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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 19 — Sep. 12, 2011
  • pp: 17973–17978

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Midinfrared optical rogue waves in soft glass photonic crystal fiber

Daniel Buccoliero, Henrik Steffensen, Heike Ebendorff-Heidepriem, Tanya M. Monro, and Ole Bang  »View Author Affiliations


Optics Express, Vol. 19, Issue 19, pp. 17973-17978 (2011)
http://dx.doi.org/10.1364/OE.19.017973


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Abstract

We investigate numerically the formation of extreme events or rogue waves in soft glass tellurite fibers and demonstrate that optical loss drastically diminishes shot-to-shot fluctuations characteristic of picosecond pumped supercontinuum (SC). When loss is neglected these fluctuations include extreme events such as formation of highly energetic pulses located at the red end of the spectrum and we obtain right-skewed heavy-tailed distributions characteristic of extreme events statistics. On the other hand, when loss is included bandwidth fluctuations follow Gaussian-like statistical distributions. Our results thus implicitly show that rogue waves will not occur in any SC spectrum that is limited by loss, such as commercial silica fiber based SC sources.

© 2011 OSA

1. Introduction

Ever since it was first reported [1

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970). [CrossRef]

], SC generation has been widely investigated and demonstrated in several pump regimes, from continuous wave to ultrashort femtosecond [2

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

]. Invariably, novel physics are uncovered and in particular an intriguing phenomenon in the form of statistically rare highly energetic pulses located at the red end of the spectra has been predicted numerically [3

M. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. Gordon, and D. Chemla, “Femtosecond distributed soliton spectrum in fibers,” J. Opt. Soc. Am. B 6, 1149–1158 (1989). [CrossRef]

, 4

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]

] and observed experimentally [5

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1058 (2007). [CrossRef] [PubMed]

] in optical fibers.

Such extreme events occur since pulses or solitons generated in a fiber can interact and transfer energy between each other during collisions. On average there will be a net energy transfer to the larger soliton and the mechanism can generate very high amplitude solitons [3

M. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. Gordon, and D. Chemla, “Femtosecond distributed soliton spectrum in fibers,” J. Opt. Soc. Am. B 6, 1149–1158 (1989). [CrossRef]

,6

A. V. Buryak and N. Akhmediev, “Internal friction between solitons in near integrable systems,” Phys. Rev. E 50, 3126–3133 (1994). [CrossRef]

, 7

O. Bang and M. Peyrard, “Generation of high-energy localized vibrational modes in nonlinear Klein–Gordon lattices,” Phys. Rev. E 53, 4143–4152 (1996). [CrossRef]

]. This energy transfer and generation of strong nonlinear excitations, is in fact a generic nonlinear phenomenon well-known from biology [7

O. Bang and M. Peyrard, “Generation of high-energy localized vibrational modes in nonlinear Klein–Gordon lattices,” Phys. Rev. E 53, 4143–4152 (1996). [CrossRef]

–9

T. Dauxois and M. Peyrard, “Energy localization in nonlinear lattices,” Phys. Rev. Lett. 70, 3935–3938 (1993). [CrossRef] [PubMed]

], optical waveguide arrays [10

O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. 21, 1105–1107 (1996). [CrossRef] [PubMed]

], and discrete lattices [11

A. Maluckov, Lj. Hadžievski, N. Lazarides, and G. P. Tsironis, “Extreme events in discrete nonlinear lattices,” Phys. Rev. E 79, 025601 (2009). [CrossRef]

].

The term optical ”rogue wave” (RW) was coined [5

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1058 (2007). [CrossRef] [PubMed]

] due to the high resemblance to the infamous so-called oceanic RWs. Moreover, these statistically rare events exhibited a shifting towards longer wavelengths and they distinctly appeared in the red end of the spectrum as large bandwidth fluctuations following right-skewed heavy-tailed statistics [5

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1058 (2007). [CrossRef] [PubMed]

].

It should be noted, that in recent studies [12

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]

, 13

A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17, 17010–17015 (2009). [CrossRef] [PubMed]

] the authors pointed to the possibility of a RW occuring inside the SC spectrum, however in the following we limit the investigations to RWs as giant solitons located on the long wavelength side of the SC.

