1. Introduction

Soliton self-frequency shift (SSFS) [1,2] allows for continuously and widely tunable light sources of femtosecond pulses [3–7]. The continuous frequency redshift is induced by the Raman effect, where inelastic scattering of light by optical molecular vibrations produces a delayed nonlinear response. Wavelength tuning of the Raman shifted solitons (in short, Raman solitons) is commonly done by varying their input power [8]. Controlling the pulse central wavelength, pulse shape, fiber length, or working with different types of fibers are also possible [3,8,9].

SSFS is enhanced using highly nonlinear fibers of favorable dispersion, and using powerful short pulses [8], e.g. through soliton fission – the breaking of a higher-order soliton into multiple fundamental solitons of different widths, under higher-order dispersion and the Raman effect [8,10,11]. Using high power few-cycle pulses improves frequency shift rates and reduces output pulse durations [12].

Fundamental solitons maximize their frequency shift when they are transform limited or compress due to slightly positive chirp, depending on the fiber length [13]. Higher-order solitons undergo a complex fission process, defying such a general relation – especially for few-cycle pulses, where the analysis is even more complicated. Despite extensive work on SSFS and soliton fission, experimental data are lacking for the few-cycle limit, and their optimal prechirp remains elusive.

Some studies have addressed this issue for supercontinuum generation [14–18], but they typically involve much higher soliton orders launched at the zero dispersion point. Different pulse and fiber parameters were found to have different optimal input prechirps, from negative to positive, that widen the output spectra by extending it to longer wavelengths. Initial prechirp was also suggested to change the power distribution (and wavelength separation) between the fundamental solitons after soliton fission [19]. The simulations in [19] imply that using a transform limited input maximizes the frequency shift of the primary Raman soliton. However, this analysis was done for longer pulses and its assumptions are not valid in the few-cycle regime.

In this work, we experimentally study the Raman induced frequency shift of few-cycle higher-order solitons. We use different lengths of nonlinear photonic crystal fibers (PCFs), and systematically compare how the input power and chirp affects the output spectra. By repeated measurements, we track the pulse dynamics as it propagates along the fiber, starting from the first millimeter. We find the optimal input chirp that maximizes the shift of the shortest soliton after fission. Small core silica PCFs with suitable dispersion have severe $\textrm {OH}^-$ absorption [20] that limits the wavelength tuning range. We show how few-cycle pulses of optimal chirp can overcome this absorption. Improving the frequency shift allows using shorter fibers, where input power fluctuations might result in reduced timing jitter [21]. This is important when synchronization is required, such as in few-cycle pump-probe experiments [22].

2. Experimental setup

Figure 1 shows the setup used to directly control the input power and chirp of the pulses, and to measure their spectra after propagation in nonlinear PCFs. A mode-locked Ti:Sapphire oscillator (Thorlabs Octavious) generated pulses of full-width at half-maximum duration 7.5 fs around 800 nm wavelength (see inset in Fig. 1 for their full spectrum), at a repetition rate of 80 MHz. We scanned the chirp using a pair of motorized glass wedges (Spectra-Physics OA325) and fixed dispersion compensating mirrors (Spectra-Physics GSM022). To control the pulse power we used a variable attenuator made of a motorized half-wave plate and a polarizer transmitting linear horizontal polarization. A flip mirror directed the beam into a SPIDER device (Spectral Phase Interferometry for Direct Electric-field Reconstruction, model: A.P.E FC Spider NIR) that measured the pulses duration and chirp.

 

Fig. 1. Experimental setup. A 7.5 fs pulse is injected into one of two PCFs (photonic crystal fibers) types of variable length, while scanning its chirp and power. A SPIDER (Spectral Phase Interferometry for Direct Electric-field Reconstruction) analyzes the input beam, and two spectrometers analyze the output beam via an integrating sphere and a spliced multi-mode fiber. Insets show initial spectrum (bottom left), an output spectrum (top right), and measured group velocity dispersion (GVD, bottom right) of "fiber A": NKT Photonics NL-1.9-765. DCM – Dispersion compensating mirrors. $\frac {\lambda }{2}$ - Half-wave plate. P – Polarizer. NIR – Near infrared.

