1. Introduction

Femtosecond laser filamentation has been extensively studied for two decades since its first report, primarily due to its rich nonlinear physics and wide range of potential applications, including remote spectroscopy [1,2], optical waveguide [3,4], lightning control [5], air lasing [6,7], fuel ignition [8], filament induced breakdown spectroscopy [9], fabrication of micro-nano structures [10,11], combustion diagnostics [12], and so on. Filamentation of the vortex femtosecond pulse in particular attracts great attention due to its unique characteristics, such as postponed filamentation at a remote distance [13], helical filaments array in a ring [14], and stable filamentation in turbulence [15]. It is widely accepted that the femtosecond laser filamentation primarily appears from the effects of Kerr self-focusing and defocusing of the self-generated plasma. This complex nonlinear propagation can be described as follows. When the femtosecond pulse has a peak power slightly higher than the critical power for self-focusing P_cr, the self-focusing of the laser beam will exceed the diffraction effect, and the laser intensity keeps increasing until ionization occurs [16], resulting in the formation of a relatively weak filament. This type of filament can be called immature filament. As the peak power of the incident femtosecond pulse further increases, the intensity of the immature filament will increase obviously. When the peak power reaches a certain power, the intensity in the filament is saturated, a phenomenon commonly referred to as intensity clamping [17,18]. As the laser power continues to increase, the intensity of the filament is hardly changed. The filament in this stage is called mature filament [19,20]. The laser power required to drive the intensity clamping is usually regarded as the critical power for mature filamentation (named as P_fil for simplicity). The two kinds of critical powers, which can distinguish femtosecond laser nonlinear propagation regimes, are usually used to give a rough estimate of the filamentation dynamics for the femtosecond laser pulse with different input powers. However, for the self-focusing critical power of the vortex femtosecond pulse, a significant difference exists between the experimental and theoretical results [21–23]. For the filamentation critical power of the vortex femtosecond pulse, there is only a rough measurement by defining the appearance of interference of 589 nm spectral component as the occurrence of filamentation [24]. This determination is somehow arbitrary. It is necessary to measure the two kinds of critical powers and distinguish the nonlinear propagation regimes for the vortex femtosecond pulse.

Many experiments and simulations have been performed to determine one critical power of the nonlinear propagation regimes for the femtosecond Gaussian beam in various media [25–27]. But the critical powers for self-focusing and mature filamentation of the femtosecond Gaussian beam cannot be applied in the vortex femtosecond pulse due to the different filamentation dynamics. For the vortex femtosecond pulse, multiple filaments will be formed in a ring around the central singularity once the filamentation is triggered, unlike the formation of a single filament in the beam center for the femtosecond Gaussian beam [4,14,21,28]. Due to this different filamentation dynamics, the methods of measuring the two critical powers of the femtosecond Gaussian beam cannot be used in the measurement for the vortex femtosecond pulse. More specifically, the focus-shifting method is not suitable, because the peak position of fluorescence intensity for the vortex femtosecond pulse does not monotonically change with the increase of energy, which should be due to the formation of non-single hot spots and their competition during the vortex beam collapsing, resulting in the indistinguishable propagation regimes [29]. For the same reason, the P-scan method [30], where an aperture is placed in the beam center in the far field to monitor the transmitted signal as the laser energy is increased, cannot be applied either. Most recently, we experimentally determined the self-focusing critical power of vortex femtosecond pulse in air by measuring fluorescence of the whole ionization region using a photomultiplier tube [29]. But this method is not suitable for measuring the filamentation critical power, because the fluorescence intensity continues to increase almost linearly and the critical point of the mature filamentation cannot be distinguished. Up to date, there is no suitable method to distinguish the nonlinear propagation regimes of the vortex femtosecond pulse and determine the two kinds of critical powers.

In this work, we adopt the method of spectral broadening evaluation to distinguish the nonlinear propagation regimes of self-focusing and mature filamentation for femtosecond vortex laser pulses in fused silica, based on the dependence of the maximum positive frequency shift of the supercontinuum spectrum on the clamped laser intensity [18]. By gradually increasing laser pulse energy, the spectral evolution of the blue-side cut-off wavelength is analyzed, and different propagation regimes of linear, self-focusing, and filamentation of vortex femtosecond pulses can be well distinguished (we call this method S-scan). In addition, a relation between critical powers and topological charges (TCs) is obtained. The results of this study demonstrate that the S-scan technology can be applied to distinguish the propagation regimes of both Gaussian beams and structured beams, regardless of their initial intensity distribution.

