Abstract

Efforts to understand the physics of rogue waves have motivated the study of mechanisms that produce rare, extreme events, often through analogous optical setups. As many studies have reported nonlinear generation mechanisms, recent work has explored whether optical rogue events can be produced in linear systems. Here we report the observation of linear rogue events with tunable height, generated from light imprinted with a non-Markovian wavefront. Moreover, if the non-Markovian wavefront is allowed to propagate through a nonlinear medium, extraordinarily long-tailed intensity distributions are produced, which do not conform to the statistics previously observed in optical rogue wave experiments.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

S. Coulibaly, M. Taki, A. Bendahmane, G. Millot, B. Kibler, and M. G. Clerc, “Turbulence-induced rogue waves in Kerr resonators,” Phys. Rev. X 9, 011054 (2019).
[Crossref]

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[Crossref]

2018 (1)

2017 (2)

A. Safari, R. Fickler, M. J. Padgett, and R. W. Boyd, “Generation of caustics and rogue waves from nonlinear instability,” Phys. Rev. Lett. 119, 203901 (2017).
[Crossref]

H. Frostig, E. Small, A. Daniel, P. Oulevey, S. Derevyanko, and Y. Silberberg, “Focusing light by wavefront shaping through disorder and nonlinearity,” Optica 4, 1073–1079 (2017).
[Crossref]

2016 (4)

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. M. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. L. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

D. Borycki, O. Kholiqov, and V. J. Srinivasan, “Interferometric near-infrared spectroscopy directly quantifies optical field dynamics in turbid media,” Optica 3, 1471–1476 (2016).
[Crossref]

P. Suret, R. El Koussaifi, A. Tikan, C. Evain, S. Randoux, C. Szwaj, and S. Bielawski, “Single-shot observation of optical rogue waves in integrable turbulence using time microscopy,” Nat. Commun. 7, 13136 (2016).
[Crossref]

M. Liu, A.-P. Luo, W.-C. Xu, and Z.-C. Luo, “Dissipative rogue waves induced by soliton explosions in an ultrafast fiber laser,” Opt. Lett. 41, 3912–3915 (2016).
[Crossref]

2015 (8)

C. Liu, R. E. C. van der Wel, N. Rotenberg, L. Kuipers, T. F. Krauss, A. Di Falco, and A. Fratalocchi, “Triggering extreme events at the nanoscale in photonic seas,” Nat. Phys. 11, 358–363 (2015).
[Crossref]

A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, “Caustics and rogue waves in an optical sea,” Sci. Rep. 5, 12822 (2015).
[Crossref]

D. S. Agafontsev and V. E. Zakharov, “Integrable turbulence and formation of rogue waves,” Nonlinearity. 28, 2791–2821 (2015).
[Crossref]

S. Toenger, T. Godin, C. Billet, F. Dias, M. Erkintalo, G. Genty, and J. M. Dudley, “Emergent rogue wave structures and statistics in spontaneous modulation instability,” Sci. Rep. 5, 10380 (2015).
[Crossref]

D. Pierangeli, F. Di Mei, C. Conti, J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref]

D. A. Keen and A. L. Goodwin, “The crystallography of correlated disorder,” Nature 521, 303–309 (2015).
[Crossref]

R. Fischer, I. Vidal, D. Gilboa, B. Correia, R. A. C. Ribeiro-Teixeira, S. D. Prado, J. Hickman, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115, 073901 (2015).
[Crossref]

T. Eichelkraut and A. Szameit, “Photonics: random sudoku light,” Nature 526, 643–644 (2015).
[Crossref]

2014 (5)

W. F. Thompson, A. H. Monahan, and D. Crommelin, “Parametric estimation of the stochastic dynamics of sea surface winds,” J. Atmos. Sci. 71, 3465–3483 (2014).
[Crossref]

