Energy deposition in air by moderately focused femtosecond laser filaments Open Access

Self-focusing and filamentation of high-power femtosecond laser pulses in air have been actively studied since the mid-1990s.^1–5 The growing attention given to this problem is due to the fact that laser filamentation is a prominent manifestation of nonlinear optics and possesses great practical value for various applications, such as higher harmonic generation,⁶ atmospheric LiDAR applications,^7–9 optical energy delivery to the remote targets,^10,11 and remote fuel ignition.^12,13 Self-focusing of optical radiation is realized due to the optical Kerr effect at the peak pulse power exceeding a certain critical value. In this case, a self-induced convex lens is formed in the medium that permanently focuses the optical wave as it propagates. As a result of such nonlinear self-focusing, a narrow high-intensity light beamlet, or a group of beamlets, usually called laser filaments, is formed within the mother beam. During filamentation, pulse spectral composition is significantly enriched as a result of strong self-phase modulation in the nonlinear medium. This is manifested in the formation of broad supercontinuum spectral bands covering in some cases several octaves of initial pulse spectrum.⁶ High intensity optical field inside the filaments causes multiphoton (MPA) and tunnel ionization of medium molecules, which produces plasma channels accompanying laser radiation with a characteristic density of free electrons 10¹⁵–10¹⁸ cm⁻³.² 

Recently, plasma filaments were used to create airborne optical waveguides formed by rarefied regions of air emerging after the passage of a laser filament and persisting for tens of milliseconds depending on the gas dynamic state of the surrounding air.¹⁴ The study of such laser-induced air thermodynamics seems important because it paves the way for the development of remotely generated long-lived virtual waveguiding optical structures, which in the long term can be used for optical communication.^15–17 The efficiency of these gas dynamic structures essentially depends on the amplitude of laser-induced effects, namely, on the pulse energy locally deposited in the gas medium by the filament, because the magnitude and spatial distribution of energy release determine the depth and axial extent of air refractive index perturbations. Therefore, it becomes critical to gain control on the amount of this energy release in the required region of pulse propagation path.

In previous studies, it was found that the amount of pulse energy deposition in a medium depends not only on optical energy but also on pulse temporal duration^15,18,19 as well as on the conditions of external focusing.^13,15,20 In the latter case, it was shown that in general, the laser energy deposition is larger as the pulse is focused more tightly (at fixed pulse energy). However, at very sharp pulse focusing (relative aperture f/3) when a propagation regime close to optical breakdown is realized, the value of the deposited optical energy on the contrary is reduced, which was attributed to strong plasma defocusing of laser pulse in the nonlinear focal area¹⁵ due to the formation of a “superfilament” structure.²¹ 

The present work is aimed at the determination of energy deposition value in air by focused filamentation of high-power femtosecond laser pulses. Unlike previously published works, in our study we consider the range of moderate pulse focusing with a relative aperture less than f/40 and two spectral bands of laser radiation with the carrier at 740 and 470 nm. By means of numerical simulations, we analyze the physical reasons leading to the dependencies observed in the experiment. It turns out that even under these conditions, one can distinguish a certain region of focal distances, when laser pulse energy losses for femtosecond plasma generation are significantly reduced and optical energy deposition in air considerably drops. We attribute this drop with the competition of nonlinear self-focusing and linear geometric divergence of initially focused laser radiation.

In our experiments, the optical radiation delivered from two separate titanium–sapphire laser oscillators is used, generating pulses with central wavelengths of 740 and 470 nm (second harmonic). The laser pulse energy can vary from 2 to 4 mJ using a diffraction attenuator. The pulse duration for both wavelengths is t_p = 100 fs. External focusing of laser radiation is performed by a spherical mirror and silica lenses with different focal distances f. Usually, the filamentation of a laser pulse occurred near the geometrical focus with the formation of an extended filamentary plasma channel. The side luminescence of this plasma channel is projected with a collecting lens onto a CCD matrix, and then the images are digitized and processed on a computer.

Representative images of plasma fluorescence in optical filament created by focused femtosecond laser pulses in air are shown in Fig. 1(a) for different focusing conditions. In the following, it is practical to use the dimensionless parameter of the numerical aperture, NA = D/2f (here D is the laser beam diameter measured at the half-height of the energy density distribution). Note, because the real shape of the beam cross section was slightly elliptical, D is considered as the effective beam diameter.

