1. Introduction

Femtosecond laser filamentation is a propagation regime of intense laser pulses in nonlinear Kerr media, when the radiation forms extended and thin light channels, which length greatly exceeds the corresponding Rayleigh length [1]. In the case of near-infrared radiation propagating in air, this regime arises mostly from the dynamic interplay between Kerr self-focusing and plasma defocusing. High intensity preserving for long distances (up to kilometers [2,3]), extended plasma channels and large nonlinear interaction length make femtosecond filaments a promising medium for a number of applications: terahertz generation [4–7], remote atmosphere sensing and lightning control [8,9], ‘virtual’ waveguiding [10,11], pulse shortening [12], aircrafts aerodynamics enhancement [13], light induced breakdown spectroscopy [14]. Although the femtosecond filamentation phenomenon has been extensively investigated during the last two decades (see, for example, reviews [15–18]), the problem of filament parameters tailoring for applications still remains. One of the most important challenges here is to obtain multifilament arrays with regular transverse structure, which is particularly important for waveguiding and THz generation (as it allows to increase THz intensity and direct THz radiation into a narrow cone [7]). A number of approaches to obtain such structures has been proposed: multifilament pattern control by amplitude meshes and masks [19–21], beam ellipticity [22], stigmatic focusing [23], focusing with axicon [24] and by modulating the phase of the initial beam using diffractive optical elements (DOEs) or a spatial light modulator [25–28]. However, the approaches employing beam ellipticity, stigmatic focusing and axicones do not allow to obtain hollow-core structures suitable for waveguiding. Amplitude regularization techniques also fail to address this task if the laser beam is additionally focused by a lens, because of the zero-order diffraction maximum, formed at the center of the beam in the lens focal plane.

Using DOEs, one can obtain hollow-core multifilament structures even in the case of additional external focusing [26,29]. Due to the high spatial resolution and energy efficiency of the transmitting binary-phase DOEs, it is convenient to use them for generation of a given inhomogeneous spatial energy distribution of the focused laser beam. Methods of modeling and fabrication of such DOEs are well developed now [30–32]. It has been recently shown [33,34] that binary-phase DOEs forming Hermite-Gaussian modes introduce low distortions for the broadband radiation, hence such DOEs are suitable for spatial modulation of the high-power femtosecond laser beam [35]. However, this approach has also a substantial drawback. It is its reliance on the beam quality. As it has been recently shown [36,37], even the beam ellipticity can easily destroy the hollow-core multifilament pattern, leading to a new filament appearing at the beam center.

In this paper, we demonstrate hollow-core multifilament arrays obtained using another type of DOEs — a diffraction Dammann grating [31,38]. We experimentally compare arrays created by this DOE and a TEM$_{11}$ phase plate, which was previously used for regular multifilament arrays production in a number of works [26,27,36,37], discussing the effect of beam quality on the resulting multifilament pattern. Shot-to-shot reproducibility of the pattern affected by pulse energy fluctuations is another point we pay attention to. Plasma concentration and its evolution along the filament is an another parameter important for applications. We treat it by tracing energy density absorbed by the medium along the multifilament propagation axis. These measurements, albeit indirectly, provide us with the data on the relative plasma concentration in both two cases under investigation, along with the information about the filament length.

2. Experiment setup and methods

The experimental scheme is shown in Fig. 1. Laser radiation with up to 25 mJ pulse energy (if measured after the pulse compressor without a DOE), $50\pm 5$ fs pulse duration, 812 nm central wavelength, 7.5 mm beam diameter FWHM and 10 Hz repetition rate was produced by the Ti:Sa laser system. Additional focusing was obtained by introducing the lens L$\mathbf {_1}$ into the optical path. We used a lens with (267$\pm$5) cm focal length, corresponding to $\textrm {NA} \approx 2.5 \cdot 10^{-3}$. The choice is motivated by availability of a large volume of data on stochastic and amplitude-regularized multifilamentation at similar NA [39–42]. These results can be used as a reference for qualitative comparison between the phase and amplitude regularized multifilaments.

