The filamentation of femtosecond laser pulses [1] leads to a high concentration of the light field [2, 3], which is maintained at a distance much longer than the Rayleigh length because of the dynamic balance between the self-focusing of radiation in a medium with the Kerr nonlinearity and its defocusing on free electrons from the ionization of the medium [1, 4]. The control of this process accompanying the propagation of laser pulses in gases in the femtosecond filamentation regime [5] is of great interest for applications in the transmission of the light energy to long distances, in particular, the laser probing of the atmosphere and the fabrication of remote sources of white light in the atmosphere. One of the possible tools for such a control is the alignment of molecules with the anisotropic polarizability by a laser pulse, which can change the conditions of filamentation for a pulse propagating in the same direction with some delay [6, 7]. A short laser pulse produces a quantum-mechanical wave packet, which is a coherent superposition of many rotational states of gas molecules. In terms of gas molecules, the axes of molecules along which the polarizability is maximal acquire a rotational moment and tend to be aligned in the direction of the polarization of laser pulses [7–11]. If the duration of the laser pulse is much shorter than the typical rotational period of the molecule, the process is a nonadiabatic (field-free) alignment [9, 10], and the characteristic time evolution of this rotational wave packet (“phasing” and “dephasing”), which is determined only by the inertia of excited molecules, occurs already after the end of the laser pulse [11]. In contrast to liquids [8], in molecular gases with the anisotropic polarizability, the rephasing or revivals of rotational wave packets are observed [10, 11] with the period Trev from units to tens of picoseconds, which is determined by the rotational constant of gas molecules B (Trev = 1/(2Bc), where c is the speed of light in vacuum). The revival of wave packets, sometimes called quantum wake, can also occur at the quarter, half, and three quarters of the period, changing the refractive index in the cross section of the already transmitted pulse with the corresponding time delays. Since changes in the refractive index are different for molecules aligned along and across the polarization of laser radiation, the initially isotropic medium in the quantum wake region becomes birefringent. This region propagates in the gas behind the first pulse aligning molecules and its velocity can be treated as its group velocity. Therefore, the second femtosecond pulse can propagate synchronously with the quantum wake region for a sufficiently long time so that the effect of the first-pulse induced change in the refractive index on the filamentation of the second pulse is strongly enhanced.

Thus, the laser-induced nonadiabatic alignment of molecules in gases opens the possibility for control of the state of atmospheric lines for transmission of laser radiation [12], for the creation of lasing conditions on nitrogen ions upon filamentation in air [13], and for control of the generation of the supercontinuum and high order harmonics for an increase in the electron density in plasma channels of the filament and the further compression of pulses [7, 14]. The experimental studies of the effect of the preliminary alignment of molecules on the features of filamentation showed that this process can increase the length of the filament [15–17] and change the duration and spectrum of the pulse [7, 14, 16, 17]. Almost all cited works were performed with radiation at a wavelength of 800 or 400 nm (second harmonic). However, the parameters of the experiments in these works (intensity of the laser pulse, focusing, the length of the filament, the pressure and type of the gas) were significantly different. Furthermore, as shown below, the methods used in [17–22] to detect the dynamics of the refractive index caused by the alignment of molecules were inappropriate and gave contradictory results.

In this work, we experimentally study the effect of the alignment of nitrogen molecules by a femtosecond laser pulse with a central wavelength of 1400 nm on the parameters of a probe laser pulse with a central wavelength of 800 nm propagating through the quantum wake region with an increase in the energy of the probe pulse up to the beginning of multiple filamentation. Using pulses with two significantly different wavelengths, we clearly separated a signal from the probe pulse at the same polarization of two laser pulses when the alignment effect was maximal. Moreover, the 1400-nm pump pulse allowed us to obtain a longer filament and to increase the energy deposition in the filament through an increase in filamentation threshold and a decrease in the ionization probability, which are due to an increase in the central wavelength of the pulse. Thus, the interaction length of the probe pulse with the quantum wake region increases and the influence of nonlinear effects appearing in the generated laser plasma decreases [23, 24]. The alignment of molecules by short-wavelength infrared pulses has not yet been studied experimentally. Some features for propagation in air were revealed in the numerical simulation in [25].

