1. Introduction

Femtosecond laser induced filamentation provides a unique platform for high-energy physics and for studying nonlinear interactions between light and matter [1]. It has been extended to various applications, such as flow field velocimetry [2], control of chemical reactions [3], and remote sensing [4]. Femtosecond laser filamentation in air originates from Kerr self-focusing and defocusing of the plasma generated from the ionization of air molecules [5]. The phenomenon of filamentation is accompanied by a series of complex nonlinear optical effects, such as multi-photon ionization, intensity clamped and supercontinuum radiation, etc. [5,6]. The interaction between filaments is a topic of interest because it realizes the regulation of the intensity, length, and transmission direction of the filaments and imposes great impacts on the energy transfer [7,8], harmonic generation [9], spectral distribution [10–12] as well as fluorescence enhancement [13]. Among them, how to promote fluorescence intensity by filaments interaction is of great significance for improving detection sensitivity in applications, such as remote sensing. In general, a number of theories and experiments have been carried out for investigating the interaction of filaments which propagate in the same direction. The phenomena, including mutual attraction and repulsion (mainly related to modulated phase shift and cross angle [14]), energy exchange [15], spectrum broadening [16], and their impacts on fluorescence energy [13] and lifetime [17] have been studied. In our previous publications [18], it has been demonstrated that the interaction of filaments can enhance the fluorescence by raising the threshold of the clamped intensity which sets an upper limit to the self-focus laser intensity [5]. As a result, the plasma density in the filamentation zone is increased, consequently causing stronger fluorescence radiation.

We notice that prior studies are predominantly performed with co-propagating beams and very few works are related to counter-propagating filaments, apart from the one using counter-propagating filaments for terahertz wave generation [19]. The results in Ref. [19] showed that the plasmas generated by counter-propagating filaments affected the motion of the electrons. However, the physics inside the interaction of counter-propagating filaments and their effects on fluorescence properties still remain puzzling.

In this paper, we experimentally investigate the fluorescence enhancement as well as the interaction of two collinearly counter-propagating filaments (CPF) in air. Two contrast experiments were performed by propagating one filament and two non-collinear filaments (NCF) with a certain crossing angle. We find that the fluorescence of the excited air molecules is enhanced and their lifetime is prolonged in both cases of CPF and NCF as compared with those of the single beam filament. The enhancement is attributed to the increase of the clamped intensity, which yields more highly excited molecules. The fluorescence spectra, the output energy at the end of a filament, and the spectra of the excitation beams are measured by changing the relative time delay between counter-propagating beams. Our experimental results add new knowledge for understanding the interaction of filaments.

2. Experimental setup

The experimental setup is shown in Fig. 1(a). A Ti:Sapphire laser system (Legend Cryo PA, Coherent Inc.) emitted a train of 100 fs laser pulses at a repetition rate of 1 kHz. The laser spectrum was centered at 800 nm with a spectral width of 30 nm. The laser was equally split into two arms (beam 1 and 2, respectively), and the pulse energy for each of them was 1 mJ. For the convenience of explanation, we used pulse 1 and pulse 2 to indicate the pulses of beam 1 and beam 2 in the following text, respectively. Two collinearly counter-propagating filaments geometrically overlapped by two identical fused silica lenses (ƒ=10 cm), respectively (forming two spatially overlapped counter-propagating filaments in air). In the meantime, two contrast experiments were performed. One was conducted with NCF (Fig. 1(b)) and the other with only one filament (Fig. 1(c)). In Fig. 1(b), the two beams, each with 1 mJ pulse energy, were separately focused by two fused silica lenses (ƒ =10 cm) to produce two non-collinear filaments in air simultaneously. The crossing angle of the two beams was α (=7°). In Fig. 1(c), the excitation beam with pulse energy of 2 mJ was focused by a lens (ƒ =10 cm). For all the three setups, the total input laser pulse energies remain constant (2 mJ) and the beams are horizontally polarized.

 

Fig. 1. Schematic diagrams for the experimental setups for (a) CPF. (b) NCF. (c) one filament. BS: beam splitter; COL: collimator; L1, L2: fused silica lens with ƒ = 10 cm; L3, L4, L5: fused silica lens with ƒ = 5 cm; HWP: half-wave plate.