 

Fig. 1. Experimental setup. DOE — diffraction optical element, M$\mathbf {_1}$ and M$\mathbf {_2}$ — dielectric mirrors, L$\mathbf {_1}$, L$\mathbf {_2}$ and L$\mathbf {_3}$ — focusing lenses, $\boldsymbol{\mathrm{\lambda}}\mathbf{/2}$ — half-wave plate. In the insert: ultrashort-pulse-measurement experimental scheme. CL1–3 — cylindrical lenses; KDP — 4 mm thick Potassium Dihydrogen Phosphate crystal.

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A small portion of radiation transmitted through the dielectric mirror M$\mathbf {_2}$ was focused onto the photodiode by the lens L$\mathbf {_3}$ to measure pulse energy within each laser shot. This allowed to compare data obtained at different laser shots and to trace multifilament transverse structure and energy deposition evolution along the propagation axis. It also enabled to get energy deposition scaling with the laser pulse energy.

The DOEs were inserted into the stretched laser pulse before the compressor in order to eliminate nonlinear effects inside the DOE substrate. The spatial energy distributions of the original beam and the beam spatially-modulated with different DOEs are shown in Fig. 2. These energy distributions were recorded before the lens L$\mathbf {_1}$ (i.e. at the distance more than 8 meters after the DOE).

 

Fig. 2. Spatial energy distribution of the laser beam after the pulse compressor: a) without DOEs, b) with the Dammann grating, c) with TEM$_{11}$ phase plate, d) with TEM$_{03}$ phase plate.

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To examine, whether the binary-phase DOEs placed before the pulse compressor distorts the temporal shape of the pulse we used the single-shot SHG-FROG [43] technique (the insert to Fig. 1) with a thick KDP crystal working as a spectrometer, like in the GRENOUILLE method [44].

In this particular experiment, we used a binary-phase DOE forming the Hermite-Gaussian mode TEM$_{03}$ [35]. The DOE was oriented so that it formed 4 closely spaced spots in the vertical direction along the y-axis (see Fig. 2-d). The y-axis is orthogonal to the picture plane in the insert to Fig. 1 (the figure is depicted in the plane $y = 0$). We chose the TEM$_{03}$-like energy distribution due to the homogeneity of this energy distribution in the x-direction (i.e. in the time domain of the FROG-trace). Hence, we have closely spaced spots in the y-direction (i.e. in the spectral domain). This influences the resulting FROG-trace non-critically and allows for the pulse retrieving.

To investigate multifilament transverse structure and energy deposition into the medium, we applied two complementary techniques: radiation fluence transverse distribution measurements (which will be hereinafter referred to as mode measurements, by analogy with radiation modes formed in a laser cavity or a waveguide) and broadband acoustic diagnostics [45]. To measure radiation modes the imaging system of the wedges W$\mathbf {_1}$, W$\mathbf {_2}$, the lens L$\mathbf {_2}$ and the CCD (MindVision MV-UB130GM-T) was used. As every few laser shots damaged the wedge W$\mathbf {_1}$ surface, the wedge was being continuously moved to avoid imaging of the damaged areas. Broadband acoustic diagnostics allows to reveal the transverse structure of multifilament plasma channels and to retrieve density of the energy absorbed within the filaments. The detector employed ensured the transverse resolution of about $60\,\mu m$ and the longitudinal one of 7 mm. The diameter of the detector case is 12 cm that prevented us from placing it directly near the wedge W$\mathbf {_1}$. Therefore, the detector was placed 10 cm before the wedge. The lens L$\mathbf {_1}$ was moved with respect to the diagnostics system in order to simplify the measurement process. The diagnostic scheme is almost identical to the one used in our experiments on amplitude regularized multi- and superfilamentation [41].

