## Abstract

We numerically study the filamentation of ultrashort laser pulses at 2 *µ*m carrier wavelength in noble gases (argon, xenon) and in air. Compared with filamentation in the near-visible domain (800 nm), mid-infrared optical sources with durations close to a single cycle can be generically produced at various pressures and powers near the self-focusing threshold. The mechanism by which self-compression takes place mainly involves optical self-focusing, pulse steepening and plasma defocusing. On-axis spectra and spectral phases are discussed. Delivering single-cycled pulses at long wavelengths has important applications in the generation of high-order harmonics and isolated attosecond pulses.

©2008 Optical Society of America

## 1. Introduction

Laser filaments in gases result from the competition between Kerr self-focusing and defocusing by a laser-induced electron plasma [1]. Besides their promising applications for remote sensing [2, 3, 4, 5], ultrashort pulses with input powers close to the self-focusing threshold, *P*
_{cr}, produce a single filament whose spectrum broadens and duration can significantly shrink along the propagation axis. Many studies have been devoted to this remarkable property, in order to generate few-cycle optical sources [6, 7, 8, 9, 10]. Recently, a couple of papers [11, 12] identified that, at 800 nm wavelength, maximum shortening in time happens when the filament diffracts beyond the self-focus point, after the shortest wavelength has been attained in the supercontinuum. Compression efficiency varies according to the propagation medium. From the experimental point of view, 800 nm pulses with about 5 fs full-width at half-maximum (FWHM) durations have been produced in argon at pressures close to atmospheric, with subsequent chirped mirror compression and spatial selection of the beam core [10]. Without any post-compression stage, ouput filaments with 8 fs durations have been measured from a gas cell filled with krypton [8]. From the numerical point of view, FWHM durations comprised between 5 and 7 fs have been obtained by propagating pulses in atom gases with moderate ionization potentials (*U _{i}*≤16 eV) at uniform pressure. As the compression rate increases with the saturation intensity fixed by the ionization threshold, filaments of 3–4 fs have been simulated in a neon gas, whose ionisation potential exceeds 21 eV [13]. Sub-cycle (<2 fs) pulses were even expected by self-channeling 5 fs pulses centered at 750 nm in helium (

*U*=24.6 eV) [14]. At smaller wavelengths, ultraviolet pulses (266 nm) compressed through the four-wave mixing of a 400 nm pump filament with a 800 nm idler beam have been shown to keep durations as short as 1.5 fs over 30 cm propagation ranges [15].

_{i}The possibility to generate intense few-cycle pulses nowadays opens new avenues in strong field physics and its applications. Among those, high-order harmonic generation (HHG) rises as the key process giving access to the production of single attosecond pulses. HHG is usually achieved by focusing an intense driving pulse into a noble gas. Initiated by optical field ionization, HHG results from the coherent frequency conversion of the pump wave into numerous harmonics extending into the XUV spectrum. Whereas many-cycle pump pulses yield attosecond pulse trains [16], few-cycle pulses with stable carrier-envelope phase allow to sort out isolated attosecond wave-packets [17]. Few-cycle pulses have often been produced by exploiting self-phase modulation in hollow fibers filled with noble gases [18]. The alternative procedure employing filamentation, however, avoids the delicate coupling between an intense pulse and a long waveguide, and it currently preserves the carrier-envelope phase [19].

So far, most of the HHG experiments have been performed using titanium:sapphire (Ti:Sa) sources operating at near-visible wavelengths (~800 nm). Nonetheless, scaling laser-atom interactions towards longer wavelengths should yield more energetic particles and shorter bursts of attosecond light [20, 21, 22, 23]. First, the cutoff photon energy of HHG is described by *E*
_{max}=*U _{i}*+3.17

*U*,

_{p}*U*being the ponderomotive energy varying like

_{p}*I/ω*

^{2}

_{0}where

*I*and ω

_{0}are the laser intensity and central frequency, respectively. This cutoff is thus proportional to the square of the wavelength and bounds a wider harmonic spectrum at long wavelengths. Second, the maximum energy gained by electrons when they return to the parent ion is 10

*U*. So, at constant intensity, longer wavelengths generate more energetic particles, opening the route to the production of multi-keV electrons. Third, depending on the electron trajectories, attosecond bursts carry a chirp (attochirp), which is inversely proportional to the laser wavelength. Longer wavelengths reduce this chirp, which increases coherence of the harmonics and the phase matching with the pump wave. Driving laser-atom interactions at mid-infrared wavelengths should thus help in generating energetic and extremely short x-ray pulses [23].