2. Experimental setup

The laser source is an amplified Ti:sapphire femtosecond laser system (Spectra-Physics, Solstice Ace) with a central wavelength of 795 nm, pulse duration of 65 fs, a repetition rate of 1 kHz, and maximum pulse energy of 5 mJ. The laser beam has a nearly perfect Gaussian shape with a 10 mm diameter (1/e²). A telescope system (not shown in Fig. 1) consisting of a convex lens (with a focal length f = 200 mm) and a concave lens (f = −100 mm) is used to reduce the beam width to ∼5 mm (1/e²). Then a spatial light modulator (SLM, HAMAMATSU, X12138-02) is used to generate vortex beams. Different vortex phase maps are encoded on the SLM to generate vortex beams with different TCs. The vortex beam is focused with a lens (f = 200 mm) into fused silica (with a size of 50 mm × 40 mm × 30 mm). The fused silica is placed 180 mm behind the lens. Incident laser power can be continuously adjusted by an energy adjustment system consisting of a half-wave plate and a wire-grid polarizer (Thorlabs, WP25L-UB). The input laser energy is monitored by a power meter with a photosensitive probe (Thorlabs, S130VC). The output laser pulses from fused silica are collected into an integrating sphere and recorded by a spectrometer (USB-4000, Ocean Optics) which has a detectable spectral range from 200 nm to 1100 nm. The spectra variations are analyzed to distinguish the nonlinear propagation regimes of vortex femtosecond pulses in fused silica. The cut-off wavelength on the blue side is estimated from the wavelength at which the spectral intensity decreases close to the background in our work. The side-view fluorescence under different experimental conditions is also recorded by a CCD camera (Andor, iKon).

 

Fig. 1. Experimental setup. EA: energy attenuator; M1 and M2: high reflection mirrors; SLM: spatial light modulator; L1 and L2: lens; IS: integrating sphere; SM: spectrometer.

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3. Results and discussion

The spectral broadening of femtosecond Gaussian beams with different pulse energies is studied using the S-scan method. The energy is gradually increased, starting from very low energy to a high level. Figure 2 (a) shows the spectra as a function of input laser energy. As can be seen, the spectrum is slightly broadened at the lower energy. When the laser energy is below 0.34 µJ, we notice little and symmetrical spectral broadening in the blue and red sides. As the laser energy increases, the blue-side spectrum is getting broader and broader. A noticeable spectrum extension is observed as the laser energy increases from 0.34 µJ to 0.44 µJ. Beyond 0.46 µJ, the spectral broadening on the blue side remains almost unchanged. These results indicate that the spectral broadening is sensitive to the input laser energy, which also indicates different evolutions of the laser propagation.

 

Fig. 2. (a) The spectra generated by the femtosecond Gaussian beam with different input laser energies and (b) the corresponding evolution of cut-off wavelength as a function of the input energy.

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The blue-side cut-off wavelength as a function of laser energy is studied to quantitatively evaluate the spectral broadening. In this work, the cut-off wavelength is defined as the spectral intensity being obviously higher than the background. Each spectrum is integrated over a total of 100 laser shots and averaged 20 times. Even so, the spectrum around the cut-off wavelength is still very fluctuant. Therefore, polynomial fitting is used to further increase the precision of the analysis. For the Gaussian case (Fig. 2(a)), we choose the wavelength at 10¹ level as the cut-off wavelength. The result is plotted in Fig. 2(b). From the evolution of the cut-off wavelength with the increasing input laser energy, we observe three distinct regimes in the curve. First, when the input laser energy is below ∼0.34 µJ, the cut-off wavelength shifts slowly towards the blue side. The symmetrical spectral broadening indicates the dominated mechanism is the self-phase modulation in this regime. Second, there is a rapid change when the input laser energy is further increased from ∼0.34µJ to ∼0.44µJ. The rapid broadening of the spectrum should be due to the occurrence and strengthening of ionization. Third, as the laser energy exceeds ∼0.46 µJ, the cut-off wavelength does not change significantly with further increase in input laser energy, which indicates the intensity clamping in filament and the formation of mature filament [18]. To find the two deviation positions during the three-regime evolution of the cut-off wavelength, linear fittings to the curve are made and plotted in the Fig. 2(b). The first cross-point of two fitted lines highlights a deviation position of 0.35 µJ and can be considered as the self-focusing critical threshold, calculated to be approximately 5.38 MW. The value of the second cross-point of two fitted lines can be considered as the critical power for mature filamentation, which is 6.77 MW. Thus, we can see that the S-scan method provides a straightforward, quick, and convenient way to distinguish the different propagation regimes of femtosecond laser pulse.

