1. Introduction

Supercontinuum (SC) generation in nonlinear media has attracted much attention due to its potential applications in ultrafast optical switching, spectroscopy, optical coherent tomography, optical clocks, etc. The rise of photonic crystal fiber (PCF) has opened new horizons for effectively SC generation mainly because of their “endlessly single-mode” behavior, controllable group velocity dispersion characteristics and high nonlinear properties [1–4]. In particular, the flexible design freedom of PCFs can be used to fabricate a photonic crystal fibers with two zero-dispersion wavelengths (ZDWs) within the optical spectrum. The efficiency of SC generation in PCFs with two ZDWs is highly enhanced with respect to standard optical fibers because the range of anomalous dispersion region which located between two ZDWs can be adjusted effectively [5–8]. For example, the anomalous dispersion region can be shifted towards much lower wavelengths than those attainable with standard fibers, allowing soliton propagation into the visible range. Therefore, this combination of the rich dispersion characteristics and enhanced nonlinear properties enable PCFs with two or more ZDWs to be potential candidates for SC generation and other nonlinear phenomena [5–15].

SC generation with anomalous dispersion regime pumping is dominated by soliton-related propagation effects in the femtosecond regime, where higher-order dispersion and Raman scattering are the two most significant effects that can perturb the ideal periodic evolution of a high-order soliton [4,16]. Consequently, the higher-order soliton can break up into red-shifted fundamental solitons through soliton fission. During the soliton fission process, dispersive wave (DW) is emitted due to the energy transfer from soliton to narrow-band resonance in the normal dispersion regime [4,7,8,10]. The so-called DW whose wavelength can be either blue shift or red shift with respect to the central wavelength of soliton, which is determined by the phase-matching condition between the soliton and DW [6–8]. Besides, the emergence of new spectral components is also known as the nonsolitonic radiation [17,18], or as the Cherenkov radiation [19–22] whose strength relates to the overlap with the soliton spectrum. It has been well understood that the formation and amplification of DW is one of the important physical mechanism underlying SC generation in PCFs. A detailed discussion has been devoted to the investigation of the dynamics about soliton fission and DW generation [4]. The effects of input pulse parameters such as pulse energy, peak power, pulse duration, and central wavelength on the SC generation have been investigated thoroughly [4].

Nonetheless, the research on the interaction between solitons and coexisting DWs is one of the long-standing problems in nonlinear wave dynamics, which may open new opportunities for effectively manipulating the properties of supercontinuum generation and spectral shaping. Up to now, there were a lot of theoretical and experimental research have been done on this subject about the interaction of solitons with DWs [23–30]. It has been demonstrated that the interactions between solitons and DWs are of particular interest in SC generation, as the shedding and trapping of DW across a ZDW can lead to the extension of the SC to shorter wavelengths [25,26,30–36]. It is a fascinating phenomenon accompanying formation of the spectral peaks at the blue edge of output spectrum that despite residing deep in the normal dispersion regime they can form quasi-nondispersive wave packets [26,30,31,35]. This intriguing effect of pulse trapping originally has been suggested in the series of papers by Nishizawa and Goto [37–40]. It has been proved both experimentally and numerically that the blue spectral peaks is trapped by a co-propagating soliton on the other side of the ZDW [37–39]. Besides, it has also been proposed that the physical mechanism behind the above effects is the cross-phase modulation (XPM) [8,31–33]. Subsequently, Skryabin et al, have explained the trapping effect of DW in fibers using the theory of four-wave mixing between the solitons and DWs [23–26]. The analytical theory of generation of new frequencies by mixing of solitons and DWs has been developed in [25]. Moreover, it has showed that the nature of the trapping effect is analogous to the gravity-like inertial force [41–43]. The decelerating soliton creates a gravity-like potential to act on DW which propagates in the normal dispersion regime, and accordingly the DW constantly blue shift to satisfy the group-velocity matching with the soliton. Recently, such trapping effect have been attracting more and more attention [44–47]. However, most of the previous investigations of pulse trapping is usually about B-DW trapped by soliton in optic fiber which has only one ZDW. It is encouraging that the advent of PCFs with two ZDWs provides an opportunity to observe the B-DW and R-DW at the same time. Although a number of papers have examined the SC generation in PCFs with two ZDWs [5–12,14,45], there were few studies showed the soliton trapping of R-DW. Here, the soliton trapping of B-DW and R-DW in PCFs with two ZDWs are studied in our work, respectively. We found a novel phenomenon that not only B-DWs but also R-DWs can be trapped by soliton across the ZDW when the temporal overlap between soliton and DW occurs. What’s more, our numerical simulation results show that the new generated spectral component by the trapping effect of DW contribute to spectrum width and spectrum flatness of SC.