_{p}For this purpose, laser sources operating in the mid-IR are now available. Femtosecond optical parametric amplifiers (OPAs) pumped by multi-mJ Ti:Sa sources deliver multicycle pulses in the spectral range 1.3–2 µm with sufficient peak power to trigger the filamentation process. Hauri *et al*. [24] recently examined long-wavelength filamentation by using phase-stabilized 330 µJ, 55 fs pulses at 2 *µ*m, produced by difference frequency generation in a Ti:Sa-pumped OPA. Pulses launched in loosely focused geometry into a xenon cell at 2 bar pressure under-went self-channeling and they shrank in time down to 3 optical cycles (17 fs) while keeping 270 µJ energy. Similar durations with higher pulse energy have also been reported at ~1.4 *µ*m wavelengths by Vozzi *et al*. [25], who applied difference frequency generation to a filament broad supercontinuum boosted by a two-stage OPA. Generation of 13 fs pulses centered at 3 *µ*m wavelength was demonstrated by Fuji and Suzuki [26], who exploited four-wave mixing through filamentation in air. By means of more classical compression techniques, sub-two-cycle (8.5 fs) pulses have been produced at 1.6 *µ*m wavelength from a white-light seeded, 800 nm pumped degenerate OPA, using a deformable mirror compressor [27].

In this paper, we numerically investigate the nonlinear dynamics of femtosecond filaments formed in different gases (argon, xenon and air) at the 2 *µ*m laser wavelength. Emphasis will be given to filamentation regimes leading to robust self-compressed pulses. By “robust”, we signify that maximum compression rates will be preserved by singly-peaked filaments over distances exceeding far their Rayleigh length. Self-compression will be discussed only in terms of on-axis temporal distributions reaching the maximum, plasma-mediated intensity. Indeed, FWHM pulse durations averaged through a pinhole with diameter comparable with the filament core size routinely remain close to the duration of the on-axis pulse profile [9, 11]. Self-compressed filaments created in argon, xenon and air at 800 nm are compared for different pressures with their counterparts at 2 *µ*m. We show that, for interaction media with moderate ionization potentials, mid-IR filaments easily reach the single-cyle limit, unlike near-visible pulses whose durations approach 2–3 optical cycles. In connection, an impressive spectral broadening takes place owing to self-phase modulation and to a sustained action of pulse steepening. The major result is that pulse self-compression monitored by filamentation works at long wavelengths and can achieve pulse durations much less than those reported so far in the mid-infrared.

## 2. Model equations

The propagation model is elaborated on the nonlinear envelope equation earlier derived by Brabec and Krausz [28]. This model governs the complex envelope of the laser electric field $E=U{e}^{i{k}_{0}z-i{\omega}_{0}t}+c.c.$ around a central frequency *ω*
_{0}=2*πc/λ*
_{0}. A new time variable *t→t-k ^{′}z* is utilized to replace the pulse into the frame moving with the group velocity

*k*

^{′-1}=(

*∂k/∂ω*|

_{ω=ω0})

^{-1}. The equation for the forward pump envelope

*U*then expands as [4]

$$-i\frac{{k}_{0}}{2{n}_{0}^{2}{\rho}_{c}}{T}^{-1}\rho U-\frac{\sigma}{2}\rho U-\left({\rho}_{\mathrm{nt}}-\rho \right)\frac{{U}_{i}W\left(I\right)}{2I}U,$$

where $I\equiv {\mid U\mid}^{2},D\equiv \sum _{n\ge 2}^{+\infty}({{\partial}^{n}k\u2044\partial {\omega}^{n}\mid}_{\omega ={\omega}_{0}}\u2044n!){\left(i{\partial}_{t}\right)}^{n},T=\left(1+\frac{i}{{\omega}_{0}}{\partial}_{t}\right),{k}_{0}={n}_{0}{\omega}_{0}\u2044c,{n}_{0}\simeq 1$ is the linear refractive index and c is the speed of light in vacuum. The first term of the operator *𝓓* corresponds to group-velocity dispersion with coefficient *k*
^{″}=*∂ ^{2}k/∂ω^{2}*|