The above results were obtained in silica fibers at low powers, with spectra far from the loss edge, however here we investigate midinfrared (MIR) RW formation in soft glass fibers. Soft glasses, which among others include chalcogenide (n 2∼0.8×10−17 m2/W) [14

A. Tuniz, G. Brawley, D. J. Moss, and B. J. Eggleton, “Two-photon absorption effects on Raman gain in single mode As2Se3 chalcogenide glass fiber,” Opt. Express 16, 18524–18534 (2008). [CrossRef] [PubMed]

] and tellurite (n 2∼0.6×10−18 m2/W) [15

W. Q. Zhang, S. Afshar V., and T. M. Monro, “A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation,” Opt. Express 17, 19311–19327 (2009). [CrossRef]

], exhibit much higher nonlinearities than silica (n 2∼0.3×10−19 m2/W) [16

G. P. Agrawal, Nonlinear Fiber Optics , 4th ed. (Academic Press, 2007).

], and thus offer a great reduction in the required fiber length or pump power. In addition soft glass fibers allow low loss guidance at much longer wavelengths, well into the MIR regime.

Hence we consider a 60 cm tellurite fiber with pitch (Λ=3 μm), relative hole size (d/Λ=0.4) and zero-dispersion wavelength (ZDW) ∼1780 nm [17

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010). [CrossRef]

] see Fig. 1(a), in which we show the measured loss and calculated dispersion. Now, since thulium mode-locked lasers exhibit a broad and high gain spectrum in the 1.8 to 2.1 μm wavelength regime [18

Q. Wang, J. Geng, T. Luo, and S. Jiang, “Mode-locked 2 μm laser with highly thulium-doped silicate fiber,” Opt. Lett. 34, 3616–3618 (2009). [CrossRef] [PubMed]

], we consider pumping at the thulium wavelength λ =1930 nm, i.e., anomalous dispersion region to generate a large number of solitons as in the aforementioned studies in silica.

Fig. 1 (a) Optical loss for tellurite fiber with Λ=3 μm and d/Λ=0.4. Shown are the combined contribution from both material and confinement loss, as well as confinement loss alone. (b) Raman gain spectrum h̃R (Ω) of tellurite with maximum peak at Ω R ∼22.5 THz. and silica with peak at Ω R ∼13.2 THz. The dashed curve shows the approximate Raman gain spectrum of tellurite.

As seen in Fig. 1(a), this fiber entails substantial loss and we demonstrate that generation of RWs can be greatly reduced when the band edge of the generated SC approaches the loss edge at ∼3 μm, similar to SC fluctuations limited by a photonic bandgap in a solid-core fiber [19

O. Vanvincq, B. Barviau, A. Mussot, G. Bouwmans, Y. Quiquempois, and A. Kudlinski, “Significant reduction of power fluctuations at the long-wavelength edge of a supercontinuum generated in solid-core photonic bandgap fibers,” Opt. Express 18, 24352–24360 (2010). [CrossRef] [PubMed]

]. We stress here however, that in a bandgap fiber the dispersion deviates to infinity as it approaches the edge of the gap. The rapid increase in dispersion alone will halt the solitons and this is exactly what stops them red-shifting in this case, before entering the actual regime of loss.

Therefore this has no resemblance whatsoever with the material loss edge of a conventional fiber, at which the dispersion is still finite and the solitons are still allowed to red-shift into the loss edge. Hence the dynamics is different in many ways, e.g., solitons/rogue waves approaching the loss edge will decay in power, whereas the rogue waves in a bandgap fiber will maintain their power, but spread out due to the increasing dispersion.

The broad absorption in the region 2.5–5 μm is due to water in the glass, note however that this glass was melted in dry atmosphere and thus has already a considerably reduced water content compared with glasses melted in ambient atmosphere [17

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010). [CrossRef]

]. In addition, improvement of the glass melting procedure is expected to further reduce the water content, where complete removal of water would shift the loss edge to ∼4 μm, at which confinement loss becomes significant. In comparison, the material IR edge loss is <1 dB/m for λ <4.9 μm [20

H. Ebendorff-Heidepriem, T.-C. Foo, Y. Li, M. Oermann, and T. M. Monro, “New tellurite glasses for erbium fibre lasers,” in Australian Conference on Optical Fibre Technology (ACOFT’2008), Sydney, 2008.

] which is negligible compared to confinement loss.

