1. Introduction

Filamentation of femtosecond laser pulse [1–42] has been intensively investigated after the invention of chirped pulse amplification, and the filament dynamics were well controlled by the alignment and the rotational states of molecules [5–9]. Filamentation with self-stabilized intensity clamping in self-guided channels owing to the dynamic balance between Kerr self-focusing and plasma defocusing, induces plenty of nonlinear optical phenomena, such as self-compression [10–15], supercontinuum generation [16, 17], third harmonic (TH) generation [18–23], and so on. In particular, it was reported that an efficient conversion about 0.2% from a femtosecond laser pulse to its TH generation could be achieved in air [18]. Recently, the interactions between two femtosecond filaments have attracted more and more attentions since it shows distinctive features, such as attraction, fusion, repulsion, and spiral propagation, depending on the relative phase shift and the crossing angle between them [24, 25]. The TH generation in air by interaction of two noncollinear filaments could be increased by two orders of magnitude with quasi-vertical crossing angle [26–28] and more than two orders of magnitude with small crossing angle [29]. The enhancement of TH generation may also be induced by an increase of the nonlinear optical susceptibility of the medium correlated with the presence of the plasma or a neutral air-plasma interface effect, which were named as bulk effect and neutral air-plasma interface [28]. Very recently, it was reported that the enhancement of TH generation originates from the breaking of the large cancellation of TH in a filament perturbed by a pump pulse [30, 31].

The plasma grating [32–35] formed by noncollinear interaction of two femtosecond filaments may tolerate ultra-high-intensity laser fields beyond the clamping intensity limit [36, 37] for the free electron generation and acceleration, and the peak electron density inside plasma grating was demonstrated to be two orders of magnitude higher than that inside a single femtosecond filament [38]. By using UV plasma grating, efficient nonlinear Bragg diffraction was observed, and free electron density was estimated by Bragg diffraction efficiency. Plasma grating was applied to study laser energy exchange between two femtosecond filaments used to form the plasma grating [39–41]. The magnitude and direction of energy flow from one beam to the other could be controlled by detuning the frequency of two pulses.

In this paper, a probe femtosecond beam is focused on a plasma grating formed by two noncollinear filaments with 3.8° crossing angle, and the influences of the probe filament on the plasma grating are revealed by the fluorescence distributions of plasma grating. By using plasma grating, the TH generation enhancement from the probe filament is investigated. The crossing angles between the probe filament and the plasma grating is set to be 3.5° or quasi-vertical. With this two crossing angles, the dependences of TH generation on the time delay τ_d, the relative polarization between the probe beam and the two pump beams, and the spatial period of plasma grating are investigated. Meanwhile, the spectral broadening of TH induced by the interaction between the probe filament and the plasma grating is also studied.

2. Experimental setup

The experiments are conducted using a Ti:Sapphire chirped-pulse amplification laser system operating at 1 kHz repetition rate (Quantronix, Odin-II). The entire laser system is able to provide 3.0 mJ pulse centered around 810 nm with laser pulse width about 33 fs. Schematic of the experimental setup is shown in Fig. 1 . The fundamental laser beam is split into three arms, Pump1, Pump2 and Probe by using two beamsplitters, BS1 and BS2. Pump1 beam with pulse energy up to 0.7 mJ and Pump2 beam up to 0.9 mJ are focused to form two filaments with length about 1.5 cm by using two 60 cm-focal-length lens, L1 and L2. Probe beam with pulse energy up to 0.6 mJ is focused to form a longer filament with length about 3 cm by using a 117 cm-focal-length lens L3. The time delay Δt between the two pump beams can be adjusted with a delay line, Delay1, and a plasma grating can be formed when the time delay Δt is set to 0 fs. And the time delay τ_d between the probe beam and the two pump beams can be adjusted with the second delay line, Delay2. The crossing angle between the filaments formed by the two pump beams is ø, and the crossing angle between the two filaments formed by Probe beam and Pump2 beam is θ. The angle θ can be adjusted to be 3.5° or quasi-vertical as shown in Fig. 1. M4 and L1 mounted together on a stage are movable to change the value of crossing angle θ, and M10 is adjusted to let the probe filament cross the center of plasma grating.