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The laser beam was focused into the small PCF core using a reflective telescope that expanded the beam, followed by an off-axis parabolic mirror. Coupling efficiency of $50\pm 3\%$ was measured using low power (to eliminate nonlinear effects). A neutral density filter reduced the power without affecting the spatial mode and polarization. The maximal peak power coupled into the fibers was over 100 kW.

We worked with two types of fibers: NKT Photonics NL-1.9-765 ("fiber A"), which was cut back from a length $L\!=\! {154}\;{\textrm{cm}}$ to $L\!=\! {1}\;{\textrm{mm}}$; and NKT Photonics NL-1.8-715 ("fiber B") with $L\!=\! {200}\;{\textrm{cm}}$. Both fibers had significant attenuation around the $\textrm {OH}^-$ absorption band, with over $ {5}\;{\textrm{dB/m}}$ for "fiber A" and up to $ {30}\;{\textrm{dB/m}}$ for "fiber B". This was directly measured using a supercontinuum source (NKT Photonics SuperK compact), and found to be consistent with the power loss of the Raman solitons during propagation. "Fiber A" was spliced to a multi-mode fiber (Thorlabs FG105LCA) for lengths $L< {16}\;{\textrm{cm}}$. The splice loss was less than a few percent power, within the coupling efficiency variability. The multi-mode fiber length was about ${30}\;{\textrm{cm}}$.

Birefringent effects were reduced by rotating the fibers to maximize the polarization extinction ratio at their output. The shifted pulses were measured using two spectrometers (Ocean-optics HR4000 & Avantes NIR 1700) connected to an integrating sphere containing the fiber end.

We measured the group velocity dispersion of the fiber (GVD, see Fig. 1) with a white-light interferometer following [23], using the supercontinuum source, a 117 cm fiber, and a translation stage with a home-built encoder.

3. Results

Figure 2 presents data of measured spectra after propagation through 28 cm of "fiber A". It shows clear evidence for the optimal prechirp and implies the underlying mechanism. The output spectra of Fig. 2(a) depict representative results for different chirps and fixed higher-order soliton power. The spectra contain three main contributions: the primary Raman soliton, of shortest duration (in spectral region $\textrm {P}_1$); secondary pulses and residual input radiation (in $\textrm {P}_2$); and bluer non-solitonic radiation, mainly dispersive waves (also referred to as resonant radiation or Cherenkov radiation [8,24,25]), which might further interact with the pulses (in $\textrm {P}_3$). The input chirp is seen to affect the measured spectra in all three regions, and appears to maximize the frequency shift of the Raman soliton for chirp parameter $\textrm {C}=-0.75$. We define the chirp parameter as $\textrm {C}=\textrm {GDD}/\tau ^2$, where GDD is the excess group-delay dispersion introduced by the glass wedges (Fig. 1), and $\tau =\sqrt {2\sigma }$ is the Gaussian temporal width parameter, in agreement with Eq. 20 in [13] ($\sigma$ is the standard deviation of the Fourier transform-limited pulse amplitude).

 

Fig. 2. Results for 28 cm "fiber A". (a): Representative output spectra for various chirp parameters (C, see text) and input peak power of $ {81}\;{\textrm{kw}}$ ($\textrm {N}\approx 3$). Dashed lines and $\textrm {P}_1, \textrm {P}_2, \textrm {P}_3$ mark the spectral regions used to obtain the corresponding curves in (c). (b): Central wavelength of the shortest soliton as a function of input chirp and input power (color corresponds to input soliton number (N) at 800 nm wavelength. Peak power difference between consecutive curves is about 4.3 kW). Larger redshifts are obtained for increasing power, regardless of initial chirp. The optimal chirp changes from slightly positive to about $\textrm {C}=-0.75$ for $\textrm {N}>2.5$, enabling to pass the $\textrm {OH}^-$ absorption barrier (black dotted line). (c): Total spectral power as a fraction of overall power (solid curves) and shortest soliton central wavelength (dash-dotted curve) for the same input power as (a). Secondary pulses power ($\textrm {P}_2$) are inversely related to shortest soliton wavelength. Overall power is constant up to few percents.

