1. Introduction

Ultrashort pulses have been widely used in spectroscopy [1–3], microscopy [4], near-field imaging [5], and high-harmonic generation [6], which have numerous applications across different disciplines such as physics, chemistry, and biomedicine. They can be obtained by several methods including self-compression [7], post-compression [8], chirped pulse amplification [9], as well as optical parametric amplification [10]. Among these, soliton self-compression, which relies on the interplay between nonlinearity and anomalous dispersion [11,12], is a simple technique that requires only a single stage of nonlinear anomalously dispersive medium. The compressed pulse is almost chirp-free without needing the complex chirp compensation system.

With the development of gas-based nonlinear optics, there is a growing interest in generating high-energy ultrashort pulses in gas-filled capillaries using the soliton self-compression [13,14]. Gas-filled capillaries have many advantages, such as high power-damage threshold, adjustable dispersion and nonlinearity, as well as wide transmission bandwidth. As an inherently multimode waveguide, capillary supports various linearly polarized leaky modes denoted as $\textrm{H}{\textrm{E}_{1p}}$ modes [15]. One interesting direction that calls for further investigations is the soliton self-compression in the higher-order modes of gas-filled capillaries. There are several attractive aspects about staging the soliton self-compression in the higher-order modes of gas-filled capillaries. Firstly, in a capillary, $\textrm{H}{\textrm{E}_{1p}}$ mode (with p $\ge 2$) has a smaller spatial dimension than the fundamental mode, leading to stronger nonlinearity. Secondly, the contribution of negative waveguide dispersion increases with the mode order. This provides wider selection of gas species and pressure range that facilitate the anomalous dispersion regime for the soliton self-compression. Thirdly, the larger anomalous dispersion also means the self-compression can be done within a shorter length [13], which is critical for large-core capillary compressors, since their bending-induced loss is one of the major limiting factors.

The soliton self-compression in higher-order modes of gas-filled capillaries has emerged as a topic of interest in the last couple of years. Numerical investigations showed that compressed pulses approaching the single-cycle regime can be obtained in the higher-order modes [16,17]. These studies assumed constant gas pressure along the length of the capillary. On the other hand, there are some major benefits of keeping the exiting end evacuated in the gas-filled hollow waveguide-based compressors [18]. Doing so can avoid the dispersion and nonlinearity induced distortion of the compressed pulse when it passes through the gas chamber attached at the output end. Furthermore, by applying a negative pressure gradient, we can minimize the nonlinear mixing between the modes and improve the mode purity of the output beam.

In this work, we apply negative pressure gradient along the longitudinal direction in the argon-filled capillary pulse compressor by evacuating its output end. We numerically investigate the soliton self-compression by launching the pump in different higher-order modes. We show that, in addition to the distortion-free pulse delivery at the output, applying the decreasing pressure enhances the compression in $\textrm{H}{\textrm{E}_{12}}$ mode by a factor of four compared to the case where an equivalent constant pressure is used. The outcome is a highly compressed pulse having a sub-cycle duration of 1.85 fs and the peak power of 7.96 GW achieved from a single-stage compressor. Additionally, high mode purities can be realized by suppressing the intermodal coupling through the decreasing pressure.

2. Model

The model is based on the multimode generalized nonlinear Schrödinger equation (MM-GNLSE) [19]. This mode-resolved equation in the frequency domain is given by:

The first term on the right-hand side of Eq. (1) describes the linear effects in $\textrm{H}{\textrm{E}_{1p}}$ mode. From the analytical model derived by Marcatili and Schmeltzer for a dielectric capillary, the wave vector, ${\beta ^{(p )}}$, for $\textrm{H}{\textrm{E}_{1p}}$ mode is given by [15]:

Here, ${n_g}$ is the pressure dependent refractive index of the gas and we take the Sellmeier coefficients reported in [20]. ${u_p}$ is the $p$-th zero of the first-order Bessel function, c is the speed of light in vacuum, and a is the bore radius of the dielectric capillary. $\beta _0^{(1 )}$ and $\beta _1^{(1 )}$ are the first two Taylor series expansion coefficients of ${\beta ^{(1 )}}$ at the pump frequency, ${\omega _0}$, and are related to the phase and group velocities of the pump in $\textrm{H}{\textrm{E}_{11}}$ mode. ${\alpha ^{(p )}}$ is responsible for the power loss in $\textrm{H}{\textrm{E}_{1p}}$ mode, and is given by [15]:

