## Abstract

We numerically investigate the generation of wavelength-tunable few-cycle pulses in the visible spectral region through soliton-plasma interactions. We found that in a He-filled single-ring photonic crystal fiber (SR-PCF), soliton-plasma interactions could shift the optical spectra of pulses propagating in the fiber to shorter wavelengths. Through adjusting the single pulse energy launched into the fiber, the central wavelength of these blueshifting pulses could be continuously tuned over hundreds of nanometers, while maintaining a high energy conversion efficiency of >57%. Moreover, we observed that during the nonlinear pulse propagation in the SR-PCF, soliton self-compression effects enhanced the plasma density in the fiber at high pulse energies, which could modulate the phase-matching condition of ultraviolet (UV) dispersive wave (DW) generation. Furthermore, we employed the recently-developed model to study numerically the loss and dispersion of the SR-PCF in its resonant and anti-resonant spectral regions, and demonstrated the remarkable influence of the core-cladding resonance on the process of soliton-plasma interactions. The numerical results demonstrated here pave the way to develop wavelength-tunable, few-cycle light sources in the visible region, which may have considerable application potential in pump-probe spectroscopy and strong-field physics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Few-cycle (only a few optical cycles) optical pulses with femtosecond durations have been successfully applied in many research fields such as perovskite nanocuboid laser [1], pump-probe spectroscopy [2], photochemical relaxation process [3], high-harmonic generation (HHG) [4] and attosecond science [5], etc. Direct generation of few-cycle, ultrafast pulses in solid-state or fiber-based lasers has, however, proved difficult to achieve, since the finite bandwidth of the gain medium in these lasers restricts the spectral widths of the laser pulses. In order to overcome this bandwidth limitation in lasers, a number of pulse-compression (spectral-broadening) schemes have been demonstrated for generating ultrafast optical pulses with broad spectral bandwidths. For example, nonlinear spectral broadening through self-phase modulation (SPM) has been extensively studied in conventional glass fibers [6] and bulk materials [7]. In such systems the applicable pulse peak powers are, however, limited by low damage thresholds of these nonlinear media. To solve this problem, the gas-filled hollow-core fibers (HCFs), with much higher damage thresholds, have recently been used as the medium for nonlinear pulse propagation, leading to the generation of intense few-cycle pulses with energies ranging from hundreds of μJ to a few mJ [8]. In these systems based on nonlinear spectral broadening, the generation of ultrafast pulses usually relies on some additional components (such as chirped mirrors) for compensating residual spectral chirps of the output pulses, which restrict, in practice, the minimum achievable pulse durations in such systems due to imperfect compensation of high-order dispersion.

Photonic crystal fibers (PCFs) can offer more flexibilities in their structure design, leading to remarkable fiber performance such as low confinement loss, tailorable nonlinearity and flexible dispersion management. A variety of PCFs have been widely used in multiple fields including optical manipulation [9], temperature sensor [10–12], optomechanical interactions [13], supercontinuum generation [14,15] and ultrafast nonlinear processes [16–19]. In particular, compared with conventional step-index fibers, gas-filled hollow-core photonic crystal fibers (HC-PCFs) [20–22] show considerable flexibilities on the dispersion management. Through adjusting the gas-pressure in their hollow core, the zero-dispersion wavelength (ZDW) of such fibers could be tuned across the ultraviolet (UV), visible and near-infrared (NIR) spectral regions, enabling controllable soliton dynamics [23,24]. Moreover, different from solid-core PCFs [19], the gas-filled HC-PCFs generally offer higher optical-damage thresholds, which can be greater than 10^{14} W/cm^{2}. The combination of anomalous dispersion and SPM effect in these gas-filled HC-PCFs results in the pulse self-compression. Self-compressed pulses with few-cycle or even sub-cycle durations have been obtained using such schemes, avoiding additional chirp-compensation components [25,26].

