1. Introduction

Nonlinear pulse propagation in photonic crystal fibers (PCFs) with two zero-dispersion wavelengths (ZDW) is now well-known to exhibit a range of dynamics that is typically not observed in single-ZDW fibers [1–5]. A particular effect that has been the subject of a recent numerical study is the generation of dispersive waves (DWs) with very large amplitude and/or enhanced wavelength shift into the normal dispersion regime due to the collision of two solitons of different wavelengths [6]. These large amplitude DWs can be considered as a class of extreme-value or “rogue” events [7] because they populate the extreme long tails of the corresponding statistical distribution of their peak power and/or wavelength.

In this paper, we report the first experimental evidence for the generation of collision-induced dispersive waves with an enhanced redshift compared to those which are generated by single solitons in the absence of collision. We initiate controlled soliton collisions through the injection of two time-delayed pulses in a PCF with two ZDWs and, by scanning the relative delay of the two input pulses we are able to vary the point of collision in the fiber to observe both single soliton DW generation as well as collision-induced DWs. In addition to the studies of controlled collisions, we also report experimental studies of spontaneous collision induced DWs in supercontinuum (SC) generation. Our experimental results are in very good agreement with numerical simulations. As well as providing direct evidence for these unique DW events, our results also show how evidence for soliton collision events in nonlinear fiber optics can be inferred from indirect measurements even when direct characterization is not possible along the propagation direction

2. Experimental setup and illustrative simulations

The collision dynamics and DW characterization were studied using the setup in Fig. 1 . Here, 200 fs pulses at 980 nm from a mode-locked Ti:Sapphire oscillator are divided between the two arms of a Michelson interferometer in which a variable attenuator is placed in the moveable arm. The interferometer output then consists of a pulse pair of variable relative energy and variable temporal delay. The pulse pair is injected into 1 m of PCF with two ZDWs located at 750 nm and 1260 nm (Crystal-fibre NL-PM-750). The group velocity dispersion and nonlinearity at the pump wavelength of 980 nm are β ₂ = −2.08 10-26 ps²m⁻¹ and γ = 0.1 W-1m-1 respectively.

 

Fig. 1 Schematic of the experimental setup for controlled soliton collision studies. BS: beamsplitter, VA: variable attenuator, OSA: optical spectrum analyzer.

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To illustrate how this setup can be used to study soliton collision dynamics, Fig. 2 shows nonlinear Schrödinger equation (NLSE) simulation results of the temporal and spectral evolution of an injected pulse pair in the PCF. Our simulations use experimental parameters, the full Raman response of fused silica, the wavelength dependent effective area, and input noise at the one photon per mode level [8]. The input consists of pulses A and B at 980 nm with peak powers of 310 W and 210 W respectively. The pulses have identical durations of 200 fs, and we consider initial separation of 0.8 ps with pulse B lagging behind pulse A (note that positive time corresponds to greater temporal delay). The initial propagation dynamics are dominated by the independent break up of pulses A and B and the orderly ejection of fundamental solitons through soliton fission [8]. Because pulses A and B have different peak power, they are associated with different soliton orders N _A ~4 and N _B ~3.5, and this leads to different initial ejected soliton durations of Δτ _Α ~29 fs and Δτ _Β ~34 fs respectively. Note that for low soliton orders, soliton ejection is coherent and deterministic [7]. A significant feature of this dynamics is that the ejected solitons propagate in a regime of negative dispersion slope where both the GVD |β ₂| and the group velocity itself decrease with increasing wavelength. Thus, because of the strong Δτ⁻⁴ dependence of the Raman soliton self-frequency shift, the soliton ejected from pulse A shifts to longer wavelengths significantly faster and thus decelerates relative to the soliton ejected from pulse B. Figure 2(a) shows clearly how soliton A slows down relative to soliton B, eventually leading to a soliton collision as their trajectories cross. The key aspect here from an experimental point of view is that by varying the initial delay, the point in the fiber where this collision occurs can be controlled. This allows investigation of different classes of dynamics depending on where the wavelengths of the frequency-shifted solitons lie relative to the long wavelength ZDW at the point of collision.

 

Fig. 2 Illustrative simulation results of (a) temporal and (b) spectral evolution to show how a pulse pair injected into a PCF can be used to study collision dynamics. (c) shows details of the temporal intensity profiles at selected distances as shown.

