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Abstract
We numerically investigate the nonlinear propagation of long-wavelength, higher order Bessel pulses in the atmosphere. We show that 10 micron higher order Bessel beams can generate more homogeneous plasma channels than their 800 nm counterparts. We utilize 4th-order 10.23 μm Bessel wavepackets to create tunable transient plasma tubes in air, and show that they are well suited for the guiding of THz radiation with exceptionally low losses.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The use of higher order Bessel beams (HOBB) for plasma generation, especially for UV to near-IR lasers, is well documented in the literature [1–3]. However, long-wave infrared (LWIR) HOBB and their distinct advantages when it comes to nonlinear propagation in transparent media are largely unexplored.
Higher order optical Bessel beams, like in the 0-order case, consist of a series of co-centric rings, with their conical energy flux sustaining their invariant spatial profile over a limited distance called the Bessel zone (BZ). The BZ is defined by the Bessel scaling parameter $w_0$ (center ring thickness), the apodizer radius $\alpha$, and the laser wavelength $\lambda$ [4–6]. The obvious difference between a HOBB and 0-order Bessel beam is that the former exhibits an intensity zero instead of the maximum at the center. Critically for waveguiding, the main ring diameter depends on the Bessel order and $w_0$. Experimentally, HOBB can be generated with the use of an axicon together with a $n^{th}$-order spiral phase plate (matching the order of the HOBB) inducing the needed vorticity [1].
The use of LWIR HOBB for the generation of plasma waveguides has many advantages when compared to short wavelengths. As has been repeatedly reported, mid-IR and LWIR high power lasers produce filaments and plasma channels that are much thicker, typically of the order of 0.5 - 1 mm in diameter [7,8], when compared to shorter wavelengths such as 800 nm Ti:Sapphire systems [9]. This allows, as will be shown later, for the production of thicker plasma waveguides that are predicted to more efficiently confine longer-wavelength guided pulses [10,11]. Because of the much wider plasma channel, LWIR HOBB allow for the generation of homogeneous plasma rings, instead of the multi-filamentation collapse of hundreds of filaments demonstrated in [12]. For a similar guiding channel length, longer wavelengths HOBB have larger ring diameters, which allows for the waveguiding of longer wavelengths [11].
In this letter we will show that 10.23 $\mu m$ HOBB retain their advantages over their 800 nm counterparts, i.e. i) being able to propagate at higher peak intensities before initiating filamentation, and ii) their ability for the generation of much more homogeneous and thicker plasma channel in the atmosphere. We then utilize a 4th order HOBB in order to generate a tubular plasma waveguiding structure in air that is able to guide a 10 THz, 2 $\mu$m-wide beam with extremely low confinement losses over 60 Rayleigh ranges in the atmosphere, essentially only limited by the BZ length. We expect our approach to scale to longer distances given enough power in the 10 $\mu$m pump laser.
2. Numerical model and input wavepackets
The laser-pulse propagation in air is simulated with the help of the Unidirectional Pulse Propagation Equation (UPPE) solver [13] in the radially symmetric geometry $(r, t, z)$. The numerical model includes: diffraction, self focusing through the instantaneous part of the optical Kerr effect for nonlinear refractive indexes $n_{2,10.23 \mu m}=4\times 10^{-23}$ $m^2/W$ [14] and $n_{2,800 nm}=3.2\times 10^{-23}$ $m^2/W$ [15]. Plasma generation is calculated using tunneling ionization for 10.23 $\mu m$ [16], and multi-photon ionization(for 800 nm) [9] as discussed in [4], as well as two-temperature avalanche ionization [17]. For the sake of simplicity we are taking into account oxygen as the only ionizing species for both wavelengths, with ionization potential U$_i=12.03$ eV with a multi-photon order of $K=8$. Dispersion and absorption of air is modeled with the help of the HITRAN database for 30% relative humidity, 25 C and 1 atm pressure.
The wavepacket of choice is a 4th-order Bessel beam with a duration of 350 fs at FHWM and a central wavelength of 10.23 $\mu m$. The Bessel scaling parameter is $w_0=1.2$ mm for the center ring thickness, and a central ring radius of 2.57 mm (from $r=0$ to the middle of the main ring). The whole beam consists of 12 rings apodized with a 2 cm radius circular iris (5th-order super-Gaussian profile), in order to limit the Bessel to the finite size and energy. The necessary spiral phase is accounted for by utilizing the Hankel transform for the radial axis. By selecting the order of the Hankel transform equal to the vorticity of the desired Bessel beam, the phase vortex is automatically enforced on the numerical solution.