From a practical point of view, the broadband wavelength tunability of ultrafast light sources is highly demanded for many applications. For example, few-cycle pulses with wavelength tunability over the visible spectral range are very useful for studying excitonic relaxation and vibrational dynamics in photosynthetic systems, since many molecules have absorption bands in this spectral range [27]. However, the commercial mode-locked lasers usually operate at some fixed wavelengths, for example, the widely-used Ti:Sapphire lasers work at a wavelength range around 800 nm. Most of schemes for wavelength tuning of ultrafast pulses are based on nonlinear frequency conversion, including optical parametric amplification (OPA) [28,29], Raman-induced self-frequency shift [30,31] and dispersive wave (DW) generation [23,32,33], etc. While the OPA scheme is a reliable technique for generating tunable ultrafast pulses, the frequency down-conversion processes of OPA systems usually produce tunable pulses at mid-infrared (MIR) wavelengths. The Raman effects, originating from inelastic scattering of photons by molecule vibrations, usually lead to the shift of the pulse spectrum to longer wavelengths, limiting their applications in visible spectral region. The DW emission, as a result of nonlinear energy conversion from the optical pulses and some linear waves, have recently been demonstrated in gas-filled HC-PCFs [23,32], generating wavelength-tunable light in the UV and MIR regions. The energy conversion efficiency of such schemes is, however, quite low. By contrast, plasma-induced frequency blueshift of optical pulses in gas-filled HC-PCFs [34], offers a possibility to obtain few-cycle pulses with simultaneous wavelength tunability over visible region and high energy-conversion efficiency. Chang et al. [35] theoretically investigated a fundamental soliton propagation and showed the optical pulse can experience a blueshift of its central wavelength from 1500 nm to 815 nm due to soliton-plasma interactions, with an energy-conversion efficiency of 30%. However, few people studied the generation of wavelength-tunable, few-cycle pulses in the visible spectral region through soliton-plasma interactions in the gas-filled HC-PCFs.

In this paper, we demonstrate the first, to the best of our knowledge, study of wavelength-tunable few-cycle pulses in the visible spectral region based on soliton-plasma interactions in a gas-filled single-ring photonic crystal fiber (SR-PCF). Using the recently-developed model [36,37], we calculated the loss and dispersion properties of this SR-PCF at its resonant and anti-resonant spectral regions, revealing the remarkable influence of the core-cladding resonance on the blueshifting soliton. We also studied the competition between SPM and plasma effects in the frequency-chirping phenomenon, and demonstrated the influence of the multiple soliton-compression stages on the generation of UV dispersive waves at high pulse energies. The numerical results show that the interactions of self-compressed soliton and plasma can lead to the generation of few-cycle pulses whose wavelengths can be tuned over hundreds of nanometers through adjusting input pulse energy, while the energy conversion efficiency remains to be >57%. The demonstrated wavelength-tunable, few-cycle light source in visible spectral region may have many potential applications in pump-probe experiments, attosecond science, and HHG [38].

## 2. Physical model

The physical model describing the nonlinear propagation of ultrafast optical pulses in a gas-filled SR-PCF is based on the well-known single-mode unidirectional field equation that has been validated in a wide range of experiments. This full-field equation is given by [39,40]:

In general, the nonlinear polarization in the right-hand side of Eq. (1) includes the contributions from three (Kerr, Raman and plasma) effects. It should be noted that the Raman effects are absent in the noble gas and the glass-induced Raman effects can be negligible since the light-glass overlap in the SR-PCF is quite small. Thus the nonlinear polarization can be expressed as a combination of Kerr and plasma effects:

where ${\chi}^{(3)}$ is the third-order nonlinear susceptibility that is relative to the Kerr effect, $E(z,t)$ is the electric field in the time domain, and ${P}_{ion}(z,t)$ is the ionization-induced plasma polarization given by:Here we use the helium gas since its large ionization potential can support a high laser intensity, which effectively enhances the process of soliton self-compression. The calculations show that ions He^{+} and He^{2+} appear at high intensity levels euqal to 10^{15} W/cm^{2} and 6 × 10^{15} W/cm^{2}, respectively [43]. For the situation we consider here, the peak intensity of optical pulses launched into the SR-PCF is higher than 5 × 10^{13} W/cm^{2}. After soliton self-compression in the fiber, the pulse peak intensity can be enhanced to ~10^{15} W/cm^{2} which is still lower than 6 × 10^{15} W/cm^{2}. Therefore, in the simulations we only consider the first ioniztion process, and the plasma density variation can be described as [44]:

A number of approaches have been developed for calculating the OFI ionization rate [46–48]. These approaches are generally catagorized into two regimes by the Keldysh parameter described by $\gamma ={({U}_{i}/2{U}_{p})}^{1/2}$. In general, when $\gamma \gg 1$, the ionization can be expressed as a multiphoton process concerned with low intensities. When $\gamma <1$, the ionization can be described as a tunneling process concerned with high intensities. The Ammonsov, Delone, and Krainov (ADK) model [48], based on earlier work by Perelomov, Popov, and Terent’ev (PPT) [46,47], is used to calculate the OFI ionization rate in the tunneling regime. But for the 800-nm pump wavelength and the laser intensities ranging from 5 × 10^{13} W/cm^{2} to 10^{15} W/cm^{2}, the Keldysh parameters correspond to $\gamma \approx 1$. For this situation, the PPT model describes not only the multiphoton ionization (MPI) process but also the tunneling ionization process. Therefore, we employ in our simulations the PPT method modified with the ADK coefficients to calculate the ionization rate.

The propagation constant is very important parpameter for ultrashort pulses propagation in the gas-filled SR-PCF. Despite its anti-resonant cladding structures, the propagation constant of ARR SR-PCF closely follows that of a capillary fiber created by Marcatili and Schmeltzer, given by [16,49]:

*m*th zero of the Bessel function ${J}_{n-1}$, where

*m*=

*n*= 1 corresponds to the fundamental mode HE

_{11}of fiber, and $a$ is the core radius of the fiber. The first two terms in the right-hand side of Eq. (5) describe the gas contribution to the propagation constant, and the last term accounts for the waveguide contribution that plays a significant role for anomalous dispersion of the fiber. It should be noted that this dispersion expression is obtained from the characteristic equation starting from the boundary conditions under two approximations: (a) the transversal contribution of the wave number is small, (b) the core radius is much larger than wavelength. This capillary-like dispersion works well up to the NIR region. But for wavelengths beyond 1100 nm, a wavelength-dependent effective core radius is required to extend the valid range of the Marcatili-Schmeltzer (MS) capillary model [51].

Figure 1(a) shows the cross section of SR-PCF with core radius a = 18 µm and wall thickness d = 150 nm, and the outer diameter of the anti-resonant tubes D = 24.4 µm and the perimeter gap g = 1.8 µm, respectively. The color map plotted in fiber core represents the intensity distribution of the fundamental core mode in the SR-PCF, calculated by the finite-element modeling (FEM) at 800 nm. Figure 1(b) shows dispersion of the fundamental mode in the SR-PCF filled with 6 bar helium gas, calculated through Eq. (5). As shown in Fig. 1(b) the zero dispersion point (ZDP) is at 293 nm. It should be noted that the MS capillary model provides a very useful tool for numerical simulations of the ultrafast nonlinear dynamics in gas-filled SR-PCFs. The MS capillary model will be used to study plasma-induced blueshifting soliton in Secs. 4 and 5. However, the MS model is invalid in the spectral regions corresponding to the core-cladding resonances of the SR-PCF. In order to make a comparison, in Sec. 6 we will use the recently-developed model to calculate the loss and dispersion of the SR-PCF in its resonant and anti-resonant regions and discuss the influence of the core-cladding resonance on the plasma-induced blueshifting soliton.

