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# Laguerre–Gaussian laser filamentation in ambient air

Silin Fu, Baptiste Groussin, Nicolas Cantonnet-Paloque, André Mysyrowicz, Aurélien Houard

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## ► To cite this version:

Silin Fu, Baptiste Groussin, Nicolas Cantonnet-Paloque, André Mysyrowicz, Aurélien Houard. Laguerre–Gaussian laser filamentation in ambient air. APL Photonics, 2025, 10 (5), pp.056108. 10.1063/5.0255740 . hal-05060129

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Submitted on 7 May 2025

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# Laguerre-Gaussian Laser Filamentation in Ambient Air

Silin Fu, Baptiste Groussin, Nicolas Cantonnet-Paloque, André Mysyrowicz, and Aurélien Houard*

Laboratoire d'Optique Appliquée, ENSTA Paris, Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, Palaiseau, France
(*Author to whom correspondence should be addressed: aurelien.houard@ensta.fr)

The filamentation of ultrashort laser pulses in air with Laguerre-Gaussian beams opens up numerous possibilities for atmospheric applications. In this work we demonstrate a novel method to measure the critical power for self-focusing of structured laser beam based on acoustic detection of laser ionization. Using this method, we determine the critical power of different Laguerre-Gaussian beams at 800 nm with ultrashort (50 fs) and sub-picosecond (500 fs) pulse durations. We also discuss the effect of the numerical aperture of the beam on the critical power and present the first measurements of the energy deposited by Laguerre-Gaussian filaments in air. Our results reveal an unexpected influence of the laser pulse duration on the filamentation of vortex beams.

## I. Introduction

Over the past decade, Laguerre-Gaussian (LG) beams have attracted significant attention due to their unique optical properties [1-4]. In the field of nonlinear optics, filamentation generated by LG beams in air have demonstrated remarkable application potential. The hollow structure of LG beams provides inherent advantages to create air waveguides through filament-induced thermal heating [5-7]. Such beams can also be used to optimize optical communication through fog and clouds thanks to the cloud clearing effect produced by high repetition rate filamentation [8,9]. Furthermore, LG filaments can homogenize the energy deposition along filamentation [6] and extend the length of electric discharges guiding when compared to a Gaussian beam [10], opening new possibilities for optimizing laser-controlled lightning [11]. Vortex laser beams can transfer their orbital angular momentum (OAM) to the THz radiation [12,13] or to the UV lasing of nitrogen produced during filamentation [14,15]. Moreover, vortex beams have been employed to enhance the stability of laser induced breakdown spectroscopy [16].

Despite their proven importance, knowledge of the properties of LG beam-generated filaments in air remains relatively scarce. One of the most critical parameters of filaments is the critical power, defined as the minimum peak power required to initiate self-focusing $P_{cr}$. For a beam with a gaussian or supergaussian profile the critical power can be expressed as $P_{cr}^{(0)} = \alpha(\lambda^2/4n_2n_0)$, where $\alpha$ is a constant dependant upon the initial beam shape, $\lambda$ is the central wavelength of the laser pulse, $n_0$ is the linear refractive index, and $n_2$ is the nonlinear refractive index of the medium [17,18]. In 2008, Fibich et al. developed a simple expression of the critical power for self-similar collapse of a CW optical vortex of topological charge $m$ as $P_{cr}^{(m)} \approx 2\sqrt{3}mP_{cr}^{(0)}$ [19]. This approximate formula predicted that the critical power should scale linearly with the topological charge $m$. In 2013, Polynkin et al. experimentally measured the critical power of focused vortex beams with 38 fs duration using a capacitive device to detect the plasma formation [20]. To fit their results with the formula of Fibich et al. they had to introduce a correction factor of 1.7 resulting in the formula $P_{cr}^{(m)} \approx (2\sqrt{3}m/1.7) P_{cr}^{(0)}$. Finally, in 2023, Liang et al. measured the critical power of vortex beams with 50 fs duration with $m = 1, 2$, and 3 through side fluorescence measurements [21]. They found that the critical power was scaling approximately as $P_{cr}^{(m)} \approx (1.6m + 1) P_{cr}^{(0)}$, which is very close to the result of [20]. They also observed that the ratio $P_{cr}^{(m)}/P_{cr}^{(0)}$ was not very sensitive to the numerical aperture of the beam.

