Magneto-hydrodynamics (MHD) provides effective mechanisms for increasing the electron density within high-current pulsed discharges. In the plasma channel the self-generated magnetic field may act to radially focus the electron fluence to near-thermonuclear current densities, with the z-pinch [1, 2] being one of the most prominent examples. In contrast, laser pulse compression [3–5] has traditionally pursued energy concentration along the longitudinal axis rather than radial contraction. In the following we show that inside a light filament, the combination of only three effects, namely diffraction, Kerr self-focusing, and plasma-induced self-defocusing, holds for a radial contraction mechanism acting on the photon fluence. In analogy to the z-pinching in MHD, we call this mechanism self-pinching. This phenomenon gives rise to spatio-temporally inhomogeneous configurations of the optical field, implying strong temporal variations of the beam radius [6, 7]. In contrast to previous explanations (see, e.g., [7–9]) of the self-compression in filaments that indicated a complex interplay of some ten effects, we show that only the above-mentioned three spatial effects suffice for self-compression.

Propagation of short laser pulses in a filament involves numerous linear and nonlinear optical processes that are typically modeled in the framework of a Nonlinear Schrödinger Equation (NLSE). It is quite remarkable that MHD bears a very similar NLSE for the magnetic field, which may give rise to ionospheric filaments and mechanisms analogous to self-pinching [10]. As all these scenarios exhibit a complex interplay of linear and nonlinear processes it is generally difficult to isolate the primary processes leading to the observed phenomena. For the optical case, however, one can compute characteristic lengths of the participating processes [12] to sort out group-velocity dispersion, absorption, Kerr-type self-phase modulation and self-steepening, leaving mainly plasma effects and transverse self-focusing and -defocusing as suspected drivers behind the experimentally observed self-compression. Such analysis, in particular neglection of dispersion, is indicative of vanishing energy flow along the temporal axis of a pulse in the filament. This essentially leaves particle densities and respective fluences as key parameters, similar to the situation in MHD. Let us therefore restrict ourselves to analyzing radial energy flow, for which we use an extended NLSE in cylindrical coordinates (r, t) [13]. This extended NLSE effectively couples the photon density to the electron density ρ. Compared to the full model equations [7], we neglect energy flow along the t-axis and dissipation. These effects have been proven unimportant in gaseous media at low or atmospheric pressure both, in previous theoretical and experimental studies [6, 7]. Consequently, we employ a reduced model considering only Kerr-type self-focusing and plasma defocusing as main dynamic effects during filament formation in gases:

Here, z is the propagation variable, t the retarded time, and ω₀ is the central laser frequency at λ ₀=2πn ₀/k ₀=800 nm. n ₂ is the nonlinear refraction index. Photon densities are described via the complex optical field envelope 𝓔, with I=|𝓔|². The wavelength-dependent critical plasma density is calculated from the Drude model according to ρ_c≡ω ² ₀ m_e ε ₀/q ² _e, where q_e and m_e are electron charge and mass, respectively, ε ₀ is the dielectric constant, c the speed of light, and ρ _nt denotes the neutral density. Plasma generation is driven by the ionization rate W[I], which is suitably described by Perelomov-Popov-Terent’ev (PPT) theory [14]. For our investigations, we use data for argon [7] at atmospheric pressure.

In the following, we search for a field configuration that represents a stationary state in regimes where a Kerr-induced optical collapse is saturated by plasma defocusing. The corresponding temporal intensity profiles that maintain a balance between competing nonlinear effects in every temporal point are derived from a time-dependent variational approach, with the following trial function

The quadratic phase guarantees preservation of continuity equations through self-similar substitutions, and the pulse radius R≡R(z, t) depends on both the longitudinal and temporal variables. For conservative systems preserving the power P(t)≡2π∫∞₀I(t)rdr along z, straightforward algebra provides the virial-type identity [15]

Inserting the trial function (3) with R(z, t)=w(z, t)/√2 being related to the Gaussian spot size w(z, t), one obtains a dynamical equation governing the evolution of the pulse radius R along z [15]. For the derivation of analytical expressions for the plasma term on the r.h.s. of Eq. (4), we approximate the PPT ionization rate by a power law dependence $W\left[I\right]={\sigma }_{N^{*}}{I}^{N^{*}}$, with parameters N*=6.13 and ${\sigma }_{N^{*}}=1.94\times {10}^{-74}{s}^{-1}{\mathrm{cm}}^{2N^{*}}{W}^{-N^{*}}$ fitted to the PPT rate for the intensity range of 80TW/cm². Using I(t)=P(t)/πR²(t), we impose a Gaussian power profile P(t)=P _in exp(-2t ²/t ² _p) with duration t _p and peak input power P _in as a boundary condition, which results in the following integral equation for steady state solutions,

 

Fig. 1. (a) Spectrum of solutions I(t) of Eq. (5). (b) Spatio-temporal representation of the solution of Eq. (5) marked in red [see (a)], obtained by rotating the line segment generated by $R\left(t\right)=\sqrt{P\left(t\right)/\pi I\left(t\right)}$ around the t-axis. Color corresponds to on-axis intensity.