Regular multifilament structure was obtained with two types of DOEs: a TEM$_{11}$ phase plate (phase mask) and a Dammann grating with 4 mm period producing a 4-maxima pattern in the focal plane (Fig. 3). The DOEs’ efficiency (fraction of the initial beam energy converted to the four main maxima) was assessed as 0.56 and 0.59, correspondingly. The phase mask introduced a $\pi$ phase lag between the neighboring maxima while the Dammann grating produced maxima with a zero phase lag with respect to each other. Notably, the fluence pattern produced by the Dammann grating exhibits significantly better contrast between the maxima and their surrounding compared to the phase plate. The causes and consequences of this feature will be discussed below.

 

Fig. 3. Patterns of the phase shifts, introduced into the beam by the DOEs (a, b), radiation modes produced by the DOEs at the focal plane at $P < P_{cr}$ (c, d), modes cross sections along the dashed lines (e,f). The first row (a,c,e) corresponds to the phase mask, the second (b,d,f) — to the Dammann grating.

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3. Results and discussion

3.1 Ultrashort pulse measurements of the spatially-modulated pulse

We measured and reconstructed FROG-traces for the original and the spatially-modulated pulse, Fig. 4. The FROG error (RMS) for the pulse retrieval from the spatially-distorted with DOE 512$\times$512 trace (Fig. 4-c) was 0.92%. This error is much larger than the one (0.58%) for the non-distorted trace (Fig. 4-a) measured with the original pulse. Nevertheless, the reconstructed trace for the spatially-modulated pulse (Fig. 4-d) is quite similar to the trace reconstructed for the original pulse (Fig. 4-b).

 

Fig. 4. The measured and reconstructed FROG-traces of the original (a, b) and the spatially-modulated (c, d) pulse.

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Pulse half-maximum duration, estimated from these traces, is 70 fs for the original pulse and about 75 fs for the spatially-modulated one. The DOE substrate is a 2-mm thick fused silica plate. The group delay dispersion (GDD) of such a plate is about +71 fs$^2$ at 810 nm. This results in less than 1-fs pulse broadening for the initial pulse longer than 50 fs. Thereby both the larger pulse duration and the positive chirp revealed in the reconstructed spatially-modulated pulse should be attributed to the relatively large FROG error but not to the dispersive pulse broadening in the DOE substrate.

Thus, an ultrashort laser pulse spatially modulated with the DOE does not experience strong distortion and retains the shape close to the shape of the original pulse. Therefore, it is appropriate to use transmitting binary-phase DOEs placed in a stretched pulse to generate spatially-modulated ultrashort laser pulses.

3.2 Regularized multifilamentation: TEM$_{11}$ phase plate

An overall picture of multifilament transverse structure evolution can be seamlessly revealed from a comparative analysis of modes and acoustic signal variations along the multifilament propagation axis. The left part of Fig. 5 presents the typical mode and acoustic signal samples obtained at various points along the propagation axis (for multifilamentation with the phase plate). The data in the left-hand column (1-8a) represent beam propagation without filamentation when $P < P_{cr}$ for each of the four maxima and provides a good reference to compare with filamentation mode. The middle and right columns depict modes (1-8b) and acoustic signals (1-8c) in filamentation mode ($P > P_{cr}$ for each of the four maxima).

 

Fig. 5. Multifilament modes and acoustic signal evolution along the propagation axis measured with the TEM$_{11}$ phase plate (columns (a)-(c)) and with the Dammann grating (columns (d)–(f)). $W\leq 0.5$ mJ ($P < P_{cr}$ in each maximum) in left columns (1-8a, 1-8d), $W = 3$ mJ ($P > P_{cr}$ in each maximum) in the middle and right columns (1-8b,c, 1-8e,f). Modes in the rows 3–5 for the phase plate (marked with magnifying glass signs) are enlarged twice for better visibility.

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Using the phase plate, we obtain the 4-maxima pattern right after the pulse compressor. Self-focusing leads to independent collapse of the maxima, forming filaments well before the linear focus. So, at 235 cm from the lens we already have the 4-filament pattern, as it can be seen from Fig. 5(1b,c). Here, the fluence maxima radii are visibly shrunk by self-focusing compared to the quasi-linear propagation in Fig. 5(1a), while the acoustic signal features two peaks corresponding to the plasma channels. It should be noted that the 4-filament pattern is represented by 2-maxima acoustic signal in our setup geometry. The acoustic detector was placed below the beam, so the signals from the bottom pair of the filaments reach it simultaneously producing a single maximum, the same applies to the upper pair.