Note that these studies were conducted with ultrashort laser pulses (less than 50 fs), which are not well suited for long distance propagation due to group velocity dispersion [22, 23]. It is
therefore interesting to study LG beams with longer pulse duration (> 100 fs) that are typically used for atmospheric applications [6,7,10,11,24,25]. In this case we expect the rotational effects of oxygen and nitrogen molecules in the air to increase the nonlinear refractive index ($n_2$) and to modify the filamentation critical power [18, 26].

To characterize filaments, another important parameter is the deposited energy [27, 28]. The primary mechanism for energy deposition is the generation of plasma, through multiphoton and impact ionization. Additionally, Raman absorption by air molecules can also produce energy loss of laser pulses [29]. The energy stored in the kinetic and potential energy of the plasma free electrons and the rotational energy of air molecules is ultimately converted into thermal energy after plasma recombination and rotational thermalization. Depending on the experimental conditions, this heating can range from 100 K to over 1000 K. Heating expels matter from the center of the plasma channel, leaving behind a low-density air channel that can be used to produce long-distance air waveguides [5-7], to guide high-voltage discharges [10, 11] and to decrease the drag of supersonic vehicles [30]. All these applications depend on the magnitude of the air fluid dynamics induced by the filament, which is directly related to the density of deposited energy.

In this work, we characterize the filamentation properties of ultra-short Laguerre-Gaussian beams at 800 nm in air. We measure the critical power $P_{cr}$ and energy deposition of Laguerre-Gaussian filaments for different pulse durations and beam numerical apertures. We confirm that the critical power for self-focusing scales linearly with the topological charge of vortex beams, and increases with the pulse duration while it decreases with the numerical aperture. Our results also reveal an unexpected sensitivity of $P_{cr}^{(m)}/P_{cr}^{(0)}$ to the pulse duration, which cannot be simply explained by the delayed contribution to the nonlinear refractive index ($n_2$). For sector beams ($LG_{0x}$), the critical power is found to be proportional to the number of sectors, while reducing the numerical aperture of the beam also lowers the critical power. Regarding energy deposition, we observe that the total deposited energy decreases with increasing topological charge of the Laguerre-Gaussian beam.

<figure><img src="image_4.png" /><figcaption>FIG. 1. Generation of the Laguerre-Gaussian beams. The supergaussian mode from the laser output is sent through a phase mask to produce a vortex beam, corresponding to the $LG_{40}$ mode (a), or a sectors beam, corresponding to the $LG_{03}$ mode (b). The first index refers to the topological charge and the second to half the number of sectors.</figcaption></figure>

## II. Experimental Setup

The laser used for filament generation is an Amplitude Ti:Sa CPA laser system, emitting pulses at a repetition rate of 10 Hz with a central wavelength of 800 nm. The maximum pulse energy is 250 mJ, and the pulse duration can be varied from 50 fs to 8 ps. The output beam profile is in a supergaussian mode (as illustrated in Fig. 1). LG beams are generated using phase plates manufactured by Holo/Or. Due to the size limitation of the phase plates, our beam was limited to a diameter of 1.5 cm. Filaments were generated in air by focusing the beam with a convex lens with $f = 1000$ mm corresponding to a numerical aperture of 0.015 or with a lens $f = 5000$ mm corresponding to a numerical aperture of 0.003.

For high charge vortex beams ($m > 2$, corresponding to $LG_{m0}$ modes), the most
common phase plates are spatial phase plates, which exhibit step phase changes due to fabrication limitations. This introduces fine diffraction structures in the beam profile, as shown in Fig. 1(a). For sector beams (corresponding to $LG_{0x}$ modes), the phase pattern alternates between $\pi$ and 0, resulting in the beam profile being divided into 2x sectors, as shown in Fig. 1(b).

The low energy laser intensity profile produced by the different vortex beams has been measured at the focus with a CMOS camera for the two focusing conditions. The result is presented in Fig. 2 (a) and (b). The impact of the beams at high energy on photosensitive paper is also shown in Fig. 2(c) revealing the presence of distinct filaments whose number increases with the vortex charge. Note that in the case of the supergaussian beam we expect to form multiple filaments, but they are too close to be distinguished on the burn photosensitive paper.