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with P _cr=λ ² ₀/(2πn ₀ n ₂) and $\mu =k_0^2{N}^{*}{\sigma }_{N^{*}}{\rho }_{\mathrm{nt}}/\pi {\rho }_{\text{c}}$.

Equation (5) is basically a generalization of a Volterra-Urysohn integral equation [16], with a kernel depending not only on I(t′) but also on I(t). Using additional simplifying assumptions, steady-state solutions with soliton-like qualities have been previously discussed [13]. Here we solve Eq. (5) without the approximations made in Ref. [13]. Taking into account that the integral term of Eq. (5) is strictly positive, it immediately follows that nontrivial solutions only exist on the temporal interval -t _*<t<t _* where P(t)>P _cr, with $t_{*}{=(\ln \sqrt{P_{\mathrm{in}}}/P_{\mathrm{cr}})}^{1/2}{t}_{\text{p}}$. From a physical point of view, Kerr self-focusing can compensate for diffraction only on this interval, enabling the existence of a stationary state. For computing a stationary solution I(t) of the integral equation, we use the method presented in [17] which combines a Chebyshev approximation of the unknown I(t) with a Clenshaw-Curtis quadrature formula [18] for the integral term. As the laser beam parameters, we choose a ratio P _in/P _cr=2 and a pulse duration $t_{\mathrm{FWHM}}=\sqrt{2\ln {2t}_{\text{p}}}\approx 100 \mathrm{fs},$, leading to t _*≈50 fs. The spectrum of solutions thus obtained is depicted in Fig. 1(a). As 1-P(t)/P_cr vanishes at the boundaries, there exists a continuum of multiple roots. All solutions show a strongly asymmetric temporal shape, with an intense leading subpulse localized at t=-t _* [13] and a minimum [dashed line in Fig. 1(a)] localized near zero delay, followed by a region of rapid intensity increase, suggesting singular behavior of the solutions. Filamentation is known to proceed from a dynamical balance between the Kerr and plasma responses, and a steady-state solution cannot strictly be reached by the physical system. Nevertheless, Eq. (5) provides deep insight into the configuration that the pulse profile tends to achieve in the filamentary regime. The structure of the emerging solutions [Fig. 1(b)] indeed indicates the formation of two areas of high on-axis intensity being separated by an approximately 20 fs wide defocused zone of strongly reduced intensity. While similar double-peaked on-axis intensities have already been observed in numerical simulations and experiments [6, 21–24] many authors considered a parasitic dispersive break-up in bulk media or optical fibers. Despite its superficial similarity, however, such a break-up cannot be exploited for the compression of isolated femtosecond pulses as will be done below. Interestingly, we observe a comparable dynamical behavior as reported for condensed media, where temporal break-up around zero-delay and the subsequent emergence of nonlinear X-waves occurs. These X-waves were recently proposed to constitute attractors of the filament dynamics [19, 20].

For a deeper substantiation of our analytical model, we perform direct numerical simulations using the reduced radially symmetric evolution model Eq. (1) for the envelope of the optical field. The incident field is modeled as a Gaussian in space and time with w ₀ :=w(z=0, t)=2.5mm and identical peak input power and pulse duration as used for the solutions of Eq. (5). The field is focused into the medium with an f=1.5m lens. The result of these simulations can be considered as prototypical for the pulse shaping effect inside filaments. These simulations also demonstrate that spatial effects alone already suffice for filamentary self-compression. As the evolution of the on-axis temporal intensity profile in Fig. 2(a) reveals, filamentary compression always undergoes two distinct phases. Initially, while z approaches the nonlinear focus (z=1.4-1.5 m), a dominant leading peak is observed. When the trailing part of the pulse refocuses in the efficiently ionized zone (ρ _max≈5×10¹⁶cm^-3) a double-spiked structure emerges. This transient double pulse structure confirms the pulse break-up predicted from the analysis of Eq. (5), see Figs. 3(a) and (c), and is compatible with the stationary shapes detailed in Fig. 1(a). Subsequently, only one of the emerging peaks survives and experiences further pulse shaping in the filamentary channel.

 

Fig. 2. (a) Evolution of the on-axis temporal intensity profile along z for the reduced numerical model governed by Eq. (1). (b) Same for the simulation of the full model equations [7].

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At z=1.7m the leading subpulse has already been reduced to a fraction of its original on-axis intensity. This effective attenuation of the leading pulse isolates the trailing pulse that now exhibits a duration t _FWHM=27 fs. The combined split-up and isolation during the first phase therefore already provides an about fourfold compression of the 100 fs input pulses. In the subsequent weakly ionized zone of the filament (z>1.6m), the surviving, trailing subpulse is then subject to additional temporal compression. At z=2.5m our simulations indicate pulses as short as t _FWHM=13 fs [Fig. 3(b)], which agrees favorably with the experimental results in Ref [6]. In contrast to the plasma-mediated self-compression in the strongly ionized zone, compression in the second zone is solely driven by the Kerr nonlinearity (ρ<10¹³cm^-3). With time slices of higher optical powers being able to compensate diffraction by Kerr selffocusing, these portions of the pulse diffract less rapidly than time slices with less optical power. Compared to a linear optical diffraction-ruled scenario, the nonlinear optical effects therefore lead to the formation of a characteristic pinch. This emerging spatial structure is depicted in Fig. 3(d). This clearly sets the self-pinching mechanism apart from numerous previous reports on seemingly similar dynamics.