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Figure 2(b) shows how the optimal chirp shifts from zero to negative, as the pulse power increases. The central soliton wavelength is plotted as a function of the input chirp parameter for various input powers, given in terms of the soliton number $\textrm {N}=\sqrt {E_{i}/E_s}$, with $E_{i}$ being the actual input pulse energy and $E_s$ the theoretical energy of a single soliton having the duration of a transform-limited pulse. The input chirp has a dramatic effect on the Raman shift for high input powers when $\textrm {N}>2.5$. At these powers, the solitons reach the $\textrm {OH}^-$ absorption band, which is most significant between 1380 and 1410 nm.

The power fissioned into the various components provide insights on the effect of the prechirp. The solid curves in Fig. 2(c) present the total spectral power that was measured in each wavelength range $\textrm {P}_1 - \textrm {P}_3$ (Fig. 2(a)), as noted in the legend. Their units are normalized to the average of their sum, which was constant up to few percents. The central wavelength of the corresponding Raman soliton is given by the dash-dotted line, with blue circles marking the measurements that are shown in Fig. 2(a). We argue that changes in power distribution into the different components during soliton fission explain why negative prechirping enhances the Raman shift. We find a striking anti-correlation between the central wavelength of the soliton and the power of the secondary pulses. Conservation of energy suggest that more power was directed into the short solitons around the optimal prechirp, enhancing their frequency shift. We measure a weaker correlation between the wavelength and power of the short solitons, which were prone to loss around the $\textrm {OH}^-$ absorption band. Although the dispersive waves interacted with the pulses and differed in shape, their total power remained similar.

Extending the measurements up to $\textrm {C}=4$ gave similar results (not shown), with clear frequency shift maxima around $\textrm {C}=0$ for low input power, and the negative prechirp peak for high input power. An additional but much smaller peak was found around $\textrm {C}=3$ for high input power.

We gain more understanding about the system by repeating the experiment using different fiber lengths and following how the spectra evolve with propagation. Figures 3(a) and 3(b) show such measurements for a single power and fiber lengths in the range 0.1-154 cm and for $\textrm {C}=-0.75$ and $\textrm {C}=0.07$, respectively. Figure 3(c) compares their 0.1 cm spectrum (black and light blue curves) with the one for $\textrm {C}=-1.5$ (blue curve) and the input spectra (dash-dotted red curve). These measurements show how the initial chirp affects the fission process as it occurs. We measure a delayed soliton fission for negative prechirps, postponing the appearance and build-up of dispersive waves. The transform limited case yields a much wider spectrum relative to the negative chirp, and its dispersive waves are almost fully developed at 1 mm (see Figs. 3(a) and 3(b) for comparison). Increasing the amount of negative prechirp increased the fission’s delay, allowing to directly observe its origin – spectral compression. The spectrum was narrowing down to 65% of the input bandwidth for high input power, where the spectral compression monotonically increased to the edge of our measurements, at $\textrm {C}=-1.5$.

 

Fig. 3. Measured spectra for different lengths of "fiber A" and pulse parameters. Top figures are for the same input peak power (P) as in Figs. 2(a) and 2(c), but different input chirp parameter (C). The first two spectra are for fiber lengths of 1 mm and 1.2 cm. Bottom figure compares the input spectrum to those after 1 mm, for the same power and the given chirp parameters. Spectral compression is seen for $\textrm {C}=-1.5$.

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Spectra at 1.2 and 5.5 cm (second and third spectra in Figs. 3(a) and 3(b)), after the fission process has ended, show a rapid frequency shift of the pulse that later becomes a Raman soliton (with a $\textrm {sech}^2$ intensity). At greater fiber lengths, spectral features in the visible region seem to interact with the shifting soliton, while the secondary pulses slowly redshift. Throughout the propagation, the optimal chirp produced secondary pulses of lower power and less redshift, fewer features in the visible range, and redder dispersive waves that interacted more significantly with the Raman soliton. The frequency shift saturated at its theoretical limit [26], close to the zero dispersion point ($\sim {1700}\;{\textrm{nm}}$) for high power pulses in long fibers. This spectral recoil came with energy transfer from the Raman soliton to the long wavelength dispersive waves, beyond the long zero-dispersion point. Similar results were obtained for other input powers of higher-order solitons.