Then, in subsequent points along the propagation axis, the maxima are converging to each other under the action of the external focusing (Fig. 5(2-3)). The 4-maxima pattern is preserved until the close proximity of the focus, where an additional filament is formed in the center of the beam (Fig. 5(4-6)), breaking the desired hollow-core multifilament structure. This maximum is absent in quasi-linear mode (Fig. 5(4-6)), appearing at energy $\gtrsim 2$ mJ, when the power in each maximum is sufficient for the filamentation start. This behavior can be explained as follows. Due to the beam imperfections, significant part of the initial radiation energy is not converted to the TEM$_{11}$ mode. This part is observed in Fig. 3(c,e) as a pedestal surrounding the maxima which reduces the contrast between the maxima and ’zero’ intensity lines to approximately 3:1. At the same time, this component has the same radius before the conversion (approximately 2 times greater then each of the maxima). This means that the non-converted radiation contains energy comparable to the energy of a single TEM$_{11}$ mode maximum. When the pulse energy is high enough, this component forms its own filament in the beam center. It starts much closer to the lens focus than the filaments formed by TEM$_{11}$ mode maxima, because of its greater radius. This implies greater self-focusing distance, according to the Marburger formula [46]. This fact is also in agreement with [47], where the the transition between linear (high NA, geometric focusing prevails) and nonlinear (low NA, self-focusing dominates) focusing regimes have been investigated. We have $\textrm {NA} = 2.4\cdot 10^{-3}$ for the beam as a whole, which is close to the transitional range between the two modes. NA for each of TEM$_{11}$ maxima can be roughly assessed as $10^{-3}$, which apparently falls within nonlinear focusing range. The nonlinear focusing predominance for the filaments originated from the TEM$_{11}$ maxima is clearly seen from the early start of the plasma channels in our experimental results.

The plasma channels remain for about 30 cm after the focus and then dissipate. After the plasma channels end, the radiation maxima in the beam center persist for another 30 cm (see Fig. 5(7b)), and then disappear. Far after the focal plane, the 4-maxima pattern restores, although the maxima become somewhat farther apart, compared to quasi-linear propagation mode (Fig. 5(8a,b)). Thus, we experimentally demonstrated the $\pi$-shifted filaments repelling that has been theoretically predicted in [48]. It is important to note that there is no maximum in the beam center, which would appear if the filament in the beam center resulted from the initial filaments merging. The data observed is in contrast to the multifilamentation with an amplitude mask, where due to filaments nonlinear interaction and merging, a new intense on-axis filament is formed [29,41]. In that case, the maximum on the axis remained in the far field, being much brighter than the maxima produced by the initial filaments.

3.3 Regularized multifilamentation: Dammann grating

Multifilament mode and acoustic signal evolution along the propagation axis with the Dammann grating is shown in the right part of Fig. 5. Here again the left-hand column depicts the beam evolution in quasi-linear regime, when the pulse energy is not sufficient for filamentation, while the middle and the right columns present modes and acoustic signals in multifilamentation regime.

In the far field after the pulse compressor, the Dammann grating creates a multitude of maxima, spreading the pulse energy between them (see Fig. 2(b)). As a result, each of the maxima contains power less than the critical self-focusing power if the pulse power does not exceed 20$P_{\textrm {cr}}$ (the highest power used in the experiment). That is why the beam evolution apart from the focus is driven mostly by diffraction in this case. It can be seen from the Fig. 5(1-3): there is almost no difference between quasi-linear propagation regime at $P < P_{\textrm {cr}}$ and the propagation at $P \approx 6 P_{\textrm {cr}}$ up to the distance about 265 cm. Only in the close proximity to the focus, when diffraction creates the 4-maxima pattern, self-focusing ‘turns on’, producing filaments from each of the maxima. This brings us to the first distinction between the two regularization approaches: while the TEM$_{11}$ phase plate forms prolonged filaments well before the focus, the Dammann grating gives them a rapid start almost in the focal plane, resulting in their shortening. However, this feature can enable one to control the filament start more precisely as it is largely determined by diffraction, becoming less dependent on the pulse energy fluctuations.

