Laser beam self-symmetrization in air in the multifilamentation regime

The collimated output 50 fs pulse from a linearly polarized multi-Terawatt Ti:Sa CPA laser (Enstamobile) at ${{\lambda }_{0}}=800$ nm is weakly focused with an f = 5 m lens in air at atmospheric pressure. Burn patterns from the laser beam are recorded on calibrated presolarized Kodak photographic plates at different distances from the focusing lens. This simple technique reliably detects in a single shot the stage of evolution of multiple filaments formed out of a single laser beam [2]. Plasma channels give rise to characteristic ∼50–100 μm wide circular burns on the photographic plate, which are easily distinguished from the ∼1 mm circular dark spots from bright channels and the less intense energy reservoir. Similar measurements were performed by intentionally adding chirp to the pulse to adjust the peak intensity. The same self-symmetrization effect of multiple filaments was observed for the transform limited 50 fs pulses and for pre-chirped pulses (see figure captions). This allows us to dismiss temporal effects as the origin of symmetrization.

For each recorded beam intensity profile, we computed the center of mass (CM) on the $xy$- (transverse) plane and then considered the radial intensity traces (with fixed polar angle ϕ), ${{I}_{\phi }}(r)$ (r = 0 at the CM). We define the degree of symmetry, able to resolve the internal structure (intensity fluctuations) of the beam profiles, as the weighted average of the intensity fluctuations: $0\leqslant \Phi \leqslant 1$ (see appendix)

$$\begin{eqnarray}&&\Phi \equiv 1-\frac{1}{2{{\pi }^{2}}}\int _{0}^{2\pi }{\rm d}{{\phi }_{1}}\int _{0}^{{{\phi }_{1}}}{\rm d}{{\phi }_{2}}\left| \frac{{{\int }_{r}}\{{{I}_{{{\phi }_{1}}}}(r)-{{I}_{{{\phi }_{2}}}}(r)\}}{{{\int }_{r}}\{{{I}_{{{\phi }_{1}}}}(r)+{{I}_{{{\phi }_{2}}}}(r)\}} \right|,\end{eqnarray} \tag{ 1 }$$

which measures the similarity in between all pairs of the radial intensity traces, $\{{{I}_{\phi }}(r),{{I}_{\phi ^{\prime} }}(r)\}$. Perfect circularly symmetric profiles are characterized by $\Phi =1$, whereas beams with a large asymmetry are characterized by a relatively low symmetry degree.

Regarding the beam quality factor, M², we use the widely used definition by Potemkin et al [32] for a beam with complex field $\mathcal{E}(x,y)\equiv A(x,y){\rm exp} ({\rm i}\phi )$ and intensity $I(x,y)\equiv |\mathcal{E}(x,y){{|}^{2}}$:

$$\begin{eqnarray}&&{{M}^{2}}={{(\langle {{r}^{2}}\rangle \langle {{k}^{2}}\rangle -{{\langle {\bf r}\cdot {\bf k}\rangle }^{2}})}^{1/2}},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&P=\int {\rm d}x{\rm d}y\;I(x,y),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\langle {{r}^{2}}\rangle ={{P}^{-1}}\int {\rm d}x{\rm d}y({{x}^{2}}+{{y}^{2}})I(x,y),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\langle {{k}^{2}}\rangle ={{P}^{-1}}\int {\rm d}x{\rm d}y\;[{{(\nabla A)}^{2}}+I{{(\nabla \phi )}^{2}}],\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\langle {\bf r}\cdot {\bf k}\rangle ={{P}^{-1}}\int {\rm d}x{\rm d}y\;[{\bf r}\cdot \nabla \phi (x,y)]I(x,y)\end{eqnarray} \tag{ 6 }$$

and ${\bf r}\equiv (x,y)$. Since the calculation of the quality factor requires knowledge not only of the beam intensity but also of the spatial phase, we characterized beams in terms of M² only for the results of our numerical simulations.