<figure><img src="image_5.png" /><figcaption>FIG. 2. Low energy beam profiles measured at the focus of the beam by a CMOS camera for the different vortex beams with a focusing lens $f= 1000$ mm (a) and $f= 5000$ mm (b). Impact on photosensitive paper at high energy (25 mJ) with a focusing lens $f= 5000$ mm (c). The green part corresponds to the presence of filaments.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIG. 3. Setup used to measure the critical power of the filaments and the deposited energy.</figcaption></figure>

### III. Methods

For the measurement of the critical power $P_{cr}$, we used both fluorescence and acoustic measurements. The fluorescence method relies on the detection of the Nitrogen fluorescence from the plasma, close to the focus of the beam [21]. The side fluorescence was collected and focused on a Photomultiplier (PMT). Before the PMT, we used a color bandpass filter BG39 to
<figure><img src="image_7.png" /></figure>

remove the fundamental frequency at 800 nm. This signal was measured as a function of the input laser energy. A typical curve with two different slopes is observed. A linear fit of these two slopes allow the determination of the threshold for filamentation at the crossing of the two lines (see Fig. 4). In this study we chose to compare this method with a second detection technique based on the measurement of the pressure wave emitted by the plasma. This pressure wave is essentially proportional to the density of deposited energy in the plasma. Acoustic emission from filaments have been widely used to determine the filament extension in air [6,28,31-33]. It presents the advantages to have a very low noise level and a very high sensitivity. It is also much easier to use and to align than the PMT detector. In the acoustic measurement method, we used a wide-bandwidth microphone (GRAS Model 46BH 1/4") connected to an amplifier (GRAS model 12PK) placed 30 mm away from the filament. The experimental setup is presented in Figure 3. For fluorescence measurements, we used the same setup as in [21]. For the measurement of energy deposited in the filaments, we used a Joulemeter with a 10 µJ precision and a detecting surface of 50*50 mm² (Gentec model QE50LP-H-P) to measure the energy of the beam before and after the filament generation. The difference between these measurements represents the total deposited energy [28].

IV. Results and discussion

First, we measured the fluorescence and acoustic signal emitted sideways as a function of the input laser energy in the case where a 50 fs pulse was focused with a f= 1000 mm lens. We consider the cases of the supergaussian beam (SG) and the vortex m = 1. As shown in Fig. 4, both signals increase gradually at lower energies but rises sharply after reaching a certain threshold. By applying two linear fits, this threshold for self-focusing can be identified as the change in slope in the dependence of fluorescence and acoustic energy on input energy. This result is fully consistent with fluorescence measurement [21], where this rapid increase signifies that the plasma quickly reaches the clamping intensity, corresponding to the critical power. Using fluorescence, for SG and vortex m = 1 beams, the critical power was measured to be 4.9 GW and 11.3 GW respectively. By comparison, the acoustic signal measurements yielded 4.7 GW and 11 GW. In this case, the acoustic method gives a critical power value 5 % lower than the fluorescence measurement. We attribute this difference to the fact that the acoustic detection is less sensitive to the longitudinal alignment of the detector with respect to the focus position of the beam. It can therefore detect the formation of plasma, even if it not perfectly centered in the detection setup.

<figure><img src="image_8.png" /><figcaption>FIG. 4. Comparison of the side fluorescence method and the acoustic method for the determination of critical power: Fluorescence signal (back points) and acoustic signal (violet point) measured as a function of the laser input energy in the case of the supergaussian beam (SG) and vortex beam with *m* = 1. The black and pink dashed lines indicate the fitting curves used to determine the critical power.</figcaption></figure>
We then measured the acoustic signal generated by different pulse energies for two pulse durations: the shortest pulse duration of 50 fs provided by our laser system and a sub-picosecond duration of 500 fs. Figure 5 shows the acoustic signals produced by vortex beams, as well as 4-sectors and 6-sectors beams, at varying energies.