C. Lecaplain and P. Grelu, “Rogue waves among noiselike-pulse laser emission: an experimental investigation,” Phys. Rev. A 90, 013805 (2014).
[Crossref]

Y. Bromberg and H. Cao, “Generating non-Rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
[Crossref]

J. J. Metzger, R. Fleischmann, and T. Geisel, “Statistics of extreme waves in random media,” Phys. Rev. Lett. 112, 203903 (2014).
[Crossref]

J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8, 755–764 (2014).
[Crossref]

2013 (3)

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

S. Birkholz, E. T. J. Nibbering, C. Brée, S. Skupin, A. Demircan, G. Genty, and G. Steinmeyer, “Spatiotemporal rogue events in optical multiple filamentation,” Phys. Rev. Lett. 111, 243903 (2013).
[Crossref]

T. Strudley, T. Zehender, C. Blejean, E. P. A. M. Bakkers, and O. L. Muskens, “Mesoscopic light transport by very strong collective multiple scattering in nanowire mats,” Nat. Photonics 7, 413–418 (2013).
[Crossref]

2012 (4)

S. Derevyanko and E. Small, “Nonlinear propagation of an optical speckle field,” Phys. Rev. A 85, 053816 (2012).
[Crossref]

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. W. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 470–474 (2012).
[Crossref]

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

F. Baronio, A. Degasperis, M. Conforti, and S. Wabnitz, “Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves,” Phys. Rev. Lett. 109, 044102 (2012).
[Crossref]

2011 (3)

D. Majus, V. Jukna, G. Valiulis, D. Faccio, and A. Dubietis, “Spatiotemporal rogue events in femtosecond filamentation,” Phys. Rev. A 83, 025802 (2011).
[Crossref]

J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (2011).
[Crossref]

F. T. Arecchi, U. Bortolozzo, A. Montina, and S. Residori, “Granularity and inhomogeneity are the joint generators of optical rogue waves,” Phys. Rev. Lett. 106, 153901 (2011).
[Crossref]

2010 (3)

R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: a microwave study,” Phys. Rev. Lett. 104, 093901 (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]

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721–726 (2010).
[Crossref]

2009 (3)

2008 (2)

2007 (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[Crossref]

2003 (3)

A. Apostol and A. Dogariu, “Spatial correlations in the near field of random media,” Phys. Rev. Lett. 91, 093901 (2003).
[Crossref]

R. Schorr and H. Rieger, “Universal properties of shortest paths in isotropically correlated random potentials,” Eur. Phys. J. B 33, 347–354 (2003).
[Crossref]

P. Sura, “Stochastic analysis of southern and pacific ocean sea surface winds,” J. Atmos. Sci. 60, 654–666 (2003).
[Crossref]

2001 (1)

D. S. Dean and S. N. Majumdar, “Extreme-value statistics of hierarchically correlated variables deviation from Gumbel statistics and anomalous persistence,” Phys. Rev. E 64, 046121 (2001).
[Crossref]

2000 (1)

T. DelSole, “A fundamental limitation of Markov models,” J. Atmos. Sci. 57, 2158–2168 (2000).
[Crossref]

1997 (2)

S. Bian, J. Frejlich, and K. H. Ringhofer, “Photorefractive saturable Kerr-type nonlinearity in photovoltaic crystals,” Phys. Rev. Lett. 78, 4035–4038 (1997).
[Crossref]

M. Horowitz, Y. Barad, and Y. Silberberg, “Noiselike pulses with a broadband spectrum generated from an erbium-doped fiber laser,” Opt. Lett. 22, 799–801 (1997).
[Crossref]

1995 (1)

T. M. Nieuwenhuizen and M. C. W. Van Rossum, “Intensity distributions of waves transmitted through a multiple scattering medium,” Phys. Rev. E 74, 2674 (1995).
[Crossref]

1992 (1)

L. Weng, J. M. Reid, P. M. Shankar, K. Soetanto, and X. M. Lu, “Nonuniform phase distribution in ultrasound speckle analysis. I. Background and experimental demonstration,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 39, 352–359 (1992).
[Crossref]

Agafontsev, D. S.