The second term on the right-hand side of Eq. (1) is the nonlinear term. ${n_2}$ is the pressure dependent nonlinear index of the gas which we obtain from [21]. The summation denotes the intra- and inter-modal optical Kerr effects, where ${A_l}({z,t} )$, ${A_m}({z,t} )$ and ${A_n}({z,t} )$ refer to the complex envelopes in the interacting modes l, m and n. Namely, $l$ = $m$ = $n$ = $p$ represents the self-phase modulation, while any other combinations of l, m and n imply intermodal cross-phase modulations and four-wave mixing processes [22]. ${S_{plmn}}$ is the overlap integral between $\textrm{H}{\textrm{E}_{1p}}$ and the other interacting modes, given as the inverse of the effective mode area for the nonlinear coupling. It is expressed as:

The filling gas pressure influences two parameters in Eq. (1), i.e., ${\beta ^{(p )}}$ and ${n_2}$, through the change of atomic density. We create a pressure gradient along the waveguide of length L by setting different pressures at the input and output ends of the capillary, ${P_0}$ and ${P_L}$, respectively. The pressure at position z is then given by [23]:

3. Numerical simulations

In our numerical study, we set the pump at 800 nm wavelength where many lasers generating high-energy femtosecond pulses are readily available. We use a transform-limited 100 µJ Gaussian pulse with 35 fs full-width at the half-maximum (FWHM) duration. Such pulses can be obtained from a commercial mode-locked laser system. Plotted in Fig. 1(a) are the loss and effective area versus the bore diameter $2a$, evaluated at 800 nm for $\textrm{H}{\textrm{E}_{11}}$, $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes in a silica capillary fiber. The loss depends strongly on the bore diameter in all modes. A large bore gives much lower loss but at the expense of increased effective mode area that leads to reduced nonlinearity. We choose the bore diameter of 300 µm, which strike a good balance between the loss and mode size. At 300 µm bore diameter, the losses at 800 nm are 0.36 dB·m⁻¹, 1.88 dB·m⁻¹, 4.61 dB·m⁻¹, and 8.56 dB·m⁻¹ in $\textrm{H}{\textrm{E}_{11}}$, $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes, respectively. These values are tolerable for experiments requiring a relatively short fiber, e.g., <2 m. Likewise, for the pump pulse considered in the study, the effective areas of these modes are small enough to induce sufficient nonlinear phase shifts within the short length at moderate argon pressure.

 

Fig. 1. (a) Loss (solid lines) and effective mode area ($A_{\textrm{eff}}^{(p )}$, dashed lines) versus bore diameter evaluated at 800 nm for $\textrm{H}{\textrm{E}_{11}}$, $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes. (b) GVD parameter ($\beta _2^{(p )}$, solid lines) and nonlinear coefficient (${\gamma ^{(p )}}$, dashed lines) at 800 nm versus argon pressure for $\textrm{H}{\textrm{E}_{11}}$, $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes. The bore diameter of the capillary fiber is set at 300 µm. Shown in the left-hand side panel are the intensity profiles of the four modes with the color of each surrounding box indicating the line color for the corresponding mode in (a) and (b).

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Figure 1(b) shows the group-velocity dispersion (GVD) parameter, $\beta _2^{(p )}$, and nonlinear coefficient, ${\gamma ^{(p )}}$, defined as ${n_2}{\omega _0}/cA_{\textrm{eff}}^{(p )}$, at 800 nm versus argon pressure for $\textrm{H}{\textrm{E}_{11}}$, $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes in a capillary fiber with the bore diameter of 300 µm. We note that for the given set of system parameters, the pump lies in the normal dispersion regime only in $\textrm{H}{\textrm{E}_{11}}$ mode, while all the higher-order modes place it in the anomalous dispersion regime. This opens the opportunity to excite soliton-driven nonlinear effects, including the soliton self-compression, in the higher-order modes of the capillary fiber. From Fig. 1(b), we note that the magnitude of the anomalous dispersion increases as the mode number is increased, while it decreases with pressure in each higher-order mode.

In our simulations, we include the first ten $\textrm{H}{\textrm{E}_{1p}}$ modes that are coupled through the nonlinear term in Eq. (1). Only $\textrm{H}{\textrm{E}_{1p}}$ modes are assumed in the calculations since a particular input mode results only in output modes of the same azimuthal symmetry [24]. This choice provides a good compromise between the computational accuracy and resource requirement. In each numerical simulation, we choose and excite at the launch only a single higher-order mode, i.e., $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, or $\textrm{H}{\textrm{E}_{14}}$. The predominant excitation of a selected higher-order mode can be realized experimentally by, for example, using a prism-coupling method [25]. As we shall see, the pulse in the higher-order mode, upon propagation, couples with other spatial modes in the gas-filled capillary fiber through nonlinear interaction, shedding some of its energy into these modes.