Then, 4 well separated filaments are formed (Fig. 5(4-6)). In contrast to the phase plate case, the separation is very clear for all distances where the plasma channels exist, and there is no additional plasma channel between them. This transverse structure was being reproduced steadily in all laser shots. The multifilament pattern created by the Dammann grating appears thereby more resistant to the initial beam imperfections, and demonstrates better shot-to-shot stability.

The total length of the plasma channels amounts to approximately 10–15 cm with 1-order increase in the acoustic signal amplitude. This behavior results from the prevalence of linear focusing making this regime close to filamentation at higher NA.

The 4-maxima fluence pattern disappears approximately 20 cm after the filamentation start. The beam transverse structure returns to a complex pattern of multiple small-scale maxima (Fig. 5)). This pattern, however, qualitatively reproduces the beam shape in quasi-linear propagation mode. This reaffirms that there are no any significant interactions between the filaments in this regime and diffraction plays the dominant role in the multifilament transverse structure evolution.

Grating-driven multifilament arrays robustness manifests also in lower susceptibility of their structure to the pulse energy variation. Changing the pulse energy from 3 to 12 mJ ($24 P_{\textrm {cr}}$ approximately), we did not observe any significant alterations in the multifilament pattern. On the other hand, radiation modes produced by the phase plate change from the markedly distorted 4-maxima pattern at 2 mJ to a mode where the central maximum dominates and the neighboring maxima are small compared to the central peak at 5 mJ. Using the phase plate, we could not set energies higher than 5–6 mJ, as the self-focusing within the initial 4 maxima led to the damage of lens L$\mathbf {_1}$ and mirror M$\mathbf {_2}$. Since the Dammann grating splits the beam into a couple dozen maxima, each of them contains less or just a bit more than one self-focusing power. It prevents them from rapid self-focusing and enabled us to put more power into the resulting filaments without damaging of the guiding optics.

It can be noticed that the filaments separation near the focus is more than twice larger for the Dammann grating. The distance between the maxima is defined by the grating period and focal length of the lens used [49]. So, to reduce the distance, one should increase the grating period that would decline the contrast of the maxima pattern. The minimum possible distance between the maxima can be assessed as the diffraction limit $l_{\textrm {min}} = \frac {\lambda f}{2 R}$, where $\lambda$ is the laser wavelength, $f$ — the focal length and $R$ is the DOE radius [49]. However, the distance at which the contrast is still acceptable would be much greater. That is why to create a closely spaced multifilament array one should prefer a phase plate.

3.4 Energy deposition into the medium

Graphs (a) and (b) in Fig. 6 present energy density absorbed by the medium at various points along the propagation axis and at different pulse energies for the phase plate and the Dammann grating, correspondingly. The results are retrieved from the acoustic signals according to the model described in [45]. Each point in the plots corresponds to an average over few tens of laser shots within the given pulse energy range. For a typical 2-maxima acoustic signal corresponding to the 4-maxima multifilament pattern (see Fig. 5), we picked the highest of the acoustic maxima. Since our acoustic measurements are one-dimensional, they do not allow to reveal whether one or two plasma channels impacted the signal maximum. (To distinguish these cases, one should use the second acoustic detector placed perpendicularly to the first one.) Hence, we did not correct the energy densities values to account for the changing filament number. However, the densities should be proportional to the number of channels (as the signals from filaments add up linearly). Thereby, the real energy deposition for the most of the points on the graphs in Fig. 6 (except for ones in the range 270–290 cm in Fig. 6(a), influenced by an additional plasma channel formed in the beam center) are approximately twice lower than it is shown in the graph.

