1. INTRODUCTION

Nonlinear optics in waveguides has been phenomenally successful over the last 40 years as a result of the availability of glass optical fibers [1,2]. Although the early fibers were multimode, the majority of experiments and analyses concentrated on the fundamental mode, higher order modes being regarded as a nuisance. The advent of single-mode optical fibers, in particular the experimental and theoretical simplicity of endlessly single-mode photonic-crystal fiber (PCF) [3], has continued this trend. The generalized nonlinear Schrödinger equation (GNLSE) has been highly successful at explaining the nonlinear dynamics of such systems [4,5], providing accurate quantitative agreement even with extremely broad supercontinuum spectra [6]. In cases where it breaks down, for example in the description of third-harmonic generation or intense cycle-dependent photoionization, a uni-directional full-field equation can be used instead [7–11]. Full-field equations are also more convenient when studying nonlinear processes leading to multiple-octave or subcycle electric fields [12].

Recently there has been a resurgence of interest in multimode fibers for studying effects such as spatial-temporal solitons [13], four-wave mixing (FWM) [14], and enhanced supercontinuum generation, as well as exploring telecom applications [15,16]. Additionally, hollow-core capillaries, which are widely used for nonlinear spectral broadening of mJ scale ultrashort pulses, are also inherently multimode. Accurate modeling of nonlinear processes in these systems requires a more sophisticated treatment.

Of particular interest over the past few years is gas-filled kagomé-style hollow-core PCF (kagomé-PCF), which has turned out to be a highly versatile vehicle for ultrafast nonlinear optical experiments [17]. Its broad transmission window, relatively low loss, and weak anomalous dispersion (which can balance that of a filling-gas) are essential features for the propagation of short pulses. Most importantly, the ability to tune the anomalous dispersion by changing the gas pressure allows one to control soliton dynamics in gases, permitting soliton self-compression to few-cycle pulses [18]. This has been exploited to generate, with efficiencies of several percent, deep and vacuum ultraviolet (UV) light through dispersive-wave emission [19,20] from self-compressed fs pulses of a few μJ at 800 nm. It has also allowed studies of photoionization-driven effects such as the soliton self-frequency blue shift [21–23], at peak powers exceeding 2 GW (intensities exceeding ${10}^{14}\mathrm{W}/{\mathrm{cm}}^2$) inside the relatively small (30 μm diameter) fiber core. Although these experimental results are quantitatively supported by numerical simulations using single-mode unidirectional full-field simulations, the high intensities reached, the extreme spectral broadening, and the presence of significant photoionization raise the question of whether spatial effects, such as self-focusing and plasma-defocusing, can become important. An example is third-harmonic generation into a higher-order mode for parameters similar to those used in the generation of deep-UV light (mentioned above) [24]. It is also not hard to imagine other experimental situations in which consideration of spatial effects is important.

In Section 2 of this paper we introduce a model to describe multimode pulse propagation using the full-field equation. The approach used is applicable to any waveguiding system. We use it to investigate nonlinear spatial effects in hollow-core kagomé-PCF (described in Section 3) and compare the results with experiments in subsequent sections. In Section 4 we validate the model by numerically modeling experimental results on third-harmonic generation to a higher order mode in kagomé-PCF. Next, in Section 5, we reproduce recent experimental results on intermodal FWM in high-pressure gas. In Section 6, we show that, after soliton-effect compression of few-μJ, $\sim 40\mathrm{fs}$ pump pulses at 800 nm in gas-filled kagomé-PCF, many dispersive waves can be emitted in the visible and UV spectral region, at different wavelengths, as a consequence of the exchange of energy among different guided modes. Dispersive waves carried by higher order modes are emitted at shorter wavelengths compared to those that appear in the fundamental mode, because their dispersion is more anomalous, resulting in a shifted phase-matching condition. The model reproduces the experimentally observed results, and is in excellent agreement with analytical considerations based on phase-matching conditions. Having thoroughly validated the model, it is used in Section 7 to explore further spatial effects in kagomé-PCF, in particular possible self-focusing and defocusing effects.