We consider pulse propagation both with and without including loss and demonstrate, for the first time to our knowledge, MIR RW formation, combining a tellurite fiber with a thulium pump. We base our simulations on the generalized nonlinear Schrödinger equation (GNLSE) [16

G. P. Agrawal, Nonlinear Fiber Optics , 4th ed. (Academic Press, 2007).

] which we solve in the so-called interaction picture [21

J. C. Travers, M. H. Frosz, and J. M. Dudley, in Supercontinuum Generation in Optical Fibers , J. M. Dudley and J. R. Taylor, eds. (Cambridge University Press, 2010).

, 22

J. Hult, “A fourth-order RungeKutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. 25, 3770–3775 (2007). [CrossRef]

], and we consider both higher order dispersion, loss, and stimulated Raman scattering, see [17

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010). [CrossRef]

] for details. In optical fibers the generation of RWs is assisted by the Raman effect [23

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010). [CrossRef]

], leading to an on-average energy transfer to the larger soliton, which is also the one red-shifting the fastest [22

J. Hult, “A fourth-order RungeKutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. 25, 3770–3775 (2007). [CrossRef]

]. The fact that the Raman red shift increases with soliton power is exactly what enables spectral measurements of RWs [5

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1058 (2007). [CrossRef] [PubMed]

].

We implement the Raman response function R(t) = (1 – fR )δ(t) + fR hR (t) in the GNLSE, where fR =0.064 and hR (t) is derived from the measured Raman gain spectrum h̃R (Ω) [15

W. Q. Zhang, S. Afshar V., and T. M. Monro, “A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation,” Opt. Express 17, 19311–19327 (2009). [CrossRef]

], see Fig. 1(b). The tellurite Raman profile is markedly different from silica and the width (bandwidth) is substantially larger. Moreover, it consists of several peaks and thus the Raman profile hR (t) should be approximated as a sum of decaying harmonic oscillators and we use two oscillators. The approximated gain profile is shown in Fig. 1(b) (dashed curve) which is more physical since it is non-zero for all Ω>0, whereas the measured profile, due to filtering of the pump, drops off abruptly at Ω∼5 THz. The approximated Raman gain thus allows us to model small frequency shifts, which is important for narrowband pulses, e.g., ps pulses.

We pump at λ =1930 nm, with a 5 ps pulse duration, and various peak powers. The nonlinear coefficient is ∼0.14 W−1m−1 and noise is included by adding one random phase photon per mode in the frequency domain. To capture the statistics of the RWs, we perform 200 simulations for each peak power, i.e., 1000, 1500 and 2000 W and the resulting spectra are shown in Fig. 2. In all cases a SC is developed. However, the detrimental effect of loss is evident, giving rise to much narrower SC than in the corresponding lossless case. In particular, when pumping with a peak power of 1000 W, the SC is not formed at all when loss is included (therefore not shown in Fig. 2). Without loss the SC are very broad, and the ensemble of each peak power contains several MIR RWs at ∼3.4 μm, ∼3.8 μm, and ∼4.2 μm for peaks power of 1000, 1500 and 2000 W, respectively. From the inserts on the right it follows that shot-to-shot fluctuations can be substantial, which is manifested in noticeable bandwidth deviations from the most red-shifted RW, to the narrowest spectra.

Fig. 2 Output spectra for various peak powers with and without including optical loss. In each ensemble the black lined spectrum depicts the ensemble median and the inserts on the right show (top, bottom) a RW and the narrowest spectrum and (middle) a spectrum close to the median.

When loss is included, the bandwidth fluctuations are much smaller. Moreover, the fluctuations decrease with peak power, since increasing the peak power, forces a larger fraction of the pulses to fully develop into the maximum allowed SC bandwidth, effectively limited by optical loss.

To investigate the statistics of each ensemble, we use a long-pass filter and select components above a particular wavelength, depending on the spectral extent of the ensemble median. In particular, cut-off wavelengths from top to bottom in Fig. 2 are 3.2 μm, 3.6 μm, 2.65 μm, 4 μm, and 2.7 μm. Fourier transforming the remaining spectral components reveals a series of short pulses of varying peak powers, see Fig. 3.

Fig. 3 (left column) Series of short pulses corresponding to the filtered components of the spectra. (right column) Statistical distributions of peak powers in the series of pulses. Inserts depict the RWs and their absolute frequency.

As expected, when loss is neglected, peak powers of the pulses follow the characteristic heavy-tailed statistics. Conversely, when loss is included, the distributions become more Gaussian-like. Note that shifting the cut-off wavelengths within a certain range, does not alter the statistical distributions noticeably.

