## Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation

Optics Express, Vol. 17, Issue 24, pp. 21497-21508 (2009)

http://dx.doi.org/10.1364/OE.17.021497

Acrobat PDF (1100 KB)

### Abstract

Numerical simulations of the onset phase of continuous wave supercontinuum generation from modulation instability show that the structure of the field as it develops can be interpreted in terms of the properties of Akhmediev Breathers. Numerical and analytical results are compared with experimental measurements of spectral broadening in photonic crystal fiber using nanosecond pulses.

© 2009 OSA

## 1. Introduction

3. G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. **87**(3), 033902 (
2001). [CrossRef] [PubMed]

6. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. **56**(2), 135–138 (
1986). [CrossRef] [PubMed]

8. G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. **59**(8), 880–883 (
1987). [CrossRef] [PubMed]

9. A. Demircan and U. Bandelow, “Supercontinuum generation by the modulation instability,” Opt. Commun. **244**(1-6), 181–185 (
2005). [CrossRef]

11. J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400W continuous wave fiber laser,” Opt. Express **16**(19), 14435–14447 (
2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14435. [CrossRef] [PubMed]

12. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature **450**(7172), 1054–1057 (
2007). [CrossRef] [PubMed]

13. J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express **16**(6), 3644–3651 (
2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-6-3644U. [CrossRef] [PubMed]

3. G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. **87**(3), 033902 (
2001). [CrossRef] [PubMed]

26. D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave-mixing processes in an optical fiber,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **57**(4), 4757–4774 (
1998). [CrossRef]

## 2. Theoretical summary and the breather dynamics of modulation instability

*A*|

^{2}has dimensions of instantaneous power in W and the dispersion and nonlinearity coefficients

*β*(< 0) and

_{2}*γ*have dimensions of ps

^{2}km

^{−1}and W

^{−1}km

^{−1}respectively. The Akhmediev Breather is an exact analytic solution describing the evolution with

*z*of a wave with initial constant amplitude on which is superimposed a small periodic perturbation taking the form of a

*T*-dependent modulation with amplitude that is a fraction of that of the CW field. The solution consists of an evolving train of ultrashort pulses that is periodic in time (

*T*) and that exhibits a FPU-like growth-return cycle in propagation distance (

*z*). In contrast to soliton solutions which are localized in

*T*, the ideal FPU growth-return behavior localizes the breather solution in the

*z*direction. As we shall see below, certain initial conditions can also yield periodic evolution along the

*z*-direction, but even in this case, the pulse train characteristics nonetheless remain well-described by the analytic AB solution.

14. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. **69**(2), 1089–1093 (
1986). [CrossRef]

*z*< ∞. Equation (2) represents a family of solutions with a variable independent parameter

*ω*

_{mod}which is a perturbation frequency (of the initial temporal modulation). The coefficients

*a*and

*b*depend on

*ω*

_{mod}and are defined by: 2

*a*= [1 – (

*ω*

_{mod}/

*ω*

_{c})

^{2}] and

*b*= [8

*a*(1-2

*a*)]

^{1/2}with

*ω*

_{c}

^{2}= 4

*γ P*

_{0}/ |

*β*

_{2}| and

*P*

_{0}the power of the CW field at large |

*z*|. The solution is valid over the range of modulation frequencies that experience MI gain:

*ω*

_{c}>

*ω*

_{mod}> 0. The coefficient

*a*then varies in the interval 0 <

*a*< 1/2 while the parameter

*b*> 0 governs the MI growth. The maximum gain condition

*b*= 1 occurs for

*a*= 1/4, i.e.

*ω*

_{mod}=

*ω*

_{c}/2. Note that these conditions that follow from the properties of Eq. (2) are identical to the well-known results obtained using linear stability analysis [2,27]. The solution in Eq. (2) describes an evolving periodic train of ultrashort pulses with temporal period

*T*

_{mod}= 2π/

*ω*

_{mod}. The individual temporal peaks or sub-pulses have maximum amplitude and minimum temporal width at z = 0. The solution at this point describes the “maximally-compressed” AB, and is given by:

*A*(

*z = z*

_{0},

*T*) =

*α*

_{mod}cos(

*ω*

_{mod}

*T*)], where the modulation parameter

*α*

_{mod}is in general a small complex number. When

*α*

_{mod}depends on

*ω*

_{mod}and takes the particular form

*α*

_{mod}=

*μ*[1 + i(2

*b*/

*ω*

_{mod}

^{2})] with

*μ*real, the evolution is described by the exact result in Eq. (2) which sweeps a heteroclinic orbit in terms of nonlinear dynamics. Any other form for the parameter

*α*

_{mod}does not lead to an ideal return to steady state and decay to a plane wave, but rather yields a more complex solution that lies near the heteroclinic orbit with periodicity along the

*z*-direction; that is, we see multiple growth-return cycles and periodic energy exchange between sidebands. Albeit more complex, this evolution can still be fully described analytically in terms of Jacobi elliptic functions (see Eq. (18) of Ref [14

14. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. **69**(2), 1089–1093 (
1986). [CrossRef]

28. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. **72**(2), 809–818 (
1987). [CrossRef]

*α*

_{mod}as initial condition show that Eqs. (2) and (3) nonetheless accurately describe the evolution over at least the first growth-return cycle and, moreover, the pulse train at the point of maximum initial temporal compression can be predicted near-exactly by the analytic solution in Eq. (2) for modulation amplitudes over a large range 0 <

*α*

_{mod}< 0.25. This is an important observation, because real-valued initial modulation amplitudes correspond to conditions that are more straightforward to achieve experimentally using dual-frequency excitation [32

32. J. M. Dudley, F. Gutty, S. Pitois, and G. Millot, “Complete characterization of THz pulse trains generated from nonlinear processes in optical fibers,” IEEE J. Quantum Electron. **37**(4), 587–594 (
2001). [CrossRef]

*α*. The simulations use standard single mode fiber parameters at 1550 nm with

_{mod}*β*

_{2}= –20 ps

^{2}km

^{−1},

*γ*= 1.1 W

^{−1}km

^{−1}, and we assume

*P*

_{0}= 30 W. With these parameters, MI gain is observed for modulation frequencies

*f*

_{mod}=

*ω*

_{mod}/2π in the range 0 –

*f*

_{c}where

*f*

_{c}=

*ω*

_{c}/2π = 408.88 GHz. Maximum gain at

*a*= 0.25 corresponds to modulation frequency

*f*

_{mod}

*= f*

_{c}/2 = 289.12 GHz. In contrast to Eqs. (2) and (3) above, these figures are plotted with the origin

*z*= 0 corresponding to the point of injection of the initial field, and the propagation coordinate

*z*is scaled relative to the characteristic nonlinear length

*L*

_{NL}=

*(γP*

_{0})

^{−1}= 30.3 m.

*f*

_{mod}= 289.12 GHz at the peak of the MI gain, but we vary the modulation parameter

*α*

_{mod}. We use a grayscale representation to show the evolution of both the temporal intensity |

*A(z,T)*|

^{2}and spectral intensity

*z*and the graphs below the grayscale figures compare the shape of the pulse train at the point of maximal temporal compression and spectral expansion (indicated by arrows) with the predicted maximally-compressed AB solution given by Eq. (3). The horizontal axes on the grayscale plots are normalized relative to

*f*

_{mod}. The results show how increasing

*α*

_{mod}decreases the characteristic distance over which periodic growth-return occurs. Figure 1(c) also illustrates how disagreement between numerical simulations and the predictions of the AB theory begins to become apparent as the modulation amplitude increases beyond

*α*

_{mod}= 0.25.

*α*= 0.01 (where the AB theory provides an accurate description of the dynamics), but we vary the parameter

_{mod}*a*to span a range of modulation frequencies

*f*

_{mod}_{.}We can thus see how the qualitative form of the AB solution changes across the MI gain band. Specifically, Fig. 2 (a) shows the minimal reshaping observed as the frequency approaches the upper limit of MI gain, whilst Figs. 2(b) and (c) show cases of decreasing modulation frequency, which would approach the Peregrine soliton in the limit

*a*= 0.5 [31]. The evolution is plotted so as to highlight the growth up to the first point of maximal temporal compression. In interpreting Fig. 2, note that the characteristic distance of the initial growth phase up to this point itself depends on

*f*, asymptotically increasing to ∞ at

_{mod}*a*= 0 and

*a*= 0.5, whilst attaining its minimal value at the peak of the MI gain when

*a*= 0.25; recall that the evolution for this latter case is shown in Fig. 1(a).