4. Results and discussions

The calculations are carried out for decreasing argon pressure along the length of the capillary fiber with 1.8 bar at the input end and evacuated at the other end. In such a case, the pulse propagation dynamics is subject to the fiber length, L, as it determines the pressure change, and therefore the changes in the system’s dispersion and nonlinearity along the propagation.

We assume the nonlinear effect initially dominates that of the dispersion, which is a valid assumption for launching a higher-order soliton. Moreover, we consider the intermodal interaction is weak such that only a small fraction, i.e., <10%, of the total energy launched end up being coupled into other modes. Then, the total nonlinear phase shift, $\phi _{\textrm{NL}}^{(p )}$, which is also known as the B-integral, experienced by the pulse launched into $\textrm{H}{\textrm{E}_{\textrm{1p}}}$ mode in decreasing gas pressure from ${P_0}$ at the input to ${P_L} = 0$ at the output can be approximated by [26]:

Figures 2(a), 2(b), and 2(c) show the evolutions of the FWHM durations when the pump is launched into $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes, respectively, for the decreasing pressure (solid lines) and constant pressure (dotted lines). The black dash-dotted lines in Fig. 2 are evolutions of the compression rate figure-of-merit, $\textrm{FO}{\textrm{M}_{\textrm{comp}}}$, values which we introduce to estimate the compression dynamics as we shall discuss later. The pulse, in each case shown, undergoes soliton self-compression due to the interplay between anomalous dispersion and nonlinear effect. In $\textrm{H}{\textrm{E}_{11}}$ mode, which is not presented here, is in the normal dispersion regime and therefore no soliton self-compression takes place.

 

Fig. 2. Evolutions of the FWHM durations in (a) $\textrm{H}{\textrm{E}_{12}}$, (b) $\textrm{H}{\textrm{E}_{13}}$, and (c) $\textrm{H}{\textrm{E}_{14}}$ modes in the argon-filled capillary with the decreasing pressure (solid lines). The evolutions under the equivalent constant pressure system (dotted lines) are plotted together for reference. The changes of the $\textrm{FO}{\textrm{M}_{\textrm{comp}}}$ values along the propagation are shown in black dash-dotted lines.

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We notice substantial enhancement in the pulse compression in $\textrm{H}{\textrm{E}_{12}}$ mode when the decreasing pressure is applied. Namely, the compressed pulse at the capillary exit is 1.85 fs in duration reaching the sub-cycle regime under the decreasing pressure. This corresponds to the compression ratio, ${F_c} = 19$, which is four times that achievable in the constant pressure system where the output duration is 7.48 fs amounting to ${F_c} = 4.68$. On the contrary, the decreasing pressure deteriorates the compression in $\textrm{H}{\textrm{E}_{13}}$ and $\textrm{H}{\textrm{E}_{14}}$ modes, resulting in the output pulse duration of 14.85 fs and 5.7 fs, respectively, as oppose to 2 fs and 4.3 fs when the equivalent constant pressure is used.

In the initial stage of soliton self-compression, the effect of nonlinearity dominates that of the anomalous dispersion where the relative strength between nonlinearity and dispersion is governed by the soliton order. The self-phase modulation broadens the spectrum and induces positive chirp, which gets compensated by the negative chirp from the anomalous dispersion. This leads to pulse compression that in turn reinforces the nonlinearity. The process repeats until the pulse spectrum broadens to the extent that the effects of the dispersion and nonlinearity are at a balance.

The wide spectral bandwidth increases the importance of the higher-order dispersion, which is particularly strong when the pulse is in the vicinity of zero dispersion [27]. The strong perturbation by the higher-order dispersion causes pulse to break, limiting the compression performance. This is what happens in $\textrm{H}{\textrm{E}_{12}}$ mode when the constant pressure is applied, because the anomalous dispersion in this case is very weak at 1.2 bar as shown in Fig. 1(b). However, when the decreasing pressure is used, the anomalous dispersion that starts near zero increases in its magnitude along the length, and surpasses that of the constant pressure at $z$ = 0.65 m. The increased anomalous dispersion effectively mitigates the perturbation caused by the higher-order dispersion in the later stage of the compression. Here, the spectral bandwidth increases as the pulse compresses down the capillary. Considering that the higher-order dispersion has more severe impact when the spectrum is broader, having larger anomalous dispersion towards the output end where the spectrum is broader works in favor of mitigating the effect. This substantially improves the compression, reducing the output pulse duration down to a quarter of that achievable under the constant pressure. As we shall see in Fig. 3(a), this improvement can be realized even with similar amount of total spectral broadening. On the contrary, in $\textrm{H}{\textrm{E}_{13}}$ and $\textrm{H}{\textrm{E}_{14}}$ modes, the waveguide dispersion becomes stronger, shifting the overall dispersion further away from the zero dispersion. This effectively weakens the influence of the higher-order dispersion. Hence, the suppression of the higher-order dispersion effect introduced by the decreasing pressure does not play a prominent role, and no enhancement is seen in $\textrm{H}{\textrm{E}_{13}}$ and $\textrm{H}{\textrm{E}_{14}}$ modes.

