We demonstrate that the fission of higher-order N-solitons with a subsequent ejection of fundamental quasi-solitons creates cavities formed by a pair of solitary waves with dispersive light trapped between them. As a result of multiple reflections of the trapped light from the bounding solitons which act as mirrors, they bend their trajectories and collide. In the spectral domain, the two solitons receive blue and red wavelength shifts, and the spectrum of the trapped light alters as well. This phenomenon strongly affects spectral characteristics of the generated supercontinuum. Consideration of the system's parameters which affect the creation of the cavity reveals possibilities of predicting and controlling soliton-soliton collisions induced by multiple reflections of the trapped light.
©2013 Optical Society of America
Various aspects of the dynamics of ultrashort pulses in photonic crystal fibers (PCFs), such as higher-order soliton fission and interaction of solitons with radiation, are the subject of profound interest for fundamental studies and technological applications in photonics. In particular, the fission of higher-order solitons [1–3] is a key mechanism for the generation of ultrashort frequency-tuned fundamental solitons  and ultra-broadband optical supercontinuum (SC) [3, 5–10]. Recently, a model based on an optical-soliton counterpart of the Newton's cradle (NC) was introduced for understanding the mechanism of N-soliton fission under the action of the higher-order dispersion . The mechanism, which remains relevant in the presence of the Raman and self-steepening effects, explains the discrete nature of ejections of fundamental quasi-solitons from the parental N-soliton temporal slot, the delay between the ejections being controlled by the strength of the third-order dispersion. Quasi-solitons with higher peak powers are ejected before their weaker counterparts, and they experience stronger Raman self-frequency shifts . Along with the ejected solitons, the N-soliton fission gives rise to emission of strong dispersive radiation. Due to interaction with the radiation, weaker solitons can acquire an additional acceleration, which leads to their collisions with the stronger solitons ejected earlier . As these interactions strongly affect the spectral broadening of the pulse at all the stages of the SC formation, a challenging objective is to develop reliable control over these phenomena. Among particular types of the interactions involving solitons and radiation, notable are various forms of the four-wave mixing (FWM) [14–18], trapping of dispersive waves by solitons , formation of bound states of solitons [20–22], soliton fusion , etc. Interactions between dispersive waves and solitons were also studied in the context of all-optical switching and rogue-wave formation [24,25]. In a recent work , dispersive waves trapped in an effective predesigned cavity created by two solitons acting like mirrors, were considered, and it was shown that the FWM strongly affects the dynamics of both the solitons and the trapped waves. In particular, this interaction may result in broadening or narrowing of the spectrum of the trapped waves . Furthermore, the radiation-induced attraction (“Casimir force”) between the solitons eventually causes them to collide. Interestingly, the trapping of light in a moving area with high refractive index and the transformation of the pulse spectrum was considered in .
The main objective of the present work is to investigate how the FWM between the solitons and dispersive waves in two-soliton cavities emerging from the fission of N-solitons, affects the ensuing SC generation.
2. Light trapping by the solitonic cavity following the fission of the N-soliton
For the sake of clarity, we chiefly disregard the Raman self-frequency shift (which is a relevant assumption for a hollow-core PCFs filled with Raman-inactive xenon ) and dispersion terms above the third order. It is shown in the last part of the paper that the the cavity dynamics reported remains valid if those effects are taken into regard.
In the present context, the evolution of amplitude of the electromagnetic waves is governed by the generalized nonlinear Schrödinger equation (NLSE) [7,28], which includes the second- and third-order dispersion (TOD) terms with respective coefficients β2 and β3, and the cubic nonlinear term, with coefficient , along with its self-steepening part:Figure 1 demonstrates the fission of the N-soliton with and 800-nm central wavelength, as produced by simulations of Eq. (1). The input is taken, accordingly, as with T0 = 50 fs (TFWHM ~90 fs) and , where P0 = 56 kW. The fiber parameters are β2 = −0.0021 ps2/m, β3 = 5.24∙10−6 ps3/m and = 1.5∙10−5 W−1m−1.
After the breakup of the injected N-soliton, a strong Cherenkov dispersive wave packet  is emitted at Z = 0.1 m, which appears at the blue edge of the respective spectrum around 450 nm, and is not involved in the process described below (see Fig. 1). It is the field in the periodic chain of pulses  propagating in a bound state, between the ejections of fundamental quasi-solitons, that gets partially trapped bouncing from the cavity-building solitons, starting from Z = 0.16 m [Fig. 1 (a)]. Although peak intensities of the trapped wave packets are 5-6 times smaller than those of the solitons, the packets play a major role in the subsequent evolution of the cavity and, thus, in the spectral evolution of the generated SC [Fig. 1(b)]. Multiple quasi-elastic  collisions of the trapped dispersive wave with the cavity-building solitons cause their acceleration and deceleration, manifested in bending of their trajectories [Fig. 1(a)], i.e., effective tilting of the mirrors in the respective cavity picture.
In the spectral domain, the curvature of the trajectories is represented by a blue shift for one soliton (from 1010 nm to 930nm), and a red shift for the other one (from 920 nm to 1030nm), as spectral filtering of individual solitons reveals [the dotted black and solid magenta curves in Fig. 1(b) demonstrate the evolution of the central wavelengths of the two solitons]. The spectral region between 460 nm and 620 nm, where the reflections of the trapped waves occur, is shown by a dashed white rectangle in Fig. 1(b), and it is discussed inmore detail below. To prove the critical role of the interaction with the dispersive waves in the bending of the solitons' trajectories and their spectral shifts, an additional simulation was performed, in which we filter out all the radiation between the two solitons at Z = 0.16 m. In this case, the solitons propagate along undistorted trajectories, as shown in Fig. 1(a) by the dotted blue lines. As a result of the deceleration of the first soliton and acceleration of the second one, the two collide at z = 0.56 m. At some distance before the collision, the “soliton mirrors” degrade, allowing the trapped radiation to escape. The shrinkage of the cavity created by the two solitons resembles the structure of tapered waveguides .