_{ω=ω0}. Equation (1) describes wave diffraction, Kerr self-focusing, plasma generation and related losses, chromatic dispersion with space-time focusing [

*T*

^{-1}∇

^{2}

_{⊥}

*U*] and self-steepening [~

*T*(|

*U*|

^{2}

*U*)]. We assume a linearly polarized field and consider a cubic susceptibility tensor ${\chi}_{{\omega}_{0}}^{\left(3\right)}$ constant around ω

_{0}. The Kerr response with nonlinear index ${n}_{2}=3{\chi}_{\omega 0}^{(3)}/4{n}_{0}^{2}c{\epsilon}_{0}$, where

*ε*

_{0}denotes the electric permittivity, is exclusively instantaneous for atom gases such as Ar and Xe (

*x*=0). For the diatomic molecules of air, the phenomenon of Raman scattering comes into play, so that the Kerr nonlinearity acting on the front pulse is partly diminished by the non-local response function

_{K}in the ratio *x _{K}*=1/2.

*τ*

_{1}=62.5 fs is the inverse of the fundamental rotational frequency and

*τ*

_{2}=76.9 fs denotes the dipole dephasing time.

Besides, assuming electrons born at rest, the growth of the electron density, *ρ*, is only governed by external source terms,

that include photo-ionization processes with rate *W(I)* and collisional ionization with cross-section *σ*=*q ^{2}_{e}*/[

*m*(1+

_{e}ε_{0}n_{0}cν_{e}*ω*

^{2}

_{0}/

*ν*)]. Here,

^{2}_{e}*ρ*is the density of neutral species;

_{nt}*q*and νe are the electron charge, mass and collision time, respectively. Electron recombination in gases is efficient over ns time scales, and therefore neglected. In Eq. (3), the rate for photoionization

_{e}, m_{e}*W(I)*follows from Perelomov, Popov and Terent’ev (PPT)’s theory [29] (see also [30]) yielding

$$\times {U}_{i}\frac{{\gamma}^{2}}{1+{\gamma}^{2}}\sum _{\kappa \ge {\nu}_{0}}^{+\infty}{e}^{-\alpha \left(\kappa -\nu \right)}{\Phi}_{m}\left(\sqrt{\beta \left(\kappa -\nu \right)}\right),$$

where, expressed in atomic units, *E _{p}*~

*√I*denotes the peak optical amplitude, ${E}_{0}={(2{U}_{i})}^{3/2},\gamma ={\omega}_{0}\sqrt{2{U}_{i}}/{E}_{p},\nu ={U}_{i}/\overline{h}{\omega}_{0},\beta =2\gamma /\sqrt{1+{\gamma}^{2}},\alpha =2[{\mathrm{sinh}}^{-1}(\gamma )-\gamma /\sqrt{1+{\gamma}^{2}}],{v}_{0}=<v+1>$ and ${\Phi}_{m}(x)={e}^{-x2}{\displaystyle {\int}_{0}^{x}{({x}^{2}-{y}^{2})}^{\left|m\right|}{e}^{{y}^{2}}dy.\text{\hspace{0.17em}}n*=Z/\sqrt{2Ui}}$ is the effective quantum number,

*Z*the residual ion charge,

*l**=

*n**-1 and

*n,l,m*are the principal quantum number, the orbital momentum and the magnetic quantum number, respectively. The pre-exponential factors

are extracted from the tunneling theory derived by Ammosov, Delone and Krainov [31]. This model reproduces with high fidelity (5% relative variations) the single ionization of Ar atoms by 800 nm driving pulses, compared with numerical solutions of the time-dependent Schrödinger equation [32, 33]. The Keldysh parameter *γ* separates the tunneling (*γ*≪1) and multiphoton (*γ*≫1) regimes and it increases with ω_{0} at constant intensity. Longer wavelengths should thus drive the interaction preferably in the tunnel regime, closer to the semiclassical description of the three-step model [23].

