Transition from linear- to nonlinear-focusing regime of laser filament plasma dynamics

A laser filament in air is a dynamic structure consisting of a nearly non-diffracting beam with a high intensity core, clamped to ∼1.8 × 10¹³ W/cm² (Ref. 1) surrounded by an energy reservoir, leaving a plasma channel in its wake. Filament formation occurs as a result of the nonlinear propagation of ultrashort laser pulses, usually of femtosecond duration. When the laser peak power is above a particular threshold (∼10 GW in air at 800 nm, measured experimentally), the beam self-focusing overcomes diffraction and collapses into a filament due to a dynamic balance between Kerr self-focusing and the defocusing effect of the plasma created along its propagation.¹ The unique properties of laser filaments, such as their diffraction-balanced long-range propagation, the intensity clamping, plasma channel formation, supercontinuum generation, and THz radiation, make them ideal for use in a wide range of applications.^2–10 

During filamentation, photoionization of the medium produces a plasma column with densities ∼10¹⁶ cm⁻³, which can dynamically extend for several meters in length. The weakly ionized plasma channel is crucial to many of the applications for filamentation, most notably for guiding of long wavelength electromagnetic waves^6–8 and inducing and guiding electrical discharges.^9,10 The efficacy of these applications depends upon the filament plasma properties, particularly the temporal and spatial distribution of the plasma electron density, which also defines the bandwidth supported by the plasma waveguides.¹¹ Experiments conducted to develop these applications are performed at varying energies and pulse durations, frequently with lens-assisted focusing. These initial conditions are known to markedly influence the properties of the plasma generated by the filamentation process, and thus the applicability of the results to other systems. For example, the NA condition used to facilitate filamentation may result in a beam in which geometrical focusing and plasma effects exceed that due to Kerr nonlinearity, producing a filament with higher than typical plasma densities and spectral broadening primarily towards the visible.²⁸ Measurements performed in this linear focusing regime may have limited relevance to applications that require long-distance propagation. In this paper, we investigate directly the interplay between the initial laser conditions (energy and pulse duration) and the NA of the focusing optics used to initiate filamentation. Through spatial and temporal measurements of the electron density in a single filament, we differentiate between the linear and non-linear filamentation regimes.

Several methods have been used to characterize the plasma in air.^12–17 The interferometric method^17–20 is a standard technique, which like the holographic method^21–23 requires no secondary means of calibration and provides sufficient sensitivity for the low densities in filament plasma. In this approach, a similar format to a pump-probe scheme is used. The filamenting beam modifies the refractive index of the air due to the plasma formation, according to the following equation:

where $n2$ is the nonlinear refractive index, $ρ$ is the electron density, and $ρc$ is the critical electron density. The secondary beam (probe) then traverses the perturbed region, accumulating a phase shift, which can be measured by an interferometer. With a small angle of interaction between the probe and the filamenting beam, interferometry enables detection of the low plasma densities in a filament, providing high resolution axial, radial, and temporal information about the plasma, without the need for additional calibration.

These experiments were conducted using the MTFL (Multi-Terawatt Filamentation Laser) facility at the University of Central Florida, which is capable of delivering pulses with energy up to 500 mJ, a 1/e² half-width of 11.9 mm, and a pulse duration of 45 fs (full width at half-maximum, FWHM) at 800 nm and a 10 Hz repetition rate. A beam splitter (BS) was used to separate the beam between the filamenting and the probe beams in a 9:1 ratio (Fig. 1). Additional neutral density filters in the probe beam path were used in order to lower its energy (<12 μJ) and thus ensure it would not interact with the medium. In order to guarantee single filament generation, the laser in these experiments was operated at a power level just above the critical power for filamentation and below that for multi-filamentation. This meant operating the laser in these experiments with energies in the range of 0.8–5 mJ per pulse. The beam was then focused with a lens to create plasma and induce a gradient of index of refraction [Eq. (1)]. A beam profiler, consisting of a series of wedges at grazing incidence and an imaging lens,²⁴ was used to measure the focused beam profile. Prior to interaction, the probe beam was spatially cleaned to enhance sensitivity to minute phase perturbations, on the order of milliradians. A shallow angle (⁠$θ$ < 6°) was introduced between the probe and the focused beam in order to increase the interaction length between the two and thus the sensitivity of the measurements. A variable delay line in the probe beam path was used to change the relative arrival time of the probe to the interaction region in order to provide temporal information about the plasma dynamics. For each experiment performed, the delay was changed from approximately $−$50 ps to 300 ps in increments of $≥$6.7 ps relative to filament creation. Since the perturbation was one order of magnitude smaller in diameter than the probe, only a portion of the probe beam profile was modified as it propagated through the perturbed air. A folded wavefront interferometer was then used to determine the phase shift experienced by the probe beam after traversing the plasma channel induced by the focused beam. The region of interest was relay imaged by a pair of lenses through the interferometer to a CCD, resulting in a spatial resolution of 6.74 μm/pixel. For each delay, 200 interferograms as well as 10 background images (interferograms taken with the focused beam blocked) were captured. The interferometer had a phase sensitivity of 9 mrad, which corresponds to a minimum detectable (2σ) electron density of 1.4 × 10¹⁵ cm⁻³, assuming a 1° crossing geometry and 100 μm diameter plasma. The beam profiler was used to confirm that the middle of the plasma channel was imaged at the focal plane of the interferometer imaging system.