2. MODELING SPATIALLY RESOLVED ULTRASHORT PULSE PROPAGATION IN WAVEGUIDES

Modal decomposition of the electric field using some appropriate basis can be a very powerful tool for the investigation of spatially dependent pulse propagation. This method, compared to other approaches (e.g., fully modeling diffraction), can be both computationally very efficient—if only a few order modes are required—and easier to interpret.

In the early 1980s, nonlinear propagation of higher modes in fibers and capillaries was initially investigated using the nonlinear Schrödinger equation [25–27]. The models developed were however not suitable for treating ultrashort pulses, because they considered only the Kerr nonlinearity, neglecting higher order dispersion and shock effects. A more complete description of propagation for ultrashort pulses in optical fibers is given by the multimode generalized nonlinear Schrödinger equation (MM-GNLSE), derived by Poletti and Horak in 2008 [28]. This equation is based on the unidirectional field equation derived by Kolesik and Moloney [9], which describes pulse propagation in homogeneous nonlinear media. Along with the inclusion of all higher order terms and intermodal coupling, the MM-GNLSE has the advantage of being vectorial (i.e., accounting for polarization).

Even though the MM model works very well in all the cases where the scalar GNLSE is valid, it becomes less accurate for pulses in the single-cycle regime, because the third harmonic term is not included. Additionally, cycle-dependent effects such as photoionization cannot be accurately modeled by this equation. Finally, when dealing with spectra spanning multiple octaves, models based on the slowly varying envelope approximation are less efficient than full-field models. In this paper we develop a field-resolved multimode propagation equation. Several somewhat similar models, derived in different ways or written down without derivation, have previously been used to treat a range of different physical problems [29–32].

A. Modal Expansion

The full 3D electric field vector, $\mathbf{E}(r,\theta ,z,t)$ can be expanded, without approximation, as a superposition of a complete set of orthogonal modes in both frequency and space (not necessarily waveguide modes; for example in free space one could use plane waves). We start by expanding in the temporal domain using the Fourier transform

The electric field expansion above is formally over both forward and backward propagating modes. Forward propagation is not a necessary assumption for what follows, but as it is the most common case, we implicitly consider only forward propagating modes from now on. This is a very reasonable assumption in most experimental situations [34].

B. Modal Propagation

General propagation equations for the modal amplitudes can be derived without approximation; for the monochromatic case the derivation can be found in textbooks [33]. A derivation explicitly for nonlinear broadband pulse propagation leads to [8,31]

When plane waves are used as the modal basis, Eq. (8) can be reduced to the well known spatially resolved unidirectional pulse propagation equation (UPPE) [8], or the forward Maxwell equation (FME) [7]. For waveguides, however, it is more natural and efficient to use the actual modes of the waveguide geometry. In particular, depending on the nature of the nonlinear contributions to ${\mathbf{P}}^{\mathrm{NL}}(r,\theta ,z,t)$, significant simplifications can then be made to Eq. (9). We review these below, starting from general forms of the nonlinear polarization.

Regardless of the form of ${\mathbf{P}}^{\mathrm{NL}}(r,\theta ,z,t)$, it is possible to monitor the accuracy of the modal decomposition in Eq. (2) by reversing the projection and calculating the norm of the error

We can compare the above multimode equation with the single mode case. If we assume only a single waveguide mode in the expansion in Eq. (2) and assume that the nonlinear polarization exactly preserves the spatial mode shape, then Eq. (9) becomes

The explicit appearance of ${\beta }_j(\omega )$ in the nonlinear polarization term in Eq. (13) raises an important point. Although the frequency dependence of this term (which causes self-steepening or optical-shock formation and is critical for accurate modeling of few-cycle pulses) is principally governed by the ${\omega }^2$ factor, there is an additional contribution from modal dispersion. In the single-mode case this is explicitly accounted for by the presence of ${\beta }_j(\omega )$ in the nonlinear term of Eq. (13). In the general multimode case, it arises from the normalization constants $N_j(\omega )$ and the polarization projection, Eq. (9).

