# On the Possibility of Intense Unipolar THz Pulses Formation in Nonhomogeneous Nonequilibrium Nitrogen Plasma Channels

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. 3D Model

_{2}molecules ${\sigma}_{tr}\left(v\right)$ (see Figure 1). As for nitrogen molecules (ionization potential 15.58 eV), one needs four photons for ionization. For moderate fields with the laser intensity ∼10

^{12}W/cm

^{2}in accordance with the perturbation theory, the probability of three-photon ionization is significantly larger than the four-photon one. Hence, the plasma channel is mainly produced by the three-photon ionization of O

_{2}molecules.

_{2}molecule is plotted in Figure 1. The vibrational cross section grows up rapidly for the absolute values of electron velocities above $7.5\times {10}^{7}$ cm/s (~1.7 eV). This fact additionally reduces the desirable initial energy of the photoelectron peak for the effective amplification of the THz radiation. On the other hand, rapid relaxation of the EVDF in photoionized nitrogen plasma makes the length of amplification behind the leading UV pulse quite narrow, so only extremely short THz pulses (up to single-cycle and even unipolar ones) can be generated in such channels.

^{12}–10

^{13}W/cm

^{2}, pulse duration ~100 fs) that are necessary to form the channel with electron density up to ${10}^{14}$ cm

^{−3}the propagation length will be much larger than the spatial size $L=30$ cm which is considered below.

^{−1}. Two values, namely 1.0 and 1.5 cm are taken for the plasma channel radii. The parameter ${z}_{p}$ is chosen ${z}_{p}=0.025$ cm, that corresponds to the seed half-pulse duration ${\tau}_{p}={z}_{p}/c\approx 8.38\times {10}^{-13}$ s. The pulse initial position was ${z}_{0}=-5{z}_{p}\approx -0.125$ cm. For the above parameters the pulse given by (9) is nearly a single cycle pulse. Such pulse shapes are rather typical for modern experiments [23,33,34]. Also, it is important that the plasma channel radius is much greater than the seed THz pulse wavelength $\lambda =2\pi c/{\omega}_{0}$.

^{−3}is the gas density. In comparison with [18] here there is the inelastic collisional integral ${Q}^{*}$ that leads to rapid (in picosecond time scale) relaxation of the EVDF in the nitrogen plasma channel. As a result, the possibility to produce pulses with a nonzero integral $U$ appears to exist for low-intensity THz pulse as well.

_{2}$\left(X{}_{}{}^{1}\mathsf{\Sigma},v\right)$ with the threshold ${I}_{i}^{\left(v\right)}$, velocity $V$ can be expressed through $v$ via the relation:

## 3. Results and Discussion

#### 3.1. Amplification in the Nonequilibrium Plasma Channel and Formation of the Unipolar THz pulses

^{−3}the gain factor is larger than the unity and exceeds $g\left(L=30\mathrm{cm}\right)~1000$ for ${N}_{e}^{\left(0\right)}\cong {10}^{14}$ cm

^{−3}for ${\rho}_{0}=$ 1.5 cm (see Figure 3a). For channel radius ${\rho}_{0}=$ 1.0 cm the diffraction divergence is more significant, therefore the gain factor is lower. Below the electron density ${N}_{e}^{*}$ we observe the absorption of the THz pulse energy. Such absorption occurs due to the fact that a significant part of the seed pulse spectrum is located in the range $\omega >{\nu}_{tr}$. As a result, the high energy spectrum part of the pulse is absorbed, while the low-energy part is not amplified enough for electron densities below the critical value. Such situation is shown in Figure 3b, where the temporal evolution of pulse spectrum for ${N}_{e}^{\left(0\right)}=2\times {10}^{13}$ cm

^{−3}indicates the dramatic spectrum reconstruction. The significant pulse amplification is accompanied by the pulse transformation to the quasi-unipolar one (see Figure 4). Similar transformation was observed for the process of amplification of a single cycle THz pulse in Xe plasma channel (see [16]), where such an effect becomes possible only for strong THz fields of the order of ${10}^{7}$ W/cm

^{2}, that are able to rapidly destroy the initial peak-like EVDF structure. The situation is quite different for the case of the nonequilibrium plasma channel in nitrogen. Due to the effective vibrational excitations of N

_{2}molecules in the energy range ~2–3 eV the typical time of EVDF relaxation for the plasma at atmospheric pressure is of order of several picoseconds [27]. As a result, the leading front of the THz pulse is located in the amplification zone behind the UV pulse while its trailing edge is not amplified. So, the unipolar pulses formation becomes possible for any value of the seed THz pulse intensity.

