1. Introduction

Nowadays, there is growing hope and effort for substantial upgrading, both in efficiency and precision, of the current remote sensing techniques in the atmosphere, known as LIDAR techniques (Light Detection and Ranging) [1]. Commonly, monostatic LIDAR systems rely on the backscattered incoherent light coming from targets in the atmosphere, and the signal-to-noise ratio of these systems makes it difficult to detect traces of chemical compounds diluted in the atmosphere. Bistatic systems might alleviate this problem, but the separation between emisor and receiver makes extremely difficult to scan great areas of the sky, which is key for environmental pollution and national security measurements.

A relatively new proposed scheme capable of overcoming these difficulties is the so-called atmospheric backward lasing. In this scheme, an intense infrared laser, typically of $\lambda =$ 800 nm and I $\sim 10^{13}$ W/cm$^{2}$, is focused in a particular region of the atmosphere, creating a plasma filament [2–4]. If the appropriate conditions are satisfied, optical gain is generated within the filament. Both forward and backward UV coherent radiation is produced, the latter directed to the same point in the surface of Earth where the IR laser and detectors are situated. Since the signal detected is intense and directional, not relying anymore in backscatering, the expected signal-to-noise ratio will be improved, allowing to detect minimal concentrations of chemical species.

One of the first possibilities that was explored to achieve backward lasing in the atmosphere was molecular oxygen, $\mathrm {O_2}$, with simultaneous resonant two-photon dissociation of $\mathrm {O_2}$ and resonant two-photon excitation (2p$\mathrm {^{3}P \rightarrow 3p ^{3}P}$, $\lambda =$ 226 nm) as the pumping mechanism [5]. Atomic oxygen deexcitation to 3s$^{3}S$ leads to emission at 845 nm. However, the fact that UV pulses are required for $\mathrm {O_2}$ dissociation, which are strongly absorbed and dispersed in the atmosphere, is a great drawback for its implementation.

On the other side, it has been also reported stimulated emission in the remaining major component of the atmosphere, $\mathrm {N_2}$, both in neutral [6–10] and singly ionized molecules ($\mathrm {N_2^{+}}$) [11–14]. In the former case, of interest for the current study, lasing was observed in the $\lambda = 337$ nm line corresponding to triplet states transition $C ^{3}\Pi _{u}^{+} \rightarrow B ^{3}\Pi _{g}^{+}~(\nu = 0 \rightarrow 0)$, where $\nu$ is the vibrational quantum number. The population of the upper level, $C ^{3}\Pi _{u}^{+}$, was attributed to several mechanisms, such as resonant excitation from excited argon atoms to nitrogen molecules (at high pressures) [10,15], collision-assisted recombination process (at low pump laser energy) [6,16] or electron-molecule inelastic collisions:

This process is currently credited for being responsible of population inversion in $\mathrm {N_2}$ plasmas. First, emission sensitivity to pump polarization, and therefore to electronic population energy distribution [17], is a clear indicator of this claim, with a resonant peak at 14.1 eV [6,7]. Second, in [18] it was revealed the ultrafast formation dynamics of excited $\mathrm {N_{2}}$ molecules and its dependence with pressure (less than 4 ps at 1 bar, 120 ps at 30 mbar ), incompatible with the other mechanisms.

It is important to note here the great quenching effect of oxygen at concentrations from 10 to 15 $\%$, due either to intensity reduction inside plasma filament [19], to electron density reduction [8] or collisions with $\mathrm {N_2}$ molecules [6]. However, optimized pumping conditions may alleviate this problem, for example, by using longer pump pulses of the order of picoseconds [20]. In order to overcome this difficulty, we can take advantage of the decades-long knowledge generated in the field of plasma-based soft X-ray lasers [21–25], since both lasers rely in electron-collision-pumped population inversion mechanism.