The second important factor that determines the output pulse duration in a decreasing pressure is the mismatch between the maximum compression point and the fiber end. The compression length in the soliton self-compression, ${L_{\textrm{comp}}}$, is related approximately to $\gamma \; $ and $|{{\beta_2}} |$ as below [13]:

A higher $\gamma |{{\beta_2}} |$ value leads to a shorter compression length and a faster compression process. In decreasing pressure, γ is continuously decreasing while $|{{\beta_2}} |$ is increasing. To qualitatively describe the impact of these two parameters on the self-compression dynamics, we use the compression rate figure-of-merit, $\textrm{FO}{\textrm{M}_{\textrm{comp}}}$, denoted as $\sqrt {({\gamma (z )|{{\beta_2}(z )} |} )/({\gamma^{\prime}|{{\beta_2}^{\prime}} |} )} $, where $\gamma (z )\; $ and $|{{\beta_2}(z )} |$ are the z dependent nonlinear and GVD parameters for the decreasing pressure at the given mode, while $\mathrm{\gamma }^{{\prime}}$ and $|{{\beta_2}^{\prime}} |$ are those at the pressure of 1.2 bar. We note that in the regions where $\textrm{FO}{\textrm{M}_{\textrm{comp}}}$ is greater than 1—shaded in gray in Fig. 2—the reduction in the FWHM duration is more rapid when the decreasing pressure is applied. This is in line with Eq. (6) where a larger value of $\gamma |{{\beta_2}} |$ shortens the compression length and accelerates the rate of compression. It shows that the combined effect of $\gamma $ and $|{{\beta_2}} |\; $ has significant impact on the self-compression dynamics. $\textrm{FO}{\textrm{M}_{\textrm{comp}}}$ can be used to estimate how the rate of compression in the decreasing pressure changes with respect to the case involving the equivalent constant pressure.

As the rate of compression changes in accordance with $\textrm{FO}{\textrm{M}_{\textrm{comp}}}$ in the decreasing pressure, the compression length is continuously adjusted. For $\textrm{H}{\textrm{E}_{12}}$ mode, the compression length in the decreasing pressure happens to roughly coincide with that of the constant pressure in our numerical example, yielding the maximally compressed pulse at the evacuated fiber end and significant gain in the compression factor. In $\textrm{H}{\textrm{E}_{13}}$ and $\textrm{H}{\textrm{E}_{14}}$ modes, on the other hand, the compression points under the decreasing pressure are brought forward. In $\textrm{H}{\textrm{E}_{13}}$ mode, the pulse reaches the maximum compression at around $z$ = 0.9 m. This is followed by a soliton fission as reflected in the red-solid line in Fig. 2(b) with a dramatic increase in the pulse duration. In $\textrm{H}{\textrm{E}_{14}}$ mode, the pulse compresses to the shortest duration at $z$ = 0.59 m, after which the duration increases slowly as indicated by the orange-solid line in Fig. 2(c). This is because further compression is prohibited due to relatively high loss, as well as the diminishing nonlinearity. In both cases, the pulses temporally broaden and peak powers decay thereafter before reaching the fiber end, and the outcomes are the weaker compressions.

Figure 3 shows the temporal and spectral profiles of the pulses at the output end of the fibers for the three higher-order mode launches, i.e., (a) $\textrm{H}{\textrm{E}_{12}}$, (b) $\textrm{H}{\textrm{E}_{13}}$, and (c) $\textrm{H}{\textrm{E}_{14}}$ modes. We observe that by introducing the decreasing pressure, peak power of the compressed pulse in $\textrm{H}{\textrm{E}_{12}}$ mode is enhanced by 2.7 times, from 2.9 GW to 7.96 GW. On the other hand, peak powers in the case of the decreasing pressure are reduced for $\textrm{H}{\textrm{E}_{13}}$ and $\textrm{H}{\textrm{E}_{14}}$ modes. It is evident in Fig. 3 that the change in the pressure profile does not influence much the bandwidth of the output spectrum. This is expected as the equivalent constant pressure is determined based on the assumption that the two systems accumulate approximately the same amount of total nonlinear phase shift by matching the B-integrals. The oscillations seen in the trailing edge of the compressed pulses in Fig. 3(a) is due to the higher-order dispersion. In $\textrm{H}{\textrm{E}_{12}}$ mode, GVD parameter is closer to zero, and hence the effect of the higher-order dispersion becomes more prominent and the oscillation is relatively large. Under the constant pressure, the oscillation is even stronger and the pulse profile is severely perturbed, resulting in poor compression performance.