We measure the spectrum, duration, and spatial distribution of the probe pulse passing through the gas as functions of its time delay from the first pulse aligning molecules. The used Ti:sapphire femtosecond system with a Spectra Physics Spitfire HP regenerative amplifier generated output pulses with a duration of 45 fs, a wavelength of 800 nm, an energy of 4 mJ, and a repetition frequency of 1 kHz. A fraction of radiation converted in a Light Conversion TOPAS-C optical parametric amplifier to 1400-nm pulses was used to align nitrogen molecules in the gas cell. An 800-nm pulse behind an achromatic half-wave plate, which matched the polarizations of both pulses, was used to probe the quantum wake region of the 1400-nm pulse. Both pulses were joined by a dichroic mirror and were focused in a 60-cm-long gas cell by a lens which had a focal length of 30 cm and was placed immediately near the input window of the cell. The initial positions of filaments formed in the gas were controlled through the side window of the cell by a digital camera and were matched for both pulses by means of a controlled telescope placed in the probe channel. The time delay between the pulses was varied using a computer-controlled optical delay line. The spectral characteristics of radiation behind the cell were determined by a FLAME-S-XR1-ES spectrometer with an Ocean Optics FOIS-1 integrating sphere with the simultaneous control of the delay line, which made it possible to record spectral–temporal characteristics. The pulse durations were measured by an Avesta ASF-20 single-pulse autocorrelator. The pulse repetition frequency in the experiment was reduced to 50 Hz in order to exclude the influence of accumulation effects caused by, e.g., the formation of ions in the process of filamentation. The pressure in the cell was increased to 3 atm (optimal for the supercontinuum generation [26]) to overcome the filamentation threshold for the energy of the 1400-nm pulse achievable in our experiment and, thereby, to increase its intensity, as well as the degree of alignment of molecules in the quantum wake region, which is proportional to this intensity. The pressure was chosen for the stable generation of the visible supercontinuum spectrum in the filament at the maximum achievable energy of 220 μJ of 1400‑nm pulses with an FWHM duration of 45 fs behind the input window of the cell. The length and diameter of the filament determined from the photograph of its glowing plasma channel in the gas were 2 cm and 90 μm for 800-nm pulses and 5 cm and 105 μm for 1400-nm pulses, respectively. Consequently, the filamentation of the 800-nm probe pulse occurred inside the quantum wake of the filament induced by the 1400-nm pulse. At time delays at which the probe pulse reached the revival region of alignment of molecules (see below), the length of the filament included by this pulse was doubled because of the focusing action of the rotational quantum wake [7, 15, 16].

The revival of rotational wave packets in our experiments was observed at times multiple to the period, as well as to the quarter, half, and three quarters of the period. It is known that processes occurring in these cases are similar. For this reason, following most of the studies in this field, we performed such measurements at time delays near 4 ps, which is approximately the half period of the revival of the rotational wave packet for nitrogen molecules. The spectra of the probe pulse with different energies in this time domain are presented in Figs. 1а–1d. The observed spectral shift of this pulse with an energy of 5 μJ, which is insufficient for the formation of the filament, is due to only the sign-alternating addition to the refractive index of the gas appearing in the quantum wake region. The characteristic cyclic spectral shift qualitatively reproduces a tendency predicted in [14] (see Figs. 1a, 1b). In the region where the rate of variation of the refractive index is maximal, a small (~5 nm) broadening of the spectrum is observed compared to that calculated in [14] because the length of the filament in our experiments was an order of magnitude smaller. The total redshift and blueshift of the wavelength are much larger (more than 40 nm). This value was interpreted in [18, 19] as the spectral broadening of the pulse apparently because of an insufficient time resolution. The 800-nm probe pulse with an energy of 65 μJ produces a filament. In this case, the frequency shift caused by the phase modulation because of a change in the refractive index in the quantum wake dominates in agreement with estimates in [27–31]. A further increase in the intensity results in an increase in the effect of the Kerr nonlinearity leading to the broadening (primarily in the direction of short wavelengths) of the spectrum by more than 100 nm with characteristic strong modulation (Fig. 1e).

Fig. 1.
figure 1

(Color online) (а–d) Spectra of the probe pulse at different time delays with respect to the 1400-nm pulse in the revival region of the rotational wave packet near the half period of the complete revival at different energies indicated in the panels. (e) Maximum spectral width of (black line) the input pulse at energies of (blue stars) 5 and (red circles) 260 μJ. (f) (Circles) Four-wave mixing signal from the pump and probe pulses and (triangles) the response in the entire spectral range under study near zero time delay.

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It is noteworthy that the observed revival times of the rotational wave packet in different works under the same conditions are rather different apparently because of different choices of zero time delay. In our experiments, zero time delay was determined by the four-wave mixing signal from the pump and probe pulses (see Fig. 1f). A time delay of about 150 fs in the rotational response is seen in the signal including the response in the entire studied spectral range (Fig. 1f). This time is slightly longer than that calculated in [29, 30] and measured in [32, 33] for the 800-nm pulse but is in good agreement with the estimate in [25] for the near infrared range.