In this paper, we numerically show the soliton trapping of DW in PCFs. The paper is organized as follows. In Sec. II, the theoretical model with high-order dispersion and higher-order nonlinear effects for ultrashort pulse propagation in PCF is introduced. In Sec. III, a detailed description of the DW trapped by soliton in PCFs with two ZDWs is made. Finally, In Sec. IV, we summarize our results and conclusions.

2. Numerical model for ultrashort pulse propagation in PCFs

For the numerical model of the nonlinear pulse propagation in the PCF, a well-known generalized nonlinear Schrödinger equation (NLSE) is used:

3. Numerical results and discussions

Based on the generalized nonlinear Schrödinger equation (NLSE), the trapping effect of DW in two types of PCFs has been analyzed by pumping with femtosecond pulse in the anomalous dispersion region, where soliton fission mechanisms play an important role [4,18]. The two types of PCFs under investigation exhibit two different ZDWs. The dispersion profiles and relative group delay as a function of the wavelength for two types of PCFs are presented on Figs. 1(a) and 1(b), respectively. The dispersion between the two ZDWs is anomalous and outside the region is normal. For the PCF of the first type, it has two ZDWs around 655 and 930 nm and provides a dispersion profile with both positive and negative dispersion slopes. The dispersion properties of the second type displays two closely lying ZDWs at 640 nm and 760 nm and anomalous dispersion in the narrow range between the two ZDWs. In this letter, the same input pulse parameters in our numerical simulations was used: the pump wavelength ${\lambda }_0$ = 725nm, initial pulse width $T_0$ = 50fs, the peak power $P_0$ = 200W, and the fiber length is 45 cm.

 

Fig. 1 (a) Dispersion profiles as a function of the wavelength for PCF1 (black curve) and PCF2 (red curve). (b) Relative group delay as a function of the wavelength for PCF1 (black curve) and PCF2 (red curve).

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Provided a large spectral overlap exists, the pump efficiently sheds away energy to DW which fall in the normal dispersion region of the PCF. The center frequencies of DW are determined by the phase-matching condition [4, 7, 9]:

3.1 Trapping effect of DW in PCF1

We first present a detailed numerical study on the trapping effect of DW in PCF1. Figure 2 shows the temporal(a) and spectral(b) evolution of pulse along with propagation distance for PCF1. In the frequency domain, a B-DW as well as a R-DW in the normal dispersion regime can be clearly identified simultaneously and the SC is bounded by the two branches of DWs in this case, see Fig. 2(b). The distance at which B-DW occurs generally corresponds to the point at which the injected pulse attains its maximum bandwidth after strong temporal compression and then higher-order soliton breaks up into a train of constituent fundamental solitons with different width and peak intensity. After the initial fission, the ejected fundamental soliton experiences a continuous shift to longer wavelengths from the effect of Raman induced frequency shift (RIFS). The soliton that is ejected first has higher intensity and shorter duration and propagate with faster group velocities relative to the pump wavelength. A consequence of this is that the soliton that is ejected earlier in the fission process experience greater downshifts of frequency. This can be confirmed by numerical simulation results in Fig. 2(b). Particularly, there is an important feature that the very efficient conversion of energy from the soliton to the R-DW and stabilized soliton self-frequency shift was observed in Fig. 2(b). As the central frequency of red-shift soliton gradually approaches the second ZDW with a negative dispersion slope, the wavelength of DW is red shift with respect to the soliton central wavelength according to the phase-matching condition. With further propagation, the R-DW will shift toward the blue side as the soliton shift toward the red side due to the RIFS. Hence, the superposition of the soliton spectrum with the R-DW spectrum is expected to be enhanced in these regions, where energy is efficiently transferred from the most intense central part of the soliton spectrum to the R-DW in the normal dispersion regime. The high efficient conversion is related to the prerequisite for the energy transfer from the soliton to the R-DW which is given by the overlap between the long-wavelength side of the red-shift soliton spectrum and the wavelength determined by the resonance condition for the generation of DW [7,8,10]. Finally, the red-shift soliton suddenly cease to shift in the vicinity of the second ZDW due to the balance between the Raman self-frequency shift and the effects of spectral recoil from the amplification of R-DW. As is shown in Fig. 2(b), such a balance will lead to the formation of frequency-locked solitary pulses with a growing tail of red-shifted radiation. In the time domain, two clear red-shifted Raman solitons can be observed and its delays increase with the propagation distance due to a decrease of the group velocity as the solitons shift towards the infrared. As expected, the soliton ejected earlier exhibits stronger intensity and experiences larger delay. Meanwhile, the DW is associated with the development of a low amplitude temporal pedestal due to the normal fiber dispersion in the process of field evolution, which can be seen clearly in Fig. 2(a).

 

Fig. 2 Evolution of the temporal (a) and spectral (b) signatures as a function of fiber length. (c) The corresponding numerical X-FROG trace at the fiber output. The vertical dashed lines in (b) mark the two ZDWs of the fiber, separating regions of normal and anomalous dispersion.

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Figure 2(c) shows the corresponding numerical X-FROG trace which simultaneously describes spectral and temporal content of the signal at the end of the fiber. From the simulated spectrogram in Fig. 2(c), we can observe the temporal distribution of the spectral components in the output pulse. In addition, we can clearly see how the input pump pulse has broken up into a number of red-shifted fundamental solitons which are very narrow in the time axis, but broad in frequency. As clearly shown in the picture, two obvious Raman red-shift solitons are formed and the first ejected soliton features a larger spectral width and higher peak power compared to the second one, which agrees with the characteristics of constituent fundamental soliton ejected through soliton fission. The energy exchange between the first soliton and the R-DW is also clearly evident in Fig. 2(c). As for B-DW, it appears on the blue side of the spectrum and has smaller group velocity than the first ejected red-shift soliton, so it lags behind the red-shift soliton in the reference frame [30]. However, the R-DW emitted at wavelength longer than that of the soliton which needs less time than the red-shift soliton to reach a given distance z in the fiber, so the R-DW propagates ahead of the soliton [10]. As shown in Fig. 2(c), both B-DW and R-DW experience dispersive spreading due to the propagation in the normal dispersion regime, while the red-shift soliton can keep its shape as it shift towards longer wavelength side. Comparing Fig. 1(b) and Fig. 2(c), one can observe that the external appearance of numerical X-FROG trace is similar to the shape of the letter “Z”, which is consistent with the wavelength dependence of the relative group delay.

Another important phenomenon, which we can observe from Fig. 2(c) is that, apart from the excitation of dispersive radiation created by the red-shift soliton, there are other new spectrum peak in the output spectrum. Firstly, we can discern a rather sharp blue peak at the blue edge of the output spectrum. It have been demonstrated that such a striking result could be connected to the soliton trapping of DW which is related to the nonliner phenomena of cross phase modulation (XPM) and four-wave mixing (FWM) between the solitons and DW components [8,26,31,32]. The trapping effect of DW by soliton is triggered by the group-velocity matching between the soliton and the DW, which critically depends on the walk-off dynamics between the two components [30,31,35]. The initial group velocity of the B-DW is considerably slower than that of the first ejected soliton, so they quickly fall behind and the XPM between soliton and DW is weak. However, the soliton frequency is continuously red shifting in the presence of intrapulse Raman scattering, which for anomalous GVD implies continuous decrease of the group velocity. Therefore, we need to realize that, even though the red-shifted Raman solitons and DW have separated optical spectra, nevertheless, they may still overlap in the time domain. It has been shown that the B-DW in the normal dispersion region can be trapped by the potential barrier presented by the decelerating soliton in the anomalous dispersion region and then the two pulses copropagate along the fiber [30,41–43]. When the leading part of the DW is overlapped with the trailing edge of the red-shifting soliton pulse, the soliton can cause a modulated refractive index on the leading edge of the DWs, leading to a blue-shift up to a wavelength that makes the group velocities of the two waves equal. Hence, the sharp spectral peaks at the blue edge of supercontinuum can be explained by the trapping of B-DW across the zero-dispersion wavelength.