The propagation model is constituted by Eq. (1) and Eq. (3). It will be integrated numerically by using initially collimated Gaussian pulses,

with input power *P*
_{in}, beam waist *w*
_{0} and 1/e^{2} pulse half-width *t _{p}*. In linear regime, Gaussian pulses are expected to diffract over the Rayleigh distance

*z*

_{0}=

*π n*

_{0}w^{2}_{0}/λ_{0}. The values of the physical parameters used in Eq. (1), including the GVD coefficient k″, are specified in Table 1 for the atmospheric pressure

*p*=1 bar. Dispersion curves follow Dalgarno and Kingston’s refractive indices [34] for argon and xenon, and that of Peck and Reeder [35] for air. Nonlinear susceptibilities ${\chi}_{{\omega}_{0}}^{\left(3\right)}$ allowing to evaluate

*n*

_{2}at different wavelengths are taken from Shelton’s Ref. 36. The selected Kerr index for air lies in orders of magnitude consistent with those of [37, 38, 39]. Ar and Xe atoms have the ionization potentials

*U*=16 eV and

_{i}*U*=12.1 eV, respectively. Because the ionization potential of

_{i}*N*

_{2}is higher than that of O

_{2}molecules, we only consider the latter specy as generating an electron plasma with again

*U*=12.1 eV. All ionization models used in the present work are illustrated in Fig. 1. Although different rates have been proposed for describing air ionization [40], these currently differ from each other by less than 2 decades, which should not significantly modify the saturation intensity in the filaments at large photon numbers

_{i}*K*≡

*ν*

_{0}=<

*Ui/h̄ω*

_{0}+1>. For the sake of clarity, all curves referring to argon are plotted in blue; those corresponding to xenon are in red, while curves describing air are plotted in green.

To understand the propagation dynamics at different carrier wavelengths, we find it convenient to fix the same ratio of input power over critical, whatever the local pressure may be. In

all cases we choose input peak powers close to critical, i. e., *P*
_{cr}≤*P*
_{in}≤3*P*
_{cr} for atom gases (*P*
_{cr}≃*λ*
^{2}
_{0}/2*πn _{0}n*

_{2}). In air, increasing

*P*

_{in}up to 5

*P*

_{cr}will overcome the reduction of the Kerr nonlinearity due to Raman scattering. The input beam waist is

*w*

_{0}=500

*µ*m and the initial FWHM pulse extent is $\sqrt{2\mathrm{ln}2}{t}_{p}=30$. Numerical snapshots of temporal field distributions are collected every 10 cm. All numerical simulations have been performed in radial symmetry with 6144×6144 points in ($r=\sqrt{{x}^{2}+{y}^{2}},t$) with respective resolutions of 2.44

*µ*m and 0.25 fs, for an adaptive step along

*z*. We cross-checked that the dynamics remained similar when the pulses were simulated from a more complete model, such as the frequency-dependent unidirectional propagation equation describing the forward spectral amplitude of the total field (see [4] and references therein). When neglecting third harmonic generation in noble gases [43], the propagation pattern was found unchanged, yielding identical compression rates. When taking third-harmonic generation into account, as expected in air [4], the coupling between the pump and the harmonic wave creates oscillations in the temporal structure. However, the maximum intensity remains of same magnitude and the supercontinuum extends over comparable band-with. Minimum durations are analogous to those given by the envelope model [Eq. (1)] and they are attained at comparable propagation distances.

From Table 1, it is clear that dispersion and collisional ionization are weak. Self-channeling then mainly relies on the dynamic balance between Kerr self-focusing and plasma defocusing, so that estimates for the filament intensities (*I*
_{fil}), electron densities (*ρ*
_{fil}) and mean filament waist (*w*
_{fil}) can be deduced from equating diffraction, Kerr and ionization responses in Eq. (1). This yields the simple relations [4]

where

represents the maximum effective Kerr index over the initial pulse profile. For the input (6), one thus has *n*̄_{2}=*n*
_{2} for noble gases and *n*̄_{2}=0.6545×*n*
_{2} in air. Δ*T* represents theFWHM temporal extent of the pulse shaped by the first ionization front. Despite the lack of knowledge on this time interval, we can fix it roughly to ~10 fs on the basis of antecedent numerical investigations addressing a broad spectrum of carrier wavelengths [38]. The values of *I*
_{fil} inferred from Eq. (7) using the rate *W(I)*, Δ*T*=10 fs and the previous data have been indicated in the last line of Table 1. Note the similarity in the saturation intensities reached at the two wavelengths. Despite much larger photon numbers involved in the ionization process, 2 *µ*m laser filaments have Kerr contributions *n*̄_{2}
*I*
_{fil} clamped at similar levels to those attained at 800 nm. The critical plasma density *ρ _{c}* indeed decreases in 1/

*λ*

^{2}

_{0}. However, the ionization rate

*W(I)*decreases in turn by about one decade in the tunnel regime at intensity close to saturation (Fig. 1).