Another notable feature in Fig. 3(a) is the delay of the pulse peak with respect to the pedestal in the leading edge. This is caused by the extreme self-steepening which occurs when the pulse self-compresses to sub-cycle duration [14]. In the self-compression process, the pulse peak experiences the strongest compression while the tails stay relatively unchanged. As compression continues, the pulse peak lags due to the intensity dependence of the group velocity, resulting in the high-intensity peak to trail behind the low-intensity pedestal. In Fig. 3(b), we can see the solitonic structure in $\textrm{H}{\textrm{E}_{13}}$ mode is distorted at the output end due to soliton fission. In $\textrm{H}{\textrm{E}_{14}}$ mode, the self-compression process ceases too early due to the relatively high loss, as well as the reducing nonlinearity. Even though the soliton is still intact, the pulse is temporally dispersed and the peak power decays by 17% as it reaches the output end, as shown in Fig. 3(c).

 

Fig. 3. Temporal (left) and spectral (right) profiles of the compressed pulses in (a) $\textrm{H}{\textrm{E}_{12}}$, (b) $\textrm{H}{\textrm{E}_{13}}$, and (c) $\textrm{H}{\textrm{E}_{14}}$ modes. Red-solid lines are for the decreasing pressure systems while blue-solid lines are for the equivalent constant pressure cases. The initial profiles are plotted in gray-dashed lines for reference.

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An additional feature to note in the spectra presented in Fig. 3 is the dispersive wave generation [28]. The linear wave is emitted as the higher-order soliton undergoes fission after the maximum compression is reached. The location of the dispersive wave can be estimated using the phase-matching condition below:

The dispersive wave starts to appear close to the output end where the spectrum is broad enough to seed the energy conversion to the phase-matching point. This means Eq. (8) needs to be solved for a lower pressure value in the decreasing pressure case which can have markedly different phase-matching point from that of the equivalent constant pressure system. To illustrate this, we show the dephasing diagram in Fig. 4 for $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes at two different pressures, i.e. 1.2 bar for the equivalent constant pressure case and 0.6 bar to exemplify a lower pressure value near the output end in the decreasing pressure capillary. At 1.2 bar, the phase-matching occurs at around 460, 320, and 240 nm in $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes, respectively, decreasing with the mode order. This is in good agreement with the results shown in blue lines in Figs. 3(a) and 3(b), where spectral features are prominent at these wavelengths in the corresponding constant pressure cases. We note due to high loss in $\textrm{H}{\textrm{E}_{14}}$ mode, spectral broadening is not sufficient to warrant dispersive wave radiation. At 0.6 bar, the phase-matching points blue-shift, which is also the case as shown with red lines in Figs. 3(a) and 3(b) where the short-wavelength bands are shifted towards shorter wavelength.

 

Fig. 4. Dispersive wave dephasing diagram for $\textrm{H}{\textrm{E}_{12}}$ (blue), $\textrm{H}{\textrm{E}_{13}}$ (red), and $\textrm{H}{\textrm{E}_{14}}$ (orange) modes. Solid lines are for 1.2 bar and dashed lines are for 0.6 bar to illustrate the equivalent constant pressure and decreasing pressure cases, respectively. Vertical-dotted lines mark the phase-matching wavelengths.

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At this point, we check the accuracy of our results and corroborate MM-GNLSE for the $\textrm{H}{\textrm{E}_{12}}$ mode launch cases considered above. For the validation, we solve the uni-directional pulse propagation equation (UPPE) for the same set of parameters with and without ionization term [29]. Note that this field-based model considers only a single mode propagation in the launch mode, and therefore lacks intermodal effects. Figure 5(a) shows evolutions of the FWHM durations obtained from three different models, i.e., MM-GNLSE, UPPE with ionization term, and UPPE without ionization term, for the identical cases studied in Fig. 2(a). Qualitatively the same evolutions are observed in all three models, except for intermodal coupling-mediated small oscillations that appear in MM-GNLSE but not in the single-mode models. In particular, the significant compression enhancement in the case of decreasing pressure is reproduced with UPPE, which indicates that the single-mode configuration is sufficient to describe the improved compression in the decreasing pressure. We also highlight that results from the models with and without ionization term are practically indistinguishable. We stress that slightly higher peak powers are reached in UPPE models compared to MM-GNLSE, yet ionization does not play any important role. Therefore, ionization can be safely omitted in our study. When decreasing pressure is used, the gas density diminishes as the pulse compresses down the capillary, effectively eliminating the impact of ionization. Figure 5(b) is the comparison of the output pulse profiles for the same decreasing pressure case computed using the three models. Likewise, no qualitatively significant differences are observed.