It is important to stress that there is another spectral peak around 980nm in the numerical X-FROG trace and the corresponding fine substructure in the frequency domain is indicated by a red rectangle in Fig. 2(b), which have not been described in previous studies. The underlying mechanism can contribute to the formation of this new spectral peak is the soliton-dispersive wave interaction through FWM process. As above said, the Raman-induced spectral downshifting of the first ejected soliton is cancelled in the vicinity of the second ZDW due to the effects of spectral recoil resulting from the amplification of R-DW. However, if the pump pulse is adequate to give birth to more than one soliton, then the second and subsequent solitons are always born inside the tail of the DW emitted by the first ejected soliton, as shown in Fig. 3, where the pump power has increased up to 350W. This new generated solitons propagate ahead of the first strong soliton and thus can effectively trap the R-DW emitted by the first ejected soliton. The trapping of R-DW is due to the temporal overlap between subsequent solitons and R-DW emitted by the strongest primary soliton. Furthermore, the mixing of the soliton and the R-DW then leads to energy transfer into a new resonance, with a wavelength lying between the soliton and the R-DW. As we can see from Fig. 3, new trapped wave packet are efficiently excited from their resonant interaction, and the number of trapped wave packet is increased as the fiber input power of the pump pulse is increased. It is interesting to note that the interaction of R-DW with the subsequent solitons leads to the appearance of a clearly observable spectral peak which can efficiently fill the spectral gap between the solitons and the R-DW, see Fig. 3. Therefore, mixing of solitons with R-DW may provide a method to make the supercontinuum become flatter. As mentioned above, the trapping of the B-DW, by contrast, can lead to the enhanced supercontinuum generation at blue edge of output spectrum. That is to say, trapping effect of DW by solitons can help to shape not only the edge of the supercontinuum, but also its central part, thus leading to a wider and smoother supercontinuum generation.

 

Fig. 3 The numerical X-FROG trace at the fiber output for pump power is 350 W.

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In order to clearly show the onset of soliton trapping of DW along with the pulse evolution process, we choose only a short propagation distance in our numerical simulations. The corresponding numerical X-FROG traces for the six selected distances are displayed in Fig. 4. At the initial stage of pulse evolution from z = 0 to about 3cm, the initial pulse undergoes SPM-induced spectral broadening, which results in a roughly symmetric spectral expansion of the pulse (see Fig. 4(a)-4(b)). For slightly longer propagation distances, between 3 and 10 cm, the SPM dominated stage of the supercontinuum development ends, see Fig. 4(c). The first ejected soliton is not stable due to higher dispersion and nonlinear effects, then sheds away energy to B-DW located at the short-wavelength edge of spectrum. The wavelength of the pump pulse is initially close to lower ZDW, which give rise to a strong amplification of the B-DW. However, there is few energy contained in the R-DW due to it emerges at larger frequency shifts and a much weaker intensity relative to B-DW, see Fig. 4(c). For propagation distances between 10 and 25 cm, the first ejected soliton continually experience SSFS under the action of Raman scattering while the bandwidth of the B-DW remains nearly constant. As a consequence, the red-shifted soliton accumulates a time delay of about 1.5 ps due to a decrease of the group velocity as the soliton shift towards longer wavelength side. When the center wavelength of the soliton shift into the vicinity of higher ZDW, the R-DW start to be amplified (see Fig. 4(c)-4(d)). Indeed, the intensity of the R-DW increases as the soliton pulse shifts to the longer wavelength, which indicates the intensity of the R-DW is related to the energy transfer from the red-shift Raman soliton. Besides, it can be seen from Fig. 4(d) that the R-DW spread quickly in time due to their dispersive nature. With further propagation the central wavelength of the soliton remains unchanged because a balance is achieved between the red shift due to Stimulated Raman scattering (SRS) and the blue shift due to spectral recoil. Furthermore, the trapping of B-DW can be related to the formation of non-dispersive wave packet on the left side of B-DW (see Fig. 4(e)). It can also be noticed in Fig. 4(e) that R-DW exhibits some spectral modulations due to the interaction with the second soliton. For z > 35cm, the increase of pump power allows formation of multiple solitons which carries less energy than the first strong soliton. These new solitons propagate ahead of the first strong soliton and thus effectively interacts with the R-DW emitted by the first ejected soliton. From Fig. 4(f), it is apparent that new spectral components located between the solitons and the R-DW can be observed, which is closely related to the trapping of R-DW by subsequent solitons.

