRef]

Uchida, A.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Viswanathan, G. M.

G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley, “Levy flight search patterns of wandering albatrosses,” Nature381, 413–415 (1996).
[CrossRef]

Wang, Y.-C.

Weiss, G. H.

G. H. Weiss and R. J. Rubin, “Random-walks - Theory and selected applications,” Adv. Chem. Phys.52, 363–505 (1983).
[CrossRef]

Wiersma, D. S.

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature453, 495–498 (2008).
[CrossRef] [PubMed]

Williams, C. R. S.

Wittmann, C.

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

Yoshimori, S.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Yoshimura, K.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

Zhang, J.-Z.

Zumofen, G.

M. F. Shlesinger, J. Klafter, and G. Zumofen, “Above, below and beyond Brownian motion,” Am. J. Phys.67, 1253–1259 (1999).
[CrossRef]

Adv. Chem. Phys.

G. H. Weiss and R. J. Rubin, “Random-walks - Theory and selected applications,” Adv. Chem. Phys.52, 363–505 (1983).
[CrossRef]

Am. J. Phys.

M. F. Shlesinger, J. Klafter, and G. Zumofen, “Above, below and beyond Brownian motion,” Am. J. Phys.67, 1253–1259 (1999).
[CrossRef]

Appl. Opt.

Electron. Lett.

C. Lafargue, J. Bolger, G. Genty, F. Dias, J. M. Dudley, and B. J. Eggleton, “Direct detection of optical rogue wave energy statistics in supercontinuum generation,” Electron. Lett.45, 217–218 (2009).
[CrossRef]

Eur. Phys. J. Spec. Top.

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top.185, 135–144 (2010).
[CrossRef]

N. Akhmediev and E. Pelinovsky, “Editorial - Introductory remarks on Discussion & Debate: Rogue waves -Towards a unifying concept?,” Eur. Phys. J. Spec. Top.185, 1–4 (2010).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

M. N. Islam and O. Boyraz, “Fiber parametric amplifiers for wavelength band conversion,” IEEE J. Sel. Top. Quantum Electron.8, 527–537 (2002).
[CrossRef]

J. R. Soc. Interface

E. A. Codling, M. J. Plank, and S. Benhamou, “Random walk models in biology,” J. R. Soc. Interface5, 813–834 (2008).
[CrossRef] [PubMed]

Nat. Photonics

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics3, 85–90 (2009).
[CrossRef]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008).
[CrossRef]

I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4, 58–61 (2010).
[CrossRef]

C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, “A generator for unique quantum random numbers based on vacuum states,” Nat. Photonics4, 711–715 (2010).
[CrossRef]

Nat. Phys.

B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, “The Peregrine soliton in nonlinear fibre optics,” Nat. Phys.6, 790–795 (2010).
[CrossRef]

N. Mercadier, W. Guerin, M. Chevrollier, and R. Kaiser, “Lévy flights of photons in hot atomic vapours,” Nat. Phys.5, 602–605 (2009).
[CrossRef]

Nature

P. Barthelemy, J. Bertolotti, and D. S. Wiersma, “A Lévy flight for light,” Nature453, 495–498 (2008).
[CrossRef] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature450, 1054–1058 (2007).
[CrossRef] [PubMed]

G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, and H. E. Stanley, “Levy flight search patterns of wandering albatrosses,” Nature381, 413–415 (1996).
[CrossRef]

K. Pearson, “The problem of the random walk,” Nature72, 294 (1905).
[CrossRef]

Opt. Commun.

K. Hammani, A. Picozzi, and C. Finot, “Extreme statistics in Raman fiber amplifiers: From analytical description to experiments,” Opt. Commun.284, 2594–2603 (2011).
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Opt. Lett.

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[CrossRef]

J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers (Cambridge University Press, 2010).
[CrossRef]

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A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, A. Heckert, J. Dray, and S. Vo, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. rep., National Institute Of Standards And Technology (NIST) Special Publication 800-22 Revision 1a (2010) http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html .

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Figures (5)

Fig. 1
Fig. 1

(a) Individual output spectra from each of 1000 individual realizations (gray) and calculated mean (black). The mutual coherence is plotted in the upper subfigure. Histograms from all 200000 events calculated from the energy filtered from a 20 nm bandpass filter with central wavelength (b) λ =880 nm; (c) λ =1100 nm; (d) λ =1270 nm.

Fig. 2
Fig. 2

(a) Representation of 20 walks of 1000 unit steps in the complex plane based only on phase fluctuations in the SC using a fixed imposed unit step length. The phase is extracted from the SC at 1100 nm. (b) Mean squared displacement (MSD) plotted as a function of step number calculated from an ensemble average of 200 realizations. Results for the phase extracted from three wavelengths are shown: 880 nm (black diamonds), 1100 nm (blue circles), 1270 nm (red squares). The black line shows the expected MSD 〈|r(n)|2〉 = n for an ideal random walk of unit steps.

Fig. 3
Fig. 3

For wavelengths of (a) 880 nm, (b) 1100 nm, and (c) 1270 nm, the upper panels show results of 20 walks of 1000 steps. The lower panels show the corresponding mean square displacement (MSD) plotted as a function of step number calculated from an ensemble average of 200 realizations.

Fig. 4
Fig. 4

Plotting successive zooms about turning and clustering points as shown for trajectories at (a) 1270 nm and (b) 880 nm reveals the qualitative features of fractal scale invariance.

Fig. 5
Fig. 5

Schematic showing random number number generation from spectral instabilities. (a) Results from 50000 simulations (gray) and calculated mean (black). The red line shows a particular sample wavelength of 1140 nm. (b) For this wavelength, we construct a time series from individual realizations in the ensemble and calculate a rolling median to determine a threshold over the first 50000 realizations. (c) Subsequent intensity values at this wavelength are compared to this threshold to yield a binary sequence.

Tables (1)

Tables Icon

Table 1 NIST benchmark tests results for 200 sequences of 106 bits. For 200 sequences and a significance level α = 0.01, the P-value (uniformity of p-values) should be larger than 0.001 and a proportion of 193 test success for the whole benchmark (115 for random excursion (variant) tests) is required to succeed statistical tests. Note: In case of a test producing multiple result outputs, the worst case is shown.

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