Comparing measurements with similar Raman soliton wavelength, but different input parameters, we found great similarity in the Raman soliton dynamics – their power, spectral bandwidth and rate of frequency shift are almost identical (see e.g. Fig. 4(a), where fiber length starts at 5.5 cm). Their spectra fitted well to a soliton shape that differed by less than 5% in their transform limited durations. Similar Raman soliton dynamics were also found for high input powers, such as $\textrm {P} = {103}\;{\textrm{kW}}\textrm {,}\,\textrm {C} = 0.07$ that matched a 21% weaker input, with $\textrm {P}= {81}\;{\textrm{kW}}\textrm {,}\, \textrm {C} = -0.78$. The optimal chirp was observed to slightly shift with propagation. An example can be seen in Fig. 4(b) that presents the Raman soliton dynamics for pulses of the same input power, but with optimal chirps for different lengths. We interpret this as originating from small differences in the Raman soliton chirp.

 

Fig. 4. Comparison of measured Raman soliton spectra for different lengths of "fiber A" and input pulse parameters as specified in each figure. (a): We find high similarity throughout for transform limited pulses (dark colored with blue diamonds) and negative chirp with reduced power (uncolored with black crosses). (b): The optimal negative chirp for $L< {30}\;{\textrm{cm}}$ produces spectra marked with red stars. The optimal chirp slightly shifts for longer fibers, generating the light colored spectra with black circles.

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We find similar results for "fiber B", which has a prominent $\textrm {OH}^-$ absorption band around 1400 nm wavelength (Fig. 5): the optimal chirp shifts to (the same) negative values with increasing power, and an inverse relation exist between secondary pulses power and short soliton wavelength. The $\textrm {OH}^-$ band causes severe attenuation for pulses entering it, extinguishing them, unless they continue to redshift fast and escape further loss. Accordingly, we see that pulses pass the $\textrm {OH}^-$ band only for high input power and close to the optimal chirp.

 

Fig. 5. Measured spectra after 2 m of "fiber B" for various input powers. Chirp parameters are $\textrm {C}=-0.75$ and $\textrm {C}=-0.08$ for figures (a) and (b), respectively. Dashed line at 1400 nm marks the severe $\textrm {OH}^-$ absorption band.

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We also note the spectral features that appear for high powers, and are slightly redder than the dispersive waves (Fig. 5(b)). Similar features were seen in "fiber A" (see Figs. 2(a) and 3(b)), and are absent for the optimal chirp (Figs. 2(a), 3(a) and 5(a)). We believe these are additional dispersive waves shed from the secondary pulses: it qualitatively explains the features appearance during the second pulse ejection, after the ejected main soliton has already created the strong dispersive waves [25]; their wavelength agrees with the phase matching conditions [8,24], with dispersion (Fig. 1 bottom right inset) expanded to fourth order, pulse wavelength of $ {800}\;{\textrm{nm}}$ and neglecting the nonlinear term; and this conjecture qualitatively agrees with the features’ strength, and their appearance and blueshift with increasing input power. Cases of optimal chirp and cases of low input power result in weaker secondary pulses that don’t excite such dispersive waves.

We obtained numerical solutions of the pulse dynamics based on the generalized nonlinear Schrödinger equation (GNLSE) [8,27], and experimental parameters such as the measured input pulses, full dispersion relation $\beta (\omega )$, effective mode-area $A_{\textrm {eff} (\omega )}$, and absorption $\alpha (\omega )$. We set the initial prechirp by propagating the pulses through BK7 glass, simulating the experiment. Our numerical solutions (not shown) were only qualitatively consistent with the measurements, due to the difficulty in correctly accounting for the Raman scattering in the few-cycle regime. Mainly, we found that nearly transform-limited pulses were best for low power, while slight negative chirp was best for high power. Also, there was a remarkable fit to a ${\textrm {sech}}^2$ for the main soliton, even during its frequency shifting. The higher-order soliton split into three spectral regions [25], in accordance with the experimental results.