In this article we show, for the first time up to our knowledge, the full 3D modelling of the amplification of UV light in a nitrogen plasma filament, extending the research carried out in [9]. The main objective is to demonstrate the potential in this field of our three-dimensional, time-dependent Maxwell-Bloch code, Dagon [26], previously applied to plasma-based soft X-ray lasers and emphasize and enhance the role that the 3D structure of probe pulses has in the diagnosis of cavity-free lasers. Since the geometry assumed for the plasma filament has cylindrical symmetry, a 2D axisymmetric model (as the one reported in [27]) should save computational time and memory while capturing the essential physics of the problem. However, as the final objective of our model is to study in future works backwards lasing from amplified spontaneous emission, we opted for the 3D model to fully tackle the 3D stochastic profile of ASE, which has no symmetries.

The layout of the article is as follows. In section 2, we explain the Maxwell-Bloch model solved by this code and the model used for computing the temporal evolution of plasma electron temperature and density. The validation of these models against available experimental data and 1D modelling results is presented in section 3. In addition to this, the importance of proper modelling of the depolarization rate and the consequences derived from the adiabatic approximation are also analysed in this section. Finally, section 4 highlights the strong potential of using the wavefront of an UV probe pulse as a new diagnostic method to unveil the plasma filament inner dynamics.

2. Maxwell-Bloch modelling

Propagation and amplification of UV light throughout a plasma filament are modelled with our three-dimensional, time-dependent Maxwell-Bloch code, Dagon [26]. With a widely proven ability to simulate plasma-based X-ray lasers, we have applied Dagon to model nitrogen atmospheric lasing. For this purpose, it solves the so-called Maxwell-Bloch equations (Eqs. (3–5), resulting from the coupling of Maxwell equations for the electromagnetic field in a plasma and a quantum mechanical model for the calculation of the elements of the density matrix of a two-level system.

The starting point is Maxwell wave equation for the propagation of an electric field in a neutral plasma:

The constitutive relation required to compute the polarization of the plasma is deduced from the dynamics of the non-diagonal terms of the density matrix of the two-level system. The resulting equation, solved within Dagon, is

The creation and evolution of the plasma conditions during propagation and amplification is computed using the model introduced in [28,29]. First, Eq. (6) gives the variation of electron thermal energy density:

The last term in Eq. (6), $(1 - \frac {T_{\nu }}{T_e})$ accounts for the relaxation process between $T_e$ and $\mathrm {N_2}$ vibrational temperature, $T_{\nu }$, such that its rate of change can be expressed as:

The second part of the model is intended to calculate electron and ion densities from the following equations:

Our plasma evolution and Maxwell-Bloch models are coupled as follows. The plasma model computes the temporal evolution of electron density and temperature. These variables are used to compute the temporal evolution of variables such as collisional (de)excitation rates, the electron-neutral collision frequency and plasma frequency. All this data is stored in files and fed to our Maxwell-Bloch model.

With this model we study the creation of a population inversion inside the plasma, via electron collisional excitation, and the posterior amplification of the UV seed. The pumping mechanism, fully described in [18], is as follows. An intense (I $\approx 10^{14}$ W/cm$^2$), circularly polarized infrared pulse generates free electrons with enough energy ($\approx 11$ eV) to collisionally excite nitrogen molecules to the C$^3\Pi _u^+$ vibrational state, creating a population inversion between the aforementioned C$^3\Pi _u^+$ state and the B$^3\Pi _g^+$ state. The elongated nature of the filament allows for efficient amplification of a $\lambda = 337$ nm UV seed or the amplification of the spontaneous emission at the aforementioned wavelength.

3. Validation

In this section, we will compare the results of our Maxwell-Bloch code, Dagon, with previously published experimental [7,8] and 1D modelling results [9]. With this benchmark we will asses the capabilities of our 3D model, ensuring the reliability of its results and its predictive capabilities.