The magnitude of the saturation intensity directly impacts spectral broadening, which is initiated by self-phase modulation (SPM). SPM creates a strong supercontinuum as the intensity increases by self-focusing. In the limits *T,T*
^{-1}→1, frequency variations evolve like

over the longitudinal path Δ*z* and they vary with the superimposed actions of the Kerr and plasma responses. When accounting for steepening terms (*T,T*
^{-1}≠1), shock edges occur in the back of the pulse [41] and they usually blueshift the spectrum [4, 42, 43].

## 3. Filament compressors at 800 nm

Below we briefly recall scaling of the quantities in Eq. (7) when the pressure differs from 1 bar. In that case, the Kerr index *n*
_{2}, the dispersion operator 𝓓, the neutral density *ρ*
_{nt} and the cross section for avalanche ionization *σ* varies linearly with the pressure *p*, whereas the ionization rate *W(I)* is pressure-independent [44]. Using the basic ordering Eq. (7), we deduce that *I*
_{fil} keeps a constant value, as confirmed by a recent experiment [45]. The filament waist *w*
_{fil} behaves like 1/*√p* while the peak electron density *ρ*
_{fil} evolves like *p*. If we conjecture that the self-channeling length, Δ*z*
_{fil}, is driven by the Rayleigh length associated with the filament, *z ^{R}*

_{fil}≡

*πn*

_{0}w^{2}_{fil}/λ_{0}, then the self-guiding range should decrease inversely proportional to the pressure.

Figure 2 summarizes macroscopic aspects of filamentation at 800 nm in Ar, Xe for the input power *P*
_{in}=3*P*
_{cr}, and in air for *P*
_{in}=5*P*
_{cr} [note that 5*P*
_{cr}(*n*
_{2})≃3.3*P*
_{cr}(*n*̄_{2}) in the atmosphere]. The saturation intensity reached in argon at this wavelength lies in the range of 100 TW/cm^{2} for *p*=0.5,1 and 2 bar, in fair agreement with the value reported in Table 1. Of course, this evaluation does not fit the maximum peak intensity reached by the pulse, as Eq. (7) just provides a crude estimate of the mean intensity level from which plasma generation can arrest beam collapse. Since we are operating at constant ratio *P*
_{in}/*P*
_{cr}, the self-focus point is located at analogous distances, *z _{c}*~0.4 m, whatever the pressure and propagation medium may be [Figs. 2(a)–2(b)]. Slight deviations originate from the action of GVD, which increases with the pressure and retards the beam collapse to some extent. The filamentation range, evaluated at half of the saturation intensity, increases by a factor of ~2 when the pressure is decreased by the same factor, as expected above. A longer self-guiding takes place in air, resulting from moderate intensity clamping (

*n*̄

_{2}

*I*

^{air}

_{fil}~0.1

*n*

_{2}

*I*

^{Xe}

_{fil}), small dispersion compared with Xe, and refocusing of time slices in the back of the pulse owing to the non-local Raman response. Figure 2(c) shows peak electron densities in Ar and Xe. Their values do increase linearly with

*p*. At decreasing pressures, the filament waist grows like 1/

*√p*. Mean values of

*w*

_{fil}[Eq. (7)] are ~100

*µ*m for Ar, ~60 µm for Xe and ~150 µm for air at

*p*=1 bar, which supports the comparison with Fig. 2(d). Note that

*n*̄

_{2}

*I*

_{fil}reaches higher values in xenon, which explains the smaller filament sizes compared with, e. g., argon at the same pressure. In every case, the self-guiding range covers several times the Rayleigh distance of the filament, i. e., Δ

*z*

_{fil}~5×

*z*

^{R}_{fil}.

