Abstract

We present a numerical study of the evolution dynamics of “optical rogue waves”, statistically-rare extreme red-shifted soliton pulses arising from supercontinuum generation in photonic crystal fiber [D. R. Solli et al. Nature 450, 1054–1058 (2007)]. Our specific aim is to use nonlinear Schrödinger equation simulations to identify ways in which the rogue wave dynamics can be actively controlled, and we demonstrate that rogue wave generation can be enhanced by an order of magnitude through a small modulation across the input pulse envelope and effectively suppressed through the use of a sliding frequency filter.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  3. S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "White-light supercontinuum generation with 60-ps pump pulses in a photonic crystal fiber," Opt. Lett. 26, 1356-1358 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
  12. M. H. Frosz, O. Bang, and A. Bjarklev, "Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation," Opt. Express 14, 9391-9407 (2006).
    [CrossRef] [PubMed]
  13. N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  28. E. Seve, G. Millot and S. Wabnitz, "Buildup of terahertz vector dark-soliton trains from induced modulation instability in highly birefringent optical fiber," Opt. Lett. 23, 1829-1831 (1998).
    [CrossRef]
  29. L. P. Barry, J. M. Dudley, B. C. Thomsen and J. D. Harvey, "Frequency-resolved optical gating measurement of 1.4 THz beat frequencies from dual wavelength self-seeded gain-switched laser diode," Electron. Lett. 34, 988-990 (1998).
    [CrossRef]
  30. J. M. Dudley, F. Gutty, S. Pitois, G. Millot, "Complete characterization of terahertz pulse trains generated from nonlinear processes in optical fibers," IEEE J. Quantum Electron. 37, 587-594 (2001).
    [CrossRef]
  31. J. Fatome, S. Pitois and G. Millot, "20 GHz to 1 THz repetition rate pulse sources based on multiple four wave mixing in optical fibers," IEEE J. Quantum Electron. 42, 1038-1046 (2006).
    [CrossRef]
  32. A. S. Y. Hsieh, G. K. L. Wong, S. G. Murdoch, S. Coen, F. Vanholsbeeck, R. Leonhardt, and J. D. Harvey, "Combined effect of Raman and parametric gain on single-pump parametric amplifiers," Opt. Express 15, 8104-8114 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2007

2006

J. Fatome, S. Pitois and G. Millot, "20 GHz to 1 THz repetition rate pulse sources based on multiple four wave mixing in optical fibers," IEEE J. Quantum Electron. 42, 1038-1046 (2006).
[CrossRef]

S. M. Kobtsev and S. V. Smirnov, "Coherent properties of super-continuum containing clearly defined solitons," Opt. Express 14, 3968-3980 (2006).
[CrossRef] [PubMed]

M. Onorato, A. R. Osborne, and M. Serio, "Modulational instability in crossing sea states: A possible mechanism for the formation of freak waves," Phys. Rev. Lett. 96, 014503 (2006).
[CrossRef] [PubMed]

M. H. Frosz, O. Bang, and A. Bjarklev, "Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation," Opt. Express 14, 9391-9407 (2006).
[CrossRef] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

2005

2004

M. Hopkin, "Sea snapshots will map frequency of freak waves," Nature 430, 492 (2004).
[CrossRef] [PubMed]

2003

A. V. Avdokhin, S. V. Popov, and J. R. Taylor, "Continuous-wave, high-power, Raman continuum generation in holey fibers," Opt. Lett. 28, 1353-1355 (2003).
[CrossRef] [PubMed]

N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

G. Chang, T. B. Norris, and H. G. Winful, "Optimization of supercontinuum generation in photonic crystal fibers for pulse compression," Opt. Lett. 28, 546-548 (2003).
[CrossRef] [PubMed]

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78 (2003).
[CrossRef]

2002

2001

A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "White-light supercontinuum generation with 60-ps pump pulses in a photonic crystal fiber," Opt. Lett. 26, 1356-1358 (2001).
[CrossRef]

J. M. Dudley, F. Gutty, S. Pitois, G. Millot, "Complete characterization of terahertz pulse trains generated from nonlinear processes in optical fibers," IEEE J. Quantum Electron. 37, 587-594 (2001).
[CrossRef]

2000

1999

1998

E. Seve, G. Millot and S. Wabnitz, "Buildup of terahertz vector dark-soliton trains from induced modulation instability in highly birefringent optical fiber," Opt. Lett. 23, 1829-1831 (1998).
[CrossRef]

L. P. Barry, J. M. Dudley, B. C. Thomsen and J. D. Harvey, "Frequency-resolved optical gating measurement of 1.4 THz beat frequencies from dual wavelength self-seeded gain-switched laser diode," Electron. Lett. 34, 988-990 (1998).
[CrossRef]

1991

P. V. Mamyshev, S. V. Chernikov and E. M. Dianov, "Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines," IEEE J. Quantum Electron. 27, 2347-2355 (1991).
[CrossRef]

1989

E. J. Greer, D. M. Patrick, P. G. J. Wigley and J. R. Taylor, "Generation of 2 THz repetition rate pulse trains through induced modulational instability," Electron. Lett. 25, 1246-1248 (1989).
[CrossRef]

1967

T. B. Benjamin and J. E. Feir, "The disintegration of wavetrains in deep water," Part 1, J. Fluid Mech. 27,417-430 (1967).
[CrossRef]

Austin, D. R.

