Femtosecond filamentation is a self-organization phenomenon during which an ultrashort high-power laser stays confined in a very small channel over very long distances. Ultimately, however, the finite energy contained in a filament is dissipated because of losses originated from ionization, limiting thereby the filament length. In other words, ionization represents a fundamental limitation in remote applications where long-ranged filaments are required. In this paper, a low-loss Kerr-driven optical filament in krypton gas is experimentally reported in the ultraviolet. A three-photon resonantly enhanced quintic nonlinearity is identified as the underlying physical mechanism responsible for intensity saturation during the filamentation process, while ionization plays only a minor role. The resonant nature of the process creates also conducive conditions, i.e., a significant population inversion, for forward and backward infrared lasing. Preliminary experimental results suggest that such lasing emission takes place. The reported resonantly enhanced filaments are one order of magnitude longer than their off-resonant counterparts. The resonance is also accompanied by a large decrease of both ionization and nonlinear optical losses. The experimental findings are supported by ab initio quantum calculations describing the atomic optical response. Beyond its theoretical interest, resonantly enhanced filamentation could benefit all applications deriving from the filamentation process. For instance, the extension of this work to molecular gases such as oxygen and nitrogen could lead to numerous atmospheric applications such as nonlinear spectroscopy, remote sensing, and lightning protection, in which the transport of high energies over long distances is of prime importance.
© 2017 Optical Society of America
Laser filamentation [1–3] denotes the property of an ultrashort and high-power laser to self-organize in a very small structure able to transport very high intensities over distances far longer than those allowed by linear optics laws. The competition between the Kerr effect, responsible for beam self-focusing, and plasma generation (i.e., ionization), responsible for beam defocusing, was rapidly identified as the paradigm describing the filamentation process in gasses, even if other processes such as vectorial effects  or cascaded third-order nonlinearities  can, in some cases, arrest the collapse in place of ionization. The unique self-guiding characteristic of filaments has been advantageously exploited in numerous remote applications  in which the transport of high energies is of prime importance, such as nonlinear spectroscopy, remote sensing, and lightning protection, to cite a few. So far, almost all studies dealing with filamentation have been performed with 800 nm laser pulses, mainly because almost all lasers able to supply enough power to trigger the filamentation process emit close to this wavelength. In fact, only a few works [7–9] have studied the filamentation process in the case of a laser operating at other wavelengths, mainly because of the lack of appropriate laser sources. Apart from 800 nm laser pulses, the filamentation process in gaseous media has been mainly studied in the case of lasers operating at harmonic frequencies of the Ti:sapphire lasers, i.e., at 400 nm  or 266 nm . Some other experiments have also demonstrated filamentation in the case of a 3.9 μm mid-infrared optical parametric chirped-pulse-amplified laser [12–14], a 248 nm KrF laser , and more recently, a 1.03 μm picosecond laser . In fact, the laser wavelength dependence of the filamentation process has not yet been studied much. In this paper, it is experimentally shown that filamentation can be advantageously enhanced if the laser central wavelength is tuned close to a resonance with a multiphoton transition involving the ground and a bound excited state. In particular, it is shown that tuning the central wavelength of the laser on a three-photon electronic resonance of krypton  at leads to a tenfold increase of the filament length, accompanied by a strong decrease of the nonlinear losses experienced by the filament and a decrease by an order of magnitude of ionization. The role of the bound excited states is confirmed by the observation of their forward and backward lasing. The resonant nature of the process is also supported by ab initio calculations reproducing the optical response of the atom. In particular, the quantum model pinpoints the resonantly-enhanced fifth-order nonlinear susceptibility as responsible, in this case, for the intensity clamping of the filament in place of ionization.