Figure 2 represents the algorithm used to extract the spatial and temporal dynamics of the plasma. A sample interferogram is shown in Fig. 2(a), with a visible fringe shift due to interaction with the plasma. Figure 2(b) shows the extracted phase from the raw interferogram using Fourier analysis.²⁵ A single vertical lineout of the phase corresponding to where the filament is in focus is then selected and averaged over 200 images for each delay [Fig. 2(c)]. The radial profile of the change in refractive index induced by the plasma is obtained through Abel inversion²⁶ of each averaged phase lineout. A factor of $sin(θ)$ must be introduced to correct for the crossing angle between the focused beam and the probe beam.²⁷ The electron density [Fig. 2(d)] is then obtained by multiplying the corrected refractive index change by a factor of $−2ρc$ [Eq. (1)].

The decay of the on-axis plasma density was simulated using a similar model to that generally used for filament plasma decay.^17,31–33 The simulation takes into account the rate equations for three-body and dissociative electron recombination with O₂⁺ and dissociative recombination with the following ions: O₂⁺, O₄⁺, O₂⁺N₂, and O₂⁺H₂O. Of the many reactions involved in non-equilibrium plasma chemistry in air,³⁴ the code incorporates twelve electron recombination reactions demonstrated to be primarily responsible for the decay within the first 2 ns.¹⁷ The peak electron density measured experimentally was input as the initial value in the simulation. An equivalent concentration of O₂⁺ ions was input, while the initial presence of the other ions was considered negligible. A humidity of 50% was assumed, in order to determine the remaining concentration of neutral molecules (O, O₂, N₂, and H₂O) in the air. Since the rate coefficients for the reactions are highly dependent on the electron temperature, the electron energy conservation equation is solved simultaneously, taking into account the effect of electron energy relaxation in collisions and recombination heating. The initial electron temperature was set to 3 eV, which corresponds to the ponderomotive energy of an electron in the laser field, assuming an intensity of 5 × 10¹³ W/cm².³⁵ 

Three external focusing conditions were examined using lenses with focal lengths f = 40 cm, 1 m, and 2 m corresponding to numerical apertures of 0.03, 0.012, and 0.006, respectively, to cover the transition from the linear to nonlinear focusing regime. There is known to be a difference in filament formation between tightly and loosely focused beams, where loose focusing is defined as NAs <0.004 for an ideal beam (M² = 1).²⁸ For each NA tested, the laser pulse duration was fixed at 65 fs, with an input energy of 3 mJ. Additional measurements were taken for the highest NA case, at energies of 0.8 and 5 mJ, and for the lowest NA, at an energy of 5 mJ. Thus, the effects of changing energy on plasma dynamics, such as electron density clamping, could be studied under different NA conditions. The beam profiler was used in each case to ensure single focused beam/filament formation.