Next we adopt a general rule [25

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]

] to define a RW, and find that, in a purified tellurite fiber and for a broad range of peak powers, MIR RWs are formed on several occasions, see Fig. 3. In sharp contrast, with loss included, only a narrow range of peak powers allow formation of a RW.

In summary, RW formation will be significantly reduced if not eliminated in SC limited by optical loss. In combination with pushing the loss edge further in to the MIR, one can obtain a ps pumped MIR SC source with Gaussian-like bandwidth fluctuations.

Acknowledgments

Daniel Buccoliero acknowledges support from the Danish Council for Independent Research, Natural Sciences grant no. 70600 as well as the Danish National Advanced Technology Foundation. Tanya Monro acknowledges the support of an ARC Federation Fellowship.

References and links

1.

R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett. 24, 584–587 (1970). [CrossRef]

2.

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

3.

M. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. Gordon, and D. Chemla, “Femtosecond distributed soliton spectrum in fibers,” J. Opt. Soc. Am. B 6, 1149–1158 (1989). [CrossRef]

4.

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]

5.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1058 (2007). [CrossRef] [PubMed]

6.

A. V. Buryak and N. Akhmediev, “Internal friction between solitons in near integrable systems,” Phys. Rev. E 50, 3126–3133 (1994). [CrossRef]

7.

O. Bang and M. Peyrard, “Generation of high-energy localized vibrational modes in nonlinear Klein–Gordon lattices,” Phys. Rev. E 53, 4143–4152 (1996). [CrossRef]

8.

O. Bang and M. Peyrard, “High-order breather solutions to a discrete nonlinear Klein–Gordon model,” Physica D 81, 9–22 (1995). [CrossRef]

9.

T. Dauxois and M. Peyrard, “Energy localization in nonlinear lattices,” Phys. Rev. Lett. 70, 3935–3938 (1993). [CrossRef] [PubMed]

10.

O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett. 21, 1105–1107 (1996). [CrossRef] [PubMed]

11.

A. Maluckov, Lj. Hadžievski, N. Lazarides, and G. P. Tsironis, “Extreme events in discrete nonlinear lattices,” Phys. Rev. E 79, 025601 (2009). [CrossRef]

12.

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]

13.

A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17, 17010–17015 (2009). [CrossRef] [PubMed]

14.

A. Tuniz, G. Brawley, D. J. Moss, and B. J. Eggleton, “Two-photon absorption effects on Raman gain in single mode As2Se3 chalcogenide glass fiber,” Opt. Express 16, 18524–18534 (2008). [CrossRef] [PubMed]

15.

W. Q. Zhang, S. Afshar V., and T. M. Monro, “A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation,” Opt. Express 17, 19311–19327 (2009). [CrossRef]

16.

G. P. Agrawal, Nonlinear Fiber Optics , 4th ed. (Academic Press, 2007).

17.

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010). [CrossRef]

18.

Q. Wang, J. Geng, T. Luo, and S. Jiang, “Mode-locked 2 μm laser with highly thulium-doped silicate fiber,” Opt. Lett. 34, 3616–3618 (2009). [CrossRef] [PubMed]

19.

O. Vanvincq, B. Barviau, A. Mussot, G. Bouwmans, Y. Quiquempois, and A. Kudlinski, “Significant reduction of power fluctuations at the long-wavelength edge of a supercontinuum generated in solid-core photonic bandgap fibers,” Opt. Express 18, 24352–24360 (2010). [CrossRef] [PubMed]

20.

H. Ebendorff-Heidepriem, T.-C. Foo, Y. Li, M. Oermann, and T. M. Monro, “New tellurite glasses for erbium fibre lasers,” in Australian Conference on Optical Fibre Technology (ACOFT’2008), Sydney, 2008.

21.

J. C. Travers, M. H. Frosz, and J. M. Dudley, in Supercontinuum Generation in Optical Fibers , J. M. Dudley and J. R. Taylor, eds. (Cambridge University Press, 2010).

22.

J. Hult, “A fourth-order RungeKutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. 25, 3770–3775 (2007). [CrossRef]

23.

G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A 374, 989–996 (2010). [CrossRef]

24.

F. Luan, D. Skryabin, A. Yulin, and J. Knight, “Energy exchange between colliding solitons in photonic crystal fibers,” Opt. Express 14, 9844–9853 (2006). [CrossRef] [PubMed]

25.