2. EXPERIMENTAL RESULTS
Filaments are produced in a 1.5 m long cell filled with krypton at a 6 bar pressure with a femtosecond tunable ultraviolet (UV) source focused with a spherical aluminum mirror. The UV pulse is produced by frequency quadrupling the output of a noncollinear parametric amplifier operating in the 1100–2500 nm spectral range. The latter is pumped by a 100 fs, 800 nm, 1 kHz, 2 mJ chirped-pulse-amplified laser. The energy of the UV pulse before the cell is set at 30 μJ in the whole wavelength range studied during the experiment (290–315 nm). The pulse duration being estimated to be 50 fs, the peak power is , i.e., more than ten times the critical power at the considered pressure. The beam size before the lens is approximatively 4 mm full width at half-maximum (FWHM), and the beam quality factor is less than 1.1. The plasma fluorescence produced by the filament was imaged on the side of the cell with the help of a silicon camera equipped with a commercial objective. A short-pass filter rejecting wavelengths longer than 675 nm was placed on the objective in order to suppress fluorescence signal induced by the bound excited state of krypton. The collected signal is then representative of ionization induced all along the propagation of the filament. Figure 1(a) shows the longitudinal profile (in log. unit) of the plasma fluorescence, and Fig. 1(b) the maximum plasma fluorescence as a function of the laser central wavelength. Filaments produced with central wavelengths higher than 305 nm (i.e., above the resonant wavelength) share in good approximation the same longitudinal profile and lead to a similar ionization yield. An abrupt change of both the longitudinal profile [see Fig. 1(c), which compares the longitudinal profile of filaments produced at 301 nm and 310 nm] and the plasma fluorescence amplitude takes place when the filament central wavelength is close to or lower than the atomic resonance wavelength. As shown in Fig. 2, the filament length increases by a factor of ten when the central wavelength of the filament is tuned to the atomic resonance. This increase of the filament length is accompanied by a decrease of ionization by an order of magnitude. Since multiphoton resonances are known to enhance, at a given intensity, the ionization process , this necessarily implies a strong decrease of the intensity within the filament core close to the resonance. Moreover, the fact that longer filaments are obtained for a central wavelength minimizing ionization suggests that the latter is the main mechanism limiting the filament length. The energy transmission has been measured as a function of the central wavelength of the filament. The associated optical losses experienced by the filament per unit length, i.e., the total optical losses normalized by the filament length, are displayed in Fig. 2. Again, a rapid transition close to the atomic resonance is observed. In particular, a strong reduction of the absorption is recorded around the resonance, which explains why the filament can sustain high intensity over longer distances in this spectral region. Finally, it has been checked that the presented results were not due to a cumulative effect (gas heating, long term depletion of the ground state) by decreasing the repetition rate of the laser.
3. THEORETICAL MODELING
In order to better understand the underlying physics, ab initio calculations capturing the atomic optical response of krypton atoms have been performed. The simulations consist of solving the time-dependent Schrödinger equation (TDSE) determining the temporal dynamics of the electronic wavepacket when submitted to a strong laser field. More specifically, under the single-active electron and dipole approximations, the three-dimensional TDSE describing the evolution of the electron wavefunction in the presence of an electric field reads18]: 2) cannot be used as such. To obtain a potential suitable for the intended simulations, a numerical treatment similar to the one made in Ref.  was performed. Inner shells can be eliminated without distorting the other wavefunctions by imposing a hard-core boundary condition with . Finally, another modification of the potential was performed so as to eliminate the state lying below the ground state, and then ensure that no transition can take place between these states during the interaction. It was done by adding the soft-core potential to the effective potential felt by the states, 20], the knowledge of the electronic wavefunction all along the interaction between the atom and the field then allows us to evaluate all physical parameters of interest, namely the polarization, and consequently, linear and nonlinear refractive indices, the population left after the interaction in the ground and the bound excited states, and the ionization yield. The refractive index at the driving frequency is defined by 3(a) displays the nonlinear refractive index as a function of both central wavelength and peak intensity of the 7 fs laser pulse. The same behavior is noticed for every wavelengths as the intensity is increased. The nonlinear refractive index first increases almost linearly with respect to the intensity, which corresponds to the well-known Kerr effect. The associated nonlinear refractive index is estimated from the slope to at 1 bar within the whole wavelength range studied experimentally. It then saturates and finally becomes negative as the intensity reaches a few tens of terawatts per square centimeter (). The intensity at which the refractive index becomes negative [white line in Fig. 3(a)] is almost the same between 500 nm and 800 nm and lies at about . This indicates that the clamping is induced by an off-resonant process, and that the exact atomic structure of the atom plays no role, as already noticed in Ref. . Below 500 nm, however, ab initio calculations foresee that strongly decreases in three different localized wavelength regions, namely around 470, 357, and 300 nm. In particular, at these wavelengths and for the 7 fs pulse case, the clamping intensity drops to 39, 24, and , respectively. In parallel, the ionization level obtained at the clamping intensity strongly drops by an order of magnitude. The expected decrease of ionization is in excellent quantitative agreement with the experimental observations as shown in Fig. 1(b). In the case of a 50 fs laser pulse, our calculations give () at (). These values then give an estimation of the peak intensity reached within the filaments at these wavelengths. Furthermore, given the initial pulse energy and duration, one can estimate the beam waist to about 85 μm at 300 nm and the Rayleigh length to 3.8 cm. In the linear optics framework, since ionization behaves as the fourth power of the intensity, the ionization channel length (FWHM) would be , which is almost half the length experimentally observed at 300 nm. This simple consideration indicates that the pulse sustains high intensity (at the level) over distances