It is worth noting that as the focal distance decreases, the plasma channel becomes shorter, but its lateral width increases and the luminal emission becomes more intensive. With increasing pulse energy, the integral fluorescence of the plasma filament B increases according to a certain power law, B ∼ E^b, and the growth increment b depends nonmonotonically on the degree of focusing [Fig. 1(b)]. The maximal growth rate of filament luminescence with the pulse energy b ≈ 2.33 is observed for a loosely and sharply focused pulse, NA < 3 × 10⁻³ and NA = 9 × 10⁻³, while a lower rate, b ≈ 1.97, is characteristic of moderate focusing, NA = 4.5 × 10⁻³.

More clearly, the influence of focusing conditions is demonstrated in Figs. 1(c) and 1(d), where the integral plasma fluorescence is plotted for two pulse energies and different NA values. As seen, in the dependencies, one can conditionally distinguish the range of numerical apertures showing a marked change in the integral filament fluorescence. This NA range corresponds to the region of moderate pulse focusing. Within this range, the minimum plasma emission is observed at NA = 4.5 × 10⁻³ for a carrier of 740 nm, while for shorter-wavelength radiation at 470 nm, the minimum integral fluorescence corresponds to a tighter focusing, NA = 7 × 10⁻³.

It is worth emphasizing that these features have a certain universality in the sense that the shape of the curves is preserved when the pulse energy is changed. This suggests that in this range of focusing, the predominant influence in the development of filamentation is the geometric compression of laser beam due to the initial curvature of the phase front, rather than self-focusing due to the Kerr effect. This region of NA values can be conditionally associated with the, so-called, transition interval between the quasi-linear and quasi-nonlinear pulse focusing in medium as previously reported in Refs. 23 and 24. Within this focusing interval, the partial contributions to the optical wave phase due to the Kerr effect and plasma generation are compared, and the transition from the linear to the nonlinear focusing mode takes place. It was also claimed that this transition region of pulse focused propagation depends weakly on the laser pulse parameters.

Importantly, for an ultrashort laser pulse, the loss of energy deposited in air during filamentation is mainly due to the combined multiphoton/tunnel ionization in the propagation medium, while only a few amounts of pulse energy are spent for maintaining the filament plasma through the inverse bremsstrahlung absorption in the formed free-electron gas. Strictly, both these loss mechanisms are coupled to the pulse dynamics and, therefore, cannot be treated as independent of each other. Experimentally, is hard to directly separate the outcome of each of these processes. However, some conditional conclusion can be drawn about the relative strength of these mechanisms based on the filament thermodynamics, i.e., by considering the acoustic pressure after filament propagation in the medium. Indeed, the main source for acoustic wave is the pressure jump due to medium temperature increase through plasma thermalization. The latter process is governed mainly by plasma electron heating via inverse bremsstrahlung absorption, while plasma emission in the visible (UV and near-IR also) region depends almost solely on free-electron concentration and not the plasma temperature. Thus, by varying, e.g., the pulse energy or pulse duration, one can simultaneously measure and compare the rate of variation in plasma emission and acoustics and conditionally assess the relative influence of these processes in filamentation dynamics. However, such investigation is not the subject of this work.

The variation in linear plasma density n_ez along the propagation distance of optical pulse with different energies is shown in Fig. 2(a) for several values of the numerical aperture NA corresponding to conventionally weak (NA = 0.0015), moderate (NA = 0.005), and strong focusing (NA = 0.013). On the horizontal axis in this figure is plotted the reduced distance, $zf=(z−f)/Lf$⁠, showing the relative spatial remoteness from the geometric focus expressed in focal Rayleigh length, $Lf=1/k(NA)2$⁠. It should be noted that in Fig. 2(a), we intentionally choose different NA values as those in the experimental data in Fig. 1(b) to provide a more clear demonstration of different pulse filamentation regimes.