Propagation of the monochromatic field envelope $\mathcal{E}(x,y,z)$ in air (${{n}_{0}}\approx 1$), with frequency ${{\omega }_{0}}=2\pi c/{{\lambda }_{0}}$, is modeled by means of a unidirectional beam propagation equation accounting for diffraction, optical Kerr effect, multiphoton absorption, plasma absorption, and plasma defocusing, respectively [33]:

$$\begin{eqnarray}\begin{array}{rcl} \frac{\partial \mathcal{E}}{\partial z} & = & \frac{{\rm i}}{2{{k}_{0}}}(\frac{{{\partial }^{2}}}{{{x}^{2}}}+\frac{{{\partial }^{2}}}{{{y}^{2}}})\mathcal{E}+{\rm i}\frac{{{\omega }_{0}}}{c}{{n}_{2}}|\mathcal{E}{{|}^{2}}\mathcal{E} \\ {} & {} & -\frac{1}{2}({{\beta }_{8}}|\mathcal{E}{{|}^{14}}+\sigma [1+{\rm i}{{\omega }_{0}}{{\tau }_{c}}]\rho )\mathcal{E}. \\ \end{array}\end{eqnarray} \tag{ 7 }$$

Here n₂ = 2 × 10⁻¹⁹ cm² W⁻¹, ${{\beta }_{8}}=8\times {{10}^{-98}}$ cm¹³ W⁻⁷, $\sigma =5.6\times {{10}^{-20}}$ cm², and ${{\tau }_{c}}\approx 350$ fs. Our $(2+1)D$ modeling does not contain temporal dynamics. Hence, plasma effects are accounted for in the so-called frozen time approximation where the explicit standard rate equations describing electron generation by optical field ionization and avalanche are used to generate a mapping of the electric field peak intensity $\mathcal{I}\equiv |\mathcal{E}{{|}^{2}}$ to the electron–plasma density, $\rho (\mathcal{I})$. This calculation is performed for reference pre-chirped Gaussian pulses with peak intensity $\mathcal{I}$ with duration given by experimental conditions. The electron density, $\rho (\mathcal{I})$, used in the propagation model equation (7) is determined from this mapping, at the temporal center of the reference pulse. This procedure provides, as shown previously and below, numerical results in good agreement with experiments [20, 21, 34]. Simulations are initialized with 10% intensity and 0.2% phase noise in order to mimic experimental irregularities in the input beam.

In the absence of any mask, a beam with power $P\sim 550\ {{P}_{{\rm cr}}}$ undergoes typical multi-filamentation dynamics. Figure 1 shows the darkening pattern impressed by such a beam at several distances from the focusing lens on photographic plates. Already after 2 m of propagation several ionizing filaments are present. These converging filaments grow in number and connect into networks upon further propagation ($\gtrsim 4$ m). Before the geometric focus, the ionized zone fills a central circular spot of a few mm in diameter and ∼1 m long, where filaments are in close contact [21]. The number of ionized spots is significantly reduced to a few which are located near the center of the beam. Losses incurred by going through the focus have been measured with a calorimeter placed before and after the focus. 82% of the initial beam energy is found in the beam at 3.5 m after the focus. This rather high value could be connected to the idea that in certain situations the filaments are connected stronger to plasma defocusing than to plasma absorption [17]. Below we show that independently on the intensity mask used to distort the input beam profile, the beam recovers after focus the same symmetry degree, hence similar roundness, as the one shown in the final stage of figure 1. In contrast, low energy beams ($P\ll {{P}_{{\rm cr}}}$) do not exhibit any improvement of their symmetry degree along propagation since the final beam profile corresponds to the diffraction pattern induced by the mask, with identical symmetry degree as the initial distorted beam.

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In order to visualize the self symmetrization effect we have put different masks on the path of the input beam, just before the focusing lens (at 0 m). Examples are shown in figures 2 and 3 where a quarter circle (or Pac-man), square, half-plane, and slit masks are used. Similarly to the undistorted case, 89% of the input chirped pulse energy was measured 3 m beyond the focus. Inspection of the burnt papers in figures 2 and 3(a) reveals that ionizing filaments get first organized along patterns dictated by the profile of the input beam [13]. Indeed, close to the sharp intensity jumps created by the mask, large intensity modulations drive filamentation in a deterministic manner. Upon further beam propagation, the self-symmetrization effect appears around the geometric focus symmetrizing the slowly diffracting output beam, despite its initial strong distortion. Solid experimental evidence of the symmetrization effect is provided by comparing the profiles in figure 2 recorded at 1.5 m before and after the focus (i.e., at 3.5 and 6.5 m).

