Formulación del problema

Whilst the significant majority of numerical supercontinuum studies model the Raman effect using experimental gain measurements or accurate lorentzian approximations, a simplified first-order linear approximation has also been often employed. In this case, the frequency-domain Raman response function is approximated as ˜hR(ω) =

1 + iωTR/fR, where TR is referred to as the characteristic Raman time scale and can

be determined from the slope of the Raman gain curve. In Fig. 5.9(a) we compare the imaginary parts of the experimental response adopted from Ref. [187] and two distinct linear approximations with TR= 3 fs and TR= 5 fs.

In Related Publication I we show that, when modeling broadband SC generation, the use of the linear Raman gain approximation can yield severe inaccuracies and result in incorrect physical interpretations. Significantly, this approach has also been used in studies related to optical rogue waves [89, 107], and we next discuss how this simplified model can in fact artificially lead to the emergence of giant solitons with rogue-wave-like characteristics.

Figure 5.9: (a) Imaginary parts of the Raman response function (note that Im[˜hR(ω)] is directly related to the Raman gain [187]). Black curve depicts the experimental response while the red and blue lines correspond to linear approximations with TR= 3 fs and TR= 5 fs, respectively. (b,c,d) Simulated time frequency representations at selected propagation distances when using the linear Raman gain approximation withTR= 5 fs.

To study the effect of the linear Raman gain approximation in long-pulse SC gen- eration we use the simplified model and simulate the propagation of a 3 ps, 220 W pulse centered at 1550 nm in a fiber with β2 = 1.13· 10−4 ps2/m, β3 = 6.48· 10−5

ps3/m and γ = 10.66 W−1_{/km. Note that although the value T}

R = 5 fs is clearly

not as good an approximation as TR = 3 fs (see Fig. 5.9(a)), it is is this value that

has been used in the context of optical rogue wave studies [89, 107] and thus the one adopted here. Yet, it should be noted that the qualitative conclusions are similar for all TR. In Fig. 5.9(b,c,d) we plot the time-frequency representation of the evolving field

at three distinct propagation lengths. The emergence of a single, extremely intense (>6 kW) and rapidly redshifting soliton can be observed. As the soliton is delayed across the trailing edge of the pump pulse the energy in the pump trailing edge can be

seen to be quasi-depleted. This depletion occurs because, in the linear approximation, the Raman gain is unbounded and increases monotonously with frequency separation. Therefore the soliton experiences significant gain with respect to the pump trailing edge which then acts as a Raman pump for the soliton. In this way, the soliton cap- tures a significant portion of the energy in the remains of the pulse envelope and grows extremely intense. Such pump depletion dynamics are of course unrealistic because the actual Raman gain possesses a finite bandwidth of approximately 30 THz. For example, in Fig. 5.9(c) the frequency separation of the soliton and the pump residue is approximately 50 THz which would realistically result in negligible Raman coupling between the pump residue and the soliton. Yet when modeling the field propagation using the linear Raman gain approximation, the gain experienced by the soliton is not only nonzero but very large. Finally, upon subsequent propagation these giant solitons can, in the linear Raman model, also disappear abruptly through a dramatic pulse collapse effect. Such collapse, to our knowledge, has not been observed in experiments nor in simulations that do not rely on the linear Raman gain approximation. In fact, in Related Publication I we show that this collapse requires not only the use of the ap- proximated Raman model but also insufficient numerical gridding parameters. These findings clearly show that it is paramount to use the full Raman gain model when modeling broadband pulse propagation.

Optical fibers allow for the study and control of many interdisciplinary nonlinear phe- nomena. In particular, fundamental instability and wave localization phenomena that are common to many nonlinear systems can be conveniently studied both in situations where they manifest spontaneously or when they are directly stimulated. In this the- sis we have studied modulation instability and wave localization dynamics in different contexts, ranging from those where they are externally excited to those wherein the phenomena arise through spontaneous nonlinear interactions. The main results of the thesis can be broadly divided into three intertwined regimes.