C. Nonlinear Polarization

The total spatially resolved nonlinear polarization is calculated from the total field $\mathbf{E}(r,\theta ,z,t)$

1. Kerr Term

The nonlinear polarization due to the atomic third-order susceptibility tensor ${\chi }^{(3)}$ can be written in general as [36]

Note that in the above discussion we were considering only the electronic contribution to ${\chi }^{(3)}$. The Raman term, which certainly cannot be considered instantaneous, can be rigorously included by solving a suitable set of Maxwell–Bloch equations, as described in [40]. In this paper we consider materials without any Raman response (i.e., noble gases). Raman active media will be discussed in later work.

If we consider only the Kerr term and neglect photoionization (i.e., consider only intensities insufficient to ionize the medium), then, under the assumption that nonlinear effects do not act strongly on the fiber mode shapes [28], which are assumed to be frequency independent, Eq. (9) can be simplified to

Although the assumption that the transverse mode shapes are frequency independent is quite valid for hollow capillaries and kagomé PCF (see Section 3 below), it is not in general true for small core or photonic band-gap fibers. Such cases have been previously considered in detail [41].

The validity of Eqs. (17,18) requires that a sufficient number of modes are included to accurately describe all of the nonlinear spatial effects. The introduction of the overlap integral $Q_{jklm}$ helps as it allows us to reduce the number of modes based on their amplitude and symmetry rules, as fully discussed in [42]. However, the only way to be certain that a given mode expansion is sufficient is to conduct numerical convergence tests for increasing numbers of modes. This is particularly important, as the nonlinear polarization error, ${\epsilon }_{\mathrm{NL}}$, introduced in Eq. (10), is analytically zero in this case and hence does not provide a measure of the error.

The main benefit of Eqs. (17,18) is that $Q_{jklm}$ does not evolve with propagation and can be evaluated once at the beginning of the calculation (using a fine radial grid). This greatly reduces the computational cost of full polarization-dependent calculations compared to the explicitly spatially resolved case of Eq. (8).

2. Photoionization Term

As already mentioned, photoionization can become important at high intensities. The liberation of free electrons from atoms or molecules by strong electric fields is a highly nonlinear and lossy process. Additionally, the polarizability of free electrons is strong. Together, these effects can have a significant effect on pulse propagation. We can calculate the number of free electrons at a given point of time and space, $n_e(r,\theta ,z,t)$, via [43]:

The nonlinear polarization due to photoionization consists of terms accounting for the energy loss in the ionization process, and the temporal phase modulation due to the fast creation of free electrons. An intuitive derivation leads to [46]

As Eq. (21) is highly nonlinear, it does not closely follow the spatial distribution of the driving electric field, and hence a modal overlap approach does not work, and the full modal nonlinear polarization given in Eq. (9) must be used. However, if the modes are all circularly symmetric then the nonlinear modal polarization due to ionization can be calculated through

D. Numerical Solution

We model the nonlinear propagation of multiple modes using the scheme shown in Fig. 1. The initial conditions are some known pump inputs to the waveguide; for example we may excite with purely the fundamental mode with a given temporal pulse shape (intensity and phase). In the general case, the solution proceeds by (i) calculating the total spatial and temporally resolved electric field through Eq. (2), (ii) using that to calculate the total nonlinear polarization using Eq. (14) or one of its simplified forms, (iii) projecting that back onto the modes using Eq. (9), (iv) checking that the error condition Eq. (10) is small, and then (vi) advancing the spatial step using Eq. (8) for each mode. This last step is made using a Runge–Kutta method similar to the split-step RK4IP method developed for the GNLSE [47], with adaptive step-size control [48].