In all the simulations performed, except those in the amplified spontaneous emission (ASE) regime, an UV ($\lambda = 337$ nm) seed with an energy of E = 3.5 pJ, a FWHM duration of $\tau = 1.5$ ps and a spatial FWHM of 20 $\mu$m is injected in the plasma synchronized with the IR pulse that creates the plasma. Our computational domain is a 50 $\mu$m $\times$ 50 $\mu$m $\times$ 3.5 cm box composed of 50 $\times$ 50 $\times$ 2000 cells. The plasma has a length of L $= 3.5$ cm and a gaussian radial profile, with $\sigma = 10.2~\mu$m, which corresponds to a radial FWHM of 24 $\mu$m (this parameter is varied in section 4.3). These dimensions (several centimetres length and tens of micrometres width) are typical of small scale, laboratory filamentation experiments, as the ones reported in [8,30]. The peak density of $\mathrm {N_{2}}$ molecules is $n_{\mathrm {N2}} = 2.7\times 10^{19}$ cm$^{-3}$. The plasma evolution is given by the aforementioned plasma model, assuming that the IR pulse creates a plasma with $n_e(0) = 2\times 10^{16}$ cm$^{-3}$ and $T_e(0) = 16$ eV, as in our previous work [7,9]. The assumed initial electron density is well within the range of reported experimental measurements [31–34], with values of electron density between $10^{16}$ cm$^{-3}$ and $10^{17}$ cm$^{-3}$.

3.1 Comparison with experiments

As a first step to validate our model, we have compared Dagon against published experimental results, namely amplification curves at different pressures [8] and temporal profiles (duration and position of the peak) of an amplified UV seed and ASE [7].

The amplification curves are constructed as follows. The plasma filament is created and pumped by an intense infrared pulse that arrives at $t = 0$ ps. An UV pulse, tuned to the wavelength amplified in the plasma, is seeded with a variable delay with respect to the infrared pulse. The energy of the amplified seed is measured and plotted against the delay. These curves allow to find the build-up and decay characteristic times of the gain and its duration, parameters of crucial importance to understand the dynamics of gain inside these filaments and achieve backwards lasing. The amplification curves at three different $\mathrm {N_{2}}$ pressures computed by Dagon are shown in Fig. 1 (left). Consistent with previous results [8], these curves exhibit a fast built-up ($\tau _b$), as opposed to a long decay time ($\tau _p$). Furthermore, this behaviour strongly depends on the gas pressure, being $\tau _b$ = 5 ps for 1 bar (4 ps experimentally), $\tau _b$ = 8 ps for 600 mbar (7 ps) and $\tau _b$ = 14 ps for 400 mbar (11 ps). In fact, the dependence on gas pressure is further evidence of the aforementioned electronic collisional pumping mechanism, since a reduction in the pressure leads to a decrease in the electron-molecule collision frequency and, with it, the gain built-up time increases.

 

Fig. 1. Left: Amplification curves at different $\mathrm {N_2}$ pressures. Right: Intensity profile after propagating through 3 cm of plasma for seeded (blue) and ASE (red) regimes.

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In addition to the amplification curves shown, we have obtained with Dagon the intensity temporal profiles of an UV seed pulse and the ASE, shown in Fig. 1 (right), and compared them with experimental measurements [7]. Our modelling results show that, while the ASE pulse is characterized by a slow built-up time ($\tau _b$ = 15.3 ps) and long response (FWHM duration $\Delta t = 12$ ps), the amplified seed pulse (actually, the wake developed after the seed) precedes the former ($\tau _b$ = 4.26 ps) and exhibits a FWHM duration of $\Delta t\sim$4.26 ps. The experimentally measured profiles of [7] revealed a rise time of $\tau _b = 17$ ps and a FWHM duration of $\Delta t = 20$ ps for the ASE while the seeded emission showed a $\tau _b = 4.5$ ps and a FWHM duration of $\Delta t = 3.3$ ps.

In conclusion, our 3D model reproduces the observed experimental results, namely the increase of the rising time of the gain at decreasing pressures and the temporal characteristics (build-up time and FWHM duration) of the emitted UV beam, both in seeded and ASE regimes.

