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The filamentation of intense femtosecond laser pulses in transparent media is an area of increasing research activity, as it combines exciting potential applications with fundamental nonlinear optical physics [1]. The filament formation process is initiated when the electric-field strength of a propagating laser pulse is sufficient to nonlinearly distort the electron clouds of the constituent atoms, leading to an ensemble-averaged dipole moment nonlinearly increasing with field strength. This nonlinear refractive index lens propagates with the pulse. Once the laser pulse peak power exceeds a critical value, i.e., $P\ >\ P_{\rm cr}$, the self-induced lens overcomes diffraction and focuses the beam, leading to plasma generation when the gas ionization intensity threshold is exceeded. According to the “standard model” of filamentation (more on that later), the on-axis concentration of free electrons then defocuses the beam, and the dynamic interplay between self-focusing and defocusing leads to self-sustained propagation of a tightly radially confined high-intensity region (the “core”) accompanied by electron density tracks over distances greatly exceeding the optical Rayleigh range: The tracks can extend from millimeters to hundreds of meters, depending on the medium and laser parameters. The phenomenon is ubiquitous for femtosecond pulses at the millijoule level in gases and at the microjoule level in liquids and solids [1], where, in both cases, $P\ >\ P_{\rm cr}$. “ Filament” describes both the extended propagation of the intense optical core and the residual electron density track, which are typically $\sim\!100\ \mu\hbox{m}$ in diameter [2]. One of the more spectacular effects accompanying filamentation is supercontinuum generation, a coherent ultra-broadband optical beam copropagating with the filament [3], resulting from ultrafast nonlinear phase evolution of the pulse. This light can be used for white-light LIDAR of the atmosphere at large distance scales [4] or generated in sealed cells, where it can be accompanied by self-compression and spatial mode cleaning of the filamenting pulse [5], [6], [7], [8].

The long ionization tracks left by filaments have stimulated work in guided high voltage discharges [9] and the related dream of laser filament-guided atmospheric lightning [10]. However, both the low electron density in typical extended filaments (< few $10^{16}\ \hbox{cm}^{-3}$ [2]) and the nanosecond plasma lifetime [11] have made filament-guided discharges much longer than $\sim$1 m difficult. Even at a meter length, such discharges had not been reproducible or uniform. Recently, meter long, uniform discharges in air sustained by a compact low jitter tesla coil have been generated at ENSTA [12]. The same group has demonstrated that laser filaments can coax high voltage discharges to occur far from their normal path, even forcing them to follow right angle paths in free space [13]. Even longer discharges await a more continuous filament with a higher and much longer-lived electron density. The nature of filamentation, where the time slices of the pulse generating ionization in the core refract while the beam periphery of the pulse self-focuses into the core, results in temporal pulse splitting and axially nonuniform plasma channels [2], [14]. One concept for making channels longer and more uniform is to exploit the quantum rotational response of molecular gases. By repetitively kicking the molecular rotor with a sequence of pulses spaced at the alignment revival intervals, enhancements in the gas refractive index can compete strongly with the defocusing effect of the plasma to increase and extend the electron density. A preliminary two-pulse experiment in air has shown enhanced electron density and filament lengthening, as well as optical pulse shaping [15]. Related simulations [16] for a sequence of six appropriately timed pulses show a very strong enhancement of electron density and filament length, with “molecular lens” focusing dominating free-electron defocusing, with the filamenting pulse quasi-confined as if by a guiding structure (see Fig. 1).

Fig. 1. (Top) Filament electron density profile extension measured as a function of two-pulse delay. The extension is sensitive to 10-fs delays with respect to the molecular alignment revival [15]. (Bottom) Simulation of electron density profile lengthening and enhancement by six sequential pulses timed to successive molecular rotational revivals [16].

Since the first demonstration of femtosecond filamentation in air [1], filaments typically have been generated using millijoule-level $\lambda = 0.8\ \mu\hbox{m}$ pulses from Ti:Sapphire lasers. Recently, the development of a new high-power midinfrared (MIR) laser source at the Vienna University of Technology has made possible the generation of filaments using $\lambda = 3.9\ \mu\hbox{m}$ [17]. Since $P_{\rm cr} \propto \lambda^{2}/N$, where $N$ is the atomic/molecular density of the propagation medium, at 1 atm, this scaling demands MIR pulse energy exceeding the laser output. A high-pressure gas cell was used to decrease $P_{\rm cr}$ to allow filamentation with 7-mJ pulses. Supercontinuum generation from these MIR filaments produced spectra spanning an enormous range of 0.35 $\mu\hbox{m}$–5 $\mu\hbox{m}$. At a fixed gas density, an advantage of MIR over $\lambda = 0.8\ \mu\hbox{m}$ filaments for some applications is lower electron density, a wider diameter core, and longer propagation distances. These features were crucial in recent experiments where the same MIR laser driver, propagating in a hollow high-pressure gas-filled capillary, generated coherent high harmonic radiation up to 1.6 keV [18]. Simulations indicate that filamentary propagation inside the capillary is responsible for spatio-temporal compression and self-confinement of the driver pulse, increasing the conversion efficiency to coherent keV X-rays.