 

Fig. 5. (a) Evolutions of the FWHM durations for the decreasing pressure (solid lines) and equivalent constant pressure (dashed lines) cases obtained using MM-GNLSE (blue), UPPE with ionization term (orange), and UPPE without ionization term (green). Note the orange and green lines overlap indicating that the effect of ionization is negligible. (b) Output profiles for the decreasing pressure case calculated using the three models.

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There is an additional benefit of using the decreasing pressure for the pulse compression. When the pulse is highly compressed, it can give rise to strong intermodal coupling significantly degrading the spatial beam quality, if sufficient nonlinearity is present in the system. The decreasing pressure ensures that the nonlinear parameter is small closer to the output end of the fiber where the pulse has high peak power, minimizing the nonlinear interaction between modes. Hence high degree of mode purity and excellent beam quality can be achieved. Figure 6 shows the evolutions of the percentage energy weightages in the first ten $\textrm{H}{\textrm{E}_{1p}}$ modes when the pump is launched into the first three higher-order modes. The energy weightage defines the mode purity, where ideally 100% of the energy should remain in the launched mode if no intermodal coupling occurs. All three modes studied exhibit high degrees of mode purity containing 94.9%, 97.8%, and 98.8% of the total energy at the output in their corresponding main modes, i.e., $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes, respectively. The oscillations in the energy weightages observed at the beginning of each propagation are attributed to nonlinear coupling between different spatial modes. Similar mode coupling-induced oscillations show up in the FWHM durations in Fig. 2. One of the reasons that oscillations appear only at the beginning is because, in the case of decreasing pressure, the nonlinearity and therefore the interaction is large in the early part of the capillary. It also explains why the oscillations are weaker for the equivalent constant pressure systems as seen in Fig. 2. This is because the nonlinearity remains below that required to stage noticeable intermodal interaction. There are other factors that govern the strength and extent of the mode coupling which we shall discuss in detail in the later part of this section. We notice in Fig. 6(a), the energy weightage of $\textrm{H}{\textrm{E}_{12}}$ mode decreases slowly towards the end of the fiber while that in $\textrm{H}{\textrm{E}_{11}}$ mode gradually rises. This is mainly due to the loss in $\textrm{H}{\textrm{E}_{12}}$ mode being higher than that in $\textrm{H}{\textrm{E}_{11}}$ mode, and hence the energy in $\textrm{H}{\textrm{E}_{12}}$ mode decays faster compared to $\textrm{H}{\textrm{E}_{11}}$ mode. The near-field intensity profiles of the output beams are presented in the insets in Fig. 6 for the three launch cases. Their modal intensity patterns are well maintained, indicating good spatial mode qualities.

 

Fig. 6. Evolutions of the energy weightages in the first ten $\textrm{H}{\textrm{E}_{1p}}$ modes for the pump launched into (a) $\textrm{H}{\textrm{E}_{12}}$, (b) $\textrm{H}{\textrm{E}_{13}}$, and (c) $\textrm{H}{\textrm{E}_{14}}$ modes in the system with the decreasing pressure. Insets show the near-field intensity profiles of the corresponding output beams.