14. N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. **69**(2), 1089–1093 (
1986). [CrossRef]

**69**(2), 1089–1093 (
1986). [CrossRef]

## 3. Akhmediev Breather dynamics and the onset of supercontinuum generation

10. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**(4), 1135–1184 (
2006). [CrossRef]

33. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. **25**(1), 25–27 (
2000). [CrossRef] [PubMed]

10. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**(4), 1135–1184 (
2006). [CrossRef]

10. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**(4), 1135–1184 (
2006). [CrossRef]

_{0}= 43 W and fiber parameters

*β*

_{2}= −75 ps

^{2}km

^{−1}and

*γ*= 60 W

^{−1}km

^{−1}. The corresponding temporal period of 758 fs is much shorter than the 1 ns duration of the input pulses so that these experimental conditions are close to the ideal CW excitation case where AB theory would be expected to be valid. We also remark that we can neglect higher-order dispersion because the pump wavelength of 1064 nm is far from the fiber zero dispersion wavelength around 780 nm.

*f*

_{mod}= 1.32 THz. The spectral bandwidth first reaches its maximum extension at a distance of 3.5 m where we extract the corresponding temporal and spectral profiles.

*z*and

*T*is not possible, but we find that the highest amplitude peaks in the temporal profile are nonetheless fitted very well by the maximally-compressed AB sub-pulses calculated from Eq. (3) using

*f*

_{mod}= 1.32 THz at the peak of the MI gain. This is shown explicitly in the expanded view of the shaded section in Fig. 4(a). In fact, analysis of the highest-amplitude peaks at other regions of the profile (not shown) reveals similarly good agreement with the calculated maximally-compressed AB solution, and we have confirmed that simulations using different noise seeds reveal the same characteristics. Moreover, we have also checked that the maximally-compressed AB characteristics at peak MI gain fit the highest peaks of the modulated field profiles for simulations of MI over a much wider source and fiber parameter range, indicating that this is a general characteristic of the MI process. Indeed, this can be understood in physical terms because the ABs associated with the peak MI gain frequency experience a higher growth rate and thus would be expected to make a predominant contribution to the field at the point of initial spectral expansion.

*P*

_{max}= 250 W and

*Δτ*= 115 fs (FWHM), we estimate

*N*= [

*γ P*

_{0}(

*Δτ*/1.763)

^{2}/ |

*β*

_{2}]

^{1/2}= 0.92. This now allows us to propose a physical scenario which may be useful in interpreting the development of a fully developed broadband supercontinuum from a regime of extended MI. Specifically, we would expect that subsequent perturbations to the MI-generated breather structures would stimulate their evolution towards ideal fundamental solitons, subject of course to the fact that the chaotic initial conditions, higher-order dispersion, Raman effects and self-steepening result in different pulse accelerations such that collisions can also play a role in the subsequent dynamics. Of course, higher-order dispersion can also mediate the generation of dispersive wave radiation [10

**78**(4), 1135–1184 (
2006). [CrossRef]

34. J. N. Kutz, C. Lyngå, and B. J. Eggleton, “Enhanced supercontinuum generation through dispersion-management,” Opt. Express **13**(11), 3989–3998 (
2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-3989. [CrossRef] [PubMed]

35. N. Korneev, E. A. Kuzin, B. Ibarra-Escamilla, M. Bello-Jiménez, and A. Flores-Rosas, “Initial development of supercontinuum in fibers with anomalous dispersion pumped by nanosecond-long pulses,” Opt. Express **16**(4), 2636–2645 (
2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-4-2636. [CrossRef] [PubMed]

*f*

_{mod}, it is possible to obtain a general analytic description showing that the AB spectrum consists of discrete frequency sideband modes with separation

*f*

_{mod}, and intensities that decrease following a geometric progression [14

**69**(2), 1089–1093 (
1986). [CrossRef]

*S*

_{0}=

*P*

_{0}(2-1)

^{2}for the pump component, and

*S*

_{n}= 2

*P*

_{0}(2-1)

