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Abstract
The filamentation of the femtosecond vortex beam has attracted much attention because of the unique filamentation characteristics, such as annular distribution and helical propagation, and related applications. The critical power for self-focusing of the femtosecond vortex beams is a key parameter in the filamentation process and applications. But until now, there is no quantitative determination of the critical power. In this work, we experimentally determine the self-focusing critical power of femtosecond vortex beams in air by measuring fluorescence using a photomultiplier tube. The relation between the self-focusing critical power and the topological charge is further obtained. Our work provides a simple method to determine the self-focusing critical power not only for vortex beams but also for Airy, Bessel, vector, and other structured laser beams.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since the discovery of filamentation of ultrashort laser pulses propagating in air [1], the related research on filamentation has aroused great interest, due to its various applications in weather control [2,3], remote sensing [4], air lasing [5,6], and THz radiation [7]. Some nonlinear optical processes are involved in filamentation, including self-focusing, multiphoton ionization, self-phase modulation, and self-steepening [8–11]. As the incident laser power exceeds a certain value in an optical medium, the self-focusing effect can overcome the diffraction of the beam, resulting in the collapse of the beam. The specific value is called the critical power for self-focusing, which is a crucial physical parameter in filamentation research and is very useful in various filamentation applications. Several methods for measuring the critical power have been proposed. Akturk et al. proposed a P-scan approach where the transmittance through an aperture in the far field is measured while gradually increasing the input power to successfully distinguish linear, moving focus, filamentation, and multi-filamentation regimes in gases, and obtained the critical power of femtosecond Gaussian beam [12]. Liu et al. studied the evolution of the longitudinal peak position of air ionization with the increase of the input laser energy, and obtained the self-focusing critical power of femtosecond Gaussian beam, where the used method can be called focus-shifting method [13]. Furthermore, the critical power of Gaussian beam has also been measured in Helium [14] and flame [15] using the focus-shifting method.
In recent decades, optical vortices in the laser field have attracted much attention due to many associated applications, such as optical tweezers [16], optical communications [17], microscopy, and imaging [18]. Recently, some novel physical phenomena and mechanisms for femtosecond vortex beams have been explored, including helical filaments [19], postponed filamentation [20], and annular distributed filaments [21]. The filamentation formed by the vortex beam has potential applications in waveguide fabrication [22] and filamentation-induced breakdown spectroscopy [23]. More recently, Fu et al. demonstrated that femtosecond vortex filamentation in air has inherent advantages in generating long-lived optical air waveguide for guiding optical pulses [24]. For these kinds of applications, the critical power for self-focusing is particularly useful. However, there are few studies on determining the critical power of femtosecond vortex beams [25–27]. Fibich et al. obtained a theoretical relation between the self-focusing critical power of vortex beam $P_{cr}^{(m )}$ and topological charge m: $P_{cr}^{(m )} = 4\sqrt 3 mP_{cr}^{(0 )}$, where $P_{cr}^{(0 )}$ is the self-focusing critical power of the Gaussian beam [25]. However, this critical power is calculated for the continuous waves. For the femtosecond pulses, the nonlinear refractive index coefficient may vary with pulse duration, resulting in a different value of the critical power [12,13]. Furthermore, the normal group-velocity dispersion can considerably increase the self-focusing critical power [28–30]. The critical power for self-focusing of femtosecond vortex beam also needs to be experimentally determined. Polynkin et al. have proposed an air gap conductivity method to obtain the ratio of $P_{cr}^{(m )}/P_{cr}^{(0 )}$ rather than the quantification of the $P_{cr}^{(m )}$ [27]. This is because in the measurement, the onset of self-focusing is identified by a signal threshold of the capacitive plasma probe which is chosen somewhat arbitrarily and possibly dependent on researchers’ prior experiences. As stated in their study, the air gap conductivity method is not suitable for quantitatively determining the critical power. So far, there is no report on experimental determination of the critical power for self-focusing of femtosecond vortex beam in air. The relation of $P_{cr}^{(m )} \approx \left( {2\sqrt 3 /1.7} \right)mP_{cr}^{(0 )}$ given by the air gap conductivity measurement is not in accordance with the theoretical relation [25–27], further confirming the necessary of quantitative measurement of $P_{cr}^{(m )}$ of the femtosecond vortex beam.