D. S. Agafontsev and V. E. Zakharov, “Integrable turbulence and formation of rogue waves,” Nonlinearity. 28, 2791–2821 (2015).
[Crossref]

Agranat, J.

D. Pierangeli, F. Di Mei, C. Conti, J. Agranat, and E. DelRe, “Spatial rogue waves in photorefractive ferroelectrics,” Phys. Rev. Lett. 115, 093901 (2015).
[Crossref]

Akhmediev, N.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. M. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. L. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

C. Lecaplain, P. Grelu, J. M. Soto-Crespo, and N. Akhmediev, “Dissipative rogue waves generated by chaotic pulse bunching in a mode-locked laser,” Phys. Rev. Lett. 108, 233901 (2012).
[Crossref]

J. M. Soto-Crespo, P. Grelu, and N. Akhmediev, “Dissipative rogue waves: extreme pulses generated by passively mode-locked lasers,” Phys. Rev. E 84, 016604 (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]

Amiraranashvili, S.

N. Akhmediev, B. Kibler, F. Baronio, M. Belić, W.-P. Zhong, Y. Zhang, W. Chang, J. M. Soto-Crespo, P. Vouzas, P. Grelu, C. Lecaplain, K. Hammani, S. Rica, A. Picozzi, M. Tlidi, K. Panajotov, A. Mussot, A. Bendahmane, P. Szriftgiser, G. Genty, J. M. Dudley, A. Kudlinski, A. Demircan, U. Morgner, S. Amiraranashvili, C. Bree, G. Steinmeyer, C. Masoller, N. G. R. Broderick, A. F. J. Runge, M. Erkintalo, S. Residori, U. Bortolozzo, F. T. Arecchi, S. Wabnitz, C. G. L. Tiofack, S. Coulibaly, and M. Taki, “Roadmap on optical rogue waves and extreme events,” J. Opt. 18, 063001 (2016).
[Crossref]

Apostol, A.

A. Apostol and A. Dogariu, “Spatial correlations in the near field of random media,” Phys. Rev. Lett. 91, 093901 (2003).
[Crossref]

Arecchi, F.

M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. Arecchi, “Rogue waves and their generating mechanisms in different physical contexts,” Phys. Rep. 528, 47–89 (2013).
[Crossref]

Arecchi, F. T.

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Supplementary Material (1)

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» Supplement 1       Supplementary information

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Figures (6)