3.2 Validity of the adiabatic approximation

In some cases, it is possible to assume that the medium response to the external electric field (the UV seed) is immediate, i.e. transients are neglected. This is the so-called adiabatic approximation. As a result, instead of solving an ordinary differential equation (Eq. (4)), polarization density is now proportional to the electric field [35], such that:

The adiabatic approximation is widely used, since it lies underneath the well known radiative transfer equation for the intensity:

3D Dagon simulations lead to the same outcome than in previous studies [9], namely, the adiabatic approximation cannot grasp amplification dynamics accurately. Figure 2 (left) shows the 3D intensity profiles of the amplified UV beam after propagating through more than 3 cm of plasma. The differences are obvious. Figure 2 (right) shows a longitudinal cut of both profiles at the centre of the plasma. As it can be observed, the adiabatically amplified pulse is more intense and shorter than the non-adiabatically amplified pulse, in disagreement with experimental results: the FWHM duration is $\Delta t =3.3$ ps experimentally, $\Delta t = 4.26$ ps for the non-adiabatic model and $\Delta t = 0.9$ ps for the adiabatic model. Furthermore, the fact that the intensity peaks of the pulses are not aligned is a consequence of the different amplification dynamics. The adiabatically amplified pulse can be identified as the amplified seed itself, whereas the time-dependent simulation pulse comes from the amplification of the picoseconds-long wake induced by the seed [38]. Once again, the adiabatic approximation is not consistent with experiments since it predicts a build-up time of $\tau _b = 0.7$ ps while the experimentally measured time was $\tau _b = 4.5$ ps and the non-adiabatic model gives a value of $\tau _b = 4.3$ ps.

 

Fig. 2. Left: Intensity of adiabatic (blue) and time-dependent simulations (red). Right: Intensity profile along x = 25 $\mu$m, y = 25 $\mu$m.

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Following [9], this is not just a result of the selected parameters, but lies in the very nature of the adiabatic approximation. Thus, it is the mismatch between the electric field of the seed and plasma induced response, polarization, which is responsible of the divergence of results. If the plasma depolarization characteristic time, $\gamma ^{-1}$, is longer than the seed-pulse duration, polarization dynamics cannot be reproduced by the expression in Eq. (12) since the plasma reacts in a slower timescale. Adiabatic approximation is then more appropriate for cases in which the duration of the pulse is greater than $\gamma ^{-1}$, such that plasma response (i.e. polarization) is bound to seed excitation.

3.3 Influence of the depolarization rate

The amplification dynamics of UV pulses in plasmas is strongly driven by the depolarization rate ($\gamma$ in Eq. (4)). This is a crucial parameter both in plasma-based X-ray lasers and atmospheric lasing, since it determines the characteristic depolarization time ($t_p = \gamma ^{-1}$) of the plasma response to the electric field and, as a consequence, the amplification dynamics. Depending on the value of $\gamma$, the seed UV pulse will develop a longer or shorter wake that will be amplified. It is thus of crucial importance that our 3D code Dagon correctly models this process, since the temporal profile of the amplified seed can be used as a probe for the collisional processes that take place inside the plasma.

Indeed, the distinct nature of the depolarization mechanism gives rise to a different temporal evolution during amplification process. For instance, when the emitters are highly charged ions, as in plasma-based soft X-ray lasers, the depolarization rate is taken to be the electron-ion collision frequency, such that $\gamma = \langle \sigma v\rangle N_{e} \propto N_{e}T_{e}^{-3/2}$ [27,38]. But this is not the case for nitrogen atmospheric lasing. If the depolarization rate were dominated by collisions with neutral molecules [28,29], its value would be constant in time since $\gamma \approx \langle \sigma v_{n}\rangle N_{n} \propto N_nT_n^{1/2}$ and both density and temperature of neutral molecules are constant during the amplification timescale, since they evolve in slower time scales.

The assumption of depolarization by collisions with molecules does not explain experimental results [7,9], specially regarding backward amplification (the ultimate goal of these schemes). This fact should not be surprising, since in forward regime it can be assumed constant plasma conditions for propagation and amplification of the seed pulse, as the delay with the infrarred pulse is fixed. As a result, a suitable constant $\gamma$ value may be adequate to reproduce seeded forward amplification, but will fail to explain ASE and backward regimes. In these cases, the observed disagreement in the temporal profile is a strong indicator of a diminishing depolarization rate, which can be explained if $\gamma$ is actually dominated by electron-neutral collisions, such that $\gamma = \langle \sigma v\rangle N_{e} \propto N_{e}T_{e}^{1/2}$ [9]. Electrons cool faster by colliding with neutral nitrogen molecules and creating the population inversion. Thus, the collisional depolarization rate decreases also in time.