## 3. SPM- and plasma-induced frequency chirping

Intense laser pulses interacting with nonlinear optical media such as glass or gas gives rise to the new frequency components that can be seen as a direct consequence of the time dependence of the nonlinear phase. For example, most commonly, the SPM effect induced by the bound electrons imparts an intensity-dependent nonlinear phase shift to the pulses, which results in spectral broadening. The contribution of SPM effect to the nonlinear refractive index is described as ${n}_{2}I(z,t)$ and then the nonlinear phase shift can be expressed as ${\varphi}_{SPM}={k}_{0}{n}_{2}I(z,t)\Delta z$, where ${n}_{2}$ is the Kerr coefficient and $\Delta z$ is the interaction length. The frequency difference between new frequency component and pump frequency is given as [52]:

Moreover, the combination of the SPM effect and anomalous dispersion results in the soliton self-compression to few-cycle or even shorter, enhancing the pulse peak intensity. When the pulse peak intensity is high enough to ionize the gases in the SR-PCF, the plasma contribution to the refractive index can be expressed as $-\rho (z,t)/2{n}_{0}{\rho}_{cr}$, where the plasma density $\rho (z,t)$ (the maximum ionization fraction is less than 0.1% in this work) is much smaller than the critical density ${\rho}_{cr}={m}_{e}{\epsilon}_{0}{\omega}_{0}^{2}/{e}^{2}$ [34,44]. At critical density ${\rho}_{cr}$ the real part of the refractive index becomes 0. Considering the combined effects of SPM and plasma, Eq. (6) is rewritten as:

## 4. Plasma-induced blueshifting soliton

In the numerical simulations, a 30-fs Gaussian-shape pulse with 12-µJ input energy, centered at 800 nm, was launched into a 30-cm long SR-PCF with a core diameter of 36 µm and a fiber loss of 0.1 dB/m. The fiber was filled using helium gases with a gas pressure of 6 bar. Figures 2(a) and 2(b) show the temporal and spectral evolutions of the pulse as a function of positions in the fiber when the ionization effect is not included in Eq. (1). As shown in Fig. 2(b), at the beginning of the nonlinear pulse propagation the spectral broadening is mainly due to the SPM effect. As the spectrum broadens, the anomalous dispersion provided by hollow waveguide becomes more and more important, which compensates the positive nonlinear phase due to the SPM effect, resulting in soliton temporal self-compression within the fiber. The increased peak intensity in turn enhances the SPM effect which further drives spectral broadening (pulse-compression) process, as shown in Figs. 2(a) and 2(b). The maximum pulse peak power is ~2.7 GW (corresponding to peak intensity ~5.5 × 10^{14} W/cm^{2}) marked as red dashed line, corresponding to a pulse duration of ~4.5 fs. Note that the pulse shows a self-steepening phenomenon due to the optical shock effect. The main effect of the shock term results from the intensity-dependent group velocity, which leads to greater delay for the higher-intensity part of the pulse.

In Fig. 2(b), the pulse spectrum shows both blueshift and redshift that are dependent on frequency chirping calculated by Eq. (6). Figure 2(c) shows the calculated results of frequency chirping. The blue and red regions, created by the leading and trailing edges of the pulse electrical field, correspond to spectral redshift and blueshift, respectively. Noted that the optical shock effect steepens the trailing edge of the pulses, leading to the maximum positive value larger than the maximum negative value. As a result, in Fig. 2(b), the width of spectral band “1” is slight broader than spectral band “3” (this effect is more obvious in the frequency coordinate). However, the duration of the pulse leading edge is greater than that of the trailing edge, which leads to a longer time slot for generating new frequency components, resulting in that spectral band “3” has a higher intensity than spectral band “1”.