 

Fig. 1 Schematic of the experimental setup for detecting TH spectra generated from probe filament, and the plasma fluorescence distributions. Here: BS1, BS2, beamsplitters; L1, L2, L3, L4, focused lenses; M1, M2, M3, M4, M5, M6, M7, M8, M9, M10, M11, 800 nm high reflective mirrors; M12, M13, 266 nm high reflective mirrors; C1, Al coated curve mirror.

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The fluorescence of plasma grating is collected and projected onto a sensitive charge-coupled device (CCD) camera by C1, so the fluorescence distributions of plasma grating is recorded. For the measurements of TH spectra generated from the probe filament, two 266nm high reflective mirrors are used as dispersive elements to separate the TH from the fundamental and reflect the TH to a fused silica lens L4 to gather into a spectrometer (Ocean Optics, USB4000). The TH intensity could change with the propagation distance as investigated in Ref 24, and the spectrometer is placed at a position with the distance about 70cm away from the center of interaction area as shown in Fig. 1. The TH spectra are recorded by the spectrometer, and the relative TH intensity is calculated by the peak frequency component of TH spectra.

3. Experimental results and discussion

3.1 Plasma fluorescence distributions

Firstly, a plasma grating is formed by two pump filaments with $\varphi$ set to 3.8° and the time delay $\Delta t$ = 0 fs, and a probe beam is focused to form a filament crossing the center of plasma grating. The fluorescence distribution of plasma grating formed by the two pump filaments is shown in Fig. 2(a) . The plasma grating is formed with a length more than 700 $\mu m$, and its spatial period$\Lambda$can be determined by the function $\Lambda =\lambda /2\sin (\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$. When the time delay ${\tau }_d$ is adjusted to zero and the crossing angles between the probe filament and the plasma grating is set to be 3.5° and quasi-vertical, the corresponding fluorescence distributions are shown in Fig. 2(b) and Fig. 2(c), respectively.

 

Fig. 2 The fluorescence distributions with different conditions. (a), (b), and (c) are the results measured without the probe filament, with the probe filament with $\theta$ set to be 3.5° and quasi-vertical, respectively.

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As is known, for the interactions between two femtosecond filaments [24, 25] with very small crossing angles, there are attraction, fusion, repulsion, and spiral propagation in their propagating process depending on the relative phase shift. In the present work, when the probe filament with time delay ${\tau }_d$ = 0 fs impacts on the plasma grating, the attraction and repulsion appear alternately along the propagating direction, which is different from the results of noncollinear interaction of two filaments. The intense interaction between the probe filament and the plasma grating may cause periodic change of the relative phase, which means phase matching and phase mismatching, to induce the attraction and repulsion between them. Due to the laser field interference effect, the interaction among multiple filaments can enhance the clamped fundamental laser intensity [42], which may induce the increase of ionization rate in the interaction area. As shown in Fig. 2(b), the contrast ratio of plasma grating is improved a lot in the attraction areas and has no clear change in the repulsion areas. When $\theta$ is set to be quasi-vertical, the free electrons inside interaction area are bounded in a small range, and as a result their density is enhanced. As shown in Fig. 2(c), the enhancement of free electron density induces the contrast ratio in the interaction area to be improved up to ten times.

3.2 Enhancement of TH generation

Together with the interactions between the probe filament and the plasma grating, a significant enhancement of TH generation is observed along the probe beam propagating direction. As shown in Fig. 3 , black solid line, red dash line, and blue dot line denote the spectra of maximum TH measured from a single probe filament, the probe filament interacting with a pump filament, and the probe filament interacting with the plasma grating, respectively. The intensity of TH generation is enhanced by a factor about 3760 with $\theta$ set to 3.5°, and about 185 with $\theta$ set to be quasi-vertical. The enhancement factor is defined as the intensity of TH from the probe filament divided by that generated from a single probe filament.

 

Fig. 3 The TH spectra generated from probe filament without pump beam, with a single pump beam, and with two pump beams (plasma grating). (a) and (b) are the experimental results measured with $\theta$ set to be 3.5° and quasi-vertical, respectively.