Figure 3 (left) shows the 3D intensity profile as given by Dagon for three diferent models of the depolarization rate: $\gamma _{high}=8.3\times 10^{11} s^{-1}$, $\gamma (t) \propto N_{e}T_{e}^{1/2}$ [9] and $\gamma _{low}= 10^{11} s^{-1}$. Figure 3 (right) shows longitudinal cuts of the intensity profile for the three cases. The results are consistent with 1D simulations [9]. A constant, low value of $\gamma$ leads to a higher value of intensity and a strongly developed wake, presenting several oscillations. A low $\gamma$ produces higher intensity pulses since collisional broadening is less important, increasing the value of the cross-section at the line centre, as stated in expression (15). In addition to this, the characteristic oscillating pulse structure is identified with the well-known Rabi oscillations, which arise from the fluctuation between lasing level populations interacting with a resonant electromagnetic field [39,40]. The development of this structure is directly related to the rise and the amplification of the wake after the seed pulse, enabled by a low depolarization rate, as previously commented.

 

Fig. 3. Left: 3D structure of the intensity for $\gamma _{low}= 10^{11} s^{-1}$ (black), $\gamma (t) \propto N_{e}T_{e}^{1/2}$ (blue) and $\gamma _{high}=8.3\times 10^{11} s^{-1}$ (red) simulations. Right: Intensity profile along x = 25 $\mu$m, y = 25 $\mu$m.

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3.4 Remarks on the validation of the model

In this section we have shown the benchmark of our 3D, time-dependent, Maxwell-Bloch code Dagon against previously published experimental and 1D modelling results. We can conclude that Dagon is capable of accurately model the amplification of UV radiation in nitrogen plasma filaments for a wide range of parameters and cases: gain dynamics, amplification dynamics in both seeded and ASE regime, adiabatic and non-adiabatic plasmas, and different collisional rates, which strongly impact the amplification dynamics. The good performance of Dagon against all these bechmarks allows us to proceed to predictive simulations.

4. 3D results: using the phase as a diagnosis tool

The benchmarks shown in the previous section ensure the accuracy and predictive capability of Dagon when modelling amplification in plasma filaments in the forward direction, both in ASE and seeded mode. Thus, in this section we will apply Dagon to study the 3D structure of the amplified pulses. We will stress the importance of the phase of the amplified beam. It has been recently demonstrated experimentally that combined measurements of intensity and phase provide a huge amount of information about plasma dynamics (electron density profile, ionization, etc). Indeed, the radial profile of the electron density of the plasma could be extracted almost directly from the phase, as reported in [41]. In this section we will show that the phase of an UV seed amplified in a nitrogen plasma filament carries information about the plasma, thus confirming the feasibility of using the phase as a diagnostic tool.

4.1 Adiabatic and non-adiabatic amplification

First, we will study the phase of the amplified beam when using the adiabatic approximation and the full time-dependent polarization. Figure 4 (left) shows the phase profile (y = 25 $\mu$m) at intensity maximums of adiabatic and non-adiabatic simulations (z = 3.34 cm and z = 3.23 cm, respectively). Although relatively similar at pulse centre, where most of the intensity is concentrated, when moving away from this region phases begin to diverge. Different amplification dynamics leads to rather different phase profiles.

 

Fig. 4. Left: Phase profile at intensity maximums (y = 25 $\mu$m, z = 3.34 cm for the adiabatic case and z = 3.23 cm for the non-adiabatic). Right: Integrated phase profile for adiabatic (blue) and time-dependent simulations (red).