Moreover, the pulse spectrum shows a multi-peak structure (spectral bands “1” to “3”) that can be understood as a result of intra-pulse interference. In general, the same frequency chirping occurs at two values of time, generating light components with the same frequencies but different phases. These two newly-generated light components interfere constructively or destructively depending on their relative phase difference, enabling the multi-peak structure of the pulse spectrum. The narrowing tendency of the pulse spectrum after the maximum pulse peak power, can be understood as a result of both the SPM effect and anomalous dispersion. If the propagation distance is long enough, the pulses will experience a periodic cycle of spectral broadening and narrowing due to the high-order soliton effect (here the soliton order is 3.23). In addition, the emission of a weak UV dispersive wave, marked by green arrow in Fig. 2(b) is observed in the normal dispersion region. It is the transfer of radiation from the self-compressed pulse to some linear waves, when the phase matching condition is matched between the pump pulse and the dispersive waves.

Figures 2(d) and 2(e) show the temporal and spectral evolutions of the pulse along the fiber when the ionization effect is included in the simulations. The pulse duration decreases along the fiber due to the soliton self-compression but increases slightly after the maximum peak power point due to the spectral narrowing, as shown in Fig. 3(a) (blue solid line). Meanwhile the increased peak intensity of the pulses starts to ionize the gases, producing the free electrons. In Fig. 3(b), the plasma density (green solid line) shows a fast accumulation starting from the propagation length of 20 cm, and reaches its maximum value with an ionization fraction of ~0.02% at the position of ~25.8 cm (marked as red dashed line). At the same time, the pulse energy (orange solid line) decreases rapidly due to the gas-ionization process. The presence of the free electrons lowers the refractive index of the gas, resulting in an acceleration of the pulses, as shown in Fig. 2(d). The plasma effect also leads to the blueshifting soliton as marked in spectral band “1” in Fig. 2(e), and obviously broadens the shorter-wavelength side of the spectrum compared to the Fig. 2(b). These phenomena can be understood using the frequency-chirping theory demonstrated above. As shown in Fig. 2(f), at the maximum pulse peak power, the maximum positive frequency chirping is larger than the maximum negative frequency chirping, resulting in a stronger blueshift of pulse spectrum than the redshift. Figures 3(c) and 3(d) show the electric field (blue solid line) and frequency chirping (red solid line) at the point of maximum pulse peak power without and with the plasma effect, respectively. It shows that the maximum positive value is larger when ionization is considered. Moreover, in Fig. 2(e), the spectral band “2” is away from pump central wavelength (marked as white dashed line) and moves toward the shorter-wavelength side due to the plasma effect. The spectral band “3” becomes narrow after the maximum peak power point, since the SPM-induced spectral redshift is limited by the plasma effect at the leading edge of pulse envelope. Figure 3(a) shows the average wavelength (red solid line) of the pulse as a function of positions in the fiber, which is calculated using ${\overline{\omega}}_{0}={\displaystyle {\int}_{-\infty}^{\infty}\omega I(\omega )}d\omega /{\displaystyle {\int}_{-\infty}^{\infty}I(\omega )}d\omega $ [52], where $I(\omega )$ is the light intensity at frequency of $\omega $. The simulation results show that due to the plasma effect, the average wavelength of the output pulse is shifted about 130 nm at an input pulse energy of 12 µJ.

## 5. Wavelength-tunable few-cycle pulses

Here we study the bright solitons that only occur in the anomalous dispersion region. The soliton dynamics are dependent on the two parameters: the ZDW and the soliton order. In general, the ZDW is decided by the core diameter of the fiber, the gas type and the gas pressure. In this work, we employ the helium-filled SR-PCF with a core diameter of 36 µm at the pressure of 6 bar, so that the ZDW of the fiber is maintained at 293 nm. Thus, the soliton order depends only on the input pulse energy. In Fig. 2(d), we investigated the soliton self-compression induced by the combined effects of both the SPM effect and anomalous dispersion at an input pulse energy of 12 µJ (corresponding to a soliton order of 3.23). The self-compressed pulse, with increased peak intensity, could lead to a higher plasma density which restricts the process of soliton self-compression.