From our experimental data, we determined the dependence of the center of gravity of the spectrum [34] on the time delay (Fig. 2). This dependence was used to calculate the time dependence of change in the refractive index \(\Delta n(t)\) upon the nonadiabatic alignment of molecules in the quantum wake region by the formula \(\Delta n(t) \approx \frac{{{{c}_{0}}}}{{L{{\lambda }_{0}}}}\int_{ - \infty }^t {\Delta \lambda } (t{\kern 1pt} ')dt{\kern 1pt} '\) [33], where \(\Delta \lambda \) is the measured spectral shift, \({{\lambda }_{0}}\) is the central wavelength of the laser pulse, L is the length of the filament, and \({{c}_{0}}\) is the speed of light in vacuum. The results are in quite good agreement (see Fig. 2) with the calculations of the refractive index of molecular nitrogen in the quantum wake region as a function of the time delay from the excitation time [25, 33, 35]. This change in the refractive index can be described by the formula \(\Delta n(t) = 2\pi Nn_{0}^{{ - 1}}\Delta \alpha (\langle \mathop {\cos }\nolimits^2 \theta \rangle (t) - 1{\text{/}}3)\), where N is the density of the gas, \({{n}_{0}}\) is the linear refractive index, \(\Delta \alpha \) is the difference between the polarizabilities of the molecule in fields parallel and perpendicular to its axis, and θ is the angle between the polarization of the pump pulse and the axis of the molecule. The maximum of the refractive index for the probe pulse at which axes of molecules are aligned along its polarization is reached at a time delay of 4.15 ps, and its minimum is reached at a time delay of 4.32 ps; both values do not coincide with the measured extremal wavelength shifts (Fig. 2), gas transmittance, and intensity of the third harmonic (are not shown).

Fig. 2.
figure 2

(Color online) (Empty red circles) Center of gravity of the spectral distribution and (filled red circles) change in the refractive index that are determined from experimental data at the filament formation threshold (Fig. 1b) versus the time delay of the probe pulse. The dotted line is the qualitative behavior of the refractive index in the quantum wake according to [25, 27].

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The experimental time-delay dependences of the transmittance of a sample, spectral broadening of the probe pulse, or the intensity of its third harmonic are treated in many works (see, e.g., [17–22]) as the dependences of the change in the refractive index in the quantum wake. These dependences are qualitatively similar, but according to the numerical simulation in [14, 16, 31, 33], spectral shifts should be maximal at times when the derivative of the refractive index is maximal (in good agreement with our measurements). A change in the refractive index at these points is close to zero, and previous results are contradictory because these points corresponding to the observed maxima of the transmittance, frequency shift, or intensity of the third harmonic were treated by the authors as the positions corresponding to the maximum changes in the refractive index.

Figure 3 presents the measured duration of 800-nm probe pulses with durations of 55 and 110 fs passing through regions with a variable refractive index. The length of both pulses increased in the region of increasing refractive index (time delays of 4.00–4.15 ps) and decreased in the region of decreasing refractive index (time delays of 4.15–4.30 ps). This behavior can be explained by the difference between the refractive indices for the rising and falling edges of the pulse, which increases the length of the pulse in the former case and decreases it in the latter. The change in the length of the pulse is maximal when it is located entirely in the region where the derivative dn/dt has the same sign [36], i.e., at a duration close to the time of pulse passage through these regions. According to Fig. 3, the effect is much stronger for the 110-fs pulse: compression and broadening reach factors of 3 and 1.5 compared to factors of 1.4 and 1.2 for the initial 55‑fs pulse. The opposite behavior for the initial 140‑fs pulse under similar conditions was observed in [20], where the authors stated that the pulse duration decreased to 130 fs and increased to 160 fs in the quantum wake regions where the axes of molecules were oriented along and across the polarization of the pulse, respectively, which contradicts the calculations in [14]. This discrepancy can be due to the aforementioned inappropriate method used in [20] to determine the positions of the extrema of addition to the refractive index.

Fig. 3.
figure 3

(Color online) (а) Relative change in the duration of the probe pulse measured at various time delays in the revival region of the quantum wake near the half period at an initial duration of (red filled squares) 110 and (blue empty squares) 55 fs; the dotted line is the qualitative behavior of the refractive index. (b–g) Correlation functions measured for pulses with a duration of (b–d) 110 and (e–g) 55 fs at various time delays.