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3. Experimental results and discussion

With pulse powers as given above, we performed spectral measurements at the PCF output for a range of delays. We emphasize that no DW components were observed when the two arms were coupled independently. Results are shown in Fig. 3 which also shows additional simulation results to allow interpretation and discussion of our results.

 

Fig. 3 For decreasing delay between the interferometer arms as indicated, the top panels show measured (upper) and simulated (lower) spectra at the PCF output. The central panels show the simulated temporal evolution illustrating the soliton crossing trajectories and the different classes of single soliton dispersive wave (SDW) or soliton collision dispersive wave (CDW) generation. The lower panel shows the output spectrograms with the insets in (a)-(c) showing the spectrograms at point of collision. For clarity the amplitude of the DW was increased by a factor of 50 in the time-trajectory plot.

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We first discuss the results in Fig. 3(a) at the largest delay of 1.9 ps. The experimental spectra in the top panel clearly shows the presence of distinct soliton peaks at 1035 nm and 1130 nm, as well as a residual component around the pump wavelength of 980 nm. Comparison with NLSE simulations allow us to identify the solitons corresponding to the leading and trailing pulses A and B as shown. We note that the agreement between experiment and simulation is excellent. The result in the middle panel shows that for this largest choice of delay, the solitons in fact do not collide in the PCF and, in the absence of collisions both the spectra and the spectrograms show no evidence of any long wavelength DW structure.

Experimental spectral signatures of long wavelength DW generation are seen as the delay between the injected pulses decreases. At a delay of 1.3 ps, Fig. 3(b) shows a strong spectral peak around 1600 nm which is clearly reproduced in the simulated spectrum. Simulations of the time-domain evolution and spectrogram dynamics allow us to identify this DW as generated from the soliton collision occurring at 0.87 m propagation. The calculated spectrogram is shown in the lower panel, and plotted both at the fiber output and at the point of collision (see inset). The latter illustrates how the collision-induced dispersive wave (CDW) is generated from the spectrally-broadened superposition field that extends into the long wavelength normal dispersion regime, whilst the former highlights the temporal separation between the solitons and the generated DW because of walkoff with propagation. As the collision occurs only over a very short distance we note that the emission of the collision-induced DW is a localized phenomenon compared to the continuous-like radiation of a DW by a single soliton approaching the second ZDW [1].

With further reduced delay of 0.8 ps in Fig. 3(c) the collision occurs earlier in the propagation at 0.78 m and in fact the spectrum in this case shows both a signature of a collision-induced DW as well as the onset of a single soliton DW (SDW). The SDW arises because during the collision the more redshifted soliton is amplified at the expense of the other soliton. The energy increase then leads to a corresponding enhanced rate of redshift experienced by the soliton which then emits a DW near the fiber output [9]. Therefore, the collision plays an indirect role in the generation of the SDW. This scenario is in contrast with the generation of the CDW component that is emitted during the collision itself [6]. Note that a SDW component is not observed if the collision occurs at a late propagation stage because of insufficient frequency-shift (see Fig. 3(b)). Again the experimental results are in agreement with simulations, and the simulated spectrogram allows us to see both the CDW and SDW components. Interestingly, with the smallest delay of 0.5 ps as shown in Fig. 3(d), the collision occurs so early in the fiber that the spectral extent of the collision field does not extend into the normal dispersion regime and there is no CDW component observed. On the other hand, the collision still induces an enhanced frequency shift to soliton A so that its individual trajectory after propagation the full length of the fiber does yield generation of a clear SDW component around 1480 nm seen in both the experimental and simulated spectra.

The results above clearly show how controlled soliton collision from twin-pulse excitation leads to the generation of DWs with significantly enhanced wavelength shift in the long wavelength normal dispersion regime. Indeed, the wavelength shift of the CDW into the normal dispersion regime exceeds by more than 100 nm that of the SDW. These results confirm the earlier numerical predictions of the dynamics of collision-induced DW generation [6]. Interestingly, with our parameters, the pulse duration and frequency separation at the collision point are in the range where phase-sensitivity to the energy exchange process may be expected [9]. The degree to which this influences the properties of the collision-induced DW waves, however, will require further clarification, because the wavelength of the DW generated during the collision does not appear to depend on the mechanism of energy exchange, but rather on the mean wavelength (and spectral tail) of the nonlinear superposition.