To the best of our knowledge, there are no reports on the measurement of the critical powers for self-focusing and filamentation of femtosecond laser pulse in fused silica. Theoretical value calculated from Marburger formula ${P_{\textrm{cr}}} = \frac{{3.77{\mathrm{\lambda }^2}}}{{8\mathrm{\pi }{\textrm{n}_2}{\textrm{n}_0}}}$ is generally used. By choosing n₂≈2.73 × 10⁻¹⁶ cm²/W measured by using spectral Z-Scan method [31], the critical power is approximately 2.42 MW. As a comparison, our experimental obtained value is slightly higher than the theoretical calculated value. One important reason should be that the solid medium has a high group velocity dispersion, which leads to a fast broadening of the pulse width during the propagating in the medium. For the critical power for mature filamentation in fused silica, there is no report either. However, Liu et al. obtained the P_fil (∼5.72 MW) for 170 fs femtosecond laser pulse in water [18], which can be a reference for our value of 6.77 MW.

Then we use the S-scan method to distinguish the nonlinear propagation regimes of vortex femtosecond laser pulses. The vortex beams with TC m = 1, 2, 3, 4, and 5 are chosen in our experiment. Figure 3 shows that the evolutions of the cut-off wavelength with increasing laser energy are very similar to that of Gaussian beam (Fig. 2(b)). The cut-off wavelength on the blue side remains almost unchanged at relative low input laser energies, and shows a rapid broadening with the gradually increased input laser energy until reaches a nearly constant value at the final stage. As expected, with the increase of the TC, both the critical powers for self-focusing and filamentation are getting higher. As shown in Fig. 3, for m = 1, 2, 3, 4, and 5, the first cross-points of fitted lines locate at 0.76 µJ, 1.35 µJ, 1.85 µJ, 2.32 µJ, and 2.54 µJ, respectively. Correspondingly, the self-focusing critical power P_cr is calculated to be 11.69 MW, 20.77 MW, 28.46 MW, 35.69 MW, and 39.08 MW, respectively. Furthermore, we can also determine the filamentation critical power (P_fil) from the second cross-points of fitted lines, which locate at 0.87 µJ, 1.54 µJ, 2.03 µJ, 2.49 µJ, and 2.80 µJ for m = 1, 2, 3, 4, 5, respectively. The calculated critical powers are 13.38 MW, 23.69 MW, 31.23 MW, 38.30 MW, and 43.08 MW. From the above results we can see that our study highlights the versatility of the S-scan method which can conveniently distinguish nonlinear propagation regimes of Gaussian and vortex beams in fused silica.

 

Fig. 3. The evolution of the cut-off wavelength of spectrum as a function of input laser energy for different TCs of 1, 2, 3, 4, and 5, respectively.

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Note that the critical powers should be strongly dependent on external focusing conditions and incident laser pulse durations. Our previous work has demonstrated that a lower numerical aperture condition is more favorable for accurate determination of critical powers in air [32]. On the other hand, Liu et al. found that P_cr in air will change from 10 GW to 5 GW when the pulse duration increases from 42 fs to 200 fs [26]. However, there are no reports in fused silica yet, which deserves further studies. Besides, we can also notice that the blue shift of the supercontinuum spectrum shows a fluctuation, increase, or decrease behavior as the pulse energy increases in the filamentation region (corresponding power higher than P_fil). A possible mechanism can be explained as follows: Due to the competition between the filament and energy background, both the intensity and length of the filamentation will undergo changes to some extent, particularly when the input power is relatively high. Furthermore, the intensity of filamentation plays a vital role in the blue-side extension of supercontinuum spectrum, as demonstrated in our previous work [33]. Thus, the behavior of the blue shift should depend on specific filamentation conditions. On the other hand, we choose the wavelength at 20 level of spectral intensity as the cut-off wavelength for the case of vortex laser beams. The choice of the intensity level will also slightly change the shapes of the curves in Fig. 3. However, we find that it does not influence the deviation position of the curve, i.e. determination of critical power for mature filamentation.