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An increase in the intensity of the probe pulse results in its spectral broadening, which is accompanied by strong modulation (see Fig. 1e). This modulation is caused by the interference of identical frequencies appearing in Kerr self-modulation in different parts of the pulse [1, 36], which indicates an increase in the contribution of Kerr nonlinearity. Nevertheless, the spectral redshift and blueshift of the probe pulse at the corresponding time delays show the decisive role of the alignment of molecules in change in the refractive index of the gas. Moreover, our experiments showed that the control function of the quantum wake holds at the intensities of the probe pulse inducing multiple filamentation. It is known that multiple filamentation leads to the formation of bundle of daughter filaments whose distribution over the cross section of the bundle and along the propagation axis and also their parameters change chaotically from pulse to pulse [1]. The problem of regularization of this process is quite relevant for applications and is actively studied. Results obtained in previous studies of the effect of the alignment of molecules on multiple filamentation only with 800-nm pulses are rather contradictory. In particular, the authors of [21, 22] observed the disappearance of multiple filamentation at the orientation of molecules along the polarization of the laser pulse, whereas the appearance of a “ripplelike” profile with a broadened spectrum in this case was reported in [7].

In our experiments, the bundle became inhomogeneous in the cross section when the energy of the probe pulse was increased to 200 μJ, which is approximately a factor of 5 higher than the critical energy of self-focusing (Fig. 4а). Under the variation of the time delay in the quantum wake region, the brightness of individual inhomogeneities in the beam increased sharply (Figs. 4b, 4с). Under the variation of the time delay from 4.3 to 4.5 ps (in the region of an increase in the refractive index upon the alignment of molecules), different (often one) inhomogeneities in the spot became “observable,” and pictures at fixed time delays were stable and were reproduced in the inverse variation of the time delay. This allowed us to detect daughter filaments and to measure their parameters. We observed that a daughter filament captured up to 4% of the energy of the probe pulse and propagated in the form of a directional beam with a divergence of 3–4 mrad with a spectrum of 300–900 nm (Fig. 4). The daughter filaments did not leave the main beam, whose divergence was much larger, increasing in this time delay range because of the appearance of a negative lens upon the alignment of gas molecules [7]. A typical radiation spectrum of the daughter filament detected at a time delay of 4.3 ps is shown in Fig. 4d. The spectrum was broadened and its intensity increased with the energy of the pulse up to 200 μJ; after that, the increase ended. This end occurred likely because the intensity in the filament reached a value of ~5 × 1013 W/cm2, at which its further increase is limited by the formed plasma (intensity clamping [37]). In terms of these data, the diameter of the filament can be estimated as ~100 μm; therefore, the divergence of the observed beams is smaller than the diffraction divergence. When the time delay leaves the quantum wake region, the beam again acquires the typical form shown in Fig. 4а.

Fig. 4.
figure 4

(Color online) (a–c) Profiles of the 800-nm probe pulse in angular coordinates behind the cell with the gas at a time delay of (а) 3.5, (b) 4.32, and (с) 4.4 ps; the capture and transfer of the energy to the daughter filament are seen. (d) (1) Spectrum of the probe pulse beyond the quantum wake region at a time delay of 3.5 ps and (2–4) spectra of radiation trapped in the daughter filament at a time delay of 4.4 ps and a pulse energy of (2) 175, (3) 205, and (4) 336 μJ.

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We believe that the observed selectivity of the energy transfer to daughter filament is due to a decrease in the interaction between them in the divergent beam, which is formed upon the appearance of a negative lens in the gas of aligned molecules. This conclusion is confirmed by the absence of the effect at times corresponding to the formation of the waveguide propagation of the probe pulse [38] caused by the positive lens in the quantum wake when a denser packing of daughter filaments occurs, enhancing their interaction and energy exchange and leading to their merging [39]. Kerr nonlinearity is responsible for a further self-focusing and phase self-modulation resulting in the generation of the broadband supercontinuum and the formation of the plasma channel in the filament.

Our experiments have shown that the nonadiabatic alignment of molecules by the 1400-nm pulse allows one to control the filamentation of the femtosecond laser pulse in gaseous nitrogen. In the single-filament regime, we have detected spectral shifts and change in the duration of the pulse caused by changes in the refractive index in the revival regions of the rotational wave packet. In the case of multiple filamentation of the femtosecond laser pulse in the gas, the selective stimulation of single filaments has been demonstrated upon the alignment of molecules in the direction perpendicular to the polarization of the pulse with the synchronous propagation of the pulse and quantum wake region. The stable and reproducible localization of radiation into separate filaments with the subdiffraction divergence and broadening of the spectrum by more than an octave caused by a decrease in the interaction between daughter filaments in the divergent beam has been observed.