Fig. 1.
Fig. 1. Generating non-Markovian phase masks. (a) A one-dimensional non-Markovian phase mask is generated by solving overlapping one-dimensional sudoku puzzles of length 9. In each such puzzle, the numbers 1–9 should appear only once. (b) Two-dimensional non-Markovian phase masks are similarly generated by solving overlapping two-dimensional sudoku puzzles of length 9. In each such puzzle, the numbers 1–9 should appear only once in each row and each column. The cases shown in (a) and (b) are for an overlap of ${r} = {2}$. Inset: the probability distribution of the phases in the ${r} = {7}$ non-Markovian phase mask. The phase values are uniformly distributed in the range $[- \pi ,\pi]$ despite the non-Markovian properties.
Fig. 2.
Fig. 2. Generating rogue events with a non-Markovian light source. A 532 nm laser beam is expanded using a telescope to illuminate a two-dimensional SLM. The SLM is used to imprint a non-Markovian spatial phase onto the beam. The beam is then focused into a photorefractive crystal. The output facet of the crystal is imaged onto a CCD camera.
Fig. 3.
Fig. 3. Experimental results: one-dimensional cuts through the recorded two-dimensional intensity patterns. (a) A 1D cut through the intensity pattern recorded by the CCD for the ${r} = {0}$ phase mask, after propagation through a linear (blue), positive nonlinear (red), and negative nonlinear (green) medium. (b) Similarly, for the ${r} = {4}$ phase mask. (c) Similarly, for the ${r} = {7}$ phase mask. Dotted blue, dashed red, and dotted-dashed green lines are plotted in all figures at two times the significant wave height of the linear, positive nonlinear, and negative nonlinear distributions accordingly, denoting the threshold value for rogue waves. The combination of non-Markovianity and nonlinearity produces extremely large events, whose height can be tuned with the overlap parameter, ${r}$.
Fig. 4.
Fig. 4. Experimental results: intensity probability distributions of measured light patterns at the crystal output. (a) Probability distribution of the intensities generated after propagation through a linear (blue squares), positive nonlinear (red diamonds), and negative nonlinear (green circles) medium, for a beam imprinted with the ${r} = {0}$ phase mask. Dotted blue, dashed red, and dotted-dashed green vertical lines indicate two times the SWH of the linear, positive nonlinear, and negative nonlinear distributions, accordingly. (b) Similarly, for the ${r} = {4}$ imprinted beam. (c) Similarly, for the ${r} = {7}$ imprinted beam. (d) Comparison of the probability distributions for the ${r} = {0}$ (purple stars) and ${r} = {7}$ (turquoise circles) imprinted beams after linear propagation. Both distributions are well fitted by the equation ${P_{{\rm pow}}}(I) = \exp (- a{I^b} + c)$, with $b \lt 1$ for ${r} = {7}$, indicating super-Rayleigh statistics. (e) Comparison of the probability distributions for the ${r} = {0}$ (purple stars) and ${r} = {7}$ (turquoise circles) imprinted beams after positive nonlinear propagation. While the ${r} = {0}$ distribution is still well fitted by a power law of the form ${P_{{\rm pow}}}$, the ${r} = {7}$ distribution follows a more extreme trend and is well fitted by a function of the form ${P_{{\rm exp}}}(I) = \exp [a\exp (- bI) - c]$. The fit to a power law is shown for reference.
Fig. 5.
Fig. 5. Simulation results: intensity probability distributions of simulated light patterns at the crystal output. (a) Probability distribution of the intensities generated after propagation through a linear (blue squares), positive nonlinear (red diamonds), and negative nonlinear (green circles) medium, for a beam imprinted with the ${r} = {0}$ phase mask. (b) Similarly, for the ${r} = {4}$ imprinted beam. (c) Similarly, for the ${r} = {7}$ imprinted beam. Dotted blue, dashed red, and dotted-dashed green vertical lines indicate two times the SWH of the linear, positive nonlinear, and negative nonlinear distributions, accordingly. (d) Comparison of the probability distributions for the ${r} = {0}$ (purple stars) and ${r} = {7}$ (turquoise circles) imprinted beams after linear propagation. (e) Similarly, for positive nonlinear propagation. All distributions are fitted with the same equations as in Fig. 4. The simulation results show nice qualitative agreement with the experimental results.
Fig. 6.
Fig. 6. 1D simulation results: output intensity patterns for different 1D phase masks, after propagation through a linear (blue), positive nonlinear (red), and negative nonlinear (green) medium. (a) For a completely random phase mask. (b) For the ${r} = {0}$ phase mask. (c) For the ${r} = {4}$ phase mask. (d) For the ${r} = {7}$ phase mask. (e) For the ${r} = {8}$ phase mask. As the value of ${r}$ is increased, regions of high probability for high-intensity events emerge, causing the elongation of the intensity distribution (see Fig. S1 of Supplement 1). Nevertheless, when random assignment disappears completely, as in the ${r} = {8}$ phase mask, the Rayleigh intensity distribution reemerges. The nonlinear cases are not shown for ${r} = {8}$ for clarity.

Equations (1)

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i A z + 1 2 k 0 n 0 [ 2 x 2 + 2 y 2 ] A γ A 1 + | A | 2 / I bkg = 0 ,