Figure 3 illustrates evolution patterns in the (*t, z*) plane. All pulses exhibit similar distortions. The pulse starts to self-focus then triggers a plasma sequence, which defocuses the back zone (*z*~0.4 m). At later distances, the trailing pulse refocuses, which forms a double-peaked distribution in the temporal profile (*z*~0.5 m in noble gases). The trailing edge self-steepens and develops a shock dynamics. Its intensity reaches the highest peak value, which triggers in turn the highest plasma density. Consequently, the trailing edge of the pulse is rapidly depleted by plasma generation, whereas the leading edge, coupled to weaker density levels, continues to propagate within a slow diffraction stage. Ultimately, the pulse shape relaxes to a very short peak in the front zone, where the smallest durations are attained (*z*≥0.6 m in Ar and Xe). These have been indicated in Fig. 3. In Ar at *p*=1 bar, the pulse time extent goes down to 6.5 fs from *z*=0.8 m, and it never exceeds 7 fs in the range 0.6≤*z*≤1.2 m. Identical durations have been measured over longer ranges at lower pressure (0.5 bar), and over shorter ranges at higher pressure (2 bar). The same property applies to xenon and air. In Xe at 1 bar pressure, the pulse does not shrink below 7.5 fs and maintains maximum compression upon 20 cm only. In contrast, diminishing the pressure down to 0.5 bar favors shorter durations over 40 cm. In air, the pulse profile develops several peaks in the range 0.5≤*z*<1 m, which we attribute to the asymmetry in time introduced by the Raman response. FWHM durations of 7.5 fs, reached from *z*=1 m, are preserved up to the distance *z*=1.4 m.

Figure 4(a) shows the shortest temporal profiles formed at maximum compression in Figs. 3(a) and 3(d). Pulses are singly-peaked. For comparison, the dash-dotted curve represents the latter pulse configuration obtained at the same propagation distance when the operator *T* is set equal to unity. Two quasi-symmetric peaks emerge, as predicted by early self-channeling scenarios in gases [46]. These peaks have comparable intensity and they run over more than 20 cm along the filament path. Each of the spikes experiences an individual compression down to ~3 fs. However, both design a profile with overall FWHM duration of 36 fs. Pulse steepening, instead, enhances the revival of the Kerr response. It makes the trail pulse rapidly refocus and decay, so that a unique leading spike survives and slowly diffracts in the front zone. It is worth recalling that antecedent works reported shortening of the pulse in the trailing region [12]. In collimated geometry this dynamics is, however, generic if the peak power is sufficiently close to critical and avoids pulse splitting in time, which is not the case here. When pulse splitting occurs, no general rule allows us to guess *a priori* that compression will systematically take place in the rear pulse. Let us remind in this regards that when the pulse splits up through, e. g., normal GVD, self-steepening amplifies the trailing pulse near the self-focus point at powers close to critical. However, the leading pulse may become the dominant component at higher powers beyond the self-focus point [47].

Figure 4(b) details on-axis spectra and related phases for the same pulses. The phase flattens around the central frequency, but it develops jumps outside. SPM competing with the plasma response [Eq. (9)] broadens the pulse spectrum widely towards the small frequencies. All spectra were found to exhibit comparable distortions at different pressures. This is justified by Eq. (9), in which Δ*z*=Δ*z*
_{fil} behaves like 1/*p* while nonlinearities scale as *p*.

## 4. Filament compressors at 2 µm

Following Ref. [38], we briefly sketch the basic changes expected in the physics of femtosecond filaments at large wavelengths, namely,

• The supercontinuum increases with the carrier wavelength *λ*
_{0}.

• At comparable ratios *P*
_{in}/*P*
_{cr}, infrared filaments are more energetic than near-visible ones. They survive along longer distances and have larger beam waist ~λ_{0}.

• The influence of pulse steepening increases with *λ*
_{0}.