^{2|n|}(where

*n*= ± 1, ± 2, ± 3, …) for the sidebands. Thus the pump and sideband intensities follow the relative progression {

*I*

_{0},

*I*

_{1},

*I*

_{2},

*I*

_{3},

*I*

_{4}…} = {1, 2, 0.3431, 0.0589, 0.0101…} so that there is a 3 dB increase from the pump (n = 0) to the first sideband (n = ± 1) and then a constant decrease of 20 log

_{10}(2-1) = −7.66 dB between subsequent sidebands. This geometric progression describing the decrease in sideband amplitudes yields a characteristic triangular shape in the wings of the spectrum when plotted semi-logarithmically. These spectral characteristics are plotted in Fig. 5 for the case of the maximally-compressed AB described above with

*f*

_{mod}= 1.32 THz.

*P*

_{0}= 43 W where we would expect analysis in terms of NLSE propagation and AB characteristics to be valid. The experimental results here are shown as the solid black line, and we first compare these with two sets of numerical simulations: the previous simulations shown using the full GNLSE including Raman scattering and assuming a 1 ns pulse input field (blue, short dashes), and also simulations using only the NLSE with a CW input field (red, long dashes). Note that the single-shot simulations exhibit fine-structure in the spectra similar to that shown above in Fig. 4(b), but this is not apparent in Fig. 6 which plots averages over multiple realisations (10) and also includes convolution with a 0.4 nm (0.1 THz) spectral response function to match experiment.

*n*= ± 1 sideband modes of the AB spectra to the average first sideband amplitude seen in experiments, we find that the subsequent decay of spectral intensity with frequency seen in both experiment and simulations is reproduced very well by the analytic geometric progression of the maximally-compressed AB breather.

## 4. Conclusions and Discussion

**69**(2), 1089–1093 (
1986). [CrossRef]

36. T. Inoue, J. Hiroishi, T. Yagi, and Y. Mimura, “Generation of in-phase pulse train from optical beat signal,” Opt. Lett. **32**(11), 1596–1598 (
2007). [CrossRef] [PubMed]

38. F. C. Cruz, “Optical frequency combs generated by four-wave mixing in optical fibers for astrophysical spectrometer calibration and metrology,” Opt. Express **16**(17), 13267–13275 (
2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13267U. [CrossRef] [PubMed]

## Acknowledgements

## References and Links

1. | T. B. Benjamin and J. E. Feir, “The disintegration of wavetrains on deep water. Part 1: Theory,” J. Fluid Mech. |

2. | V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. |

3. | G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. |

4. | N. N. Akhmediev, “Nonlinear physics. Déjà vu in optics,” Nature |

5. | J. Beeckman, X. Hutsebaut, M. Haelterman, and K. Neyts, “Induced modulation instability and recurrence in nematic liquid crystals,” Opt. Express |

6. | K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. |

7. | 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. |

8. | G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. |

9. | A. Demircan and U. Bandelow, “Supercontinuum generation by the modulation instability,” Opt. Commun. |

10. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. |

11. | J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400W continuous wave fiber laser,” Opt. Express |

12. | D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature |

13. | J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express |

14. | N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. |

15. | M. J. Ablowitz and B. M. Herbst, “On homoclinic structure and numerically induced chaos for the nonlinear Schrödinger equation,” SIAM J. Appl. Math. |

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19. | S. Trillo, S. Wabnitz, and T. A. B. Kennedy, “Nonlinear dynamics of dual-frequency-pumped multiwave mixing in optical fibers,” Phys. Rev. A |

20. | S. Liu, “Four-wave mixing and modulation instability of continuous optical waves in single-mode optical fibers,” Appl. Phys. Lett. |

21. | S. M. Kobtsev and S. V. Smirnov, “Influence of noise amplification on generation of regular short pulse trains in optical fibre pumped by intensity-modulated CW radiation,” Opt. Express |

22. | K. B. Dysthe and K. Trulsen, “Note on Breather type Solutions of the NLS as models for freak-waves,” Phys. Scr. |

23. | D. Clamond, M. Francius, J. Grue, C. Kharif, “Long time interaction of envelope solitons and freak wave formations,” European J. Mech. B – Fluids/B |