In the simulations we observe that when the input pulse energy increases, the corresponding to higher soliton order can lead to the second and stronger soliton self-compression process in a limited propagation length. In fact, if the fiber is long enough, the soliton self-compression processes can occur several times even at low input pulse energies. Figures 4(a) and 4(b) show the temporal and spectral evolutions of the optical pulse as a function of fiber positions when the input pulse energy is 13.3 µJ (corresponding to soliton order 3.4), and the calculated frequency chirping is shown in Fig. 4(c). As shown in Fig. 4(a), the first and intense soliton self-compression process (marked as white letter “A”) occurs in the range from 20 to 26 cm, after that, the second and stronger soliton self-compression (marked as white letter “B”) starts from 27 cm to the output end of the fiber. The maximum peak power of the pulses is close to 4 GW that creates a larger ionization fraction of ~0.09% compared to the former case shown in Fig. 2, leading to a higher plasma density. As a result, in Fig. 4(b), the band “1” of the pulse spectrum shows an obvious and continuous blueshift after propagation length of about 20 cm. In particular, the strong plasma created by the second soliton self-compression process pushes further the blueshift of the pulse to the visible spectral region (already hundreds of nanometers away from the central frequency of the pump pulse). This remarkable phenomenon is also demonstrated in Fig. 4(c). The maximum positive value of the frequency chirping is 5 times larger than that in Fig. 2(f). Moreover, the maximum negative value is also 2 times larger, leading to a larger spectral redshift (marked as white numbers “4” and “5”) in the last few-cm fiber length, as shown in Fig. 4(b). This can be explained as that the SPM-induced frequency chirping on the back edge of pulse envelope is also enhanced due to the second pulse self-compression process marked as the region of “B” in Fig. 4(a).

Figures 5(a) and 5(b) show the temporal and spectral evolutions of the pulses at output end of the SR-PCF as a function of input pulse energy, the corresponding normalized temporal and spectral intensities are shown in Figs. 5(c) and 5(d). As shown in Fig. 5(c), the temporal envelopes of the pulses exhibit some oscillations (region (i)) on the back edge of the pulses after an input pulse energy of 12.9 µJ and these oscillations become more and more profound as the input pulse energy increases. It can be explained as that the increased input pulse energy can enhance the spectral broadening of the pulse due to the SPM effect. Then the combination of the SPM effect and anomalous dispersion results in stronger soliton self-compression and higher the pulse peak intensities. The stronger gas-ionization processes and therefore higher plasma densities due to higher pulse intensities, can lead to these oscillations. In particular, when the input pulse energy is high enough, the plasma-induced pulse fission occurs [24]. In Fig. 5(d), for the input pulse energy ranging from 12.9 to 13.3 µJ (marked as red double arrowed line), the dispersive wave emission at UV wavelengths can be observed on the pulse spectrum, having wavelengths at ~100 nm (region (ii)). This can be understood as that at these high pulse energies, the generated strong plasma changes the dispersion property of the fiber, which modulates the phase matching condition of the dispersive wave generation.

Moreover, the stronger soliton self-compression at higher input pulse energies leads to shorter output pulse duration after propagating the 30-cm long SR-PCF. As shown in Fig. 6(a), the pulse duration (blue square line) decreases as a function of the input pulse energy, and the results show that the pulse duration can be compressed to <1 fs when incident pulse energy is large enough. At the same time, the plasma-induced energy loss also increases, resulting in a drop in the fiber transmission (red square line). The simulation results in Fig. 4 show that the higher input pulse energy can lead to the stronger spectral blueshift. Figure 6(b) shows the wavelength (green circle line) of the blueshifting soliton marked in number “1” in Fig. 5(d) and the conversion efficiency (orange circle line) that defined as the ratio of the blueshifting soliton energy to the total output pulse energy. Note that the blueshifting soliton can be tuned by a few hundred nanometers, and its conversion efficiency can be maintained at >57%.