Using a 4th-order Bessel with the above geometry as the plasma waveguide generator is found to be a good compromise between Bessel beam diameter, ring thickness, relatively low input laser energy (up to 7.1 Joules), and a Bessel zone spanning 4 - 5 m in the atmosphere. The input radial-temporal intensity profile is depicted in Fig. 1. All of the above parameters can in principle be tuned, through different axicons and source beam parameters, in order to optimise the plasma waveguide for specific applications.
Using radial symmetry in our simulations (neglecting modulation instabilities) is a safe approximation, as Bessel beams are known to self-reconstruct after being perturbed. This resilience is further enhanced by the spiral spatial phase of HOBB. In addition, the power contained in each Bessel ring is relatively low ($\sim 4 P_{cr}$ for the most intense case). Finally, the peak intensities reached in the Bessel zones are notably 2-3 times lower than in typical filaments and lower than those found in previously reported HOBB experiments [1,2] that do not suffer from modulation instability.
We compare the LWIR wavepacket to its Self-Similar Bessel Zone Equivalent (SSBZE) at 800 nm, as was defined in [4] for 0-order Bessels. We find that that SSBZE scaling holds true for HOBB, as expected. By definition, the SSBZE 800 nm HOBB has the same Bessel zone length and the same number of rings, and for 800 nm, is uniquely defined by $w_0=336$ $\mu m$, apodizer of $5.6$ mm and a central ring radius of $0.72$ mm. This 800 nm beam is essentially a shrunk down version (by a factor $\sqrt {10.23 \mu m / 800 nm}$ in radius) of its 10.23 $\mu m$ counterpart, carrying 95.6 mJ. For consistency and in order to limit the effect of dispersion on the dynamics, both pulses have a duration of 350 fs.
3. Results
Figure 2 shows the peak intensity and peak electron density vs propagation distance z, of the 10.23 $\mu m$ and 800 nm SSBZE beams for various starting peak intensities. As depicted in Fig. 2(a) and (c), when the peak starting intensity $I_0$ is increased from the linear-regime to higher values, the dynamics start to deviate from the linear baseline (black curves) by exhibiting visible oscillations in the $I(z)$ profiles (brown dotted and blue dashed curves). As was the case for the recent 0-order Bessel study [4], both 800 nm and 10.23 $\mu m$ eventually reach filamentation intensities (above $2-3 \times 10^{17}$ $W/m^2$, red continuous curves), resulting in notable plasma generation along the Bessel zone, shown in Fig. 2(b) and (d). It is clear that for our 4th-order Bessel wavepackets, the longer wavelength 10.23 $\mu m$ beam is able to:
Fig. 2. 4th-order Bessel for 800 nm (a, b) and 10 $\mu m$ (c, d) SSBZE beams. (a, c) Peak intensity, (b, d) peak plasma densities vs propagation distance z, for various starting peak intensities.
The intensity threshold for collapse, as in the case of 0-order Bessels, is strongly dependent on $\lambda$: with $I_{Crit,800nm} \approx 3 \times 10^{16}$ $W/m^2$ and $I_{Crit,10.23 \mu m} \approx 1.75 \times 10^{17}$ $W/m^2$ respectively. This means that longer wavelength HOBB can propagate at higher intensities, while still maintaining close to linear behavior when compared to shorter wavelength SSBZE HOBB of the same order. This is particularly important for HOBB, as the stationary ring-profile has unique advantages compared to single spot beam profiles (0-order Bessel or Gaussian) that are typically used in long range atmospheric propagation. It is also notable that HOBB can have much higher energy content for the same intensity compared to 0-order Bessel, as the main ring, where the nonlinearity is highest, has a larger area compared to a single on axis-lobe with the same $w_0$, extending their near-linear behavior to even higher pulse energies.
Regarding the peak plasma density vs propagation distance, Fig. 2(b) and (d) show that LWIR HOBB, just like in the case of 0-order Bessels at longer wavelengths, exhibit a much more homogeneous channel once high intensities and plasma densities are reached (red curves). This is of key importance as we will discuss next: a multi-meter homogeneous plasma tube in air can serve as an excellent transient waveguide for much longer wavelengths (THz and microwave radiation). In addition, the 10 micron HOBB generates more ($\times 10^2$ to $\times 10^4$) overall electrons compared to the 800 nm (not shown here). These trends hold true for all higher order Bessel beams we tested (2nd, 4th, and 16th-order, results not shown).