In this work, we propose to experimentally determine the critical power for self-focusing of femtosecond vortex beam in air by measuring the fluorescence emission from the air ionization with a photomultiplier tube (PMT). The critical powers of femtosecond Gaussian and vortex beams with different topological charges are obtained. The relation between the critical power and the topological charge is further obtained for different external focusing conditions. To the best of our knowledge, it is the first experimental measurement of the self-focusing critical power of the femtosecond vortex beam in air.
2. Experimental setup
The schematic of the experimental setup is shown in Fig. 1. A Ti:sapphire chirped pulse amplification laser system with a central wavelength at 800 nm, a repetition rate of 1 kHz, a diameter (1/e2) of 9.5 mm, and a pulse duration of 65 fs is used. The laser power can be continuously adjusted by a combination of a half-wave plate (HWP20-800B, LBTEK) and a polarizer (WP25L-UB, Thorlabs). The laser beam successively passes through a quarter-wave plate (QWP20-800B. LBTEK), a vortex plate (VRx-800, LBTEK), and another quarter-wave plate to generate a vortex beam. Then the vortex beam is focused in air by a lens with a focal length of f = 500 or 750 mm. The corresponding numerical apertures (NAs) are 1 × 10−2 and 6 × 10−3, respectively. Another f = 75 mm lens is used to collect fluorescence from the whole air ionization region into a PMT (CH253-2, Hamamatsu). A band-pass filter with a central wavelength of 337 nm (#65-128, Edmund) and two filters of ZWB1 are put before the PMT. The 337 nm spectral line in the fluorescence emission is from the second positive band system (C3Πu(ν=0) → B3Πg(ν’=0) transition) associated with the ionization of N2, which is usually used to study the filamentation evolution and determine the critical power for self-focusing [31,32]. The PMT signal is monitored and collected by an oscilloscope. Each collected signal is averaged over 30 times.
Fig. 1. The schematic of the experimental setup. HWP: half-wave plate. P: polarizer. QWP1 and QWP2: quarter-wave plate. VWP: vortex wave plate. L1, L2: lens. F: filters. PMT: photomultiplier tube.
3. Results and discussion
First, the self-focusing critical power of femtosecond Gaussian beam is studied by using the PMT fluorescence measurement (named PMT method for convenience) and the often-used method of focus-shifting, which can further verify the feasibility of the PMT method we proposed in this work. For the focus-shifting method, the PMT is replaced by a CCD to image the air ionization distribution, and a ZWB1 filter is used before the CCD to integrate the fluorescence emission around 350 nm and block the scattered fundamental laser light [13]. The input laser energy is gradually increased from a very low energy to a high energy enough for filamentation. The evolution of the on-axis fluorescence intensity with gradually increased laser energy is shown in Fig. 2(a). We can see that the intensity and region of the air ionization are getting stronger and broader with the increase of laser energy, respectively. The fluorescence peak positions are obtained through Gaussian fits to the on-axis fluorescence intensities, respectively, and the peak positions as a function of the input laser energy are plotted in Fig. 2(b). The evolution shows the same tendency as that obtained by Liu et al. [13]. The peak position does not change too much when the laser energy is relatively low, and this stage should indicate the linear propagation regime. With the further increase of laser energy, the peak position shifts even closer to the focusing lens. The cross point of the two red fitted lines highlights the deviation position of ∼129.15 µJ which can be regarded as the critical power point. It is ∼1.99 GW under our experimental condition. The 337 nm fluorescence PMT measurement is then accomplished. The evolution of the fluorescence intensity with incident laser energy is shown in Fig. 2(c). First, when the laser energy is very small, the fluorescence intensity remains very weak and does have an obvious change with the increase of energy. As the laser energy is gradually increased from ∼70 to ∼110 µJ, the fluorescence intensity increases very quickly. The obvious change in intensity indicates that the critical power should locate in this range. With the further increase of the energy, the fluorescence intensity continues to increase. This tendency of evolution is almost the same as that obtained using the focus-shifting method as shown in Fig. 2(b). By using the same strategy, the critical power can be obtained from the cross point (∼90.56 µJ) of two fitted lines shown in Fig. 2(c), which is ∼1.39 GW.
Fig. 2. (a) On-axis fluorescence intensity obtained by using a CCD and (b) corresponding peak position of the air ionization induced by femtosecond Gaussian laser beam when an f = 500 mm focusing lens is used. (c) The 337 nm fluorescence intensity measured using a PMT as a function of input laser energy. The red linear fitting lines in (b) and (c) are used to find the critical power for self-focusing of the femtosecond Gaussian laser beam in air. The coordinate origin of the propagation distance noted in (a) is taken to be the starting pixel 1 of the CCD.