The measurements of the filament electron density and scale reveal some striking differences between the linear and nonlinear regimes of filament formation. The temporal evolution of the on-axis electron densities for the highest NA case (f = 40 cm) and different input energies are displayed in Fig. 3(a). Zero time on the horizontal axes corresponds to when the focused and probe beams overlap in time, while negative time indicates the probe preceding the focused beam. The peak electron densities are in the range of 10¹⁷–10¹⁸ cm⁻³, depending on the input energy (see Table I) as expected for plasma induction dominated by photoionization, a highly intensity dependent process. The rate for multiphoton ionization (MPI) is proportional to I^K, where I is the laser intensity and K is the number of photons that must be simultaneously absorbed in order to ionize the medium. Despite this change in the peak density, the plasma lifetime remains at about 2 ns. The plasma lifetime, defined here as the time over which the degree of ionization remains above $2×104$ (⁠$ρe≥5×1015$ cm⁻³), was determined from the simulation (solid lines in Figs. 3 and 5). The plasma diameter, averaged over the first 100 ps, also notably increases with energy, from 36 to 82 μm for energies from 0.8 to 5 mJ, respectively. The FWHM diameter of each focused beam is approximately 16 μm, as measured with the beam profiler. Based on these measurements, the focused beam intensity is estimated to be on the order of 10¹⁴–10¹⁵ W/cm², depending on the focused pulse temporal profile.

In contrast, the on-axis electron densities for the low NA case (f = 2 m) do not demonstrate any increase with the change in energy [Fig. 3(b)]. The densities are on the order of 10¹⁶ cm⁻³ with a constant plasma diameter of ∼43 μm, averaged over the first 100 ps and lifetime of just over 1 ns (see Table I). The focused beam FWHM diameters, measured with the beam profiler, were fixed at approximately 110 μm. From the profiles, the intensity can be estimated to be on the order of 10¹³–10¹⁴ W/cm², depending on the temporal profile of the focused beam.

There is excellent agreement for all datasets between the experiments and the simulation. From the simulation, the heating due to dissociative recombination in the electron energy conservation equation proved to be negligible, though the energy from three-body recombination was vital. All of the positive ions considered (O₂⁺, O₂⁺N₂, O₄⁺, and O₂⁺H₂O) were required for modeling of the decay to be accurate. A simulation involving only O₂⁺ would result in a 76% difference in the final electron density after only 1 ns.

The results demonstrate two clear NA regimes. High NA conditions fall in the linear focusing regime in which geometric focusing is the dominant focusing mechanism, while low NA conditions fall in the nonlinear focusing regime, in which Kerr self-focusing dominates.²⁸ Additional input energy, in the high NA case, leads to denser plasma but not in the low NA case (Fig. 5). Thus, plasma density clamping, which arises from intensity clamping, occurs for the lower NA condition but not for the higher NA case. Also, a difference between the beam size and shape is noted between the two regimes (see Fig. 4), with smaller cores and larger reservoir-to-core ratios in the high NA case. While the ratio between the plasma and beam diameter is a simple factor of $8$ for the low NA regime (corresponding to K ∼ 8), this factor fails with higher NAs. The ratio between the plasma and beam diameter indicates that MPI is the dominant ionization mechanism in the low NA regime but is insufficient to describe plasma formation in the high NA case. According to Ilkov et al., a Keldysh parameter of less than 0.5 falls in the tunneling ionization regime²⁹ corresponding to an intensity of at least $4×1014$ W/cm² under our experimental conditions. This implies that tunneling along with double ionization³⁰ should be considered in the higher NA regime. The contrast between the low and high NA cases is highlighted in Fig. 5. As the NA increases by a factor of 5 with a fixed energy and pulse duration, the peak density increases by two orders of magnitude and the lifetime doubles.

This paper reports the influence of input energy and external focusing on the laser produced plasma density, diameter, and lifetime in the linear and non-linear focusing regimes for single filament formation. The results substantiate previous distinctions between these two regimes.²⁸ Intensity clamping and thus electron density clamping do not occur in the linear focusing (or high NA) regime. Under high NA conditions, the peak electron density and plasma channel diameter increased with energy, reaching 1.5 × 10¹⁸ cm⁻³ and 80 μm, respectively, with high peak intensities. Under low NA conditions, the peak density was on the order of 10¹⁶ cm⁻³ with order of magnitude lower intensities and a plasma diameter of 43 μm, which remained constant with the increase in energy. The time over which the plasma density remained above 5 × 10¹⁵ cm⁻³ rose from 1 to 2 ns with the increase in NA, though little change was seen with energy. While MPI is sufficient to describe the density and diameter in the low NA case, it is insufficient for the higher NA conditions, at which point other ionization processes contribute. Supplementary studies of the electron energy distribution function produced by the ultrafast ionization process during propagation are needed to determine which additional process could be considered. These results demonstrate that conclusions drawn from experiments performed in the linear regime are not necessarily applicable to the long range propagation of filaments, and vice versa.

The authors wish to acknowledge funding support from HEL-JTO MRI, ARO MURI “Light Filamentation Science,” and the State of Florida.