Avdokhin, A. V.

Bandelow, U.

A. Demircan and U. Bandelow, "Analysis of the interplay between soliton fission and modulation instability in supercontinuum generation," Appl. Phys. B 86, 31-39 (2007).
[CrossRef]

Bang, O.

Barry, L. P.

L. P. Barry, J. M. Dudley, B. C. Thomsen and J. D. Harvey, "Frequency-resolved optical gating measurement of 1.4 THz beat frequencies from dual wavelength self-seeded gain-switched laser diode," Electron. Lett. 34, 988-990 (1998).
[CrossRef]

Benjamin, T. B.

T. B. Benjamin and J. E. Feir, "The disintegration of wavetrains in deep water," Part 1, J. Fluid Mech. 27,417-430 (1967).
[CrossRef]

Bjarklev, A.

Bolger, J. A.

Brown, T.

Chang, G.

Chau, A. H. L.

Chernikov, S. V.

P. V. Mamyshev, S. V. Chernikov and E. M. Dianov, "Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines," IEEE J. Quantum Electron. 27, 2347-2355 (1991).
[CrossRef]

Coen, S.

A. S. Y. Hsieh, G. K. L. Wong, S. G. Murdoch, S. Coen, F. Vanholsbeeck, R. Leonhardt, and J. D. Harvey, "Combined effect of Raman and parametric gain on single-pump parametric amplifiers," Opt. Express 15, 8104-8114 (2007).
[CrossRef] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

F. Vanholsbeeck, S. Martín-López, M. González-Herráez, and S. Coen, "The role of pump incoherence in continuous-wave supercontinuum generation," Opt. Express 13, 6615-6625 (2005).
[CrossRef] [PubMed]

B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "White-light supercontinuum generation with 60-ps pump pulses in a photonic crystal fiber," Opt. Lett. 26, 1356-1358 (2001).
[CrossRef]

Corwin, K. L.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2003).
[CrossRef] [PubMed]

de Sterke, C. M.

Demircan, A.

A. Demircan and U. Bandelow, "Analysis of the interplay between soliton fission and modulation instability in supercontinuum generation," Appl. Phys. B 86, 31-39 (2007).
[CrossRef]

Dianov, E. M.

P. V. Mamyshev, S. V. Chernikov and E. M. Dianov, "Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines," IEEE J. Quantum Electron. 27, 2347-2355 (1991).
[CrossRef]

Diddams, S. A.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

J. M. Dudley, F. Gutty, S. Pitois, G. Millot, "Complete characterization of terahertz pulse trains generated from nonlinear processes in optical fibers," IEEE J. Quantum Electron. 37, 587-594 (2001).
[CrossRef]

L. P. Barry, J. M. Dudley, B. C. Thomsen and J. D. Harvey, "Frequency-resolved optical gating measurement of 1.4 THz beat frequencies from dual wavelength self-seeded gain-switched laser diode," Electron. Lett. 34, 988-990 (1998).
[CrossRef]

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.

Eggleton, B. J.

Fatome, J.

J. Fatome, S. Pitois and G. Millot, "20 GHz to 1 THz repetition rate pulse sources based on multiple four wave mixing in optical fibers," IEEE J. Quantum Electron. 42, 1038-1046 (2006).
[CrossRef]

Feder, K.

Feir, J. E.

T. B. Benjamin and J. E. Feir, "The disintegration of wavetrains in deep water," Part 1, J. Fluid Mech. 27,417-430 (1967).
[CrossRef]

Frosz, M. H.

Gaeta, A. L.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Giessen, H.

González-Herráez, M.

Greer, E. J.

E. J. Greer, D. M. Patrick, P. G. J. Wigley and J. R. Taylor, "Generation of 2 THz repetition rate pulse trains through induced modulational instability," Electron. Lett. 25, 1246-1248 (1989).
[CrossRef]

Gutty, F.

J. M. Dudley, F. Gutty, S. Pitois, G. Millot, "Complete characterization of terahertz pulse trains generated from nonlinear processes in optical fibers," IEEE J. Quantum Electron. 37, 587-594 (2001).
[CrossRef]

Hansen, K. P.

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78 (2003).
[CrossRef]

Harvey, J. D.

Herrmann, J.

Hopkin, M.

M. Hopkin, "Sea snapshots will map frequency of freak waves," Nature 430, 492 (2004).
[CrossRef] [PubMed]

Hsieh, A. S. Y.

Husakou, A.

Husakou, A. V.

A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical Rogue Waves," Nature 450, 1054-1058 (2007).
[CrossRef] [PubMed]

Kibler, B.

B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

Knight, J. C.

Kobtsev, S. M.