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In order to have a clear understanding of the physical mechanism of the plasma grating on the TH enhancement, the dependences of TH enhancement factor on the time delay ${\tau }_d$ and normalized TH intensity on the relative polarization are shown in Fig. 4 (a) and Fig. 4(b) with $\theta$ set to 3.5°. The time delay dependences of enhancement factor obtained from the probe filament interacting with a single pump filament and the plasma grating with the polarization of probe filament perpendicular to that of the two pump filaments are also measured for comparison. With $\theta$ set to 3.5° (see inset in Fig. 4(a)), the TH signal grows fast with the increase of time delay ${\tau }_d$, reaches a maximum at ${\tau }_d$ = 250 fs, and becomes stabilized at ${\tau }_d$ = 500 fs for the probe filament interacting with a single pump filament, which is in good agreement with the results obtained in Ref 22. With longer time delay, TH signal exhibits a slow monotonic decay that can be well described by the function $a{(1+\beta {\tau }_d)}^{-1}$ with decay constant $\beta \approx 24f{s}^{-1}$. The dependence of TH enhancement factor on the time delay ${\tau }_d$ with cross-polarization between the probe beam and the two pump beams used to form the plasma grating reveals a nearly similar phenomenon. For the interactions between the probe filament and the plasma grating with co-polarization between the probe beam and the two pump beams, the TH signal grows fast with the increase of time delay ${\tau }_d$and reaches a maximum at ${\tau }_d$ = 260 fs, which is similar with the results obtained from the probe filament interacting with a single pump filament. However, it becomes stabilized at ${\tau }_d$ = 10³ fs, which is much longer than that obtained from two filaments interaction. The TH signal also displays a decay which can be described by the same function $a{(1+\beta {\tau }_d)}^{-1}$, but the decay constant becomes$\beta \approx 6f{s}^{-1}$, which means a slower decay of the free-electron density inside plasma grating.

 

Fig. 4 The dependences of (a) TH enhancement factor on the time delay ${\tau }_d$ (black solid line, red dash line, and blue dot line denote the results generated from the probe filament interacting with the plasma grating with co-polarization and cross-polarization, and a co-polarized pump filament, respectively), and (b) normalized TH intensity on the relative polarization (black circles, and red triangles denote the results obtained by turning the polarization of probe filament, respectively) measured with $\theta$ set to 3.5°.

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For the time delay ${\tau }_d$ = 260 fs, the normalized TH intensity as a function of the relative polarization shown in Fig. 4(b) exhibits an approximately exponential decay with the relative polarization. The maximum and the minimum TH generation are observed as the polarization of probe beam is parallel and orthogonal to that of the two pump beams. The TH intensity ratio $I_{cx}$, which is defined as the TH enhancement factor obtained from the probe filament interacting with the plasma grating with co-polarization divided by that with cross-polarization, is about 3. For comparison, the same measurement is done for probe filament interacting with a single pump filament. As shown in Fig. 4(b), they both have a similar tendency which is different from the results obtained from the interaction between two noncollinear filaments in Ref 24.