 

Fig. 1. Integration scheme for the waveguide mode propagation model.

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If we are modeling purely the Kerr effect, we can skip steps (i), (ii), and (iv) above and directly use Eq. (17) for step (iii).

1. Grid Sizes

To model nonlinear pulse propagation accurately in extreme regimes it is important to ensure that the spatial and temporal grids are sufficiently fine.

Convergence tests have shown that a time-step significantly finer than that required by the (linear) sampling theorem is needed to accurately calculate the nonlinear polarization. We use a temporal grid step of $<50$ attoseconds and a temporal grid width of between 1 and 50 picoseconds (for comparison, in what follows, the shortest simulated wavelength is 180 nm, which requires a linear sampling time-step of 300 attoseconds).

The kagomé PCF cores we consider below have radii $\sim 15\mathrm{\mu m}$. Convergence tests indicated that 10 to 15 radial points are sufficient for the spatial integrals and between 3 and 6 modes are required, depending on the simulation parameters.

3. PROPERTIES OF GAS-FILLED KAGOMÉ-PCF

The kagomé-PCF structure is shown in the inset of Fig. 2. As we mentioned in the introduction, the core can exhibit broadband low loss transmission across the UV, visible, and near-IR [17,49,50], even with small core sizes ($\sim 30\mathrm{\mu m}$ diameter). This enables the waveguide dispersion to be balanced by varying the filling gas species and pressure, and so the overall dispersion can be tuned to be either weakly anomalous or normal across the guidance band. In combination with its high damage threshold, kagomé-PCF provides an excellent means to study intense ultrafast soliton and related nonlinear optical phenomena [17].

 

Fig. 2. Calculated group velocity dispersion curves for the fundamental ${\mathrm{HE}}_{11}$ mode (blue) and the first two higher modes, ${\mathrm{HE}}_{12}$ (green) and ${\mathrm{HE}}_{13}$ (red). The calculations are based on a 27 μm core diameter kagomé-PCF filled with xenon at 2 bar (solid) and 7 bar (dashed). The inset shows a scanning electron micrograph of the fiber used for the experiments in Section 6.

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In the visible and UV spectral regions, the real part of the modal refractive index of gas-filled kagomé-PCF is accurately described by a simple analytical model first developed for hollow capillaries by Marcatili and Schmelzer [24,51,52]. This model works well for both fundamental and higher order modes. The imaginary part of the refractive index (describing propagation losses) is however dramatically smaller, permitting low-loss guidance at core diameters orders of magnitude smaller than is possible with hollow-capillaries [17,49].

According to the Marcatili model, the transverse component of the HE modes is zero outside the core and inside the core can be written in the following form:

The capillary approximation for the dispersion of kagomé-PCF modes has been experimentally verified in the visible and UV spectral region by Nold et al. [24], who demonstrated phase-matched third-harmonic generation to a higher order mode in an argon-filled kagomé PCF. The accuracy of the approximation has also been confirmed by finite element modeling (FEM) [17,52], which also confirms that the transverse modal distributions are approximately frequency independent for the core diameters we use (15 to 50 μm), even though the exact mode shapes do not strictly follow Eqs. (23) due to the hexagonal core structure. It should be noted, however, that using Eq. (12) with Eqs. (23) leads to ${\beta }_{nm}^2=k^2{n}_{\text{gas}}^2$, which overestimates the actual ${\beta }_{nm}$ given by Eq. (24). This is because the expressions in Eq. (23) are derived assuming a number of approximations [51].

Figure 2 shows the calculated dispersion [from Eq. (24)] of the ${\mathrm{HE}}_{11}$, ${\mathrm{HE}}_{12}$ and ${\mathrm{HE}}_{13}$ modes for a 27 μm diameter kagomé-PCF filled with Xe at 2 and 7 bar.