 

Fig. 6. Upper row. Dependency of the energy density, absorbed by the medium, on the distance from the focusing lens L$\mathbf {_1}$ a) for the phase plate, b) for the Dammann grating. Bottom row. Energy density dependency on the laser pulse energy for c) the phase plate at distance 256 cm from the lens, d) the Dammann grating at 278 cm from the lens. Red and magenta curves on graphs (c) and (d) show running average with 20 points sampling window length.

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For the phase plate (Fig. 6(a)), one can observe an extended plateau in 230–260 cm where the energy deposition is almost constant and does not change with the growth of the pulse energy. It corresponds to the distances where 4 separate filaments are formed. The constant energy deposition level can be explained as a manifestation of the plasma concentration clamping in the filaments. In 270–290 cm distance range, the plot behavior depends heavily on the pulse energy. At energies lower than approximately 2.5 mJ, the energy deposition level remains almost constant until the plasma channels fade. It happens at, approximately, 1.5 mJ, when the pulse energy becomes insufficient to create a filament from any of the 4 initial maxima. In the range between 1.5 and 2.5 mJ, less than 4 filaments are created (due to the lack of the energy and the beam position fluctuations, which result in uneven distribution between the maxima). For higher pulse energies, the energy density grows 2-4 times due to the formation of the additional plasma channel at the beam center. The unconverted radiation undergoes effectively tighter focusing than the radiation in each of the TEM$_{11}$ lobes. That explains the drastic growth in energy deposition with only one plasma channel added. After the central spot dissipates, the energy density returns to the plateau level (see points at 295–305 cm in fig 6(a)).

In the case of the Dammann grating (Fig. 6(b)), there is no similar region with almost constant energy deposition level. This can be explained as the result of predominance of linear focusing (accompanied by the diffraction-governed formation of the 4 intensity maxima) over Kerr focusing. Filamentation ‘turns on’ abruptly once the 4-maxima pattern is produced. Still, self-focusing can move the filament start towards the laser (due to faster self-focusing within the maxima). Accordingly, the energy density maximum moves closer to the focusing lens as well. This results in even tighter effective focusing for each of the maxima and leads to significant energy deposition increase with the pulse energy growth. However, in the decreasing part of the graph (280–285 cm), where, after the linear focus, self-focusing starts to play significant role in keeping the filament, the points at different energies coincide quite precisely. Another effect that can contribute to the energy deposition growth, is the fact that the diffraction-driven maxima pattern impedes filament splitting within the maxima. This effect is similar to the one observed in [41], where merging of the 4 filaments driven by diffraction of the amplitude modulated beam, resulted in a single filament with higher energy density inside it. Here, we have four such ‘single superfilaments’ formed by the Dammann grating.

Noteworthy, that the energy density in case of the Dammann grating is more than one order higher than for the phase plate (but at the expense of the filaments length). It also stems from tighter effective focusing for the filaments, generated using the Dammann grating.

Graphs (c) and (d) in Fig. 6 (bottom row) present the energy deposition dependencies on the laser pulse energy for the phase plate and the Dammann grating correspondingly. For the grating, we picked the data at distance 278 cm, close to the maxima positions of the curves in Fig. 6(b). For the phase plate we picked the values at the end of the plateau (256 cm), where the 4-maxima pattern still preserved in most of the laser shots.

For the phase plate, the points are clustered along the horizontal band with significant shot-to-shot scattering. The energy density remains constant in the whole considered pulse energy range (1.5–5 mJ) and amounts to 1 mJ/cm$^3$ on average (taking into account signal doubling due to the setup geometry). In contrast, the energy density experiences approximately 4-fold growth when the pulse energy increases from 4 to 7 mJ in case of the Dammann grating. Then the energy deposition saturates and even shows somewhat decrease with further energy growth. This graph clearly illustrates the movement of the nonlinear focus and the corresponding shift of the energy deposition maximum along the propagation axis with the pulse energy increase. At lower energies, the maximum locates at distance $L > 278$ cm, then reaches this point (at pulse energy $W \approx 7.5$ mJ), then moves further to $L < 278$ cm.