It can be seen that moderate beam focusing, in contrast to other extreme values of NA, demonstrates more rapid decrease in n_ez values after the focus. At the same time, under conditions of weak focusing, high values of electron density persist on the longest spatial scale, and as a result, the total pulse energy losses ΔE [Fig. 2(b)] during the filamentation increase sharply. These losses are associated not only with increase in the free electron concentration in the filament but also with the active escape of wave energy through the non-radiation channel to heating of the already formed plasma, which subsequently leads to heating and hydrodynamic expansion of air in the beam channel.¹⁴ 

For conditionally tight focusing, NA > 0.01, the increase in pulse energy losses ΔE is no longer due to the absorption in plasma but to an increase in the density of free electrons produced via medium ionization. As shown in Fig. 2(c), the peak values of the electron concentration n_em in this range of pulse focusing can reach 10¹⁸ cm⁻³, which is close to the density of neutral air molecules at normal pressure. The filament length L is reduced in this case. Obviously, in general, exactly the opposite tendencies to the growth of n_em and reduction of L lead to the appearance of a dip in the dependence of the total number of plasma electrons N_e on the focusing strength, as shown in Fig. 2(d). Indeed, this claim is confirmed if one multiplies the main plasma parameters, (n_em⋅L), and analyzes the dependence of this product on NA. The obtained dependence will also exhibit a dip in the range of moderate pulse focusing (not shown). Moreover, for the laser wavelength of 470 nm this, decrease in energy deposition becomes even more pronounced due to a sharper dependence of the medium ionization on the pulse intensity.

It should be noted that overall, our theoretical simulations only partially reproduce the experimental data. However, a specific feature persists both in the experiment and theory: the appearance of a dip in the dependence of the total number of free electrons in Fig. 2(d) and integral plasma fluorescence in Fig. 1(c) on the focusing strength in a certain range of NA. This evidences that our theory correctly interprets the experimental findings and confirms the physical consistency of expression (1).

Remarkably, in our paper we focus only on the influence of pulse spatial focusing on the amount of pulse energy deposition in a medium, while the effect of laser pulse duration on filament energy deposition remains beyond our consideration. As mentioned above, this topic was previously investigated in several works.^15,18,19 Usually, the authors fixed the input laser pulse energy and varied the pulse duration via wave phase modulation (chirping) from hundreds of femtoseconds to hundreds of picoseconds. As reported, this pulse lengthening leads to better plasma spatial localization but results in decreasing optical energy deposition in the medium. Clearly, by increasing the pulse duration one obtains lower peak pulse power provided that the pulse energy is fixed. Thus, the filamentation becomes more regular and the plasma distribution turns out to be less intensive and more homogeneous along the filament. In addition, for long pulses the role of plasma absorption (through the inverse bremsstrahlung mechanism) prevails over the MPA, which is responsible for new free-electron production. This results in higher Joule heating of the propagation medium but lower plasma density production by the filament. All these processes decrease the intensity of fluorescence originating from the filament plasma and, therefore, experimentally one should measure less plasma emission. This can lead to a seeming but incorrect conclusion¹⁵ that upon pulse lengthening the deposited energy value decreases. However, according to our simulation (not shown here), by fixing, not the input pulse energy but, the input peak power, one obtains a rather opposite result demonstrating an increase in energy deposition in absolute units (mJ) with the pulse duration. In summary, the optical energy deposition during pulse filamentation is undoubtedly sensitive to the pulse temporal length. However, it should be understood that the effect from pulse duration variation depends on which pulse parameter is fixed in the experiment.

In conclusion, we consider the influence of external spatial focusing of a femtosecond laser pulse during its filamentation in air on the amount of optical energy deposited in the propagation medium as a result of laser ionization and plasma generation. The results of our laboratory experiments and theoretical simulations show that in the range of pulse numerical aperture NA, approximately from 0.003 to 0.007, a pronounced decrease (about 50%) of filament energy transferred into air is observed. This focusing range is in close correlation with the discovered earlier transition region between the linear and nonlinear focused propagation of high-power laser pulse,²⁴ which is governed mainly by the action of geometric focusing or Kerr self-compression, respectively. The results obtained can serve as, for example, a roadmap for constructing a gradient dynamic optical waveguide in air after the passage of a laser filament or for optimizing the conditions of remote ignition of combustible gases by femtosecond filament plasma.

Russian Science Foundation (24-12-00056).

The authors have no conflicts to disclose.

Yury E. Geints: Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (lead); Project administration (lead); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Ilya Yu. Geints: Investigation (equal); Software (equal). Yakov V. Grudtsyn: Investigation (equal); Methodology (equal). Andrey V. Koribut: Data curation (equal); Investigation (equal); Validation (equal). Dmitrii Vladimirovich Pushkarev: Data curation (equal); Investigation (equal); Methodology (equal); Visualization (equal). Georgy E. Rizaev: Data curation (equal); Investigation (equal); Validation (equal). Leonid V. Seleznev: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.