In what follows we are interested in what happens when the launched pulse is purely in the ${\mathrm{HE}}_{11}$ mode. With appropriate and well-aligned launch optics, coupling to higher order modes from a Gaussian input beam profile is negligible and was not observed in any of the cases considered here. At much higher intensities, however, self-focusing in the in-coupling gas-cell reduces the coupling efficiency and can cause the excitation of higher order modes [35].

In what follows we also include only the ${\mathrm{HE}}_{1n}$ modes because, based on the magnitude of the overlap integral $Q_{jklm}$, they are the ones most strongly excited when starting from the ${\mathrm{HE}}_{11}$ mode [42]. Restricting the analysis to these modes is well supported by experimental evidence in the low energy limit ($<1\mathrm{\mu J}$) when ionization is insignificant and is broadly confirmed in the following comparisons between numerical and experimental results.

4. PHASE-MATCHED THIRD-HARMONIC GENERATION

To test the validity of the model, we simulated the results in [24], where a kagomé-PCF (core diameter 28 μm and filled with Ar) was used to generate a third-harmonic signal in the ${\mathrm{HE}}_{13}$ mode. By changing the pressure, the phase-matched wavelength could be tuned, in good agreement with analytical predictions. Figure 3(a) shows the experimental spectra from [24] for 1.3 μJ, 30 fs pump pulses at 800 nm. Figure 3(b) shows the results of the simulations using the model developed in this paper. There is good agreement with both the experimental results and the analytical predictions. We attribute the small discrepancies to our imperfect knowledge of the pump pulse parameters at the input—probably they were not perfectly transform-limited.

 

Fig. 3. Tunable third-harmonic generation in the ${\mathrm{HE}}_{13}$ mode (linear scale). (a) Experimental spectra from [24]. (b) Simulated spectra using the multimode model. In both cases a pulse with 1.3 μJ energy and 30 fs duration at 800 nm propagates through 20.5 cm of Ar-filled kagomé-PCF with 28 μm core diameter.

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5. INTERMODAL FOUR-WAVE MIXING IN HIGH PRESSURE XE

Recently experimental evidence has emerged of intermodal FWM in kagomé-PCF filled with Xe at higher pressures (25–35 bar) [35]. This effect was analyzed using the multimode model presented here. Specifically, when an 18 μm kagomé-PCF is filled with more than 18 bar of Xe, the pump wavelength (800 nm) lies in the normal dispersion region. The dominant nonlinear broadening mechanism is then self-phase modulation (SPM). As the pump pulse spectrally broadens, it becomes possible to pump and seed discrete frequencies that satisfy phase-matching and energy-conservation for FWM, the anti-Stokes (AS) signal being emitted in a higher order mode:

Figure 4 shows the results of numerical simulations for propagation of 0.6 μJ, 150 fs pulses at 800 nm in a kagomé-PCF (core diameter 18 μm) filled with 25 bar of Xe. From Fig. 4(a) we see that SPM dominates the initial broadening. When the spectrum crosses the zero dispersion line, the broadening increases and in the time domain we see evidence of MI on the leading edge of the pulse [Fig. 4(b)]; this was confirmed by numerical XFROG spectrograms (not shown) of the pulse evolution in the ${\mathrm{HE}}_{11}$ mode. Solutions of Eq. (25) are also shown by dashed lines in Figs. 4(a) and 4(c). When SPM and MI broaden the spectrum in the ${\mathrm{HE}}_{11}$ mode to reach the FWM pump wavelength at 590 nm, and also broaden the long wavelength side sufficiently to seed a Stokes signal at 1360 nm, as shown in Fig. 4(a), an anti-Stokes signal is emitted at 377 nm in the ${\mathrm{HE}}_{12}$ mode, as seen in Fig. 4(c). Numerical XFROG spectrograms (not shown) clearly show the emergence of the 377 nm anti-Stokes signal when the pump and Stokes signals temporally overlap. These numerical results agree quite well with the experimental observations in [35], although that experiment was complicated by strong self-focusing effects in the gas-cell before light entered the fiber. Some evidence of self-focusing effects is apparent in Fig. 4(c), where energy is transferred to the ${\mathrm{HE}}_{12}$ mode directly at 800 nm. We examine self-focusing in kagomé-PCF in more detail in Section 7.