The interaction between the dispersive waves and solitons dealt with here is different from , where the dispersive waves emitted by the first soliton collided with the second one and mostly passed through it without being reflected back to the first soliton and affecting its trajectory or central wavelength. To further clarify the present effect, we have performed another numerical experiment, by filtering out everything from Z = 0.16, except for the cavity consisting of the two solitons with the field trapped between them, see Fig. 2(a). Figure 2(b) shows the evolution of the trapped-field spectrum while the spectral lines of the solitons are located far away, between 900 and 1000nm, as seen in Fig. 1(b). It can be observed from both panels of Fig. 2 that the trapped light experiences strong spectral changes as it collides with the solitons, similar to the case of the specially designed solitonic cavity . The condition of the resonant scattering  for the case referred to as phase-insensitive in  can be written in the following form:Eq. (2) can be referred to as the dispersion characteristic of the FWM, while the right-hand side represents the dispersion of the fiber. Solutions of Eq. (2) give wavelengths at which the FWM terms are in resonance with the system's eigenmodes. For example, for the first reflection shown in Fig. 2 we deduce from Eq. (2) that the trapped wave packet, centered around λp = 490 nm, is reflected from the first soliton (centered around λs = 950 nm) into a spectral region around λ = 596 nm. The change of the soliton wavelengths in the course of the propagation explains the observation that the cascaded re-scattering of the trapped waves is not exactly periodic, leading to the modification of the trapped radiation.
To build the soliton cavity, one has to choose system parameters appropriately. For very small values of TOD parameter β3, the fission leads to a decomposition of the N-soliton into N distinct fundamental solitons without trapping any light between them, as predicted for the nearly integrable NLSE [1,2]. In this case, the peak powers of fundamental soliton emerging after the splitting of the N-soliton are [1,2]. At larger values of β3, the fission is dominated by the NC mechanism , with solitons ejected in a step-like sequence, see Fig. 3(a)-3(c). This situation is appropriate for building the soliton cavity. However, if the TOD is too strong, ejections of the fundamental quasi-solitons occur with a very large delay , leaving little opportunity for the light trapped between the ejected solitons to mediate the attractive interaction between them.
The cavity can be formed by not only the first two ejected solitons, but by another pair as well. Figure 3(a) demonstrates that, despite evident multiple reflections of the trapped field between the first two solitons, no significant bending of their trajectories is observed, due to a strong initial frequency difference imparted to these two solitons while they were ejected. However, the interaction between the third and fourth solitons via the light trapped between them overcomes their initial spectral separation, making them to build the cavity. As mentioned above, additional cavities may be built, as Fig. 3(b) reveals.
For each normalized value of the relative TOD strength, defined as δ3 = β3/(6|β2|Τ0), there exists a minimum value, Nmin, of the input-soliton's order N for which the light trapped by the ejected soliton pair induces the effective attraction and subsequent collision between them. At N < Nmin, no collisions occur [see Fig. 3(c)], as in that case the large separation between the ejected solitons in the temporal and frequency domains cannot be surmounted by the attraction induced by the bouncing light. The curve for Nmin versus δ3, plotted in Fig. 3(d) in the range of 0.0016 < δ3 < 0.11, shows that the absolute minimum of the necessary soliton order is N ≈11, at which the solitons do collide if the fission takes place at δ3 = 0.0083. The two green marks near the left segment of the curve pertain to the examples shown in Fig. 3(a), 3(b), while the red mark, which is located slightly to the left of the critical curve, pertains to panel (c), where no collision occurs.
Finally, we discuss the role of the higher-order nonlinear and dispersion terms in the generalized NLSE. The shock term, which has been included in Eq. (1), facilitates the ejection of quasi-solitons from the NC, therefore it facilitates the attraction and collisions between the solitons. If the shock term is dropped, the first “collapsing cavity” emerges at, instead of 12 in Fig. 1, for the same values of other parameters. In the presence of the standard Raman term in the extended NLSE which describes regular silica PCFs [8,10], red-shifted accelerated solitons tend to separate from each other faster. Nevertheless, the mechanism of the soliton-soliton attraction through the trapped light can overpower this tendency, as seen in Fig. 4(a). Further, the effect reported here is not specific solely for the TOD, and may be observed as well in the presence of higher-order dispersion terms. In the case when the dispersion terms up to the seventh order are included (as in Ref .), the effect looks qualitatively the same, see Fig. 4(b), 4(c). For fibers with the leading TOD term, the dependence of Nmin on the TOD strength (not shown here in detail) is similar to that presented in Fig. 3(d), where the higher-order dispersion terms are disregarded.
We have demonstrated that after the fission of the initial N-soliton, cavities consisting of two solitons are formed with light trapped between them. As a result of multiple reflections of the trapped light waves from the bounding solitons, which act as mirrors in the cavity, the solitons experience strong bending of their trajectories and eventually collide. The spectrum of the trapped waves also changes as they bounce from the solitons with varying wavelengths. This phenomenon strongly affects spectral characteristics of the generated SC (supercontinuum). The systematic study has identified the minimum order N of the input soliton, above which the mutual attraction and ensuing collision of the solitons occur. The mechanism, which is explored in detail in the Raman-free setting, remains valid in the model of regular glass fibers, with all higher-order linear and nonlinear terms included.
The work of R.D. and B.A.M. was partly supported by the Binational (US-Israel) Science Foundation through grant No. 2010239. The work of AVY was supported by the FCT grant PTDC/FIS/112624/2009 and PEst-OE/FIS/UI0618/2011.
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