The dependences of TH generation on the time delay ${\tau }_d$ and the relative polarization point out different physical mechanisms responsible for TH generation enhancement. The TH generation efficiency is correlated to the free electron density in the plasma grating. It was reported the local peak plasma density was enhanced up to about 2 orders of magnitude within the plasma grating. Since the plasma fluorescence is related closely with the plasma density, from Fig. 2 it is easy to find that the plasma density is enhanced by the probe filament crossing through the plasma grating. The enhanced plasma density can effectively enhance the third-order nonlinear optical susceptibility, which is known as bulk effect and has no relation with the relative polarization [28]. The so called bulk effect is one of the possible reasons for the enhancement of the TH generation. According to our experimental results, the TH intensity has a relation with the relative polarization. As is known, the interactions between the probe filament and the plasma grating along the propagating direction can overcome intensively the effect of Gouy phase shift, and stop the energy of TH generated from probe filament flowing back to fundamental pulse. The TH produced in the leading and trailing parts of a filament possess opposite carrier phase [30], which results in the large cancellation of TH at the end of the filament. The presence of an intercepting plasma grating can break the large cancellation of the TH at the end of the filament. And this effect has a relation with the relative polarization [30], so it is the other possible reason for the enhancement of TH generation. The plasma grating may act as a filament blocker for the energy of the two pump filaments is low, and the polarization of the pump pulse will be no longer crucial for the TH enhancement if the contribution of the relatively weaker probe pulse to the formation of the plasma blocker is negligible. For the laser pulse energies of the three arms are nearly at the same level, the TH enhancement factor exhibits an approximately exponential decay with the relative polarization. The TH generation from the probe filament could be enhanced by these two effects for the situation of the co-polarization between the probe and the two pump beams, and only by bulk effect for the situation of the cross-polarization. About 65% of the TH generation enhancement is from the arrestment of Gouy phase shift for the situation of the co-polarization between the probe and the two pump beams.

For further information, the dependences of TH enhancement factor on the time delay ${\tau }_d$ and normalized TH intensity on the relative polarization with $\theta$ set to be quasi-vertical are shown in Fig. 5 . With $\theta$ set to be quasi-vertical (see inset in Fig. 5(a)), although the TH signal also grows fast as it does with $\theta$ set to 3.5°, it reaches the maximum at ${\tau }_d$ = 10³ fs and then decays differently. The decay can also be described by $a{(1+\beta {\tau }_d)}^{-1}$ but with the larger value of decay constant $\beta \approx 16f{s}^{-1}$ for co-polarized beams interaction and $\beta \approx 71f{s}^{-1}$ for cross-polarized beams, which means a quicker decay of the free-electron density inside plasma grating, comparing with the results obtained with $\theta$ set to 3.5°. The maximum enhancement factor is only about 185, which is much lower than 3760 obtained with $\theta$ set to 3.5°, and the TH intensity ratio $I_{cx}$ obtained with $\theta$ set to be quasi-vertical is about 7 which is different from 3 obtained with $\theta$set to 3.5°. The arrestment of Gouy phase shift makes about 85% contributions to the enhancement of TH generation for the co-polarization between the probe and the two pump beams when $\theta$ is set to be quasi-vertical. It is clear that the length of interaction range is effectively extended with $\theta$ set to be 3.5° as shown in Fig. 2. The different length of interaction range should be the reason for the different experimental phenomena observed in Fig. 4 and Fig. 5. As is known, the bulk effect is related with the plasma thickness [28] and the harmonic generation increases with the length of interaction range. Although it was not reported that the arrestment of Gouy phase shift has a relation with the plasma thickness, it is reasonable to hypothesize that it may increase with the increase of plasma thickness, because our experimentally observed enhancement factor with $\theta$ set to be 3.5° is about 20 times higher than that with $\theta$ set to be quasi-vertical. Generally, it is the length of interaction range, which induces the change of the bulk effect and the arrestment of Gouy phase shift, results in the different TH intensity ratios $I_{cx}$ and the different enhancement factors when $\theta$ is set to be 3.5° and quasi-vertical.

 

Fig. 5 The dependences of (a) TH enhancement factor on the time delay ${\tau }_d$ (black solid line, and red dash line denote the results measured from the probe filament interacting with the plasma grating with co-polarization, and cross-polarization.) and (b) normalized TH intensity on the relative polarization measured with $\theta$ set to be quasi-vertical.