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A detailed comparison between experimental and numerical results is presented in figure 3 for the case of the highest input distortion. Here, a 10 mm wide slit transforms the input circular profile into a rectangle with aspect ratio $\sim 1/4$. Upon propagation the symmetrization degree increases as the beam approaches and goes through the focal region at ∼5 m. Figure 4(a) shows symmetrization of the beam along propagation for the beam profiles presented in figures 3(a)–(b). For a comparison, two additional numerical results for linear propagation and for an input intensity ${{I}_{0}}=5$ GW cm⁻² are also presented. The symmetry degree is seen to be higher by the end of the nonlinear propagations, reaching its maximum after the focus and holding it for a distance of at least ∼2–3 m. The minimum of the symmetry is observed for the linear propagation at the focal plane. This is simply due to the sharp horizontal line (y = 0) apparent in the beam profiles at this position, z = 5 m (see below, e.g., in figure 6), corresponding to the Fourier transform of the slit induced sharp profiles along x (which is a sinc(x) function at focus). One can easily see that the width of the Fourier transforms of a Gaussian, $\Delta {{\kappa }_{G}}$ (at FWHM), and a slit, $\Delta {{\kappa }_{s}}$ (full width of the cetral 'sinc' lobe), satisfy $\Delta {{\kappa }_{s}}/\Delta {{\kappa }_{G}}=w/{{L}_{s}}\times \pi /\sqrt{{\rm ln} 2}$, where $w\approx 3.5$ cm is the FWHM of the input (z = 0) Gaussian profile along y and L_s = 1 cm the slit width (along x). It therefore follows that while ${{L}_{s}}\lesssim w$, the horizontal line appearing at focus will be much longer than the spot size along y affecting substantially the symmetry degree (see the sharp minimum $\Phi \approx 0.5$ in figure 4(a)). This feature disappears gradually as the input peak intensity is increased in simulations up to ${{I}_{0}}\approx 5$ GW cm⁻², because of the novel spatial frequencies that are generated (see discussion in section 5). Also, the photographic paper used in experiments has a different definition in the low intensity parts of the beam and therefore the threshold intensity at which the horizontal line disappears is expected to be different than for simulations. Note in the linear propagations for the masked beams (not shown) the beam profiles at a fixed distance before and after the focus are almost equal to each other after a rotation of π in the $\widehat{XY}$ plane. The equality does not hold exactly due to the input noise and mask induced fluctuations, however the difference is small at sight. This is the reason why $\Phi (z)$ is (almost) symmetric from the focal plane (see figure 4(a)). In our experiments, the slit mask is the one screening a biggest fraction ($\sim 64\%$) of the input 300 mJ carried by laser pulses, which was the maximum available energy. We believe this is why the experimental symmetry degree along propagation (circles in figure 4(a)) starts to drop a bit 2–3 m after focus. This feature may be appreciated even visually in figure 3(a), conversely to what is shown in figure 2 for the other masks. Additionally, the symmetry degree presents a local minimum around focus, which is a reminiscence of the linear propagation (indeed we see similar $\Phi (z)$ trends in our monochromatic modeling for ${{I}_{0}}\lesssim 10$ GW cm⁻²). Still, the symmetry degree is remarkably high during 2–3 m and we expect (see below) that this would improve for higher input pulse energies.