By comparing the distribution of the data obtained by the two methods, we can see that the data from the focus-shifting method has a larger error and higher fluctuation which can also be seen from noisy curves in Fig. 2(a) and larger deviations of the data points from the red fitted lines in Fig. 2(b). It is because there is a high uncertainty to determine the peak position of the laser ionization when the ionization is very weak, and the laser ionization zone will fluctuate significantly from shot to shot when the laser energy is getting higher. One main reason for the large noise of the data is that the sensitivity of the CCD is not high enough, however, the situation also exists in the work by using sensitive ICCD [13]. But in the PMT method, the fluorescence of the whole air ionization region is collected into PMT, and thus the fluctuation of the fluorescence from shot to shot is much smaller. On the other hand, the PMT method is faster and more convenient than the focus-shifting method to determine the critical power. First, the PMT method does not need an expensive CCD or ICCD camera. Second, the PMT method allows more sensitive measurements of weak signals because the PMT collects the fluorescence light and amplifies it instead of imaging the air ionization zone. The points mentioned above illustrate that the proposed PMT method has many advantages over the focus-shifting method in determining the self-focusing critical power.
By using the PMT method, the critical powers of vortex beams with different topological charges are then studied. Due to the limitation of the laser energy in our experiments, the laser beams with topological charge m = 1, 2, and 3 are chosen. The results are shown in Fig. 3. It can be seen that the evolutions of the fluorescence intensity with increasing laser energy are very similar to that of Gaussian beam. For the three cases, the fluorescence intensity almost does not change when the input laser energy is relatively low, and then increases rapidly with further increase of laser energy. Furthermore, as expected, with the increase of the topological charge, the rapid increase of the fluorescence intensity occurs at a higher laser power. Obvious intensity changes happen at ∼200, ∼340, and ∼380 µJ for the three laser beams respectively, i.e., more laser energy is needed to stimulate the nonlinear increase of the fluorescence intensity. The same fitting method is used to find the critical powers of the vortex beams. The cross points of linear fitting lines locate at 230.71, 410.66, and 512.71 µJ, respectively, as shown in Fig. 3, which correspond to 3.55, 6.32, and 7.89 GW for the beams with topological charge m = 1, 2, and 3, respectively. The obtained self-focusing critical powers of the vortex beams increase with the increase of the topological charge.
Fig. 3. Fluorescence intensity as a function of input laser energy for vortex beams with topological charges of (a) 1, (b) 2, and (c) 3, respectively. (Black solid blocks show experimental data and the red lines are linear fitting lines). An f = 500 mm focusing lens is used. Insets show the fluence distributions of the femtosecond vortex beam at the focal plane for different laser energies and topological charges.
Several fluence distributions of the laser beam at the focal plane are added in the insets of Fig. 3. The distributions are imaged by using a 4f system which consists of a pair of optical wedges and lenses, neutral density filters, and the CCD, and the system is located at ∼50 cm away from the air ionization region along the laser propagation direction to avoid damage of optics. When the input laser energy is relatively low (see 300 µJ case in Fig. 3(c)), the transverse fluence maintains a ring distribution with a singularity in the center, almost the same distribution as that of input vortex beam. There are two intensity maxima around the central singularity of the beam. When the input laser energy is higher than that value corresponding to the critical power, two hot spots are observed in the ring. During the propagation of the laser beam, both the intensity ring of the beam and spot positions are stable from shot to shot. It can also be seen that the vortex ring with a singularity in the center is kept during propagation. Furthermore, by comparing the evolutions of the fluorescence intensity with increasing laser energy for Gaussian and vortex beams shown in Fig. 3, the formation of multiple spots in the case of vortex beams does not affect the evolutions. However, it may bring serious influences to the measurement using focus-shifting method which will be discussed in the latter part.
It should be noted that the critical powers are only valid under similar experimental conditions to this work, because the powers are strongly dependent on external focusing conditions and the obtained critical powers are expected to be closer to the theoretical critical powers when using a lens with a longer focal length [33]. For the high-NA condition, the tight focusing dominates the nonlinear self-focusing, leading to remarkable plasma generation when the input laser power is much lower [34–36]. It will result in measured critical power lower than that in the case of low-NA. This is an important reason that the critical powers by our measurement are lower than the theoretical values [25], in addition to the different values of nonlinear refractive index coefficient for different pulse durations. The critical power of the vortex beam in our measurement is also lower than that in Ref. [27] for the same topological charge. This difference may also come from the different NAs used in the experiments. The NA in this work is much larger than that in Ref. [27]. Besides, the input laser conditions are also different in our experiment and in Ref. [27]. For example, the laser beams used in our experiment and in Ref. [27] are Gaussian and flattop beams, respectively. The pulse durations are also different. These factors should greatly influence the measured values of the critical power.