 

Fig. 4 The numerical X-FROG traces of the pulse for different propagation lengths z . (a) z = 0 cm, (b) z = 3 cm, (c) z = 10 cm, (d) z = 25 cm, (e) z = 35 cm, (f) z = 45 cm .

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3.2 Trapping effect of DW in PCF2

In this part, the propagation characteristic of input pulse is investigated in PCF2, whose two ZDWs are closer to each other compared to PCF1. The evolution of the numerical X-FROG traces in PCF2 for various values of propagation lengths z is plotted on Fig. 5(a) to 5(f). It is important to point out that there is no observed Raman soliton in this situation because the anomalous dispersion region of the PCF2 is too narrow to form soliton. Indeed, the output spectrum is characterized by two nearly symmetric major spectral content and the “Z-shaped” intensity distribution of the numerical X-FROG traces is identified at the end of fiber, as shown in Fig. 5(f). At the initial stage, the pulse spectrum begins to broaden also due to self-phase modulation. Subsequently, the propagation of pulse is influenced by the dispersion properties of the fiber as seen from the bending of the major peaks in the spectrogram. It has been demonstrated that such a supercontinuum is generated through the combined action of self-phase modulation and phase-matched four-wave mixing, other than soliton fission as in the initial PCF [5]. The anomalous dispersion region is so narrow that the SPM has rapidly moved most of the pulse energy into the normal dispersion region and soliton dynamics plays only a minor role in the formation of the supercontinuum. Therefore, in this case, the soliton trapping of DW cannot be observed.

 

Fig. 5 The evolution of the numerical X-FROG traces in PCF2 for various values of propagation lengths z. (a) z = 0 cm, (b) z = 3 cm, (c) z = 10 cm, (d) z = 25 cm, (e) z = 35 cm, (f) z = 45 cm.

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4. Conclusion

In this work, we have numerically investigated the trapping effect of DW during femtosecond-pumped supercontinuum generation in two different PCFs characterized by two ZDWs. In order to get a clear insight on this phenomenon, we plot the numerical X-FROG traces to display the relative temporal positions of the frequency components of the pulse as it propagates along the PCFs. In particular, we have showed that both B-DW and R-DW can be trapped by soliton for properly designed PCFs with two zero-dispersion wavelengths. Furthermore, we observed that the new spectral components can come into being is closely related to the trapping effect of DW through the mixing between soliton and DW. For a given optical fiber length, more solitons can be formed as the input power is increased. Thereby, increasing the pump power would also enhance the trapping effect of DW by solitons. In addition, it is important to point out that the dynamics of the R-DW differs from the behavior of the B-DW. The trapping of B-DW can extend the supercontinuum to the blue edge, whereas the trapping of R-DW can fill the spectral gap between the red-shift soliton and the R-DW components. Therefore, the numerical simulations and analysis presented here may open up the possibility of generating broadband and flat supercontinuum output in PCF by properly tailoring the fiber dispersion profile.