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Among the three main nonlinear effects that appear in the system, self-phase modulation, cross-phase modulation, and four-wave mixing, only the intermodal four-wave mixing can induce energy exchange between the modes [22]. This is apparent when expanding the summation in Eq. (1). The nonlinear terms responsible for self-phase modulation, i.e. ${S_{pppp}}{|{{A_p}} |^2}{A_p}$, and cross-phase modulation, i.e. ${S_{pnpn}}{|{{A_n}} |^2}{A_p}$ and ${S_{ppnn}}{|{{A_n}} |^2}{A_p}$, have purely real nonlinear wave vectors, which introduce modulations on the phase alone and not any gain or loss in $\textrm{H}{\textrm{E}_{1p}}$ mode. Only four-wave mixing terms carry imaginary parts that give rise to energy exchange between $\textrm{H}{\textrm{E}_{1p}}$ and other modes. Out of these, terms that contain ${|{{A_p}} |^2}$ play major role in the mode coupling where $\textrm{H}{\textrm{E}_{1p}}$ is the dominant mode. The intermodal coupling is then proportional to ${n_2}{|{{A_p}} |^2}$ = ${n_2}I$ where ${n_2}$ and I are the pressure dependent nonlinear index and peak power in $\textrm{H}{\textrm{E}_{1p}}$ mode, respectively, at z. From this, we can adopt the four-wave mixing figure-of-merit (FOM_fwm) for determining the strength of the intermodal interaction. This is given by $({{n_2}(z )I(z )} )/({{n_2}({z = 0} )I({z = 0} )} )$. Here, $\textrm{FO}{\textrm{M}_{\textrm{fwm}}}$ of greater than 1 means increase in the intermodal coupling strength as compared to that at the input, while that less than 1 suggests otherwise. Figure 7 shows ${n_2}$ (black-dotted lines), I (red-dotted lines), as well as $\textrm{FO}{\textrm{M}_{\textrm{fwm}}}$ (solid-blue lines) along the propagation when the pump is launched into $\textrm{H}{\textrm{E}_{12}}$, $\textrm{H}{\textrm{E}_{13}}$, and $\textrm{H}{\textrm{E}_{14}}$ modes. The peak power always slightly lags the local maxima of $\textrm{FO}{\textrm{M}_{\textrm{fwm}}}$ values for all three cases presented. For the case where the pump is launched into $\textrm{H}{\textrm{E}_{12}}$ mode, $\textrm{FO}{\textrm{M}_{\textrm{fwm}}}$ is kept well below 1 even though the peak power increases by three times through the compression. It implies that the intermodal interaction is effectively suppressed by adopting the decreasing pressure. For $\textrm{H}{\textrm{E}_{13}}$ mode, $\textrm{FO}{\textrm{M}_{\textrm{fwm}}}$ first decreases but later rises beyond 1. This directly results in the increased energy transfer from $\textrm{H}{\textrm{E}_{13}}$ mode to the other modes (e.g., $\textrm{H}{\textrm{E}_{12}}$ mode) at $z$ = 0.8 m as shown in Fig. 6(b). This clearly demonstrates that $\textrm{FO}{\textrm{M}_{\textrm{fwm}}}$ provides an excellent indication of the intermodal coupling strength in the system.

 

Fig. 7. Evolutions of ${n_2}$ (black-dotted lines), I (red-dotted lines), and $\textrm{FO}{\textrm{M}_{\textrm{fwm}}}$ (blue-solid lines) in the case of the decreasing pressure systems when the pump is launched into (a) $\textrm{H}{\textrm{E}_{12}}$, (b) $\textrm{H}{\textrm{E}_{13}}$, and (c) $\textrm{H}{\textrm{E}_{14}}$ modes.

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Another important factor that affects the strength and extent of the mode coupling is temporal overlap between fields in the interacting modes, which is governed by the pulse duration and group-velocity mismatch. Namely, a shorter pulse and larger group-velocity mismatch results in weaker and shorter-lived interaction. This is yet another reason why the mode coupling-induced oscillations are seen only in the early part in Figs. 2 and 6. The pulse continues to compress as it propagates which, together with the walk-off due to modal dispersion, reduces substantially temporal mode overlap. This effect also explains the much weaker mode coupling observed when the pump is launched into $\textrm{H}{\textrm{E}_{13}}$ or $\textrm{H}{\textrm{E}_{14}}$ mode as shown in Figs. 2(b), 2(c), 6(b), and 6(c). We observe from Fig. 2 that the pulse initially compresses more rapidly in $\textrm{H}{\textrm{E}_{13}}$ or $\textrm{H}{\textrm{E}_{14}}$ mode than in $\textrm{H}{\textrm{E}_{12}}$ mode, which leads to smaller temporal mode overlap in the early section where the nonlinearity is high. The group-velocity mismatch dictates for how long the temporal overlap is sustained. Its effect on extent of the mode coupling is illustrated in Fig. 8(a). It shows temporal evolution of the total field when the system is pumped in $\textrm{H}{\textrm{E}_{12}}$ mode. Here, the time frame is moving at the group velocity of $\textrm{H}{\textrm{E}_{12}}$ mode that is changing with the pressure along the length. At the very beginning, some of the launch energy couples into other modes, notably $\textrm{H}{\textrm{E}_{11}}$ and $\textrm{H}{\textrm{E}_{13}}$ modes. The interaction shows up as interference fringe on top, which lasts until $\textrm{H}{\textrm{E}_{11}}$ and $\textrm{H}{\textrm{E}_{13}}$ modes walk off due to group-velocity mismatch. It agrees with the positions in Figs. 2(a) and 6(a) where the oscillations diminish. The group velocities of the first ten $\textrm{H}{\textrm{E}_{1p}}$ modes are plotted in Fig. 8(b). The velocity decreases as the mode order increases with the difference becoming larger as the mode order gets higher. These characteristics are reflected in Fig. 8(a). As they walk off, $\textrm{H}{\textrm{E}_{11}}$ mode advances and $\textrm{H}{\textrm{E}_{13}}$ mode lags with respect to $\textrm{H}{\textrm{E}_{12}}$ mode. Moreover, $\textrm{H}{\textrm{E}_{13}}$ mode decouples earlier than $\textrm{H}{\textrm{E}_{11}}$ mode due to bigger group-velocity mismatch. This also explains why the mode interaction is shorter lived for the case where $\textrm{H}{\textrm{E}_{13}}$ or $\textrm{H}{\textrm{E}_{14}}$ mode is excited. The non-dominant modes eject soon after the launch because of the increased group-velocity mismatch.