It is also interesting to compare the results obtained for the phase plate and the Dammann grating with multifilament parameters in the case when the regular multifilament pattern is produced by the 4-holes amplitude mask [41,42]. In the latter case, the 4-filaments structure is also formed well before the focus and is characterized by the constant energy density level. Typical energy density is about 4-5 mJ/cm$^3$ for the amplitude mask, which is 4-5 times higher than for the phase plate (in both estimations, signal amplitude doubling due to the setup geometry is taken into account). Then, the initial filaments merge, resulting in almost 20-fold energy density growth. The length of the area with enhanced plasma density spans for about 40 cm. Although there is a surge in energy density near the focus in the case of the phase plate, this increase has completely different origin (an additional filament formed by the radiation, not converted to the TEM$_{11}$ mode). The range with the increased energy density is significanly shorter and the energy density is lower than for the superfilament formed by the amplitude mask. In contrast, the energy density in filaments, obtained using the Dammann grating is only 2 times lower than in the superfilament, created by the amplitude mask ($\approx 40$ and 90 mJ/cm$^3$, correspondingly). This stems from the tighter effective focusing for the maxima created by the Dammann grating, but not only. In both cases, diffraction redistributes the beam energy in either one maximum (amplitude mask) or several maxima (Dammann grating) near the focus. The pre-defined sharp maxima pattern diminishes the role of modulation instability, making it possible to enhance plasma concentration and intensity in a single filament. By the contrast to our findings, ’stochastic’ multifilamentation is governed by the nonlinear interplay between the filaments resulting in their coalescence (see e.g. [39,41,50] ).

3.5 Beam quality impact

For better understanding of the beam quality impact on the resulting multifilament pattern, we simulated the radiation modes produced by the DOEs for different types of initial beam distortions in the linear propagation regime. The calculations have been performed basing on the Rayleigh-Sommerfeld diffraction integral using Diffractio module for Python [51].

First of all, we considered the propagation of the beam with initial intensity distribution obtained from the experiment. Initial phase was given as Zernike polynomial $Z_2^{-2}$ to reproduce astigmatism, existing in our system. Also, a quasi-stochastic phase modulation with amplitude $\lambda /8$, which corresponds to the typical imperfection of an optical surface, has been added. Focusing geometry corresponded to the experimental (focus at 267 cm with 7.5 mm FWHM beam diameter). We considered monochromatic radiation propagation with 812 nm wavelength. The resulting beam patterns are presented in Fig. 7.

 

Fig. 7. Beam diffraction simulation. Left column (1–4a) — beam without DOEs, middle column (1–4b) — beam with the TEM$_{11}$ phase plate, right column (1–4c) — beam with the Dammann grating. As the beam evolution in linear mode is symmetric relative to the focal plane, the figure represents the evolution before the focal plane only.

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For the phase plate, the beam distortions lead to a new maxima in the center of the beam near the focus (Fig. 7(4b)). This supports our assumption that the initial filament in the beam center is formed from the part of the radiation that has not been converted to the TEM$_{11}$ mode. So, the ‘seed’ for the central filament has diffraction nature and is not connected with the filaments interaction. In the nonlinear mode, self-focusing just elongates the range where this maximum exists. On the contrary, the same beam imperfections do not cause any significant deviation from the ideal case for the Dammann grating. Therefore, the higher stability and quality of the multifilament pattern, produced by the Dammann grating can be explained by the diffraction properties of the DOEs.

To reveal, how different types of amplitude and phase distortions can alter the resulting radiation pattern, produced by the DOEs, we examined (i) random amplitude noise, (ii) random phase noise, (iii) beam ellipticity and (iv) astigmatism. The results are shown in Fig. 8.

 

Fig. 8. Influence of different beam imperfection types on regularized beam propagation in linear mode (simulation). Column (a) — initial mode, column (b) and (c) — beam phase after the phase plate and the Dammann grating correspondingly, columns (d) and (e) — beam modes in the focal plane ($F = 267$ cm).