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3. Experimental results and discussions

The recorded fluorescence spectra, excited by CPF (black curve), one filament (blue), and NCF (red), are shown in Fig. 2. The filamentation-induced fluorescence was collected, in the direction perpendicular to the laser propagation, with a lens (f = 5 cm) and measured by a spectrometer (FX2000-RD). Note that the filaments are located at the focal spot of the lens, the position of which is adjusted so that the recorded fluorescence is maximized. In Fig. 2, each spectrum, spanning from 350 nm to 710 nm at a resolution of 0.3 nm, was acquired within 0.1 s. The spectral fingerprints for air molecules (particularly, N₂) are observed in the range of 400-600 nm [20]. The supercontinuum background (i.e., Bremsstrahlung) is caused by the energy loss of free electrons, which are continuously decelerated by colliding with surrounding particles in the filamentation zone. In Fig. 2, we notice that the fluorescence spectra, obtained with CPF (black) and NCF (red), are almost identical, with spectral line intensities more than 4 times stronger than those obtained with one filament. The intensities of spectral lines are derived by taking the differences between the peaks (e.g., those marked with green dotted triangles in Fig. 2) and their nearest valleys (yellow dotted circles in Fig. 2), to remove the background.

 

Fig. 2. The fluorescence spectra induced by CPF (black curve), NCF (red curve), and one filament (blue curve) in air, respectively. The green dotted triangles and the yellow dotted circles represent the peak of the fluorescence lines and its nearest valley, respectively.

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To investigate the physics inside the fluorescence enhancement, we measured the average peak intensity for CPF, NCF, and one filament. The peak intensity, ${I_0}$, in the filamentation zone can be evaluated by [21–23]

 

Fig. 3. Fluorescence spectrum measured with CPF, 337 nm (orange arrow) and 391 nm (green arrow).

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Then, the fluorescence lifetimes induced by CPF, NCF, and one filament are investigated (see Fig. 1). In the experiment, the ICCD gate width was set to 2 ns and the gate was opened 2 ns before the laser pulse arrived at the interaction zone to ensure that the beginning of a fluorescence decay was recorded. The lifetimes of fingerprint fluorescence are plotted in Fig. 4. In addition to the spectral lines at 337 and 391 nm, the fluorescence lifetimes at 427 (the ${B^2}\mathrm{\Sigma }_u^ + \to {X^2}\mathrm{\Sigma }_g^ + $ transition of $N_2^ + $), 449, and 481 nm (from the $\,\textrm{}{C^3}{\mathrm{\Pi }_u} \to {X^2}{B^3}{\mathrm{\Pi }_g}$ transition of ${N_2}$) are evaluated. As an example, the insets in Fig. 4 show the fluorescence decay curves, measured by the three schemes for the spectral lines at 391 and 481 nm, respectively. Note that the intensities are normalized for comparison. The fluorescence lifetime is defined by the time taken for the fluorescence to decay to 1/e of its initial value [29]. In Fig. 4, the fluorescence lifetimes excited by CPF and NCF in air are almost identical, which, however, are longer than those of one filament. This seems consistent with the trend of spectral enhancement in Fig. 2. It means that both CPF and NCF lead to higher filament intensity, giving rise to more free electrons [30]. Consequently, the initial plasma temperatures would be different and thus cause changes in fluorescence lifetimes.

 

Fig. 4. The fluorescence lifetime at different characteristic bands (337, 391, 427, 449, and 481 nm, respectively) for CPF (red curve), NCF (green), one filament (blue) in air. Their error bars represented the standard deviation of repeated measurements. The time dependent evolution of signal intensity at 391 nm and 481 nm are shown in the insets. In the insets, the hollow dots and the solid lines represent experimental data and exponential fitting, respectively.

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In order to provide a detailed physical picture of CPF, the fluorescence intensity and the output energy at the end of the filament were measured by changing the relative time delay of the counter-propagating beams. Here, the delay between beam 1 and beam 2 was scanned by the displacement stage with a step size of 0.01 mm, corresponding to a temporal resolution of ∼33 fs. We measured the fluorescence intensity dependence on the time delay (see Fig. 5(a)). The delay zero in Fig. 5 is determined as the time delay corresponding to the maximum of fluorescence intensity. The positive value for the time delays means that pulse 2 is in front of pulse 1, and vice versa. As expected, the fluorescence spectral evolution is nearly symmetric concerning the zero delay.

 

Fig. 5. (a) The fluorescence spectra, (b) the output power at the end of the filament, and (c) the spectrum of beam 1 varying with the time delay between counter-propagating beams. The experimental setup in Fig. 1(a) is used for the measurement.

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The output powers at the ends of the filaments from beam 1 (i.e., P₁) and beam 2 (i.e., P₂) as well as the sum of the two powers, P₁+ P₂, were measured when the time delay is changed. The time-delay-dependent output powers (Fig. 5(b)) present an opposite trend as compared with the fluorescence spectra in Fig. 5(a). It means that more energy inside the filaments is transferred to multiphoton/collision ionization as the pulses 1 and 2 are temporally approaching each other. In Fig. 5(b), there is a time-domain region from -0.8 to 0.8 ps (indicated by the black arrows in Fig. 5(b)), in which the output powers keep roughly constant with the changing time delay. For CPF, the two pulses propagated from opposite directions, in which case they would always coincide at a certain spot despite their time delay being changed. This is different with the case of NCF where the pulses met only when they approached to the zero time delay. For CPF, as long as this coincidence spot was inside a filament, the fluorescence could be enhanced (while the enhancement was reduced compared to that at the zero time delay). In Fig. 5(b), the region from -0.8 ps to 0.8 ps corresponded to the physical length (full width at half maximum of ∼0.5 mm) of a filament induced by one beam.

