1. INTRODUCTION

There are currently ongoing efforts to extend the range and transported energy of long-distance nonlinear pulse propagation in the atmosphere by moving to longer infrared wavelengths, with an additional purpose to mitigate the effects of turbulence. More recently, however, there has been increasing interest in nonlinear propagation for picosecond pulses at near infrared (NIR) to long-wave infrared (LWIR) wavelengths [1–7]. However, here avalanche ionization can become more significant since picosecond timescales exceed the doubling time for electron multiplication, and the quiver energy of the electron in the field, which scales as the wavelength squared, can lead to large freed electron kinetic energies compared to near-infrared pulses. We consider the situation of a small ionization fraction that can accompany long-distance nonlinear propagation prior to optical filamentation in atmospheric pressure gases such as air.

Recent studies in the literature [8] assume that the physical processes that control ionization and regularization in the LWIR are identical to those found at 800 nm. As has been predicted [7,9,10] and experimentally verified [11] in recent years, a new paradigm emerges at mid-wave infrared (MWIR) and LWIR wavelengths where weak ionization can occur well below the usual multiphoton or tunnel ionization thresholds at these wavelengths. In this regime, longer picosecond duration, high-energy pulses can accumulate a sufficient number of free electrons to effectively suppress the Kerr lens self-focusing action and maintain a whole self-trapped beam over multiple Rayleigh ranges. This whole beam self-trapping is much more favorable for long-range, low-loss energy delivery [9] than filamentation. The latter involves a strong spatial compression extracting close to one critical power (Pcr) from the whole beam while the remaining larger portion acts as a supporting energy reservoir. Moreover, the sudden explosive growth in peak intensity of the filament results is a rapid turnoff after some regularization mechanism (plasma defocusing/absorption at 800 nm and optical carrier self-steepening at MWIR) sets in, severely limiting the propagation length of an individual filament [12]. For multi-ps pulses, many-body excitation dephasing (MBEID) generates free electrons by interrupting nonlinear-induced phase coherences well before the high-intensity filamentation stage. Given a weak plasma subjected to the electric field long enough (long or multiple pulses), subsequent cascade ionization should be appropriately modeled by the two-temperature avalanche (TTA) model that was specifically developed for mid-IR lasers [10] and was experimentally verified in [11]. To reiterate, the ionization processes in the mid-IR are fundamentally different than in the UV to near-IR regimes as, not only are the ionization rates different (wavelength dependence) but, in addition MBEID plasma generation in the mid-IR is cumulative over multiple ps rather than instantaneous through MPI/tunnel ionization.

In the mid-IR optical regime, the net focusing or defocusing lens from the combined effects of Kerr, diffraction, and MBEID plasma defocusing from the ionization sources is strongly dependent on pulse duration. As can be seen in Fig. 1, the overall contribution of plasma defocusing and Kerr is overwhelmingly positive for pulse durations up to 500 fs, but becomes strongly negative for longer pulse durations, especially in the trailing part of the pulse. The pulse duration as well as a train of pulses, if any, is therefore expected to be of key importance in the propagation dynamics of pulses longer than 1 ps. This should be especially significant with pulse trains when memory effects from a two-temperature avalanche are incorporated.

 

Fig. 1. Contribution toward focusing of optical Kerr effect and the overall effect of both Kerr and MBEID, in terms of the rate of change with propagation distance ${\rm{z}}$ of the inverse focal length for various pulse durations ranging from 0.5 ps to 4.5 ps. Shorter pulses are dominated by Kerr, while longer pulses are dominated by plasma defocusing from MBEID.

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In this work, we present a study of the propagation dynamics of 10 µm multi-ps pulse trains in the atmosphere. We show that appropriate prepulses can be used to prepare the medium in such a way as to effectively control the filamentation of the main high-power trailing pulse over hundreds of meters in the atmosphere, and deliver a higher quality wave packet downstream by avoiding the naturally occurring pulse splitting found in the single pulse scenario.

2. NUMERICAL MODEL

Simulations are conducted in three dimensions in $ (r,z,t) $ geometry, using the carrier wave-resolved unidirectional pulse propagation equation (UPPE) solver [13]. Since UPPE is propagating the electric field, we are able to natively describe field-related effects such as carrier wave shock formation and higher harmonic generation, which are known to be dominating mid-IR atmospheric filamentation [14]. The spatial box is large enough to accommodate the large beam width with a numerical box of 30 cm resolved by 1200 radial points. A large soft radial aperture was applied every 10 m to minimize boundary reflections. In the temporal domain, a computational box of 80 ps and 65536 points was used, which is fine enough to properly resolve individual oscillations of the electric field. Propagation in ${\rm{z}}$ is controlled by an adaptive step method over a distance of up to 800 m. The spectral box was chosen wide enough to include the 3rd and 5th harmonics, spanning from 2.7 µm to 23.5 µm. All simulations where conducted on our in-house HPE Superdome Flex supercomputer, in parallel on 248 cores (Xenon Skylake 3.8 GHz) with an average simulation time of $\sim 20\,\,{\rm{h}} $.