 

Fig. 4. Numerical simulations showing intermodal FWM in kagomé-PCF (core diameter 18 μm) filled with 25 bar of Xe. (a) Spectral evolution of the ${\mathrm{HE}}_{11}$ mode. (b) The temporal intensity evolution of the ${\mathrm{HE}}_{11}$ mode. MI indicates the breakup of the leading part of the pulse in the anomalous dispersion region. (c) Spectral evolution of the ${\mathrm{HE}}_{12}$ mode. The zero dispersion wavelengths are marked by dotted lines and anomalous and normal dispersion regions indicated by A and N. Also shown are the phase-matched FWM pump and Stokes wavelengths in the ${\mathrm{HE}}_{11}$ mode and anti-Stokes wavelength in the ${\mathrm{HE}}_{12}$ mode.

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6. DISPERSIVE WAVE GENERATION IN HIGHER ORDER MODES: EXPERIMENT AND SIMULATION

Dispersive wave emission in kagomé-PCF can be observed when lunching an ultrashort pulse ($\sim 50\mathrm{fs}$, soliton order $N\sim 10$) in the anomalous region of the fundamental mode. As such pulses propagate they temporally compress through the soliton effect (balance of SPM and anomalous dispersion). At the point of strongest compression, if the pressure is correctly tuned, the phase matching condition [53,54]

It turns out that the different guided modes have rather similar dispersion curves in the UV. This makes it possible to satisfy Eq. (26) for dispersive waves in several different higher order modes, albeit at different wavelengths. Additionally, other frequencies may be generated as result of FWM between the different dispersive waves and the pump soliton [55].

Figure 5 shows the simulated evolution of a 40 fs, 0.7 μJ pulse at 800 nm in a kagomé-PCF (core diameter 27 μm) filled with Xe at 2.7 bar. At this pressure the pump pulse is in the anomalous dispersion region, and phase-matched dispersive waves in the fundamental, ${\mathrm{HE}}_{12}$ and ${\mathrm{HE}}_{13}$ modes are expected to appear in the wavelength window from 180 to 350 nm. From Fig. 5 we see that the input pulse initially experiences soliton compression, considerably broadening its spectrum and eventually phase-matching to several dispersive waves. Three different emission lines with decreasing amplitude sequentially appear in the UV after the $\sim 14\mathrm{cm}$ compression point, corresponding to dispersive waves in different modes.

 

Fig. 5. Simulation of the propagation of a 40 fs, 0.7 μJ pulse in a kagomé-PCF (core diameter 27 μm) filled with 2.7 bar of Xe. (a) Temporal and (b) spectral evolution. A and N indicate anomalous and normal dispersion. Three modes were included in the calculation.

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In an experiment we placed a 17 cm length of kagomé-PCF between two gas cells each equipped with a ${\mathrm{MgF}}_2$ window for launching and extracting light. Pulses at 800 nm with duration 40 fs and energy 0.7 μJ were launched into the fundamental mode and the output spectra recorded with a McPherson monochromator. The experiment was repeated at Xe pressures in the range 2–7 bar, corresponding to the dispersion curves in Fig. 2.

Figure 6(a) shows the experimental (blue curve) and simulated (gray curve) output spectra for 0.7 μJ input energy and 2.7 bar of Xe. This energy level was chosen to allow a clean output spectrum, minimizing distortion due to ionization of the gas at the fiber input.

 

Fig. 6. (a) Experimental spectra (blue curve) and corresponding simulations (gray curve) at the output of a 17 cm long kagomé-PCF (core diameter 27 μm) filled with 2.7 bar of Xe. (b) Simulated spectrum decomposed into its individual modes. Pulses of duration 40 fs and energy 0.7 μJ were used in both the experiment and the simulations.