An important prediction of the numerical study of Ref. [6]. was that collision-induced DWs would be generated spontaneously during supercontinuum generation under conditions of significant shot-to-shot fluctuations, and would exhibit long-tailed “rogue” statistics when spectral filtering was used to measure spectral power in the long wavelength normal dispersion regime. To test this, we performed additional experiments generating noisy SC using 5 ps pump pulses at 865 nm from a Ti:Sapphire laser with an 80 MHz repetition rate and with peak power of 500 W. With these pump pulse parameters the SC expands beyond the second ZDW and a distinct DW component is observed in the long wavelength normal dispersion regime as can be seen in Fig. 4 . Note that the apparent smoothness of the SC indicates substantial shot-to-shot variations in the output SC. The DW component at 1450 nm corresponds to the average DW generated by single-solitons, and typically occurs in every shot. On the other hand, the collisions leading to the generation of DW with enhanced wavelength shift are rare and thus they cannot be seen directly in the measured (averaged) spectrum. However, because the collision-generated DWs are associated with an enhanced redshift, their statistical signature can be readily captured by using a long-pass filter with a cutoff wavelength larger than the wavelength of the single-soliton DW seen in the spectrum.

 

Fig. 4 Measured (a) and simulated (b) average SC spectra generated by 5 ps pulses at 865 nm. Experimentally recorded (c) and simulated (d) histograms of the time-series when DW components are filtered. The long-pass filter (marked as a dashed line) is placed at a convenient position to allow single-soliton and soliton-collision-induced DW to be distinguished through the statistics. (e) Selected portion of recorded time trace containing 800 shots and showing evidence of a CDW. Insets: histograms in a log-log scale.

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To examine the statistics of the long wavelength DWs we placed a long-pass spectral filter at the fiber output to isolate the spectral content beyond 1500 nm. The corresponding time-series of the shot-to-shot DW component is detected with a fast detector (InGaAs, 2 GHz bandwidth) and oscilloscope (2 GHz bandwidth, 10 Gsample/s) [10]. The signal amplitude on the oscilloscope in fact corresponds to the filtered energy rather than peak power and therefore we are measuring the energy of the DW in the long wavelength normal dispersion regime that is located beyond the filter cutoff. The spectrum of the measured SC and the corresponding histogram of the detected time-series are illustrated in Fig. 4. The statistical distribution of the DWs energy can be seen to be highly skewed indicative of extreme-value events. Numerical simulations are in very good agreement with the experimental average spectrum and histogram of the filtered time-series. Significantly, careful inspection of multiple shots in the numerical simulations shows that the tail of the distribution is exclusively composed of DWs generated as a result of soliton-collisions that have occurred along propagation in the fiber. Indeed, the DWs generated through soliton-collision possess a larger wavelength shift than DW generated by single solitons [6]. As a result, when the cutoff wavelength is chosen to coincide with the extreme edge of the SC spectrum only collision related events are captured and, therefore, the technique employed is particularly appropriate as the energy of the collision-induced DW post-collision remains constant and they can thus be captured a posteriori. With these parameters, as the soliton collisions leading to the generation of DWs with enhanced redshift are rare, the statistics exhibit the highly skewed-shape characteristics of extreme-value events. We emphasize that this mechanism is different from that leading to non-Gaussian statistics for DWs generated on the short wavelength side by single solitons and which originates from cross-phase modulation coupling [11].

5. Conclusions

In conclusion, we have reported on experimental signatures of dispersive waves generated directly through the collision of two femtosecond solitons in a photonic crystal fiber with two zero-dispersion wavelengths. In particular, by tuning the distance at which the collision occurs we were able to distinguish our observations from ordinary dispersive waves shed by single solitons. We have further shown that, in rare cases, the DWs generated on the long wavelength normal dispersion region in a noise-driven SC exhibit a significantly enhanced redshift as a result of solitons-collision mechanism. Our experimental results are of applied interest due to the extreme wavelength associated with dispersive waves generated by collisions. The DWs with extreme redshift observed here could also be used to indirectly study the statistics of soliton collisions. Finally, we emphasize the fact that similar dynamics are expected in a fiber with a single ZDW, yet their observation is more difficult as in this case the Raman shift detunes the solitons away from the ZDW.