To demonstrate the influence of the NA on the measurement of the critical powers, the f = 500 mm focusing lens is replaced by a lens with a longer focal length of 750 mm. The PMT measurements are then repeated by using the lens. The results are shown in Fig. 4. As seen from the figure that, for both Gaussian and vortex beams, the evolutions of the fluorescence intensity with the increase of the input laser energy have the same tendency as those in the f = 500 mm case. After fitting to the experimental data, critical powers are obtained from the cross points of the fitting lines, which are 2.58, 5.17, 10.20, and 16.45 GW for Gaussian, vortex beams with m = 1, 2, and 3, respectively. The obtained critical powers are much higher than those for the f = 500 mm case, which is due to the use of lower NA and in good agreement with the analysis above and literature results [34–36].
Fig. 4. Fluorescence intensity as a function of input laser energy for (a) Gaussian and vortex beams with topological charges of (b) 1, (c) 2, and (d) 3, respectively, when an f = 750 mm focusing lens is used. (Black solid blocks show experimental data and red lines are linear fitting lines).
Finally, we can get the relation between the $P_{cr}^{(m )}$ and the topological charge m. The ratio of $P_{cr}^{(m )}$ to $P_{cr}^{(0 )}$ as a function of m for both cases of f = 500 and 750 mm are plotted in Fig. 5. As same as observed by Ploynkin et al. in their work [27], it is also evident here that the $P_{cr}^{(m )}$ grows approximately linearly with m. By referring to the format of the theoretical relation $P_{cr}^{(m )} = 4\sqrt 3 mP_{cr}^{(0 )}$ [25], and experimental one $P_{cr}^{(m )} \approx \left( {2\sqrt 3 /1.7} \right)mP_{cr}^{(0 )}$ [27], linear fitting to the data by forcing the intercept to 0 is used to determine the slope of the growth. The fitting lines for both f = 500 and 750 mm cases are plotted in Fig. 5(a). Relations of $P_{cr}^{(m )} \approx \;2.04mP_{cr}^{(0 )}$ and $P_{cr}^{(m )} \approx \;2.07mP_{cr}^{(0 )}$ for the two cases are obtained, respectively. The slopes of the linear fits are 2.04 and 2.07 with standard deviations of 0.20 and 0.16, respectively, which can be considered the same as the result in Ref. [27]. However, we notice that the points with m = 0 in Fig. 5(a) are relatively isolated, and the relation is not valid mathematically when m = 0. Therefore, a linear fitting by forcing the intercept to 1 is used to determine the relation between $P_{cr}^{(m )}$ and $P_{cr}^{(0 )}$. The fitting lines for the two cases are plotted in Fig. 5(b). Relations of $P_{cr}^{(m )} \approx \;({1.61m + 1})\;P_{cr}^{(0 )}$ and $P_{cr}^{(m )} \approx \;({1.64m + 1})\;P_{cr}^{(0 )}$ are obtained for the two cases, respectively. The two slopes of 1.61 and 1.64 have the standard deviations of 0.06 and 0.13, respectively. In this case, the equation of the relation is valid when m = 0, although the beam is not a vortex beam. From the above results of the fitting, we can see that the slopes for both case of f = 500 and 750 mm are almost the same for any one of fitting methods. Therefore, it can be concluded that the relation between $P_{cr}^{(m )}$ and $P_{cr}^{(0 )}$ as a function of m does not change with the NA conditions in our experiments. However, the vortex beams with only three topological charges are studied in our experiment, which may result in a linear fitting with less accuracy. More data are needed to find a more precise fitting. On the other hand, the two different formats of relations will have a larger and larger difference with the increase of the topological charge, which needs to be either verified by using a big enough topological charge or replaced with a new relation. Besides, the relations and critical powers are also different from the theoretical results [25,26]. The self-focusing critical power of vortex or other structured beams deserves further investigations.
Fig. 5. The ratio of the self-focusing critical power of femtosecond vortex beam to that of Gaussian beam as a function of topological charge for both cases of f = 500 and 750 mm. Two linear fitting methods by forcing the intercept to (a) 0 and (b) 1 are used for the two cases, respectively.