Femtosecond filaments in gases are a burgeoning source of coherent THz radiation with clear advantages over other laser-driven sources in biased semiconductors or nonlinear crystals. Most importantly, filaments allow very high pump pulse intensities without permanent material damage. The free-electron current generated during laser ionization and driven by the pulse envelope is in fact the primary THz source [19]. Filaments also allow generation of THz at locations distant from the source, an important feature for applications such as remote sensing of complex molecules. THz generation in filaments is becoming increasingly finely controlled. Complex control of the THz polarization state has been demonstrated using external electrodes to modify the laser-driven electron trajectories in the filament plasma [20], and the detailed THz waveform has been controlled by the carrier envelope phase of the filamenting pulse [21]. THz generation in filaments by two-color laser fields (typically the fundamental and second harmonics of Ti:Sapphire) has been found to be especially efficient, allowing even stronger laser envelope-driven electron currents. Despite the gains, THz yields are still low, with laser-to-THz conversion efficiencies of $\sim\! \!10^{-4} - 10^{-3}$. Recently, phase matching in two-color filament-based THz generation has been demonstrated [22], with the yield scaling linearly with plasma length, suggesting that further extending filaments is a promising scheme for THz generation.

Another concept for radiation generation at a distance, to be used for remote sensing applications, is a filament-initiated $\hbox{N}_{2}$ molecular ultraviolet laser in the atmosphere [23], [24]. The filament is envisioned to weakly ionize the air to $\sim\!\!10^{16}\ \hbox{cm}^{-3}$. A copropagating “heater” pulse heats these electrons, which then pump the molecular transition. Preliminary experiments are taking place using a $\lambda = 0.8\ \mu\hbox{m}$ filament and a $\lambda = 1.064\ \mu\hbox{m}$ heater pulse [25]. Meanwhile, $\hbox{N}_{2}$ lasing has been demonstrated in a high-pressure (6-atm) N2/Ar gas cell driven only by a $\lambda = 3.9\ \mu\hbox{m}$ laser pulse [26], with the laser transition upper state populated via a different mechanism than in [23] and [24]. A different remote sensing approach, with molecular specificity, has been achieved by exploiting the extreme bandwidths and short pulses generated in filaments to impulsively pump characteristic vibrational states and then probe them with a narrower-bandwidth nonfilamenting pulse to generate a stimulated Raman spectrum for analysis [27]. The same group also directly measured the extreme optical pulse shortening, which takes place inside a filament core [28].

Fundamental processes in filaments have also recently been investigated, including loss of phase of upon collapse of single and interacting filaments [29], coherent nonlinear interactions among multiple filaments in glass [30], energy exchange between crossing filaments [31], [32], and polarization state stability of filamenting optical pulses [33]. Meanwhile, the debate triggered by [34], [35], [36] continued as to whether a high-order (and negative-going) Kerr shift of the refractive index at high fields is important for filamentation or whether it even exists as a fundamental atomic response to high laser fields. As the optical field and the atoms in the filament core “live” right at the ionization threshold, this is a matter of practical and fundamental interest. A significant amount of work already had been done, in which the effects on beam propagation and harmonic generation of a possible high order Kerr effect were assessed [37]. But recent work directly measuring the atomic and molecular nonlinearities at high laser fields, using single-shot spectral interferometry, appears to settle the matter: the quadratic field strength scaling of the Kerr contribution to the refractive index, in a wide range of gases, is measured to persist all the way to the ionization threshold, well beyond the expected limits of perturbation theory [38], [39]. There is no high-order Kerr effect at 800 nm, and the negative polarizability needed to offset the nonlinear self-focusing is supplied by free electrons. The so-called “standard model” of filamentation is correct. The same measurement technique has also made possible new and accurate measurements of the nonlinear refractive index and polarizability anisotropy for a range of atomic and molecular gases [40]. This is important for realistic propagation simulations [16].

## Footnotes

Corresponding author: H. M. Milchberg (e-mail: milch@umd.edu).

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