<figure><img src="image_9.png" /><figcaption>FIG. 5. Amplitude of the acoustic signal measured as a function of the energy of the incident pulses. (a) and (b) are measured with a laser pulse duration of 50 fs, (c) and (d) with a pulse duration of 500 fs. (a) and (c) show the results for supergaussian beams and vortex beams. (b) and (d) show the results for 4-sectors and 6-sectors beams.</figcaption></figure>

Table 1 presents the critical power determined for different LG beams based on the results in Figure 5. It can be seen that for the vortex beams, the critical power increases with the increase in topological charge *m*. This overall trend is consistent with previous studies [20, 21]. For a 50 fs pulse duration, the measured critical power for sector beam corresponds approximately to the critical power for a SG beam $P_{cr}^{(0)}$ multiplied by the number of sectors. This result is easily understandable since each sector of the beam is propagating independently.

<table><caption>Table 1. Critical energy $E_{cr}$ and power $P_{cr}$ for self-focusing of LG beams determined by acoustic measurement.</caption><thead><tr><th></th><th>SG</th><th>m = 1</th><th>m = 2</th><th>m = 4</th><th>m = 8</th><th>4-sectors</th><th>6-sectors</th></tr></thead><tbody><tr><td>E<sub>cr</sub> (τ = 50 fs) in mJ</td><td>0.22</td><td>0.55</td><td>1.07</td><td>1.66</td><td>3</td><td>0.81</td><td>1.16</td></tr><tr><td>P<sub>cr</sub><sup>(m)</sup> (50 fs) in GW</td><td>4.4</td><td>11</td><td>21</td><td>33</td><td>60</td><td>16</td><td>23</td></tr><tr><td>P<sub>cr</sub><sup>(m)</sup>/P<sub>cr</sub><sup>(0)</sup> (50 fs)</td><td>1</td><td>2.5</td><td>4.77</td><td>7.5</td><td>13.6</td><td>3.64</td><td>5.27</td></tr><tr><td>E<sub>cr</sub> (τ = 500 fs) in mJ</td><td>0.55</td><td>1.53</td><td>4.4</td><td>8</td><td>13.8</td><td>4.9</td><td>5.8</td></tr><tr><td>P<sub>cr</sub><sup>(m)</sup> (500 fs) in GW</td><td>1.1</td><td>3.06</td><td>8.8</td><td>16</td><td>27.6</td><td>9.8</td><td>11.6</td></tr><tr><td>P<sub>cr</sub><sup>(m)</sup>/P<sub>cr</sub><sup>(0)</sup> (500 fs)</td><td>1</td><td>2.78</td><td>8</td><td>14.54</td><td>25</td><td>8.9</td><td>10.5</td></tr></tbody></table>

By performing a more detailed analysis of the critical power for vortex beams, we found that the increment in critical power appears to have a linear relationship with the topological number. In Figure 6 we have plotted the normalized critical power $P_{cr}^{(m)}/P_{cr}^{(0)}$ as a function of the vortex charge *m*. For comparison we also plotted the relations determined by Fibich *et al*.
for a CW beam, and by Polynkin *et al.* and Liang *et al.* for a 38 and 50 fs pulse. We can see that the measurement with the 50 fs pulse duration matches well the curves proposed in [20, 21] for similar pulse durations.

For the 500 fs case, we observe a critical power for the supergaussian beam $P_{cr}^{(0)}$ that is 4 times smaller than that measured with a duration of 50 fs. This factor of 4 can be explained by the increase of the nonlinear index $n_2$ with the pulse length, due to the rotational response of the medium [26]. However, the normalized critical power $P_{cr}^{(m)}/P_{cr}^{(0)}$ should be independent on $n_2$ and only depend on the transverse beam profile. We therefore expect to observe the same normalized critical power for 50 fs and for 500 fs. As shown in Figure 6, the normalized critical power is about two times larger with 500 fs than with 50 fs. However, this 500 fs case agrees well the analytical formula for the normalized critical power of a continuous wave initially proposed by Fibich *et al.* in ref [19]. We can then conclude that the laser pulse with pulse duration of 500 fs behaves like a continuous wave with low intensity. As suggested by Polynkin *et al.* the discrepancy with the measurement at shorter pulse durations might be due to the contribution of modulational instability that tends to reduce the critical power for self-focusing for shorter pulse duration, an effect not taken into account in the calculation of ref. [19].