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The strongest peak is the dispersive wave in the fundamental mode, while the other two peaks are associated with dispersive waves in the ${\mathrm{HE}}_{12}$ and ${\mathrm{HE}}_{13}$ modes. The weak peak in the ${\mathrm{HE}}_{13}$ mode at 330 nm can be attributed to FWM between the pump soliton and the dispersive waves [55]. The main reason for the discrepancies between experiment and simulation is transmission loss, which was not included in the simulations because it is not known precisely (it is very difficult to measure). This is particularly a problem for higher order modes and in the deep-UV region, where FEM studies show that the loss spectrum is very complex, exhibiting many narrow spectral features.

Changing the pressure alters the dispersion and causes the phase-matched wavelengths to shift. Figure 7 shows the measured central wavelength of the three dispersive waves, corresponding to the ${\mathrm{HE}}_{11}$, ${\mathrm{HE}}_{12}$, and ${\mathrm{HE}}_{13}$ modes at different pressures (dots) together with a plot of solutions of Eq. (26) with ${\varphi }_{\mathrm{NL}}$ neglected (solid lines). The experimental data points agree well with the theoretical curves for the higher order modes, while a major discrepancy appears for the fundamental mode at longer wavelengths. In this region the dispersion is weaker and dispersive wave emission is more sensitive to the nonlinear phase shift (neglected in the analytical phase-matching calculation).

 

Fig. 7. Pressure dependence of the phase-match wavelength for dispersive waves in the ${\mathrm{HE}}_{11}$, ${\mathrm{HE}}_{12}$, and ${\mathrm{HE}}_{13}$ modes (solid lines), together with the central wavelengths at which the peaks appear in the experiments (points). Also shown are the near-field mode patterns measured at the fiber endface for the ${\mathrm{HE}}_{12}$ and ${\mathrm{HE}}_{13}$ modes.

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The prediction that the dispersive wave bands are associated with higher order modes was further confirmed by measuring the near-field mode profiles. The two insets in Fig. 7 show the measured near-field intensity distributions experimentally observed at the output of the fiber at (i) 225 nm for 1.8 bar Xe and (ii) 230 nm for 2.8 bar Xe, with 1.2 μJ input energy. The first is clearly in the ${\mathrm{HE}}_{12}$ mode and the second in the ${\mathrm{HE}}_{13}$ mode. The ${\mathrm{HE}}_{13}$ mode is somewhat distorted, which we attribute to the hexagonal core and the residual presence of the ${\mathrm{HE}}_{12}$ and ${\mathrm{HE}}_{21}$ modes, which could not be completely filtered out. The ${\mathrm{HE}}_{21}$ mode was probably generated by accidental excitation of some ${\mathrm{HE}}_{21}$ mode at 800 nm, caused by beam-pointing instability and plasma formation at the focal point, which for Xe is already significant even at this low energy level.

Note that by using Ar in place of Xe and evacuating the monochromator we were able to observe dispersive waves at wavelengths as short as 184 nm. The signals were however very weak, with powers 20 d lower than the fundamental dispersive wave, which itself becomes weaker and weaker as it shifts to shorter wavelengths. For practical purposes, short wavelength dispersive waves can be generated around 180 nm in the fundamental mode, using a different gas and core size [20].

7. SELF-FOCUSING AND PLASMA-DEFOCUSING IN KAGOMÉ-PCF

It has been pointed out by Tempea and Brabec [56], and then by Chapman et al. [57], that self-focusing in hollow core waveguides can be understood as the transfer of energy to higher order modes. We have therefore used the multimode model to explore beam-width changes of the light guided in a kagomé-PCF, caused by mode-related self-focusing and defocusing effects. To this end we numerically modeled intermodal interactions along the fiber, and coherently superimposed them to estimate the overall beam shape. Cladding modes were not taken into account, but these should not play a significant role since the overall width of the guided modes is less than the core diameter.