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For more information about the effect of plasma grating on the enhancement of TH generated from probe filament, the crossing angle $\phi$ between the two pump filaments is adjusted to change the spatial period $\Lambda$ of plasma grating by the function$\Lambda =\lambda /2\sin (\raisebox{1ex}{$\theta $}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$. The dependence of normalized TH intensity on the spatial period $\Lambda$ is shown in Fig. 6(a) with $\theta$ set to 3.5°, and Figs. 6 (b)-6(f), which share the same ordinate, are the fluorescence distributions of plasma gratings corresponding to the data dots in Fig. 6(a) in sequence. It can be seen that the plasma gratings formed, when $\phi$ is set to 5.2° and 7.3°, are not stable, and no plasma grating is formed when $\phi$ is set to 9.2°. In our experiment, the obtained first two data dots from stable plasma gratings drop a hint that TH generation may increase with the decrease of spatial period $\Lambda$. Comparing the results with and without plasma grating, it can be found that plasma grating plays an important role on TH generation enhancement. When $\theta$ is set to be quasi-vertical and $\phi$ is set to be 3.8°, Fig. 7(a) shows how the normalized TH intensity changes with the relative position of probe filament on the plasma grating along the propagating direction of the two pump filaments. The light interaction area appeared in Fig. 7(b) corresponds to the third experimental data dot in Fig. 7(a). As shown in Fig. 7(a), TH generation has nearly no change at the leading part of the plasma grating (corresponding to the plasma grating length from 0 to 400$\mu m$), and decays quickly in the trailing part of plasma grating. This tendency may suggest that the free electron density inside the plasma grating has nearly no change in the leading part of plasma grating within a length of about 400$\mu m$, and decays quickly in the trailing part.

 

Fig. 6 The dependence of normalized TH intensity on spatial period of plasma grating with $\theta$ set to 3.5°, and (b)-(f) are the fluorescence distributions of plasma grating corresponding to the data dots shown above.

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Fig. 7 The dependence of normalized TH intensity on crossing position on the plasma grating measured with $\theta$ set to be quasi-vertical.

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3.3 Broadening of the TH spectrum

When the time delay ${\tau }_d$ = 0 fs, and $\theta$ is set to be 3.5° and quasi-vertical, the measured TH spectra generated from the probe filament interacting with the plasma grating are compared with those measured from a single probe filament in Fig. 8 . The width of TH spectra is about 7 nm with $\theta$ set to be quasi-vertical, and about 32 nm with $\theta$ set to 3.5°. A peak, which is close to the main peak, appears on the VUV edge of TH spectrum when $\theta$ is set to be quasi-vertical, and more than three peaks appear when $\theta$ is set to 3.5°.

 

Fig. 8 The measured TH spectra generated from the probe filament. Black solid line, red dash line, and blue dot line denote the results obtained from a single probe filament, and the probe filament interacting with the plasma grating with $\theta$ set to be 3.5° and quasi-vertical, respectively.

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The broadening of the TH spectra is due to the self-steepening effect which dominates the generation of spectral components in the UV region and the Cross-phase modulation which can induce unsymmetrical broadening of TH spectrum when more than one femtosecond laser pulses propagate together. The intense interaction of multifilament can induce the enhancement of the laser peak intensity in the interaction area due to the light filed interference [42], and the higher fundamental laser peak intensity can result in more intense nonlinear effects, such as self-modulation, self-steepening effect, cross-phase modulation effect, i. e., the spatiotemporal modulation of the laser pulses in the interaction area is enhanced. The length of interaction area is effectively elongated when the crossing angle $\theta$ is very small, as shown in Fig. 2, and consequently, the length of spatiotemporal modulation is enhanced to cause the TH spectrum broadening to the VUV edge and having a much wider width. In general, the TH spectra of the probe filament can be broadened by the interaction between the probe filament and the plasma grating, and strongly depend on their crossing angle.

4. Conclusion

In summary, TH generation from the probe filament is controlled by the plasma grating formed by two noncollinear femtosecond filaments, and the contrast ratio of plasma grating can be enhanced by the probe filament. The enhancement of TH generation can be strongly affected by the relative polarization, the spatial period $\Lambda$of plasma grating and the crossing position between probe filament and plasma grating. The length of the interaction range between the probe filament and the plasma grating dominates the enhanced efficiency of TH generation, and the physical mechanisms behind TH generation enhancement are the bulk effect and arrestment of Gouy phase shift. By using the plasma grating, the TH spectrum generated from the probe filament is considerably broadened, which can be explained by the self-steepening effect and the cross-phase modulation depending on the crossing angle between the probe filament and the plasma grating. The present work sheds light on the generation of femtosecond pulse with higher energy and shorter pulse duration in the UV range.