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical Rogue Waves," Nature 450, 1054-1058 (2007).
[CrossRef] [PubMed]

Kuhlmey, B. T.

Kutz, J. N.

Leonhardt, R.

Li, Y.

Limpert, J.

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78 (2003).
[CrossRef]

Luan, F.

Lyngå, C.

Mamyshev, P. V.

P. V. Mamyshev, S. V. Chernikov and E. M. Dianov, "Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines," IEEE J. Quantum Electron. 27, 2347-2355 (1991).
[CrossRef]

Marshall, G. D.

Martín-López, S.

Millot, G.

J. Fatome, S. Pitois and G. Millot, "20 GHz to 1 THz repetition rate pulse sources based on multiple four wave mixing in optical fibers," IEEE J. Quantum Electron. 42, 1038-1046 (2006).
[CrossRef]

J. M. Dudley, F. Gutty, S. Pitois, G. Millot, "Complete characterization of terahertz pulse trains generated from nonlinear processes in optical fibers," IEEE J. Quantum Electron. 37, 587-594 (2001).
[CrossRef]

E. Seve, G. Millot and S. Wabnitz, "Buildup of terahertz vector dark-soliton trains from induced modulation instability in highly birefringent optical fiber," Opt. Lett. 23, 1829-1831 (1998).
[CrossRef]

Murdoch, S. G.

Newbury, N. R.

N. R. Newbury, B. R. Washburn, K. L. Corwin, and R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, "Fundamental noise limitations to supercontinuum generation in microstructure fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Norris, T. B.

Onorato, M.

M. Onorato, A. R. Osborne, and M. Serio, "Modulational instability in crossing sea states: A possible mechanism for the formation of freak waves," Phys. Rev. Lett. 96, 014503 (2006).
[CrossRef] [PubMed]

Osborne, A. R.

M. Onorato, A. R. Osborne, and M. Serio, "Modulational instability in crossing sea states: A possible mechanism for the formation of freak waves," Phys. Rev. Lett. 96, 014503 (2006).
[CrossRef] [PubMed]

Patrick, D. M.

E. J. Greer, D. M. Patrick, P. G. J. Wigley and J. R. Taylor, "Generation of 2 THz repetition rate pulse trains through induced modulational instability," Electron. Lett. 25, 1246-1248 (1989).
[CrossRef]

Pitois, S.

J. Fatome, S. Pitois and G. Millot, "20 GHz to 1 THz repetition rate pulse sources based on multiple four wave mixing in optical fibers," IEEE J. Quantum Electron. 42, 1038-1046 (2006).
[CrossRef]

J. M. Dudley, F. Gutty, S. Pitois, G. Millot, "Complete characterization of terahertz pulse trains generated from nonlinear processes in optical fibers," IEEE J. Quantum Electron. 37, 587-594 (2001).
[CrossRef]

Popov, S. V.

Pricking, S.

Ranka, J. K.

Ropers, C.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical Rogue Waves," Nature 450, 1054-1058 (2007).
[CrossRef] [PubMed]

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The three parameter Weibull distribution is described by the probability density function f(x) = ?/? [(x-?)/?]??1 exp(- [(x-?)/?]?) , defined on ? <x< +¥ where ?, ?, ? are shape, scale and location parameters respectively. For this data, maximum likelihood estimation yields parameters: ?=0.4515, ?=40.11, ?=0.004719. The compatibility of this distribution with the data in Fig. 1(c) was confirmed using a Kolmogorov-Smirnov test, and the null hypothesis was accepted at the 0.05 significance level. The fit in Fig. 1(c) is normalized for comparison with the histogram.

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Figures (4)

Fig. 1.
Fig. 1.

(a) Results showing 1000 individual spectra (gray curves). The mean spectrum is shown as the solid black line. (b) Expanded view above 1210 nm. (c) Histogram of the peak power frequency distribution using 25 W bins. We plot normalized frequency such that bar height represents the proportion of data in each bin. The inset plots the results on a log-log scale, and also shows the fitted Weibull distribution (solid line) [24].

Fig. 2.
Fig. 2.

Density plots of spectral and temporal evolution for: (a) a rare event leading to a rogue soliton (RS) and (b) a result close to the distribution median.

Fig. 3.
Fig. 3.

(a) MI gain curve and (b) density plot of output spectra with a 4% envelope modulation at the frequency indicated. Subfigures (i)–(iii) show spectra at the frequencies indicated.

Fig. 4.
Fig. 4.

Results for (a) an induced modulation at 5.8 THz and (b) a sliding frequency filter. The left panels show individual spectra (gray curves) and the corresponding mean spectrum (solid line) and the middle panels shows expanded views of the long wavelength edge. The right panels show the normalized frequency distribution of the peak power after spectral filtering with the insets plotting results on a log-log scale. These histograms should be compared with Fig. 1(c) to emphasize the relative rogue wave enhancement and suppression.

Equations (1)

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A z k 2 i k + 1 k ! β k k A t k = i γ ( 1 + i τ shock t ) ( A ( z , t ) + R ( t ) A ( z , t t ) 2 d t ) .

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