4. Discussion

We found that negative prechirping optimizes soliton self-frequency shift (SSFS) in the case of intense few-cycle pulses, with soliton number $\textrm {N}\gtrsim 2.5$. The situation is different for fundamental solitons $\textrm {N}\sim 1$, where approximately zero chirp is favorable, in agreement with the literature for longer pulses [13,15]. Our results suggest that power redistribution among the constituent solitons during their fission plays the dominant role. The optimal chirp was almost constant from right after soliton fission, to 150 cm, and the soliton dynamics appeared to be mostly determined by the near infrared spectral power distribution after fission (Fig. 4). The secondary pulses had simple dynamics, allowing their total power to be used as a proxy to the main soliton power, which had more complicated dynamics. The wavelength of the main soliton was anti-correlated with the secondary pulse power (Fig. 2(c)) throughout our experiments. Negative prechirp not only reduces the pulse peak power, but counteracts its self-phase modulation (SPM) [28–30], producing the observed spectral compression and delayed soliton fission (Fig. 3). As SPM is dominant during soliton fission, these effects may be vital ingredients for explaining how the prechirp changes the power distribution after soliton fission. We note that positive prechirp counteracts the anomalous dispersion and affects the SPM mainly through peak power reduction, not causing similar spectral compression. This can explain the asymmetry seen between positive and negative chirps.

Optimal negative prechirp was correlated with enhanced interaction between the dispersive waves and the soliton. Such interaction has the potential to reinforce the frequency shift through a spectral recoil [31]. However, improved soliton frequency shift and redder dispersive waves enhance their mutual collision and interaction, reversing cause and effect. Results such as in Fig. 4(a) suggest a minor role for the interactions between the soliton and the dispersive wave after fission, as the two solitons share almost identical dynamics, despite a big difference in the corresponding dispersive-wave spectra (the visible spectra is not shown in this graph, but is similar to those seen in Figs. 3(a) and 3(b)). We note that such interactions might affect the power redistribution during or right after soliton fission [17,32], and further research is needed in this respect.

A similar trend for the optimal prechirp was found in [17], despite the great differences in system parameters. The underlying mechanisms in [17] include the transfer of power from normal to anomalous dispersion during initial propagation, reduction in the number of produced solitons, and the interaction between the soliton and the dispersive wave. In our case, such a transfer of energy is irrelevant, as nearly all the input spectrum lies in the region with anomalous dispersion. Our soliton numbers are much lower, but a similar trend exists for the power redistribution after soliton fission, and spectral recoil might be relevant for the initial propagation. Studies with $\sim {100}\;{\textrm{fs}}$ pulses in the realm of supercontinuum generation [14,16] found optimal prechirps to be positive, but conveyed the same message – more of the power should be directed into the main soliton using prechirp.

The increased efficiency of the SSFS allows to tune the solitons past severe $\textrm {OH}^-$ absorption bands (Fig. 5), which are typical for small core PCFs [7,20]. If the frequency shift is fast enough, the soliton continues to rapidly shift through the absorption band, avoiding further loss. This can be viewed as effectively shortening an absorber, which is the fiber segment propagated while the pulse is in the absorption band. If the frequency shift is too slow, a positive feedback loop diminishes the pulse power: the absorption introduces chirp and reduces the soliton power, which in turn slows the Raman shift, causing the soliton to spend more time (and propagate a greater distance) in the $\textrm {OH}^-$ band, thus further increasing absorption. The $\textrm {OH}^-$ barrier is worsened when relatively high dispersion is present near it, which is common for solid core PCFs with extended anomalous dispersion.

The high efficiency enables to shorten the fibers used, and potentially reduce the pulses timing jitter [21] – the Raman effect translates input power fluctuations into modified soliton frequencies that affect the propagation time via GVD. This is important for applications that require synchronization, such as in few-cycle pump-probe experiments [22].

5. Conclusion

We have seen the significant effects of prechirp on the dynamics of few-cycle higher-order solitons in PCFs. Few-cycle fundamental solitons Raman-shift the most when they are transform limited, but not higher-order solitons. The right amount of negative prechirp improved their Raman shift efficiency, and allowed tuning them beyond severe $\textrm {OH}^-$ absorption bands. Evidence for affecting the soliton fission process suggest a physical explanation for the prechirp effectiveness.