Next we take a look at the profile of the plasma channel left in the wake of the 10.23 $\mu m$ 4th-order Bessel of the highest intensity used in our numerical study: $I_0=1.75 \times 10^{17}$ $W/m^2$. Fig. 3 shows the radial plasma density distribution at various propagation distances along the Bessel zone in air, at $t_{max}=10$ ps after the passing of the pulse. We can see that the plasma profile exhibits very limited variations in terms of thickness, tube diameters, and peak $\rho (r)$, over the 4.5 m propagation distance.
Fig. 3. Radial plasma density generated by the 10.23 $\mu m$ 4th-order Bessel in air with starting $I_0=1.75 \times 10^{17}$ $W/m^2$ at various propagation distances along the Bessel zone.
The maximum density of the tube-shaped plasma structure remains high over the whole Bessel zone, which can also be verified in Fig. 2(d) (red curve). The maximum density variation is roughly one order of magnitude, which is not significant enough to degrade waveguiding over the length of the channel.
As the propagation is highly nonlinear in this case, the plasma profile can exhibit multiple peaks in r (as depicted at $z=3$ m and $z=4$ m). However, in all instances these structures reside on top of wide $\approx 300-500$ $\mu m$ thickness pedestals and can be considered local features of the wider pedestal structure. More importantly for guiding purposes, the contrast between the plasma tube and the inner part remains very high throughout the length of the BZ, which makes up for variations of peak $\rho (r)$ and thickness.
Notably, we observe a slight narrowing of the the radius R of the plasma tube structure, from $R_{z=0}=2.54$ mm to $R_{z=4m}=2.0$ mm. If counting the first of multiple peaks (on top of a wider pedestal), at its most narrow point, the plasma tube radius minima is $R=1.6$ mm at $z=4$ m. As the waveguide radius is of key importance, we will investigate this variation and its impact on confinement loss, in more detail below.
Finally, the peak plasma density generated by the outer rings is multiple orders of magnitude lower than that generated by the first (main) ring, and can safely be neglected for waveguiding purposes here.
To examine the feasibility of guidance in the plasma structure depicted in Fig. 3, we calculated the confinement loss of the lowest loss mode for a frequency of 10 THz ($\sim$188 $\mu m$). Fig. 4 shows the 1/e propagation distance of the mode in terms of intensity ($L_{1/e}$) for a range of inner radii $R$ and plasma densities, showing excellent confinement matching or exceeding the BZ length. The range of radii chosen corresponds to the range of the waveguide’s diameters for the optimal 10.23 $\mu m$ case (red curve in Fig. 2(d)), depicted in Fig. 3. The electron densities were chosen in the lower range of the the peak plasma density values vs z for the same case, approximately $1.0-2.0 \times 10^{15}$ $cm^{-3}$.
Fig. 4. 1/e propagation distance in z for the lowest-loss mode at a range of electron densities $N_e$ and channel radii $R$. Inset: Normalized radial profiles for each transverse polarization component of the mode at $R=2$ mm and $N_e=1.0 \times 10^{15} \mathrm {cm}^{-3}$. The radial position of the plasma channel layer is highlighted.
Though these plasma waveguide structures are often thought of as highly lossy, Fig. 4 shows that higher electron densities and radii can lead to substantially lower confinement loss. These calculations indicate that relatively low densities are sufficient to support guidance over the whole length of the BZ (roughly 60 Rayleigh ranges), and in practice the loss of the plasma channel will be mostly driven by the stability of the structure rather than the confinement potential of the plasma itself.
Numerically, the fundamental mode was simulated using a vectorial finite-difference mode solver with open boundaries. The plasma structure was implemented as an approximation of the ones depicted in Fig. 3 in the form of a uniform $w=300$ $\mu m$ thick plasma ring as depicted in the inset in Fig. 4. A comparison to the traditional boundary matching method under the scalar weakly guiding approximation, closely matched the vectorial approach for all parameters used here. The propagation distance is calculated from the imaginary part of the effective index of the simulated mode by taking the inverse of the attenuation coefficient $\alpha = 4 \pi \mathrm {Im}[n_{eff}] / \lambda$.