<figure><img src="image_10.png" /><figcaption>FIG. 6. Critical power for self-focusing measured as function of the topological charge of the vortex beams. The critical power is normalized to the critical power of the supergaussian beam. The formulas proposed in [19-21] are also plotted as dashed lines.</figcaption></figure>

We also studied the influence of the beam numerical apertures on the critical power by changing the external focusing lens from $f=1000$ mm to $f=5000$ mm. Under otherwise similar conditions, we measured the critical power for the supergaussian (SG) beam and the vortex beam with $m=4$, as shown in Figure 7. For a supergaussian beam we found a critical power of 12 GW with a focal $f=5000$ mm and 4.4 GW with a focal $f=1000$ mm. In the case of a vortex with $m=4$ we obtained a critical power of 80 GW with a focal $f=5000$ mm and 33 GW with a focal $f=1000$ mm. We can therefore conclude that the numerical aperture has a significant impact on the critical power of a beam since it is increased by a factor ~3 when the numerical aperture is reduced by a factor 5.

By contrast, the ratio $P_{cr}^{(m)}/P_{cr}^{(0)}$ appears to be almost independent from the numerical aperture. For $f=5000$ mm, $P_{cr}^{(m)}/P_{cr}^{(0)} = 6.67$, while for $f=1000$ mm it is 7.5 (as shown by the green cross in Figure 6).
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0255740

<figure><img src="image_11.png" /><figcaption>FIG. 7. Effect of the external focusing on the critical power for self-focusing of a beam. The black dots represent the case of the supergaussian beam. The green dots represent the case of the vortex beam with $m = 4$.</figcaption></figure>

We also measured the deposited energy of filaments generated by Laguerre-Gaussian beams. Firstly, we measured the deposited energy at different incident energies for pulse durations of 50 fs and 500 fs, as shown in Figure 8. Here, we used a focusing lens $f = 1000$ mm. As expected, it can be seen that the onset of energy deposition roughly corresponds to the critical power for self-focusing previously measured.

In the case of vortex beams, after an initial phase where the energy increases slowly, we observe a linear increase of the deposited energy. We also note that for a fixed input energy, the deposited energy decreases with increasing topological charge. This is not surprising; in previous sections, we found that the critical power also increases with increasing topological number, meaning that higher topological number generate fewer plasmas at the same energy, resulting in lower deposited energy.

Furthermore, for a pulse duration of 50 fs, the deposited energy of 4-sectors and 6-sectors beams appears to be similar. However, as the energy increases, the deposited energy of 4-sectors gradually surpasses that of 6-sectors, possibly due to the generation of superfilaments by 4-sectors at higher energies [34].

Under long-pulse (500 fs) conditions, the deposited energy trend for LG filaments is similar to that of short pulses, as shown in Figures 8(c) and (d). The deposited energy decreases with increasing topological number. However, it is evident that the deposited energy for long pulses is significantly lower than that for short pulses. This is because, although long pulses have a higher nonlinear Kerr refractive index ($n_2$) and thus a lower critical power, the significant difference in peak power results in much less plasma generation.
<figure><img src="image_12.png" /><figcaption>FIG. 8. Deposited energy in LG filaments for incident pulse durations of 50 fs ((a) and (b)) and 500 fs ((c) and (d)). (a) and (c) show the results for SG and vortex beams, while (b) and (d) show the results for 4-sectors and 6-sectors beams. A $f=1000$ mm focusing lens was used for external focusing.</figcaption></figure>

## Conclusion

We demonstrated a novel, simple, and reliable method to measure the critical power of filaments generated by Laguerre-Gaussian beams in ambient air, based on acoustic measurement. Using this method, we characterized the critical power for self-focusing of vortex and sector beams as a function of their topological charge, pulse duration and numerical aperture. Our findings reveal that while the numerical aperture has a minimal impact on the normalized critical power $P_{cr}^{(m)}/P_{cr}^{(0)}$, the pulse duration plays a significantly larger role, that could be explained by the contribution of modulational instability at very short pulse durations. Moreover, we present the first measurements of deposited energy by Laguerre-Gaussian beams, offering new insights into their interaction with air and paving the way for future atmospheric applications.

## Acknowledgments

This work has benefited from a grant managed by the Agence Nationale de la Recherche (ANR), as part of the program "Investissements d'Avenir" under the reference (ANR-18-EURE-0014).

## Data availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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