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However, in order to asses the usefulness of the phase as a diagnostic method, we need to integrate the phase along all the beam, since this approximates better the signal detected in an experiment (not the phase of a single slice of the amplified beam). The integration takes into account the intensity of the beam by weighing the phase at each point with its respective intensity. This integrated phase is shown in Fig. 4 (right). It is clear from the figure that both phase profiles are similar. This result is logical since the adiabatic approximation is known to compute accurately integrated quantities in which the temporal dependence does not play a role [9]. As shown in Fig. 2, the pulses mainly differ in its intensity and duration, but since there are not more complex structures, such as a decaying oscillatory tail, the integral phase is expected to be similar in both cases. Note as well that the agreement in the outer regions of the plasma is related to the initial gaussian seed, as in this region neither the seed is amplified nor the wake is developed.

4.2 Impact on the phase of different values of collisional depolarization rates

The phase as a diagnostic tool is expected to stand out when the amplified beam has complex structures induced by the interaction with the plasma filament. This is the case shown in Fig. 3. When the collisional depolarization rate is low a wake is developed and the amplified beam has a complex, oscillating temporal structure. Figure 5 shows an equivalent analysis to that of the previous section of the three different cases. While now the phase profile at intensity maximums (Fig. 5 left) are nearly coincident (z = 3.27 cm for $\gamma _{low}$, z = 3.23 cm for $\gamma (t)$ and z = 3.24 cm for $\gamma _{high}$), the $\gamma _{low}$ integral phase profile considerably differs from the other cases (Fig. 5 right). Parabole-shaped profiles of $\gamma _{high}$ and $\gamma (t)$ contrasts sharply with the wolf-ears-shaped profile of $\gamma _{low}$, with a central plateau and peaks at around 15 $\mu$m from plasma centre. Again, at the domain borders all profiles are equivalent, as explained earlier.

 

Fig. 5. Left: Phase profile at intensity maximums (y = 25 $\mu$m, z = 3.27 cm for $\gamma _{low}$; z = 3.23 cm for $\gamma (t) \propto N_{e}T_{e}^{1/2}$ and z = 3.24 cm for $\gamma _{high}$). Right: Integrated phase profiles of $\gamma _{low}=1\times 10^{11} s^{-1}$ (black), $\gamma (t) \propto N_{e}T_{e}^{1/2}$ (blue) and $\gamma _{high}=8.3\times 10^{11} s^{-1}$ (red) simulations.

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So, why a low depolarization rate triggers such a different response? As mentioned above, the UV seed amplified in a plasma with a low collisional depolarization rate $\gamma _{low}$ develops a long, oscillatory wake, such that the pulse itself is composed of several sub-pulses, shown in Fig. 6. Each of them, or at least the first three, have a non-negligible contribution to the final intensity, so they will also have a weight in the integrated phase of the pulse. While it is true that all the oscillations come from the same initial excitation, i.e. the seed, they may not face the same plasma conditions and dynamics and, therefore, the phase may be somehow affected. For instance, the medium gain, which is dependent on the population inversion, induces a phase distortion in the amplified pulse [42], such that amplification and absorption have opposite effects over the phase.

 

Fig. 6. Intensity slice in x-z plane (y = 25 $\mu$m) of $\gamma _{low}=1\times 10^{11} s^{-1}$ case (centre), confronted with the integral phase profile of the different identified sub-pulses (left) and the cumulative integral phase profile (right).

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As a result, our initial guess was that the long, low-intensity wake following the first amplified pulse (3.21 < z < 3.34 cm, as shown in Fig. 6) was disturbing the otherwise "typical" phase distribution. In order to assess the veracity of this statement and find the origin of the abnormal phase contribution, we computed the integrated phase of the different parts of the pulse, considering as well the seed. Figure 6 (centre) shows the longitudinal section of the pulse (y = 25 $\mu$m), along with the integrated phase of each sub-pulse (left, subfigures Fig. 6( I1)–(I4)) and the "cumulative" integrated phase (right, subfigures Fig. 6(C1)–(C4)), in which the integration domain comprises several sub-pulses, as marked by the arrows in the figure. In this way, it is possible to directly associate the complex structure of the pulse with the observed phase. It must be noted that the color scale has been limited so intensities greater than I$~> 10^{7}$ W/cm$^{2}$ are depicted in red to ease sub-pulse visualization and that the last sub-pulse (3rd sub-pulse, z < 3.06 cm) involves the remaining 3.06 cm of the plasma column (even other less intense pulses could be found within the tail).