Self-focusing can be viewed as occurring in waveguides when the Kerr effect is sufficiently strong to cause phase-matching, and hence energy transfer, from the fundamental to higher order modes at the same frequency. This requires high peak powers. It has been shown that even in waveguides the required power is similar to the critical power in free-space [58]. For our parameters (18 μm core diameter, filled with 10 bar of Xe) this occurs at peak powers of $\sim 140\mathrm{MW}$. However, as discussed in the above sections, in kagomé-PCF significant energy can be transferred to higher order modes at different frequencies, which can also act to reduce the beam size and cause self-focusing.

We now consider pumping this fiber with 5 μJ, 500 fs pulses (peak power 18.8 MW) at 800 nm, i.e., well within the region of modulational instability (MI) [12]. The resulting nonlinear dynamics are shown in Fig. 8. After a few cm of propagation, the input pulse breaks up into a shower of solitons, considerably broadening the spectrum. As observed in [12], in this regime the solitons have extremely short duration (a few fs) and very high peak power (tens of MW). Upon further propagation these solitons emit dispersive waves, a process that may lead to a considerable transfer of energy to higher order modes (essentially each individual soliton can undergo the same emission processes described in Section 6). This was confirmed by numerical XFROG spectrograms. Under these circumstances strong ionization of the gas is also expected, the peaks of free electron density being as high as ${10}^{18}{\mathrm{cm}}^{-3}$ [12]. Convergence tests indicated that it was necessary to include all ${\mathrm{HE}}_{1n}$ modes up to $n=6$.

 

Fig. 8. Simulations of MI dynamics in the fundamental mode. A pulse with energy 5 μJ, duration 500 fs and wavelength 800 nm propagates along a kagomé-PCF (core diameter 18 μm) filled with 10 bar of Xe. (a) Temporal and (b) spectral evolution. A and N indicate anomalous and normal dispersion. (c) Magnified image of the boxed-in region in (a), showing the fine details of the “soliton shower”.

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In Fig. 9 the beam-width of light inside the core is plotted against distance, with and without the photoionization term, and including the radial dependence of the free-electron-induced polarization. Neglecting ionization, the beam diameter becomes considerably smaller due to self-focusing. This is not, however, common beam collapse; rather, MI generates many solitons, resulting in transfer of a considerable amount of energy to higher order modes through dispersive wave emission and causing the beam waist to shrink. When ionization is included, however, plasma defocusing compensates for this beam collapse, leading to more stable transverse profiles. The fact that plasma defocusing can prevent beam collapse is at first reminiscent of free-space filamentation. It should, however, be noted that filamentation based on a balance between MI, plasma defocusing, and dispersive-wave emission into higher order modes is novel, to the best of our knowledge.

 

Fig. 9. Beam radius against propagation distance, with and without ionization, in a kagomé-PCF (core diameter 18 μm) filled with 10 bar Xe and pumped with 5 μJ, 500 fs pulses at 800 nm.

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These results are supported by previous experimental studies of MI in the presence of ionization in kagomé-PCF, where no changes in beam diameter could be observed for the same parameters [12].

8. CONCLUSION

In conclusion, nonlinear spatial effects in gas-filled kagomé-PCF can be numerically analyzed by expanding the full 3D electric field in terms of higher order modes. The analysis may be cast in the frame of a multimode full-field equation including the full linear dispersion, Kerr and ionization effects. The multimode model successfully reproduces previous experimental results on third-harmonic generation into a higher order modes and intermodal FWM. Emission of dispersive waves in higher order modes was experimentally observed to be in good agreement with the results of the simulations. The model is suitable for studying Kerr-related self-focusing and plasma-defocusing in the same system, and predicts the presence of filamentation in gas-filled kagomé-PCF.