The UPPE propagator is sourced by a nonlinear polarization and current source that incorporates all relevant physical effects such as a Kerr nonlinear response, its modification due to MBEID ionization, two-temperature avalanche memory effects, and tunnel ionization, as discussed below.

A. Linear Material Properties

We model atmospheric air with a density of $ {N_0} = 2.5 \times {10^{25}} $ particles per cubic meter at 1 atm pressure and 25 C. Atmospheric dispersion and absorption is modeled based on the full HITRAN database accounting for all relevant species ($ {{\rm{O}}_2} $, $ {{\rm{N}}_2} $, $ {{\rm{CO}}_2} $, CO, $ {{\rm{CH}}_4} $, $ {{\rm{O}}_3} $, Ar) including water vapor at a relative humidity of 30% [15], effectively taking into account approximately 800,000 vibration-rotation spectral lines. The real and imaginary part of the linear susceptibility can be seen in Fig. 2.

 

Fig. 2. Blue curves indicate (a) real and (b) linear susceptibility of air for 30% humidity at 25 C and 1 atm pressure. Black curve indicates the initial Fourier limited spectrum of the 3.5 ps input pulses centered at 10.23 µm (scaled).

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Accurately describing the atmospheric dispersive landscape is of paramount importance, since even small inaccuracies can add up to significant cumulative changes over long distances. This is the case of mid-IR filamentation where the broad and intense supercontinuum is driving nonlinear dynamics hundreds of meters downstream. To our knowledge, the HITRAN database is the most detailed way to describe the dispersion of air and, at the same time, flexible enough to include or modify the concentration of each species independently. In this way, it is capable of adjusting to realistic conditions.

B. Two-Temperature Avalanche Ionization

We model avalanche ionization using a two-temperature model (TTA) that has been recently proposed for picosecond duration LWIR pulses in air [11]. Here we briefly summarize the salient features of this model that are relevant to the present discussion. Building on a microscopic foundation, we include sources for laser-induced heating, ionization, collision loss, and impact ionization cooling terms. The three equations capturing transient avalanche ionization then become

The TTA model approaches the standard model of avalanche ionization via inverse bremsstrahlung in the limit of high pressures compared to atmospheric, so that the densities and associated plasma frequencies ${f_p}$ and scattering rates ${\nu _e}$ can be much larger, meaning that lower temperatures will be attained. Then the kinetic temperature Eq. (1) is dominated by the first two terms, and for long enough pulses ${t_p} \gt \tau $ and ${t_p} \gt 1/{\nu _e}$, this may be solved in a steady-state for $({T_{{\rm pl}}} - {T_{{\rm kin}}})$ as a function of $I(t)$. Substituting this solution into the first term of Eq. (3) then yields

C. Ionization through Many-Body Excitation-Induced Dephasing

We will consider initial peak intensities well below the threshold for tunneling ionization, and the dominant source of free electron generation in this regime, at atmospheric pressure, is excitation-induced dephasing (MBEID) due to many-body Coulomb effects that enhance the low-intensity electron densities [19]. We have previously developed a consistent fit to the microscopic source term $\frac{{dN}}{{dt}}{|_{{\rm source}}}$ in terms of the applied field to capture the MBEID effect [7]. More specifically, this accurate fit is provided by the equation

 

Fig. 3. Ionization degree of oxygen for tunneling ionization (dashed curves) and MBEID + tunneling (continuous curves) for Gaussian pulses with durations of 3 ps (black curves), 1 ps (red curves), and 300 fs (blue curves), for various intensity values. Notably, at lower intensity values ionization is dominated by MBEID while tunneling becomes dominant above 2–3$\; \times \;{\rm{1}}{{\rm{0}}^{17}}\,\,{\rm{W}}/{{\rm{m}}^2}$. Ionization degree is calculated for initial generated plasma; no propagation is taking place.

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D. Tunneling Ionization Model

Finally, we include tunnel ionization for oxygen using the model detailed in [20–22], with a rate that is given by

 

Fig. 4. Radial–temporal intensity for the single pulse case (top row) and the three pulse baseline case carrying 0.65 Pcr—1.3 Pcr—3.89 Pcr (bottom row) at ${\rm{z}}=0\,\,{\rm{m}}$ (a) and (d); ${\rm{z}} =44\,\,{\rm{m}}\;$ (d) and (e); and ${\rm{z}}=300\,\,{\rm{m}}$ (c) and (f). Movies of both cases can be found in the supplementary material (Visualization 1 and Visualization 2).