To confirm the measurement by the S-scan method, the fluorescence intensities and far-field beam images are also measured. The fluorescence intensity is recorded by a CCD camera. An additional QB21 filter is used before the CCD to block the scattered fundamental frequencies. The fluorescence spectra taken without and with the QB21 filter are plotted in Fig. 4(a), respectively. The transmission efficiency of the QB21 filter is also given in the inset. The fluorescence exhibits a continuous spectrum which is distinct different from that observed in air. Three peaks can be observed on the fluorescence spectrum at 450 nm, 650 nm, and 795 nm (fundamental frequency). The peaks at 450 nm and 650 nm are demonstrated to be generated from the complex electron dynamical processes in fused silica [34–37]. This observation of fluorescence spectra is in good agreement with previous findings [38,39]. It turns out that the use of the QB21 filter can effectively block the scattered light of fundamental frequencies. Figures 4(b1) to 4(d1) show the evolution of the on-axis maximum intensity for the vortex femtosecond laser pulses with three typical energies and topological charge m = 3. For the laser energies 1.7 µJ and 1.9µJ, there is no notable signal in the fluorescence measurements, and a distinct peak can be observed when the laser energy reaches 2.0 µJ. Correspondingly, the far-field images of the vortex femtosecond laser pulses with three typical energies and topological charge m = 3 are recorded and presented in the right column in the Fig. 4. For the laser energy 1.7 µJ, the vortex ring is still preserved, and no supercontinuum emission is observed. When the incident energy is 1.9 µJ, we can see that the color of the image is changed, which indicates the occurrence of spectral broadening. As the input laser energy increases to 2.0 µJ, the colorful far-field image indicates the generation of the supercontinuum. The far-field image then remains almost unchanged except for the increasing brightness as the energy continues to increase to 2.4 µJ (not shown). It indicates that the fluorescence induced by ionization is too weak in the linear propagation stage and the self-focus stage. Therefore, the fluorescence measurement is not suitable to distinguish the two stages. In contrast, the S-scan method provides a straightforward, quick, and convenient way to distinguish these processes (see Fig. 3 (c)).

 

Fig. 4. (a) The fluorescence spectrum and filtered by a QB21 filter (inset shows the transmission efficiency of the QB21 filter). The on-axis maximum intensity and the corresponding far-field images taken with a digital camera (Canon, ESORP) on a paper screen placed at 30 cm from the back surface of fused silica under different input laser energy of (b1)-(b2) 1.7 µJ, (c1)-(c2) 1.9 µJ, and (d1)-(d2) 2.0µJ with TC of 3.

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Finally, the relations between the two kinds of critical powers and TC are obtained from the above results, respectively, which are plotted in Fig. 5(a). As same as demonstrated by previous works [21,29], the $P_{cr}^{(m )}$ will grow nearly linearly with the TC increasing. It is shown in the Fig. 5(a) that the $P_{fil}^{(m )}$ also has the same trend of increase as the $P_{cr}^{(m )}$. By linearly fitting the critical powers, we can get the relations of $P_{cr}^{(m )} \approx 7.09m + 5.78\,({\textrm{MW}} )$, and $P_{fil}^{(m )} \approx 7.54m + 7.23({\textrm{MW}} )$, respectively. From the values of the critical powers and the two relations, we can see that the $P_{fil}^{(m )}$ is always higher than $P_{cr}^{(m )}$, which is one well-known precondition for filamentation. For Gaussian case, $P_{fil}^{(0 )}/P_{cr}^{(0 )}$=1.26, which is close to theoretical prediction values of 2–2.5 [40]. According to the relations obtained by linear fitting, the ratio will be getting smaller and smaller with increasing TC but is around 1.1, which suggests that to have filamentation of vortex beams in fused silica, the initial laser power should be higher than ∼1.1$P_{cr}^{(m )}$. Furthermore, to get the relation between the critical powers and TCs, the ratios of $P_{cr}^{(m )}/P_{cr}^{(0 )}$, $P_{fil}^{(m )}/P_{fil}^{(0 )}$ and TC are calculated and plotted in Fig. 5(b). By linearly fitting the data by forcing the intercept to 1, the relations are obtained: $P_{cr}^{(m )} \approx \,({1.34m + 1} )\,P_{cr}^{(0 )}$ and $P_{fil}^{(m )} \approx \,({1.13m + 1} )\,P_{fil}^{(0 )}$, respectively. The relations obtained in our work provide important references for the study and applications of filamentation of vortex pulses in condensed media. However, the slopes are different between the two relations, and also different with the case in air [21]. Further studies are needed to reveal the underlying physics.

 

Fig. 5. The relations between (a) $P_{cr}^{(m )}$, $P_{fil}^{(m )}$, (b) $P_{cr}^{(m )}/P_{cr}^{(0 )}$, $P_{fil}^{(m )}/P_{fil}^{(0 )}$, and TC. The black solid blocks and red dots present data of results, and correspondingly black and red lines are linear fitting lines to the data.

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4. Conclusion

In conclusion, the nonlinear propagation processes of the femtosecond laser pulse in fused silica have been studied using the S-scan method. The method effectively distinguishes the propagation regimes of the vortex femtosecond pulses, ranging from linear to self-focusing and filamentation. The critical powers for self-focusing and mature filamentation of both Gaussian and vortex beams are determined by analyzing the spectral cut-off wavelength variation. Furthermore, the relations between the critical powers for self-focusing and mature filamentation of the vortex beam and TC are obtained. The S-scan method could also be applied in gas or liquid condition, which will be of great help to related applications such as atmospheric sensing, vortex air lasing and the optical waveguide [4,7,41]. Our method could also be used to investigate the nonlinear propagation processes of Airy beams, Bessel beams, vector polarized beams, and other structured beams.