For vortex beams with donut-shaped intensity distribution, the process of self-focusing has a big difference from that for the Gaussian beam. Under the effects of modulation instability and self-focusing, the vortex beam will break up into several hot spots around its central singularity during the propagation. If the initial laser power is high enough, the hot spots will form filaments. As illustrated in the insets of Fig. 3, two or more intensity hot spots are formed around the central singularity of the vortex beams, which will form multi-filamentation when the input laser energy is high enough. As a consequence, as the input laser power increases from slightly lower than the critical power for self-focusing to higher powers, the longitudinal peak of air ionization fluorescence intensity may not move continuously closer to the focusing lens due to the hot spots formation and their competition. Concerning the applicability of the often-used focus-shifting method for measuring self-focusing critical power of vortex beams, we make further measurements by using the focus-shifting method. The results are shown in Fig. 6. The fluorescence intensity of vortex laser beams is much lower than that of Gaussian beam, which can be obviously seen from the traces of the on-axis fluorescence intensity for the cases of vortex beams, shown in Figs. 6(a1)–6(c1). The fluorescence peak positions are further obtained through Gaussian fits to the fluorescence intensities. The results are plotted in Figs. 6(a2)–6(c2). Compared with the Gaussian case, the fluorescence peak position of the air ionization of the vortex beam shows a complicated change with the increase of the input laser energy. The peak position does not monotonically change with the increase of energy. It will also move back and forth, which is evident especially for the higher order case of m = 3, as shown in Fig. 6(c2). One important reason should be that the formation of non-single hot spots and competition among them greatly influence the focus-shifting process, and another is that the traces of intensity are too noisy to have accurate and reliable Gaussian fittings. It is impossible to extract the value of the critical power by fitting the data points of peak positions shown in Figs. 6(a2)–6(c2). However, the peak positions still have a tendency of shifting closer to the focusing lens with the increase of input energy, especially for the cases of m = 1 and 2. Therefore, if a camera with high sensitivity is able to recognize clearly the low-intensity fluorescence emission from the air ionization, especially when the input energy is low, there is a possibility that the critical power could be extracted by using the same strategy of linear fittings to the data. Even so, the evolution of the peak position would still be wiggly due to the above-mentioned reason, and reliable critical power is hard to be obtained. Therefore, a unique advantage of the PMT method over the focus-shifting method lies in the reliability in the determination of the self-focusing critical power of femtosecond vortex beams.
Fig. 6. (a1)-(c1) On-axis fluorescence intensity obtained by using the CCD and (a2)-(c2) corresponding peak positions of the air ionization induced by femtosecond vortex beams with different topological charges of (a2) 1, (b2) 2, and (c2) 3 as a function of input laser energy, respectively. The coordinate origin of the propagation distance noted in (a1)-(c1) is taken to be the starting pixel 1 of the CCD. The f = 500 mm focusing lens is used.
4. Conclusion
In this work, the self-focusing critical powers of femtosecond Gaussian laser beam and vortex beams with different topological charges are obtained by measuring 337 nm fluorescence intensity of the air ionization region using a PMT. For the femtosecond vortex beams with topological charges of 1, 2, and 3, the measured critical powers are 3.55, 6.32, and 7.89 GW when using f = 500 mm focusing lens, and 5.17, 10.20, and 16.45 GW when using f = 750 mm focusing lens, respectively. Furthermore, the relation between the self-focusing critical power of vortex beams and the topological charge is obtained. Two relations of $P_{cr}^{(m )} \approx \;2mP_{cr}^{(0 )}({m \ne 0} )$ and $P_{cr}^{(m )} \approx \;({1.6m + 1})\;P_{cr}^{(0 )}$ are finally obtained by forcing the intercept to 0 and 1 in the linear fitting to experimental data, respectively, which will not change for different NAs (1 × 10−2 and 6 × 10−3) under our experimental conditions, however, further studies using larger topological charges are needed to amend the relations. Our results further demonstrate that the often-used focus-shifting method to determinate the critical power of Gaussian beam is not applicable for vortex beams by using the CCD camera, however, more deep investigations are needed. The proposed PMT method is demonstrated to be a simple and sensitive method to determine the self-focusing critical power of vortex beams and could be used in cases of Airy, Bessel, vector, and other structured beams.
Funding
National Natural Science Foundation of China (12074228, 11874056, 12204282, 11774038); Natural Science Foundation of Shandong Province (ZR2021MA023); Taishan Scholar Project of Shandong Province (tsqn201812043); Innovation Group of Jinan (2020GXRC039).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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