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As shown in Fig. 3 the ionization degree for tunneling (dashed curves) and MBEID + tunneling is dominated by MBEID for intensities below $ {10^{17}}\, \,{\rm{W}}/{{\rm{m}}^2} $. Tunnel ionization becomes dominant above $2.3\times{10^{17}}\,\,{\rm{W}}/{{\rm{m}}^2}$. We therefore anticipate that the microscopic model for the ionization degree captures the initial low-loss whole beam self-trapping, while subsequent spatiotemporal filament collapse downstream is ultimately arrested by tunnel ionization.

3. HIGH POWER ATMOSPHERIC PROPAGATION OF 10 µm PULSE TRAINS

$ {{\rm{CO}}_2} $ laser amplifiers are known to generate trains of multiple pulses separated by a time delay of about $ \sim 24\,\,{\rm{ps}} $ [23]. In the case of a high-power pulse followed by lower-power post pulses, it has been shown that the post pulses increase the detected plasma density generated by the leading pulse by roughly one order of magnitude, mainly due to avalanche ionization consistent with the MBEID+TTA model [11]. In this section, we take the inverse approach, studying the effect of prepulses on the filamentation of a trailing high-power pulse in the atmosphere. In our numerical experiment, each wave packet starts collimated with a beam waist of 1.8 cm and has a Fourier limited pulse duration of 3.5 ps at $ 1/{e^2} $ radius. The delay between the three pulses making up the pulse train is 24 ps between successive pulses. The order of pulses is from lowest power to highest. To study the dynamics between the multiple pulses, we vary the peak power (by modifying only the intensity) of the two prepulses by $ + - 20\% $ of the baseline values of 0.65 Pcr (first pulse) and 1.3 Pcr (second pulse). The main high power wave packet is launched last carrying 3.89 Pcr. ${\rm{Pcr }}=392\,\,{\rm{ GW}}$ is the critical power for air at 10 µm for an instantaneous nonlinear refractive index of $ {n_2}= 4 \times {10^{ - 23}}\,\,{{\rm{m}}^2}/{\rm{W}} $ [11]. The baseline setup of three pulse is depicted in Fig. 4(d).

 

Fig. 5. Effect of prepulses in terms of overall refractive index modulation felt by the trailing pulse at ${\rm{z}} =0\,\,{\rm{m}}$. Black continuous curves: on-axis intensity versus time. Red dashed curves: on-axis peak plasma density versus time. Blue curves: overall refractive index modulation from Kerr and plasma defocusing versus time. (a1, a2) Single high-power pulse carrying 3.89 Pcr. (b1, b2) High-power pulse with two prepulses carrying 0.65 Pcr—1.3 Pcr. (c1, c2) Prepulse intensity $ - {\rm{20}}\% $, now carrying 0.52 Pcr—1.04 Pcr. (d1, d2) Prepulse intensity $ + {\rm{20}}\% $, now carrying 0.78 Pcr—1.56 Pcr. See supplementary material for movies of all four cases as a function of propagation distance z (Visualization 3, Visualization 4, Visualization 5, and Visualization 6).

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Figure 4 show the spatiotemporal intensity for the single pulse case (top row) and our three pulse baseline case carrying 0.65 Pcr—1.3 Pcr—3.89 Pcr (bottom row) at ${\rm z} = 0\,\,{\rm{ m}} $ (a) and (d); $ {\rm z} = 44\,\,{\rm{ m}} $ (d) and (e); and $ {\rm z} = 300\,\,{\rm{ m}} $ (c) and (f). In the single pulse case depicted in the top row of Fig. 4, we can observe the typical reshaping dynamics in atmospheric propagation previously studied [9,11,12]. The single initial Gaussian wave packet is self-focusing and reshaping into the typical “arrow-like” shape, while also compressing in time over the first 44 m. The high intensity leads to strong pulse splitting at ${\rm{z}}={\rm{300}}\,\,{\rm{m}}$, as depicted in Fig. 4(c). On the other hand, in our study case, where two additional prepulses are present, we can see a profound difference in the dynamics. At ${\rm{z}} =44\,\,{\rm{m}}$, we can clearly see that the plasma generated by the prepulses are defocusing the third pulse, which reforms into a donut-like shape. Partial defocusing of wave packets by plasma generated by leading wave packets has been previously reported in the literature for various setups [8] and is expected. This reshaping causes a softening of the nonlinear dynamics in the main pulse, which significantly improves the pulse quality at ${\rm{z}} =300\,\,{\rm{m}}$ downstream. By comparing the two setups at ${\rm{z}}=300\,\,{\rm{m}}$, Fig. 4(c) versus Fig. 4(f), we can see that the three-pulse setup completely avoids pulse splitting while at the same time containing more energy in a single pulse with a peak intensity of $\sim{\times \rm{2}}$ the intensity of the split pulses in Fig. 4(c).