Figure 5 illustrates some examples of light self-guiding in argon, xenon and air at 2 *µ*m wavelength. In Fig. 5(a), filamentation starts at 1 critical power in Ar for *p*=1 bar (dash-dotted curve). However, the pulse intensity reaches saturation thresholds in the magnitude of those theoretically expected (~100 TW/cm^{2}) only at higher peak power (solid curve), or at lower pressure (dashed curve). At atmospheric pressure, power ratios strictly above unity are indeed required to realize a collapse dynamics triggering maximum electron densities. With *P*
_{in}=1*P*
_{cr}, the normalized GVD coefficient *δ*≡2*z*
_{0}
*k*
^{″}/*t*
^{2}
_{p}, although weak in argon, is still large enough to inhibit self-focusing at moderate pressures *p*≥1 bar. This limitation is overcome at 0.5 bar pressure [48]. When the pulse self-focuses until triggering a plasma sequence, the scaling with respect to the pressure *p* still holds. The location of *z _{c}* is practically unchanged at equal power ratios; the magnitude of

*I*

_{fil}remains pressure-independent [Fig. 5(b)]; plasma densities follow pressure variations [Fig. 5(c)], and the filamentation ranges and waists increase when

*p*decreases [Fig. 5(d)]. As expected in Sec. 2, filament intensities are comparable at near-visible and mid-IR wavelengths, although the peak intensity decreases to some extent at 2

*µ*m in air. The most significant changes reported in Fig. 5 concern the filamentation range and waist. Engaging smaller power ratios than in Fig. 2, filament channels extend upon equal or longer distances. This can be explained by the dependency of Δ

*z*

_{fil}over

*λ*

_{0}and the much larger values of the critical powers (see Table 1), rendering 2

*µ*m pulses more energetic even at weaker ratios

*P*

_{in}/

*P*

_{cr}. On the other hand, the filament waist increases up to 290

*µ*m in Ar, 180

*µ*m in Xe, and 600

*µ*m in air, as computed from Eq. (7) at atmospheric pressure. These estimates are again in rather good agreement with the filament sizes plotted in Fig. 5(d). Increase in the filament waist also follows from the direct proportionality between

*w*

_{fil}and

*k*

^{-1}

_{0}. On the whole, the self-guiding ranges cover at least two filament Rayleigh lengths

*z*

^{R}_{fil}. The ratio between the input (500

*µ*m) and filament waists differs in the computations done at 800 nm and 2

*µ*m, which directly modifies the evolution pattern of the pulses.

The temporal dynamics of 2 *µ*m filaments are detailed in Fig. 6. Unlike near-visible filaments, the shortest duration is not attained beyond the point of maximal intensity, but either before or along the self-guiding stage. In xenon at 0.5 bar pressure, self-steepening occurs before the self-focus point and shapes the pulse into a steep trailing edge that causes maximum compression at *z*=0.4 m [Fig. 6(a)]. Afterwards, self-focusing and self-steepening continue to amplify the trail pulse until it broadens towards the front zone, which enlarges the pulse duration. In this configuration, FWHM time extent of ~13 fs can be preserved over about 15 cm only. In contrast to this rapid compression mechanism, 2 *µ*m atmospheric propagation produces shorter pulses within a long self-channeling process forming a unique peak, that shifts towards the rear region via self-steepening. The pulse preserves FWHM durations less than 12 fs inside the range 1.4≤*z*≤1.9 m at moderate peak intensities [Fig. 6(b)]. Atmosphere thus offers a convenient medium for promoting moderately-intense, short filaments capable of keeping a robust shape with better compression rates than in xenon. Minimum durations are less than 2 optical cycles (*λ*
_{0}/*c*≃6.7 fs) and definitively below the 17 fs realized in the experiment [24]. They are comparable with the sub-two-cycle durations (~13 fs) reported for 3 *µ*m carrier wavelength by Fuji and Suzuki [26]. The most surprising results come from propagation in argon. At exactly one critical power, the pulse maximum is pushed into a narrow domain of the rear zone, where the pulse is shrunk to near single-cycle durations ~8.5 fs (not shown here). Triggering full plasma generation from supercritical beams next results in the production of almost exactly singly-cycled waveforms with 6.5 fs FWHM duration at 1 bar pressure [Fig. 6(c)]. Even sub-cycle optical distributions compressed down to 5.5 fs may emerge from critical beams in argon at 0.5 bar pressure [Fig. 6(d)]. In principle, the model equations no longer hold at such points of extreme compression. Ionization rates and non-dispersive Kerr nonlinearities indeed suppose the existence of an optical intensity, which can only make sense by averaging the squared field amplitude over at least one cycle. However, we can conjecture that the validity limits applying to envelope models for pulses approaching punctually the single cycle do not break the relevance of compression stages involving larger, but close durations. Assuming this, pulses extending within one optical period should physically be produced at long wavelengths in gases promoting high enough saturation intensities. In the situation illustrated in Fig. 6(c), the pulse conserves its smallest durations between 6.5 and 8 fs in the longitudinal interval 0.5≤*z*≤0.7 m, i. e., upon 20 cm at least. For critical pulses at *p*=0.5 bar, FWHM durations less than 9 fs have been diagnosed between *z*=0.7 m and *z*=0.9 m.