Furthermore, the transmitted spectrum of beam 1 was investigated. The spectrum was detected by taking the reflection from the BS1 in Fig. 1(a). The recorded spectra in Fig. 5(c) changed drastically when we scanned the time delay. The spectral evolution can be explained as follows. When pulse 1 is in front of pulse 2, the pulse 1 experiences a typical single-beam filamentation, which yields supercontinuum in the range of 680 to 830 nm due to self-phase modulation and plasma generation [5,6,31]. When the two pulses are temporally overlapped, the transmitted spectrum of beam 1 becomes even broader in the range from 560 to 840 nm. The additional extension of the spectrum is due to the interaction between the filaments. When pulse 1 is behind pulse 2, a filament formed by the pulse 2 leads to a modulation of the refractive index of the plasma channel [31,32]. Pulse 1 is influenced by such a modulation in space and time [32] and therefore experiences additional spectral modulations (or spectral changes) as observed in Fig. 5(c). A similar spectral evolution was also found when we detected the transmitted spectrum of beam 2 as the time delay changed. In CPF, the interaction of filaments may lead to the increase of clamped intensity and plasma density [33], and higher clamped intensity would promote the ionization rate and cause the stronger fluorescence spectra in Fig. 5(a).

However, although roughly equal fluorescence intensities are observed by the schemes of CPF and NCF, their fluorescence intensities have different dependence on the relative polarization of the interacting beams. We fixed the polarization of beam 2 in a horizontal direction, while the polarization of beam 1 was adjusted by a half-wave plate in the beam path (see Figs. 1(a) and 1(b)). Figure 6(a) shows the spectral lines at 427 and 481 nm vary with different polarizations of beam 1. The counter-propagating beams have equal amplitudes (E₁= E₂= E) due to equal pulse energies. In NCF, the fluorescence intensity is related to the field intensity which can be expressed as $I = E_1^2 + E_2^2 + 2{E_1}{E_2}|{cos\theta } |= 2|{E^2}|{ + 2|{E^2}cos\theta } |$, assuming the two excitation fields have the same phase. Also, because the focal spot of the lens for collecting fluorescence is smaller than the intersection area of the NCF beams, the influence of the angle between the two beams is ignored for this observation point. In general, the maximum fluorescence intensity can be achieved with the interacting beams of parallel polarizations (e.g., $\theta $=0), and the minimum is for the beams with perpendicular polarizations (e.g., $\theta $=90°). As expected, the ratio between the minimum and the maximum is 0.5 in NCF (marked in Fig. 6), while unexpectedly it is only 0.1 in CPF. The reason is given in the following. In NCF, the fluorescence is mainly collected from the intersection area of the interacting beams, which is therefore largely influenced by their interference. While, in CPF, the interference behavior of the counter-propagating pulses is constrained by their temporal overlapping (i.e., the time-domain convolution of the two pulses). Therefore, the impact of the interference of the two beams on the CPF fluorescence is less significant. This also implies that the interaction of two filaments is insensitive to the polarization of the excitation fields.

 

Fig. 6. (a) The intensities for the spectral lines at 427 and 481 nm, respectively, excited with NCF (squares) and CPF (triangles) as a function of the relative polarization between the two counter-propagating beams. The solid curves represent the cosine fitting of the data points. The intensities are normalized for comparison. The pictures of the filamentation zone (b) in NCF and (c) CPF, respectively, measured at θ=0 with a camera.

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Additionally, we have noticed that the spatial structures of the filaments in the three geometric configurations were different. In NCF, a typical plasma grating structure (Fig. 6(b)) was observed in the interaction zone, while it was absent in the CPF case, as shown in Fig. 6(c). Note that the total filament length of CPF was measured to be about 1.4 times longer than that of one filament. Therefore, the counter-propagating configuration was not influenced by the spatial interference of the two laser fields.

4. Summary

In conclusion, we have demonstrated that the interaction of two counter-propagating filaments can increase the peak intensity and enhance filamentation-induced fluorescence. We find that the threshold of the intensity clamped is increased in the CPF case which is less sensitive to the polarization, as compared to the case of NCF. It is also possible to control a filament or ultra-short pulses (regarding pulse spectra and energy) with another filament from the opposite direction. Our results provide new perspectives on the interaction of filaments as well as the improvement of detection sensitivity for sensing applications with filament-induced nonlinear spectroscopy.