In this work we have adopted the analytical breather formalism originally introduced by Akhmediev and Korneev for the description of modulation instability and investi- gated its applicability under realistic excitation conditions. In particular, we have derived an improved estimate for the fiber length required to generate an ultrahigh- repetition-rate pulse train from an initially weakly modulated continuous wave and also shown that the breather formalism can be intuitively extended to describe modulation instability also in the pulsed regime. Whilst the breather theory provides an accurate representation of the initial phase of modulation instability, we have further shown that there exists parameter regimes where deviations from the ideal breather evolu- tion are observed. Specifically, when the modulation frequency is sufficiently small allowing for multiple instability harmonics to experience nonzero gain the breather ex- periences complex dynamics resulting in a cascaded temporal splitting process. In this thesis we have shown experimentally, numerically and analytically that such temporal splitting dynamics can be described in terms of higher-order modulation instability corresponding to a nonlinear superposition of elementary breather solutions. To our knowledge this constitutes the first experimental identification of such a higher-order instability process in any physical system as well as the first utilization of the Darboux transformation method in the design and analysis of physical experiments.

Whilst the purely analytical breather formalism does provide an interesting per- spective to modulation instability, it is rather the temporally localized soliton that is often heralded as the central structure of nonlinear science. Soliton propagation effects also lie at the focus of the present thesis because of their crucial role in the generation of broadband supercontinua in highly nonlinear fibers. In particular, here we have demonstrated experimentally and numerically novel soliton dynamics in long- pulse supercontinuum which lead to the generation of spectral components inaccessible through other spectral broadening processes. The first effect occurs due to the fact that ultrashort solitons ejected from the broad input pulse envelope temporally overlap with the trailing edge of the pump residue. Such a nonlinear superposition gives rise to a

phase-matched four-wave-mixing-like interaction of the soliton and the pump residue leading to the generation of narrowband spectral components clearly isolated from the continuum spectrum. In the second effect, the collision of two ultrashort solitons gives rise to a dispersive wave due to a phase-matched cascaded four-wave mixing process. Significantly, we have shown experimental signatures of such collision-generated disper- sive waves by systematically exciting solitonic collisions using an interferometric setup. These signatures represent, to our knowledge, the first experimental observation of radiation emission by soliton collisions.

The initial stages of long-pulse supercontinuum generation are governed by noise- driven modulation instability which leads to the break-up of the input envelope into a train of short solitons. This intrinsically noise-sensitive process has been shown to lead to the emergence of rare, abnormally redshifted solitons that have been suggested as the optical analogues of the infamous oceanic rogue waves. In this thesis we have shown experimentally and numerically that similar optical-rogue-wave-like fluctuations can be observed in supercontinuum generation even under conditions other than those corresponding to genuine long-pulse excitation. However, we have also shown that the spectral selection technique originally used to capture optical rogue waves biases the measured statistical distributions and, in fact, does not necessarily reveal the presence of high-amplitude waves but instead quantifies SC bandwidth fluctuations exclusively. By performing a more detailed analysis in the absence of spectral selection we have, nonetheless, shown that the supercontinuum field does contain extreme events whose amplitudes fulfill criteria often used to discriminate rogue waves in a hydrodynamic environment. Such events are associated with the collision of solitons, displaying ex- treme wave localization both in time and space and exhibiting many of the features often attributed to oceanic rogue waves.

To summarize, this thesis has presented studies on nonlinear instability and wave localization dynamics in fiber optics, providing significant results both applied and fundamental that are expected to be of immediate use in the analysis of fiber-optic parametric amplifiers, in the design and implementation of ultrafast soliton-based all- optical switches and logic devices and in the design of broadband SC sources for specific applications. On a more fundamental note, this thesis contains several novel results concerning the existence and interactions of universal nonlinear structures. It is ex- pected that these results, in particular, will evolve beyond fiber-optics and stimulate research of similar nature in the more general field of nonlinear science.

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