From a practical point of view, laser sources operating in the mid-IR usually deliver pulse energies limited to the mJ level. The previous simulations predict pulse shortening to the single cycle in Ar for optical powers close to 40 GW, corresponding to ~1.3 mJ energy for 30 fs Gaussian pulses. This energy level is relatively high, so that retrieving the same compression efficiency at lower energies would require to increase the pressure. We observed, however, that augmenting the pressure at near-critical powers may not guarantee to preserve the compression, because of the increase of gas dispersion. Alternatively, exploiting filamentation in xenon and air will need smaller energies (~90–280 *µ*J), but this should lead to lesser compression rates.

The on-axis temporal profiles of the last two configurations of Fig. 6 have been detailed in Fig. 7(a). We can observe how steep the trailing edge of the pulses becomes in the vicinity of the self-focus point. Pulse shaping results from the self-steepening operator *T*=1+(i/ω0)∂t, whose impact increases at long wavelengths and sharpens temporal gradients in the filament [41]. This statement can already be seen from the dash-dotted curve representing the intensity profile formed at the same distance in 1 bar argon without steepening terms. Steepening effects start along the early self-focusing stage. They push the pulse towards the rear zone and are responsible for shock formation on the trailing pulse. Their influence increases with λ_{0} and *I*
_{fil}, which contributes to shorten the field distribution in time. In this regards, Fig. 7(b) shows the on-axis spectra relative to argon and xenon. SPM broadens the initial spectrum and pulse steepening enhances asymmetry to large frequencies [4]. Unlike the 800 nm pulses depicted in Fig. 4(b), the spectral phases associated to such extreme self-compressions are remarkably flat in the entire bandwidth around the central frequency.

To complete the comparison between 800 nm and 2 *µ*m pulses, we find it instructive to display their respective spectral broadenings over several decades in logarithmic scales. In Fig. 8(a), on-axis spectra have been plotted for 800 nm pulses at distances of maximum compression in argon, xenon and air. SPM broadens the pulse spectrum preferentially to the left, i. e., to the red wavelengths at normalized spectral intensities >0.1. This is a signature of the first plasma response that limits the beam collapse [4]. Blueshift owing to self-steepening occurs at lower spectral intensities. This blueshift is more pronounced in air, which we attribute to the combined action of plasma, self-steepening and Raman delay at higher power. In contrast, red-shifts appear limited in 2 *µ*mfilamentation [Fig. 8(b)]. The reason is that the Kerr-induced SPM is early distorted asymmetrically by self-steepening prior to plasma generation, as confirmed by the dash-dotted curve showing the on-axis spectrum with no steepening term. The sharp shocks which mark the trailing edge of the pulse then cause an important blueshift. The resulting spectrum develops a bandwidth significantly broader than that characterizing near-visible pulses.

## 5. Conclusion

In summary, numerical simulations suggest that 2 *µ*m filaments can be compressed down to durations very close to the single optical period, unlike 800 nm filaments whose best compression rates make them reach 2*–*3 optical cycles in gases with moderate ionization potentials <20 eV. At long wavelengths, pulse shrinking in time is justified by an amplified action of self-steepening for saturation intensities remaining comparable with those attained at 800 nm. Pulse compression is accompanied by a prominent spectral broadening, shifted towards the large frequencies. Further improvements of the theoretical model will be necessary to describe with more accuracy what happens when the pulses reach sub-cycle durations. However, all our numerical simulations emphasize the generic compression of 2 µm laser pulses to the single-cycle limit with no particular limitation. The remarkable compression rates achieved with long carrier wavelengths should offer novel perspectives for high-order harmonic generation and production of isolated attosecond pulses.

## Acknowledgments

The author thanks Dr. Stefan Skupin for fruitful discussion and judicious pieces of advice. Numerical simulations have been performed on the computer cluster CCRT at CEA-France.

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