For an analytic description of temporal compression during further filamentary propagation, we use the dynamical equation for the time dependent beam radius derived from Eqs. (3) and (4) [15], yet neglecting the plasma term. With the initial conditions R(z=z ₀, t)=R ₀ and ∂_zR(z=z ₀, t)≡0 the resulting problem is analytically solvable, and we find $R(z,t)=R_0\sqrt{1+{\left[\left(z-z_0\right)⁄k_0{R}_0^2\right]}^2(1-P\left(t\right)⁄P_{\mathrm{cr}})}$. This equation models the evolution of the plasma-free filamentary channel from z>1.6m, assuming P(t)≤P_cr. The profile P(t) represents the power contained in the filament core region only. For simplicity, we assume here P(t)=P_cr exp(-2t ²/t ² _p), R ₀=100 µm and t_p=23 fs. This corresponds to the duration of the pulse at z ₀=1.7m shown in Fig. 3(a). Resulting characteristic spatio-temporal shapes are shown in Fig. 4, clearly revealing the presence of self-pinching in this Kerr-dominated stage of propagation and the dominant role it plays for on-axis temporal compression.

 

Fig. 3. (a) Pulse sequence illustrating the two-stage self-compression mechanism. Shown are the on-axis intensity profiles for z=1.5m (solid line), z=1.55m (dashed line) and z=1.7m (dashed-dotted line). (b) Self-compressed few-cycle pulse at z=2.5m. (c) Spatiotemporal characteristics of the double-spiked structure at z=1.55 m. (d) Same for the fewcycle pulse at z=2.5m.

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So far our analysis has completely neglected dispersion, self-steepening and losses. To convince ourselves that dissipation and temporal coupling between time slices have only a modifying effect on the discussed self-compression scenario, we pursued full simulations of the filament propagation, including few-cycle corrections and space-time focusing [7]. As shown in Fig. 2(b), minor parameter adjustment, setting w₀=3.5mm and leaving all other laser beam parameters the same value, suffices to see pulse self-compression within the full model equations. Now self-steepening provides a much more effective compression mechanism in the trailing part. However, the comparison of Fig. 2(a) with (b) also reveals that the dynamical behavior changes only slightly upon inclusion of temporal effects. Clearly, the same two-stage compression mechanism is observed as in the reduced model. We therefore conclude that the pulse break-up dynamics in the efficiently ionized zone is already inherent to the reduced dynamical system governed by Eq. (1). Rather than relying on the interplay of self-phase modulation and dispersion as in traditional laser pulse compression, filament self-compression is essentially a spatial effect, conveyed by the interplay of Kerr self-focusing and plasma self-defocusing. This dominance of spatial effects favorably agrees with the spatial replenishment model of Mlejnek et al. [21]. However, our model indicates previously undiscussed consequences on the temporal pulse structure on axis of the filament, leading to the emergence of the pinch-like structure [Fig. 3(d)] that restricts effective self-compression to the spatial center of the filament [7, 25]. Our analysis confirms the existence of a leading subpulse, in the wake of which the short self-compressed pulse is actually shaped during the first stage of filamentary propagation. This leading structure gives rise to a pronounced temporal asymmetry of self-compressed pulses, which is confirmed in experiments [7].

While qualitatively similar break-up processes have frequently been observed in laser filaments, our analysis identifies the spatially induced temporal break-up as a first step for efficient on-axis compression of an isolated pulse. In our case, the leading break-up portion is eventually observed to diffract out and to reduce its intensity, while the trailing pulse can maintain its peak intensity. A subsequent stage, dominated by diffraction and Kerr nonlinearity, serves to further compress the emerging isolated pulse, and may give rise to almost tenfold on-axis pulse compression. The main driver behind this complex scenario is a dynamic interplay between radial effects, namely diffraction, Kerr-type self-focusing, and, exclusively close to the geometric focus, plasma defocusing. The dominance of spatial effects clearly indicates the unavoidability of a pronounced spatio-temporal pinch structure of self-compressed pulses. The frequently observed pedestals in this method are identified as remainders of the suppressed leading pulse from the original split-up. Our analysis also indicates that lower pulse energies <1mJ requiring more nonlinear gases or higher pressures will see an increased influence of dispersive coupling, which can eventually render pulse self-compression difficult to achieve. Higher energies, however, may not see such limitation, opening a perspective for future improvement of few-cycle pulse self-compression schemes.

 

Fig. 4. Sequence of pulses illustrating temporal self-compression due to Kerr-induced spatial self-pinching in the variational model corresponding to (a) z = 1.7m, (b) z = 1.9m, (c) z = 2.1m and (d) z = 2.3m.

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Financial support by the Deutsche Forschungsgemeinschaft, grants DE 1209/1-1 and STE 762/7-1, is gratefully acknowledged. We acknowledge support by the GENCI project N^o x2009106003.