In addition, in Fig. 6(c), the peak power (blue triangle line) of the output pulses is enhanced at the beginning, while for energies greater than 12.7 µJ, it shows an alternating change since the peak power of the pulses is dependent on a few factors including the pulse energy, pulse duration and its temporal profile. The generated plasma density (red triangle line) only depends on the peak intensity of the pulses based on OFI process. Furthermore, we employ a super-Gaussian function to filter the blueshifting soliton (spectral band “1” in Fig. 5(d)) for input pulse energy of 12.8 µJ, the calculated temporal envelope and electric field are shown in Fig. 6(d) with (upper plot) and without (lower plot) filter, respectively. After using the filter, the pulse duration changes from ~1.1 to ~1.5 fs and the peak power of the electric field changes from ~2.6 to 2.2 GW.

## 6. Effect of core-cladding resonance

For ARR SR-PCF, the broad transmission window is interrupted by several sharp resonance bands since the modes of the cladding features are in resonance with the core mode. These resonances can be explained as a result of the presence of a thin glass capillary in the fiber core boundary, where the loss is very high and the dispersion is considerably affected. The resonant wavelength can be expressed as [53,54]:

where ${n}_{g}$ is the refractive index of the cladding material at wavelength ${\lambda}_{q}$, $q$ is an integer number.Here we employ a recent method termed bouncing-ray (BR) model [36] that can calculate the mode loss in the resonant bands. The resulting expression through the BR model is given by:

where ${\alpha}_{BR}^{H}$ is the mode loss for hybrid modes including the fundamental mode HE_{11}, ${\alpha}_{BR}^{TE}$ and ${\alpha}_{BR}^{TM}$ represent the TE and TM modes, and the loss coefficients are:

More recently, Hasan et al. proposed an empirical formula to describe the effective mode index of the resonant and anti-resonant regions, without the time consuming numerical simulations [37]. The effective index can be given as:

_{${n}_{tube}$}is the number of the anti-resonant tubes, and ${\sigma}_{q}$ describes the resonant strength that is checked for resonance number $q$ ranging from 1 to 4, and its expression is:

Figures 7(a) and 7(b) show the calculated loss and dispersion for the wall thickness of 150 nm through Eqs. (14) and (15), respectively. The first resonant wavelength ${\lambda}_{1}$ occurs ~328 nm and this wavelength shows a high loss ~1.9 m^{−1} (corresponding to ~8.4 dB/m). We consider the influence of core-cladding resonance by introducing the loss and dispersion into Eq. (1). As shown in Figs. 7(c) and 7(d), the core-cladding resonance has a minor impact on the pulse propagation compared to the case of Figs. 4(a) and 4(b) except for the emission of narrow spectral peak (marked in (i)) near the region of the first resonance. This enhanced narrow band can be understood as a consequence of phase matching induced by parametric nonlinear process such as four-wave mixing (FWM) or/and DW generation. This effect has been confirmed by a recent experiment in [54]. In Fig. 8, when the wall thickness is increased to 250 nm, the resonance-induced loss is enhanced. At the first resonant wavelength of ~535 nm, the corresponding loss is ~3.2 m^{−1} (corresponding to ~13.7 dB/m). Note that the core-cladding resonance presents a strong and negative impact on spectral blueshift near the first resonant region. In addition, the emission of narrow spectral peak (marked in (ii)) becomes wider and stronger as the resonant wavelength is closer to the pump wavelength. Therefore, in order to minimize the impact of core-cladding resonance on the pulse propagation, one can adjust wall thickness to shift the resonance far away from the pump wavelength.