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The quantitative agreement in between our monochromatic numerical simulations and experimental results in figures 3 and 4(a) strongly suggests that the spontaneous symmetrization shown here is an effect essentially dominated by spatial dynamics of the beam. We have computed the symmetry degree at a fixed distance z = 9 m (4 m after focus) and varying input power (figure 4(b)). These measurements reveal an increase of $\Phi (P/{{P}_{{\rm cr}}})$ exhibiting saturation: $\Phi (P/{{P}_{{\rm cr}}}\gtrsim 40)\approx 0.95$. Such behavior vividly manifests the nonlinear (intensity dependent) nature of the spontaneous symmetrization. Indeed, a systematic numerical study reveals that the Kerr effect plays a major role in self-symmetrization (see section 4 below). We also characterized the beam quality via beam quality factor [32, 35]. A Gaussian beam at waist is characterized by ${{M}^{2}}=1$ and is considered as a reference for perfect quality. Figure 4(c) shows that the highly distorted beams always present a beam quality 3–5 orders of magnitude worse than that of a Gaussian beam at focus or the Townes mode. Therefore, even if there is a partial improvement of M² with propagation distance, the main effect is self-symmetrization. Beam self-cleaning occurs in the sense of an improvement of the beam quality factor but the latter effect is not so efficient as for beam self-cleaning obtained in the context of lower power beams [5, 6, 8], where $P/{{P}_{{\rm cr}}}\gtrsim 1$.

We consider below the case of an input beam with a peak intensity of ${{I}_{0}}=50$ GW cm⁻² under the distortion of the slit mask. This situation corresponds to the minimum peak intensity needed to observe a high symmetry degree (see figure 4(b)) and only a few filaments may be observed on the output beam profile, note there are only four filaments in figure 5 (marked by the squares). In order to show that the main responsible for self-symmetrization of powerful focused beams is indeed the optical Kerr effect, we made a set of simulations with identical initial conditions as those in figure 5 in which the different nonlinear terms in equation (1) are switched on or off at will (see figure 6). Below, we are only showing beam profiles at and after the focus ($z\geqslant 5$) because those before the focus are indistinguishable by sight from those in figure 3(b) for all cases.

When the only nonlinear effect is MPA (n₂, $\sigma =0$, figure 6(a)), propagation does not differ substantially from the one observed in the linear regime, except for the big absorption induced across the profile that lead to the appearance of vertical fringes. Linear propagation would lead to the observation of the diffraction pattern of the slit. Nonlinear absorption plays the role of a distributed stopper, leading to modulations in the diffraction pattern in an effect similar to the Arago spot effect. Nonlinear absorption participates to the fringe formation since it enhances the self-healing process of a beam that is known to reshape Gaussian beams into Bessel-like beams [36–38]. Addition of plasma absorption and defocusing ($\sigma \ne 0$, figure 6(b)) has a strong impact in the profile after focus: generation of plasma leads to refractive index jumps along the sharp intensity edges and the defocusing induces strong scattering towards the directions perpendicular to those edges (light propagates towards the higher index regions). As a consequence of this, the scattering of light after focus occurs predominantly at an angle $\pi /2$ from the long axis of the slit. It is only when the Kerr effect is switched on that symmetrization degree improves substantially, as shown in figure 6(c) (${{\beta }_{8}}=0$) and figure 5. Note from figure 6(d) that with Kerr effect symmetry is higher when both MPA and plasma effects are accounted for, presumably due to the fact that MPA and plasma tend to induce scattering along perpendicular directions (at least in the case of the slit mask). Remarkably, the symmetrization effect persists even for beams experiencing a large multifilamentation (see figure 3).

The second essential ingredient for symmetrization (together with Kerr) is the focusing induced by the lens. We have done extensive numerical modeling with collimated beams and controlled the density of filaments via the input power, P. None of these simulations showed traces of spontaneous symmetrization in the distances of ∼10 m. Figure 7 shows an example with the slit mask. Here, in order to decrease filament separation, the input power was four times larger than for the focused cases shown in figures 3(b) or 4 with the largest I. The multiple filaments develop along propagation and the phase gradients impinged by the mask make some of the optical energy migrate into the initially unpopulated area. However, symmetry is seen to remain rather low along the 10 m, which implies that the self-symmetrizing effect reported here is intimately related to the focusing induced interaction amongst the different parts of the beam.

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