As the plasma-tube waveguide is transient, additional control over the waveguide’s refractive index contrast could be gained by adjusting the time delay between the plasma generator and the guided THz pulse. As mentioned above, Fig. 3 depicts the radial plasma density distribution at $t_{max}=10$ ps after the passing of the driving 10.23 $\mu m$ pulse. At this point in time, all rapidly evolving temporal dynamics driving plasma generation have stabilized and the pulse has moved on. In the absence of any other intense field the plasma is expected to slowly recombine over ns-timescales as $\frac {\partial \rho }{\partial t}=-\alpha \rho ^2$, where $\alpha = 5 \times 10^{-13}$ $m^3/sec$ [18] is the recombination coefficient for air. This recombination of free electrons can potentially be used as yet another way of control over the plasma waveguide properties, as the electron density experienced by a guided pulse can easily be controlled by adjusting the relative time-delay between the two pulses. In addition, even longer timescale effects can be utilized such as: radial acoustic wave generation over $\mu s$-timescales [19], and even thermal diffusion over ms-timescales [20], to further control the waveguiding parameters and guide much longer pulses. Such a study goes beyond the scope of this work and will be presented in future studies.
Our approach also gives good control over the waveguide’s spatial profile (tube radius R and thickness), as well as length through the use of a suitable driving HOBB. Since the radius of the main Bessel ring scales with the Bessel order, as well as the scaling parameter $w_0$, and the wavelength $\sqrt {\lambda }$ for SSBZE beams, the plasma waveguide can be engineered to support the guiding of longer wavelengths than 10 THz, and over longer propagation distances.
Figure 5 showcases the scalability of our approach, depicting the peak intensity and peak plasma density vs propagation distance z for a larger version 10.23 $\mu m$ Bessel, with $w_0=3$ mm and apodizer $\alpha =7$ cm. This current geometry exhibits a much longer Bessel zone length of approximately 40 m, and as we can see in Fig. 5, is able to generate a homogeneous plasma channel over the majority of the Bessel zone. By controlling the input geometry and given enough power, higher order LWIR Bessels and their generated plasma channels can in principle be scaled to longer or shorter distances as needed. However, it is important to note that, the Bessel beam energy quickly becomes the practical limiting factor when extending to longer distances, as even for this 40 m example the input energy of the Bessel wavepackets is $\approx$ 25 Joules.
Fig. 5. Peak intensity (black continuous curve) and peak plasma density (dashed red curve) vs propagation distance z of a 10.23 $\mu m$, 4th-order Bessel with $w_0=3$ mm, $\alpha =7$ cm and $I_0 = 7 \times 10^{16}$ $W/m^2$ with a 40 m Bessel zone, showcasing the scalability of the our approach.
Collapse is still driven by the power content of the main ring. The threshold $I_0$ value at which filamentation is triggered is found to be inversely proportional $w_0$. This can be seen when comparing Fig. 2(c) to Fig. 5. On the other hand, the threshold $I_0$ value does not seem to be significantly effected by the order of the Bessel beam, e.g. 4th-order vs 0-order (see [4]).
4. Conclusion
In conclusion, we have numerically studied the nonlinear propagation dynamics of 10.23 $\mu m$ pulsed HOBB in the atmosphere, with a special focus on the plasma generation left in the pulse wake. We find that higher order LWIR Bessel wavepackets retain their advantages over equivalent shorter wavelength wavepackets recently reported for 0-order Bessels. More specifically, LWIR HOBB are able to propagate at higher intensities before departing from the linear prediction, and given enough power are able to generate more homogeneous plasma channels in the atmosphere. We show that a $300-500$ $\mu m$ thick - 2.5 mm radius plasma tube generated by a 10.23 $\mu m$ - 4th-order Bessel pulse can be employed for the guiding of 10 THz radiation, with exceptionally low confinement losses over a distance of 60 Rayleigh ranges. Our approach is in principle scalable to longer and/or wider waveguides by adjusting the Bessel beam parameters to accommodate a broad range of guided wavelengths over extended distances in the atmosphere. We expect the use of LWIR HOBB to have a significant impact on future waveguiding and energy delivery applications in the atmosphere.
Funding
Office of Naval Research; Multidisciplinary University Research Initiative (N00014-17-1-2705); Air Force Office of Scientific Research (FA9550-19-1-0032).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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