## 7. Discussion

There are a few points needed to be considered on the experiments. First, for ARR SR-PCF, the broad transmission bands are interrupted by localized high-loss regions as the resonance between core mode and cladding mode. The resonant wavelength is dependent on the wall thickness, thus we can control the wall thickness to shift the resonances out of the wavelength range we are interested in. Second, the gas types also have an influence for blueshifting soliton. For example, the first ionization potential of argon gas (~15.76 eV) is less than helium gas (~24.59 eV), which means the argon gas is easier to be ionized during soliton self-compression. By contrast, helium gas can support a larger peak intensity, thus the self-compressed pulses can reach the shorter duration when the free-electron population effectively builds up, resulting in the larger spectral blueshift based on the relation of$\Delta \omega (z,t)~{\partial}_{t}\rho (z,t)$. Third, similarly, the input pulse duration should be short to effectively achieve spectral blueshift in a limited propagation length.

In addition, the pulse duration measurements are also important. The common methods to measure ultrashort pulses such as the second-harmonic generation frequency-resolved optical gating (SHG-FROG) [55,56] and the spectral phase interferometry for direct electric-field reconstruction (SPIDER) [57], are both valid to measure the few-cycle duration or even single-cycle and sub-cycle for long-wavelength pulses [26]. However, in comparison with long-wavelength pulses, the lasting time of one optical cycle for short-wavelength pulses is quite short, which is very difficult to measure. For example, the shortest pulse duration that can be measured experimentally should be more than one optical cycle (~2.7 fs) at central wavelength of 800 nm. Furthermore, if the spectral components of the blueshifting soliton are tuned below 400 nm, these two methods will be invalid since the frequency-doubled signals based on ${\chi}^{(2)}$ process are out of the transmission window of the crystal. Another approach such as the self-diffraction frequency-resolved optical gating (SD-FROG) can be employed and the self-diffracted signals through the ${\chi}^{(3)}$ process work well in the broad spectral range including UV region. It should be noted that this method requires high pulse energy [58]. One can use a larger core diameter SR-PCF that supports intense input pulse energy, but this large core waveguide provides the limited anomalous dispersion, which is not conducive to soliton self-compression. Therefore, we need to keep a good balance between blueshifting spectral range, compressed pulse duration and the pulse energy.

## 8. Conclusions

In conclusion, we theoretically studied the soliton-plasma interactions in a 30-cm long and 36-µm core diameter SR-PCF filled with helium gas at pressure of 6 bar. The simulation results show that the system can be used for generating wavelength-tunable, few-cycle pulses in the visible spectral region. The blueshift of the pulse spectrum due to the soliton-plasma interactions can be tuned over hundreds of nanometers through adjusting input pulse energy, while maintaining a high conversion efficiency of >57%. In addition, we found at high input pulse energies, the oscillations on the trailing edges of the output pulses become more and more profound, and the second soliton self-compression process appears which further enhances soliton-plasma interactions and modulates the phase-matching condition of the dispersive wave generation at UV wavelengths. Besides, we used the BR model and empirical formulas to describe the loss and dispersion of the resonant and anti-resonant spectral regions in the SR-PCF, and we also investigated the effect of the core-cladding resonance on the blueshifting soliton. The emission of a narrow-bandwidth spectral peak was observed near the spectral resonance of the SR-PCF due to the phase matching, and such emission will be enhanced when the resonant wavelength is closer to the pump pulse wavelength. We believe the tunable few-cycle light source proposed here could be used to study both pump-probe spectroscopy such as ultrafast electronic vibrational dynamics and strong-field physics such as HHG. The numerical results demonstrated here may also provide some useful insight for understanding the soliton-plasma interactions in gas-based ultrafast nonlinear optics.

## Funding

Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB1603), International S&T Cooperation Program of China (Grant No. 2016YFE0119300), Program of Shanghai Academic/Technology Research Leader (Grant No. 18XD1404200), National Natural Science Foundation of China (Grant No. 61521093).

## Acknowledgments

The authors would like to thank Fei Yu for useful discussions.

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