24. | I. Ten, and H. Tomita, “Simulation of the ocean waves and appearance of freak waves,” Reports of RIAM Symposium No.17SP1–2, Chikushi Campus, Kyushu University, Kasuga, Fukuoka, Japan, March 10 – 11 (2006). |

25. | V. V. Voronovich, V. I. Shrira, and G. Thomas, “Can bottom friction suppress ‘freak wave’ formation?” J. Fluid Mech. |

26. | D. L. Hart, A. F. Judy, R. Roy, and J. W. Beletic, “Dynamical evolution of multiple four-wave-mixing processes in an optical fiber,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

27. | G. P. Agrawal, |

28. | N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Exact first-order solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. |

29. | N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions,” Sov. Phys. JETP |

30. | N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, “N-modulation signals in a single-mode optical waveguide under nonlinear conditions,” Sov. Phys. JETP |

31. | N. Akhmediev, and A. Ankiewicz, Solitons, |

32. | J. M. Dudley, F. Gutty, S. Pitois, and G. Millot, “Complete characterization of THz pulse trains generated from nonlinear processes in optical fibers,” IEEE J. Quantum Electron. |

33. | J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. |

34. | J. N. Kutz, C. Lyngå, and B. J. Eggleton, “Enhanced supercontinuum generation through dispersion-management,” Opt. Express |

35. | N. Korneev, E. A. Kuzin, B. Ibarra-Escamilla, M. Bello-Jiménez, and A. Flores-Rosas, “Initial development of supercontinuum in fibers with anomalous dispersion pumped by nanosecond-long pulses,” Opt. Express |

36. | T. Inoue, J. Hiroishi, T. Yagi, and Y. Mimura, “Generation of in-phase pulse train from optical beat signal,” Opt. Lett. |

37. | B. Barviau, C. Finot, J. Fatome, and G. Millot, “Generation from continuous waves of frequency combs with large overall bandwidth and tunable central wavelength,” Electron. Lett. |

38. | F. C. Cruz, “Optical frequency combs generated by four-wave mixing in optical fibers for astrophysical spectrometer calibration and metrology,” Opt. Express |

39. | J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics |

40. | D. R. Solli, C. Ropers, and B. Jalali, “Active control of rogue waves for stimulated supercontinuum generation,” Phys. Rev. Lett. |

41. | 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 |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: September 4, 2009

Revised Manuscript: October 8, 2009

Manuscript Accepted: October 8, 2009

Published: November 10, 2009

**Virtual Issues**

November 17, 2009 *Spotlight on Optics*

**Citation**

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, "Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation," Opt. Express **17**, 21497-21508 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21497

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### References

- T. B. Benjamin and J. E. Feir, “The disintegration of wavetrains on deep water. Part 1: Theory,” J. Fluid Mech. 27(03), 417–430 (1967). [CrossRef]
- V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).
- G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87(3), 033902 (2001). [CrossRef] [PubMed]
- N. N. Akhmediev, “Nonlinear physics. Déjà vu in optics,” Nature 413(6853), 267–268 (2001). [CrossRef] [PubMed]
- J. Beeckman, X. Hutsebaut, M. Haelterman, and K. Neyts, “Induced modulation instability and recurrence in nematic liquid crystals,” Opt. Express 15(18), 11185–11195 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-18-11185 . [CrossRef] [PubMed]
- K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56(2), 135–138 (1986). [CrossRef] [PubMed]
- 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(18), 1246–1248 (1989). [CrossRef]
- G. P. Agrawal, “Modulation instability induced by cross-phase modulation,” Phys. Rev. Lett. 59(8), 880–883 (1987). [CrossRef] [PubMed]
- A. Demircan and U. Bandelow, “Supercontinuum generation by the modulation instability,” Opt. Commun. 244(1-6), 181–185 (2005). [CrossRef]
- J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006). [CrossRef]
- J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400W continuous wave fiber laser,” Opt. Express 16(19), 14435–14447 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-19-14435 . [CrossRef] [PubMed]
- D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007). [CrossRef] [PubMed]
- J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16(6), 3644–3651 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-6-3644U . [CrossRef] [PubMed]
- N. Akhmediev and V. I. Korneev, “Modulation instability and periodic solutions of the nonlinear Schrodinger equation,” Theor. Math. Phys. 69(2), 1089–1093 (1986). [CrossRef]
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