 

Fig. 8. (a) Temporal evolution of the total field when the decreasing pressure system is pumped in $\textrm{H}{\textrm{E}_{12}}$ mode. The time frame is moving at the group velocity of $\textrm{H}{\textrm{E}_{12}}$ mode that is changing with pressure along the length. (b) Group velocities normalized to the speed of light versus pressure for the first ten $\textrm{H}{\textrm{E}_{1p}}$ modes.

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We observed qualitatively the same pulse compression dynamics in the simulations under different decreasing pressure gradients and pulse energies when the same capillary bore diameter of 300 µm is used. That is, when $\textrm{H}{\textrm{E}_{12}}$ mode remains close to zero dispersion at the capillary input, we yield enhanced compression in $\textrm{H}{\textrm{E}_{12}}$ mode while degradations are seen in $\textrm{H}{\textrm{E}_{13}}$ and $\textrm{H}{\textrm{E}_{14}}$ modes with respect to the equivalent constant pressure systems. When the pressure at the input is raised to 4.2 bar such that $\textrm{H}{\textrm{E}_{13}}$ mode is in the vicinity of zero dispersion at the input, the self-compression enhancement compared to its equivalent constant 2.8 bar is observed only in $\textrm{H}{\textrm{E}_{13}}$ mode. The same happens upon further increase in the input pressure that places $\textrm{H}{\textrm{E}_{14}}$ mode just next to zero dispersion. In other words, we can selectively improve the compression in different higher-order modes simply by tuning the pressure at the capillary input which, from an application point of view, is very useful. This suggests that our findings are not restricted only to the specifics of the parameters considered here, but they can be applied to broader systems. Moreover, the enhancement is expected to be more pronounced when the compression approaches the ionizing intensity, allowing further power scaling of the compression. The additional loss and defocusing effects caused by the photoionization can be largely alleviated in decreasing pressure since there is only weak presence of the gas medium where the pulse is highly compressed.

5. Conclusion

In this work, we numerically investigate the effect of decreasing pressure on soliton self-compression dynamics in higher-order modes of a gas-filled capillary. In $\textrm{H}{\textrm{E}}_{12}$ mode, we observe four times improvement in the compression ratio under the decreasing pressure with the output pulse in the sub-cycle duration of 1.85 fs, achieving ${F_c} = 19$ as compared to ${F_c} = 4.68$ when the equivalent constant pressure is applied. The marked enhancement in the decreasing pressure is realized through the effective suppression of the higher-order dispersion in the later stage where the anomalous dispersion increases with the pressure drop. In $\textrm{H}{\textrm{E}_{13}}$ and $\textrm{H}{\textrm{E}_{14}}$ modes, the waveguide contribution to the dispersion is stronger, making the higher-order dispersion less important. Hence the decreasing pressure does not make any significant impact. Moreover, the maximum compression points under the decreasing pressure do not match exactly the compression lengths in the equivalent constant pressure systems due to different compression dynamics governed by the compression rate figure-of-merits. Namely, the compression points in $\textrm{H}{\textrm{E}_{13}}$ and $\textrm{H}{\textrm{E}_{14}}$ modes are brought forward in the decreasing pressure, after which the pulse either undergoes a soliton fission ($\textrm{H}{\textrm{E}_{13}}$ mode) or decays and disperses ($\textrm{H}{\textrm{E}_{14}}$ mode) due to attenuation and diminishing nonlinearity. Both results in degraded compressions. One advantage of the negative pressure gradient is that it can effectively mitigate the intermodal nonlinear coupling in the compression process and therefore preserve the spatial beam quality with high mode purity. Additionally, with the fiber evacuated at its output end, we can minimize the pulse distortion in the gas chamber. This allows the delivery of the compressed pulse free of dispersion and nonlinear effects. We believe these findings will benefit applications that require high-intensity ultrashort pulses in non-conventional spatial modes, such as in nonlinear frequency conversion, microscopy, micromachining, and particle manipulation.