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Random small-scale amplitude fluctuations (Fig. 8, row (1)) do not lead to significant distortions of the resulting pattern for the both DOEs (although some decrease in the pattern contrast is noticeable for the phase plate). The same applies to the small-scale phase noise (Fig. 8, row (2)). It can reduce the pattern contrast (producing random low-intensity pedestal), but due to the low power contained in each of the maxima, small-scale phase noise (with typical inhomogeneity size much smaller than the beam diameter) cannot cause modifications in the multifilament pattern. The situation changes for the elliptic beams (Fig. 8, rows (3)–(4)) or when there is astigmatism in the system (Fig. 8, rows (5)–(6)). In both cases, the Dammann grating preserves the beam pattern, regardless its orientation with respect to the ellipse axes. For the phase plate, the resulting mode depends heavily on the plate orientation. For the elliptic beam, if the ellipse axes coincide with the dividing lines between the plate lobes, it does not affect the 4-maxima pattern. If the axes oriented at 45$^{\circ }$ to the plate dividing lines, the ‘waist’ between the diagonal maxima is formed. At higher pulse energies, it can lead to an additional filament formation. For the astigmatic beam, when the axes of focal ellipses are oriented at 45$^{\circ }$ to the boundaries between the phase plate lobes (zero lines of the corresponding Zernike polinomial coincide with the boundaries), an additional maximum is formed in the vicinity of the focus. However, when the focal ellipses axes coincide with the boundaries (the Zernike polinomial zero lines are 45$^{\circ }$ shifted with respect to them), astigmatism does not destroy the 4-maxima pattern.

This analysis allows to make some conclusions about the nature of mode distortions for the phase plate and high robustness of the modes, created by the Dammann grating. Only when the beam imperfections lead to substantial difference in the energy of $\pi$-shifted and not-shifted radiation components, the phase mask produces the ‘broken’ pattern. In this case, a good deal of the laser pulse energy is not transformed to the TEM$_{11}$ mode. That is why the ‘incorrect’ orientation of an elliptical beam destroys the 4-maxima pattern in the focus, while the small-scale amplitude fluctuations do not. In turn, the Dammann grating contains dozens of $\pi$-phase-shifting and 0-phase-shifting parts, which means approximately equal energy in shifted and non-shifted components for any large-scale inhomogeneities. As for the small-scale fluctuations, the high number of grating parts ensures approximately equal energy in the $\pi$-shifted and $0$-shifted components (provided that the fluctuations are random).

4. Conclusions

In this paper, we presented a novel approach to obtain regular and robust multifilament arrays using a diffraction Dammann grating. We compared the properties of this multifilamentation regime (radiation patterns and energy density evolution, their dependencies on the pulse energy, the impact of the beam quality on multifilament stability) with the using of a $\pi$-shifting large-lobes phase plate (namely, the TEM$_{11}$ phase plate) suggested for multifilamentation regularization in the literature.

Both methods are capable to create 4-filament multifilament arrays in the case of the high-quality Gaussian initial beam with a plane wavefront. However, the physics underlying this process in both cases is different. The phase mask creates a hollow-core pattern due to the phase discontinuities, which it introduces into the beam, and zero intensity lines associated with them. Large-scale deviations of the amplitude from Gaussian or the phase from plane can lead to the destruction of the desired hollow-core pattern. We experimentally demonstrated this effect for the astigmatic beam. In contrast, the Dammann grating does not necessary shifts the relative phase of the maxima in the focal plane (the one used in our experiment produced 4 unshifted beams). Instead, it creates spatially separated energy reservoirs (due to diffraction), which do exchange energy due to this separation. This automatically prevents nonlinear interaction between the 4 filaments as well. This structure is shown to be much more resistant to large scale beam inhomogeneities, being fairly the same sustainable to the small scale fluctuations, as multifilaments obtained with the phase plate. Another useful feature that can facilitate Dammann grating employment in applications is that it allows to use higher pulse energies without small-scale focusing to start in the resulting beam.

Nevertheless, a phase plate should be preferred as a tool for multifilament regularizing when the filaments length is more important than multifilament pattern stability, or when very closely spaced filaments are needed.

The approach proposed can be useful for such filament applications as THz generation and material microfabrication.