Now it is possible to better understand the relationship between the amplified beam temporal profile and the phase profile. The first sub-pulse has an integrated and cumulative phase (Fig. 6(I2) and 6(C2)) similar to that of $\gamma _{high}$ and $\gamma (t)$ integrated phase profiles (Fig. 5 right). The small influence of the UV seed (Fig. 6(I1) and 6(C1)) is also obvious. However, the second sub-pulse’s phase (3.06 < z < 3.21 cm, Fig. 6(I3)) perfectly illustrates how different two consecutive sub-pulses can be, showing an opposite profile with a sharp top-hat shape in its centre. Its contribution to the cumulative integrated phase (Fig. 6(C3)) explains the absence of a parabole-shaped phase at the centre of the pulse.

Finally, the sharp peaks that appear at r$\sim$ 15 $\mu$m in Fig. 6(C3) and 6(C4) and the valleys at r$\sim$ 10 $\mu$m in Fig. 6(C3) are still unexplained. They are not directly caused by the second and third pulse themselves but are associated with the long tail following each sub-pulse. As it can be observed in the intensity slice of Fig. 6, the previous division into separate pulses is only clear in the plasma centre. At higher radii each sub-pulse developes a continuous tail that runs in parallel with the following sub-pulses. Consequently, the aforementioned peaks are the result of the evolution of the phase, positive in this case, of the first pulse’s tail, at an approximate radius of 12.5 $\mu$m. The same reasoning can be applied to explain the valleys in the phase, which disappear in the final profile due to the second sub-pulse’s tail. These last statements can be further appreciated in the profile of the third sub-pulse, where four different contributions are distinguished: the seed, in the outer regions; the first sub-pulse, with a narrow peak around 12.5 $\mu$m from the centre; the second sub-pulse, with its strong peaks; and finally, the third sub-pulse, surrounded by the former pulses and thereby limited to a small central region of r$\sim$5 $\mu$m.

The presence of this long lasting tails can be explained from the combination of two factors: a low collisional depolarization rate and the injection of a seed beam with a gaussian profile, which rapidly depletes the population inversion at the centre of the filament (high intensity region), leading to Rabi oscillations, but does not saturate the amplifying medium at higher radii (low intensity regions) and results in longer continuous pulses, the observed tails.

Another way to visualize the influence of the sub-pulses is through the longitudinal phase profiles at plasma centre (x = 25 $\mu$m, y = 25 $\mu$m), shown in Fig. 7 along with the intensity of $\gamma _{low}$ case. While $\gamma _{high}$ and $\gamma (t)$ cases have a slow, moderated and smooth evolution of the phase, the $\gamma _{low}$ case exhibits steep variations of the phase that are precisely located in the intermediate region between sub-pulses: z$\sim$3.21 cm (first-second sub-pulse), z$\sim$3.06 cm (second-third sub-pulse) and others could be identified from even lower intensity sub-pulses. These transitions allow to understand the strong differences observed between the integrated phase profiles of Fig. 6 and again reinforces the idea that the structure of the pulse is responsible for the different phase profile of the low collisional depolarization rate case. It is not a coincidence that the phase starts to diverge and rise rapidly from the moment when the first sub-pulse decays and the second emerges.

 

Fig. 7. Longitudinal phase profile (x = 25 $\mu$m, y = 25 $\mu$m) of $\gamma _{low}=1\times 10^{11} s^{-1}$ (black), $\gamma (t) \propto N_{e}T_{e}^{1/2}$ (blue) and $\gamma _{high}=8.3\times 10^{11} s^{-1}$ (red) simulations. The intensity of $\gamma _{low}$ case is represented by the green dashed line.