significantly longer than those allowed by linear optics laws, and that filamentation does take place. The decrease of the ionization yield, noticed in both our experimental and theoretical results, then suggests that the intensity clamping is not due to ionization at resonance. In particular, when ionization drives the clamping, the stabilization of the filamentation process occurs when , where is the nonlinear refractive index of the medium, is the critical plasma density, is the laser intensity, the ionization yield, is the vacuum permittivity, and are the electron mass and charge, respectively, and is the laser pulsation. The red line in Fig. 3(b) shows the ionization yield at which the nonlinear refractive index changes its sign calculated in this framework. In very good agreement with TDSE calculations performed in the infrared region, the clamping scenario based on ionization fails to describe the strong decrease of the electron density taking place in the ultraviolet and blue regions. In particular, the calculations reveal that the ionization contributes to only 10% of the intensity clamping at , and consequently is not the mechanism responsible for the intensity stabilization occurring during the filamentation process in this spectral domain (see Fig. 4). This observation is in complete agreement with the experimental findings shown above and confirms the existence of a Kerr-driven filamentation regime. In the present situation, the clamping mechanism is in fact due to resonant three-photon transitions leading to a giant defocusing resulting from a higher-order Kerr effect. It is analogous to the sign inversion of the nonlinear refractive index occurring close to a two-photon resonance . As we see here, the same phenomenon takes place close to a three-photon transition for the higher-order nonlinear refractive index . The latter is estimated from ab initio calculations to at . Note, however, that the dynamics of the optical response are more complex than simply considering a higher-order nonlinear refractive index because of the intrinsic noninstantaneous nature of the three-photon absorption process close to the resonance. In order to illustrate this resonant phenomenon, the populations left in the excited states as a function of the central wavelength are depicted in Fig. 5 for (a) and (b) peak field intensities. At low intensity, the dynamic Stark shift is negligible so that the eigen-energies of the excited levels are not perturbed by the field. Conversely, at higher peak intensity, the dynamic Stark shift starts to play a role and displaces the wavelengths at which resonances take place all along the excitation. This is especially the case for longer wavelengths, since the energy shift varies as . As a consequence, while sharp resonances still take place in the ultraviolet at higher intensities, they are completely smeared out in the infrared region. This explains why resonance-induced effects can only be observed in the ultraviolet. In fact, because of the finite spectral extent of the laser field, two three-photon resonant transitions take place simultaneously at 300 nm, namely the and transitions. After the interaction, the two excited states involved in the resonances ( and ) are significantly populated. In particular, they are far more populated than the 5p excited state (located energetically below the and states), which cannot be populated by a three-photon process because of its parity. In other words, TDSE calculations predict a population inversion when the laser wavelength is tuned at 300 nm, potentially resulting in forward and backward lasing of krypton at wavelengths corresponding to and transitions. Note that this three-photon pumping scheme has been recently demonstrated in the case of argon . In order to confirm these theoretical predictions, the spectrum of the light propagating along the laser axis was recorded after the cell as a function of the pump wavelength. The pump beam was rejected with a dichroic mirror while the transmitted light was analyzed either with an optical spectrum analyzer between 600 and 1600 nm, or a spectrometer between 1650 to 2000 nm. As shown in Fig. 6, several lines, located at 1362.24, 1363.42, 1442.68, 1679, 1689.5, and 1800.22 nm, are emitted at resonances. As expected, these lines correspond to transitions from and states to states. All the observed lines share the same linear polarization as the pump UV laser, except for the 1442.68 nm line, which is linearly polarized perpendicularly to the pump. As it can be seen in Fig. 6, not only the intensity of the lines but also their width and shape depend on the pump wavelength. All observed lines are far broader than what can be expected in case of fluorescence-induced emission. For instance, the line centered around 1800.22 nm spans over more than 150 nm at resonance, while emission coming from fluorescence would result in linewidths of about 1 Å at the considered pressure . Even if additional measurements are needed to confirm this preliminary observation, the spectral shape and the polarized nature of the lines seem then to indicate the stimulated origin of the emitted radiations. Note that the same lines have been observed in the backward direction (data not shown). A more exhaustive study of this effect will be the subject of another publication.
In conclusion, the existence of a Kerr-driven regime of filamentation is experimentally demonstrated in krypton at 300 nm. In this regime, the filament length is greatly extended, the nonlinear optical losses strongly drop, and ionization decreases by an order of magnitude. The experimental findings are supported by ab initio quantum calculations that are in excellent agreement with the experiments. The theoretical results pinpoint the role of three-photon resonant transitions in this Kerr-driven ultraviolet filamentation. This is also confirmed by the observation of forward and backward emission of lines appearing only at resonance. Observed in krypton, the resonantly enhanced UV filamentation process should take place in other atomic systems. In particular, it is expected that this regime could also be observed in argon, where forward and backward lasing, triggered by a three-photon UV resonance, has been recently demonstrated . It could be also interesting to study the possibility to extend this work to molecular gases, in particular to those of atmospheric interest. Indeed, even if resonant effects are expected to be less efficient in molecular systems, the generalization of the present work to oxygen or nitrogen could allow the optimization of the filamentation process in the atmosphere and its underlying applications by an appropriate choice of the laser wavelength.
Agence Nationale de la Recherche (ANR) (ANR-13-BS08-0013); Labex (ANR-11-LABX-0001-01).
P.B. thanks the CRI-CCUB for CPU loan on its multiprocessor server and B. Lavorel for fruitful discussions.
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