To quantify the effect of the prepulses on the filamentation of the high power trailing pulse, we employ a simple, but intuitive, technique to calculate the pulse overall refractive index modulation Dn with respect to the linear refractive index of air $n$. On axis, the time-dependent refractive index modulation ${\rm{Dn}}({t},{r = 0})$ is given by

Figure 5 shows the effect of the prepulses on the high-power trailing pulse in terms of refractive index modulation due to the Kerr (positive Dn) and plasma defocusing (negative Dn) from all sources. As can be seen in Figs. 5(a1) and 5(a2), in the case of a single pulse without any prepulses present, the front part of the wave packet experiences a positive Dn while the trailing part Dn is negative due the plasma generation from MBEID. This leads, as shown in Fig. 6, to filamentation at ${\rm{z}}=557\,\,{\rm{m}}$ downstream. In the pulse train of 0.65 Pcr—1.3 Pcr (blue dotted curves in Fig. 6), we can see in Fig. 5(b1) that the plasma density is increasing in a stepwise fashion as each pulse is contributing a finite increment due to MBEID and TTA avalanche ionization. As can be clearly seen in Fig. 5(b2), the trailing pulse is now initially propagating in a medium with negative Dn. Decreasing or increasing the power of the prepulses to 0.52 Pcr—1.04 Pcr and 0.78 Pcr—1.56 Pcr will influence the electron density and the negative Dn felt by the third pulse accordingly. This decrease and increase by 20% in prepulse power is depicted in Figs. 5(c1) and 5(c2) and Figs. 5(d1) and 5(d2), respectively. See supplementary material for movies of all four cases as a function of propagation distance z (Visualization 1, Visualization 2, Visualization 3, and Visualization 4).

 

Fig. 6. (a) Beam waist at FWHM and (b) peak intensity versus propagation distance ${\rm{z}}$ for various pulse setups. Black continuous curve: single 3.89 Pcr pulse. Blue dotted curve: three-pulse setup carrying 0.65 Pcr—1.3 Pcr—3.89 Pcr. Red dashed curves: 0.52 Pcr—1.04 Pcr—3.89 Pcr. Gray dash–dotted curves: 0.78 Pcr—1.56 Pcr—3.89 Pcr.

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Figure 6 shows the beam waist at full width half maximum (FWHM) and peak intensity versus propagation distance z for various pulse train setups. The black continuous curves represent a single 3.89 Pcr pulse which, after an initial weak self-focusing first stage (see [12] for a study of two-stage filamentation at 10 µm), undergoes filamentation at much higher intensities, around 557 m. When the two prepulses are introduced carrying 0.65 Pcr and 1.3 Pcr represented with the blue dotted curve, we can clearly see that the filament forms at $\sim385\,\,{\rm{m}}$ with a second refocusing downstream. This at first counterintuitive behavior is a result of the prelensing coming from the leading pulse train, which is able to deliver a single high power compressed wave packet at ${\rm{z}}= 300\,\,{\rm{m}}$. In contrast, the single pulse case exhibits pulse splitting at ${\rm{z}}= 300\,\,{\rm{m}}$ and a loss of power. When comparing the wave packets at ${\rm{z}}= 300\,\,{\rm{m}}$ depicted in Figs. 4(c) and 4(f), it is expected that the latter should collapse sooner, which is indeed the case when comparing the black continuous and blue dotted curves in Fig. 6. As can be inferred by Fig. 5, further control over the filamentation process is possible through the adjustment of energy in the prepulses. A decrease by 20% in the power of the leading two pulses can move the filament even closer (red dashed curve) or, in the case of too much power involving an increment of $ + {\rm{20}}\% $ completely defocus the third pulse preventing filamentation altogether (gray dash–dotted curves). While we show that pulse trains can offer control over spatiotemporal dynamics, filamentation distance, and number of refocusing cycles of LWIR filaments, optimal pulse train power and geometry is not within the scope of this work, and will be the focus of further studies.

4. SUMMARY

In summary, we present a comprehensive and realistic model to study the optical properties of the atmosphere for high power and energy LWIR ultrashort pulse remote delivery. Our model predicts that carefully stacked pulse trains can be used to tailor the filamentation process, in terms of wave packet quality, downstream power, and filamentation onset and refocusing cycles, over hundreds of meters in the atmosphere. Many-body excitation-induced dephasing and a two-temperature-based avalanche ionization mechanism in conjunction with Kerr self-focusing are the main drivers of complex spatiotemporal dynamics in the LWIR regime. The cumulative effect of MBEID and avalanche ionization is of key importance here, enabling a graduated and incremental weak defocusing of subsequent pulses into a weak induced ring-like beam that subsequently refocuses into a single high peak intensity filament.