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]

OCIS Codes
(060.2390) Fiber optics and optical communications : Fiber optics, infrared
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 11, 2011
Manuscript Accepted: August 5, 2011
Published: August 29, 2011

Citation
Daniel Buccoliero, Henrik Steffensen, Heike Ebendorff-Heidepriem, Tanya M. Monro, and Ole Bang, "Midinfrared optical rogue waves in soft glass photonic crystal fiber," Opt. Express 19, 17973-17978 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-17973


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References

  1. R. R. Alfano and S. L. Shapiro, “Emission in the region 4000 to 7000 Å via four-photon coupling in glass,” Phys. Rev. Lett.24, 584–587 (1970). [CrossRef]
  2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006). [CrossRef]
  3. M. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. Gordon, and D. Chemla, “Femtosecond distributed soliton spectrum in fibers,” J. Opt. Soc. Am. B6, 1149–1158 (1989). [CrossRef]
  4. M. H. Frosz, O. Bang, and A. Bjarklev, “Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation,” Opt. Express14, 9391–9407 (2006). [CrossRef] [PubMed]
  5. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature450, 1054–1058 (2007). [CrossRef] [PubMed]
  6. A. V. Buryak and N. Akhmediev, “Internal friction between solitons in near integrable systems,” Phys. Rev. E50, 3126–3133 (1994). [CrossRef]
  7. O. Bang and M. Peyrard, “Generation of high-energy localized vibrational modes in nonlinear Klein–Gordon lattices,” Phys. Rev. E53, 4143–4152 (1996). [CrossRef]
  8. O. Bang and M. Peyrard, “High-order breather solutions to a discrete nonlinear Klein–Gordon model,” Physica D81, 9–22 (1995). [CrossRef]
  9. T. Dauxois and M. Peyrard, “Energy localization in nonlinear lattices,” Phys. Rev. Lett.70, 3935–3938 (1993). [CrossRef] [PubMed]
  10. O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Opt. Lett.21, 1105–1107 (1996). [CrossRef] [PubMed]
  11. A. Maluckov, Lj. Hadžievski, N. Lazarides, and G. P. Tsironis, “Extreme events in discrete nonlinear lattices,” Phys. Rev. E79, 025601 (2009). [CrossRef]
  12. 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]
  13. A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express17, 17010–17015 (2009). [CrossRef] [PubMed]
  14. A. Tuniz, G. Brawley, D. J. Moss, and B. J. Eggleton, “Two-photon absorption effects on Raman gain in single mode As2Se3 chalcogenide glass fiber,” Opt. Express16, 18524–18534 (2008). [CrossRef] [PubMed]
  15. W. Q. Zhang, S. Afshar V., and T. M. Monro, “A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation,” Opt. Express17, 19311–19327 (2009). [CrossRef]
  16. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, 2007).
  17. D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett.97, 061106 (2010). [CrossRef]
  18. Q. Wang, J. Geng, T. Luo, and S. Jiang, “Mode-locked 2 μm laser with highly thulium-doped silicate fiber,” Opt. Lett.34, 3616–3618 (2009). [CrossRef] [PubMed]
  19. O. Vanvincq, B. Barviau, A. Mussot, G. Bouwmans, Y. Quiquempois, and A. Kudlinski, “Significant reduction of power fluctuations at the long-wavelength edge of a supercontinuum generated in solid-core photonic bandgap fibers,” Opt. Express18, 24352–24360 (2010). [CrossRef] [PubMed]
  20. H. Ebendorff-Heidepriem, T.-C. Foo, Y. Li, M. Oermann, and T. M. Monro, “New tellurite glasses for erbium fibre lasers,” in Australian Conference on Optical Fibre Technology (ACOFT’2008), Sydney, 2008.
  21. J. C. Travers, M. H. Frosz, and J. M. Dudley, in Supercontinuum Generation in Optical Fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University Press, 2010).
  22. J. Hult, “A fourth-order RungeKutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol.25, 3770–3775 (2007). [CrossRef]
  23. G. Genty, C. M. de Sterke, O. Bang, F. Dias, N. Akhmediev, and J. M. Dudley, “Collisions and turbulence in optical rogue wave formation,” Phys. Lett. A374, 989–996 (2010). [CrossRef]
  24. F. Luan, D. Skryabin, A. Yulin, and J. Knight, “Energy exchange between colliding solitons in photonic crystal fibers,” Opt. Express14, 9844–9853 (2006). [CrossRef] [PubMed]
  25. 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]

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