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4.3 Diagnostics of the active region of the filament

The results presented in this article along with previously published experimental and modelling results [41] highlight the sensitivity of the phase to amplification and plasma dynamics. In this section we will study the amplification of an UV seed through plasma filaments with a gaussian radial density profile of different width and relate the phase to its width. The final objective in future works is to apply phase measurements towards unveiling quantities such as electron density, electron density gradients and ionization in plasmas. Thus, this technique would add to current techniques to measure electron density in filaments as measurements of plasma conductivity [43], terahertz spectroscopy [30], the interaction of the filament with microwaves in cavities [34] and interferometric techniques [33] among others.

Figure 8 shows the phase of the amplified UV beam when seeded in different width filaments (left) and the radial electron density profile of the filament corresponding to each width (right). From the image it is apparent that there are three differentiated phase profiles: one corresponding to the filament widths $\sigma = 50$ $\mu$m and $\sigma = 25$ $\mu$m, a second one corresponding to filament widths of $\sigma = 12.5$ $\mu$m and $\sigma = 10$ $\mu$m and a third one corresponding to the narrowest filament, $\sigma = 5$ $\mu$m ($\sigma$ denotes the standard deviation of the gaussian profile). One may conclude that the radial profile of the phase allows to estimate the width of the active region of the plasma filament. It is true that the resolution, at least in this model, is micrometres at best, since the $\sigma = 12.5$ $\mu$m and $\sigma = 10$ $\mu$m cases are barely distinguishable. Worse, the $\sigma = 50$ $\mu$m and $\sigma = 25$ $\mu$m cases present the same radial phase profile. This was expected since all the cases are indistinguishable near $r = 0$ $\mu$m, i.e. near the peak of the gaussian profile, where the density gradients are smaller. The widest cases present small density gradients in the region covered by the UV seed and thus the phase remains similar for both cases. Seeding a wider UV pulse will alleviate this problem and will allow to discriminate between the two filament widths. We can conclude that these results support our hypothesis: the radial phase profile of amplified UV beams in plasma filaments might allow to diagnose electron density profile and gradients. Assuming that nitrogen molecules are, at maximum, single ionized, we can estimate from the electron density the mean ionization and the density of $\mathrm {N_2^+}$ ions.

 

Fig. 8. Left: Integral phase profile (y = 25 $\mu$m) for different plasma widths. Right: The corresponding varying-width electron density profiles.

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5. Conclusion

In this article we show, for the first time up to our knowledge, 3D modelling of the amplification of UV beams seeded in plasma filaments of nitrogen with our 3D Maxwell-Bloch code Dagon. This code has been used in the field of plasma-based soft X-ray lasers and it has been adapted to study cavity-free nitrogen lasing. We show different benchmarks against experimental and 1D modelling results previously published that allow us to asses the predictive capabilities of Dagon. Up to now, our code has modelled laboratory scale (several centimetres long and tens of micrometres wide) filaments. Our model can be enhanced to tackle filaments of practical interest for applications (metres long, 50-100 $\mu$m wide). A change of variables to have a moving computational window that follows the IR and amplified UV pulse will suffice for forward lasing. This approach will not work for backwards lasing. In this case, Dagon can be enhanced with the adaptive mesh refinement (AMR) paradigm, using AMReX [44], for example. This paradigm allows to discretize the full computational domain with a coarse mesh and locally refine the mesh when required (i.e. the backward-propagated UV pulse). This approach is easily parallelizable and enables the modelling of disparate length scales. After this validation, we have used Dagon to propose the measurement of the radial profile of the phase of an externally seeded UV beam after amplification in the plasma filament to unveil different aspects of the amplification dynamics. We have shown that complex temporal structures of the amplified pulse (development of a long standing, oscillatory wake), directly related to the amplification dynamics and collisional processes inside the plasma, imprint the phase and can thus be diagnosed with these measures. Moreover, the radial phase profile allows to directly probe the width of the active region of the plasma filament with micrometre precision. Consequently, we can conclude that measuring (using different techniques already available in experimental facilities, such as UV wavefront sensors or ptychographic techniques) the radial profile of the phase of an amplified UV beam seeded in the plasma filament, allows to probe amplification and collisional processes, along with electron density values and gradients, mean ionization and density of $\mathrm {N_2^+}$ ions.