^{3}W/cm

^{2}the dependence of the gain versus intensity is flat, i.e., there is no influence of the amplified THz field on the EVDF. Above this value which is nearly the same for both

^{2}it becomes less than unity, i.e., the amplification is changed by the absorption. Near the value ${I}_{0}^{*}$ the amplification dominantly takes place for the rising edge of the pulse, while for the trailing edge the THz energy is absorbed. As a result, there is an increase in the unipolar factor within this intensity region (see Figure 6b). The maximum possible value of the unipolar factor reaches $U\approx 0.7$ for ${R}_{0}={\rho}_{0}=$ 1.5 cm. Thus, for considered parameters the seed bipolar pulse is not amplified but can be converted to the unipolar one.

^{2}. We see that for spatial point $z-ct=-0.1$ cm there is almost no dependence of the EVDF on the propagation length, i.e., the THz field does not contribute to the EVDF evolution. On the other hand, for the spatial point $z-ct=-0.4$ cm the more the propagation length the wider the peak of the EVDF that prevents the possibility of the amplification process.

#### 3.2. Transferring of the Unipolar THz Pulse out of Plasma Channel

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Transport cross-section (solid line) and cross-section of the vibrational level excitation ${\sigma}_{v}^{\left(1\right)}\left(v\right)$ $X{}_{}{}^{1}\mathsf{\Sigma},v=1$ (dashed line) for the N

_{2}molecule. The plot is based on data from [26].

**Figure 2.**3D spatial distribution of the absolute value of the electric field strength in the THz pulse: initial distribution (

**a**), after the 30 cm length propagation in the nonequilibrium nitrogen plasma channel with Gaussian profile of electron density with ${N}_{e}^{\left(0\right)}=$ ${10}^{13}$ cm

^{−3}(

**b**) $3\times {10}^{13}$ cm

^{−3}(

**c**) and 10

^{14}cm

^{−3}(

**d**). Radial sizes of channel and the seed THz pulse are ${R}_{0}={\rho}_{0}=1.5$ cm. The level lines indicate the certain values of the pulse electric field strength. The femtosecond UV pulse is located at the point $z-ct=0$. Initial THz pulse intensity is weak and does not contribute to the EVDF evolution.

**Figure 3.**The gain (

**a**) after 30 cm propagation in dependence on the electron density in plasma channel for initial channel and beam parameters ${R}_{0}={\rho}_{0}=$ 1.0 cm (1) and 1.5 cm (2). The field of THz pulse is weak and does not contribute to the EVDF evolution. On-axis spectra (

**b**) of the THz pulse for different propagation lengths (given in the inset in cm) for ${N}_{e}^{\left(0\right)}=2\times {10}^{13}$ cm

^{−3}and ${R}_{0}={\rho}_{0}=$ 1.5 cm.

**Figure 4.**The unipolar factor after 30 cm propagation in dependence on electron density in the plasma channel for initial channel and beam parameters ${R}_{0}={\rho}_{0}=$ 1.0 cm (1) and 1.5 cm (2). The field of THz pulse is weak and does not contribute to the EVDF evolution.

**Figure 5.**On-axis electron velocity distribution functions in the plasma channel for different spatial points behind the leading UV pulse. The point positions are given in the inset in centimeters. The field of THz pulse is weak and does not contribute to the EVDF evolution.

**Figure 6.**The gain (

**a**) and the unipolar factor (

**b**) in dependence on the seed pulse peak intensity for initial channel radii ${R}_{0}={\rho}_{0}=$ 1.0 cm (1) and 1.5 cm (2), electron density is $7\times {10}^{13}$ cm

^{−3}.

**Figure 7.**On-axis electron velocity distribution functions in dependence on propagation length in the spatial points behind the UV pulse $z-ct=-0.1$ cm (

**a**) and $-0.4$ cm (

**b**) for ${\rho}_{0}=1.5$ cm, ${I}_{0}=$ 10

^{6}W/cm

^{2}. The propagation lengths (in cm) are given in the inset.

**Figure 8.**Unipolar factors in the dependence on the propagation length for plasma channel parameters ${R}_{0}=$ 1.0 cm (

**a**) and 1.5 cm (

**b**). The boundary between the plasma channel and free space is at 30 cm. Curves correspond to rectangular (solid) and smoothed (dash) electron profiles along z-axis.

**Figure 9.**Spatial distribution of the absolute value of the electric field strength in the THz pulse: distribution at the «plasma–free space» boundary after the 30 cm propagation in nonequilibrium nitrogen plasma channel with Gaussian profile of electron densities with ${N}_{e}^{\left(0\right)}=$ $7\times {10}^{13}$ cm

^{−3}(

**a**), after additional 15 cm length propagation in the free space (

**b**). Radial sizes of channel and THz pulse are ${R}_{0}={\rho}_{0}=1.5$ cm. The level lines indicate the certain values of pulse electric field strength. The femtosecond UV pulse is located at the point $z-ct=0$. Initial THz pulse intensity is weak and does not contribute to the EVDF evolution.

**Figure 10.**On-axis distributions of the electric field strength in the THz pulse inside the plasma channel (

**a**) and after its propagation in the free space (

**b**). Propagation lengths are given in the inset in cm. All the parameters of the plasma channel and initial pulse correspond to those given in Figure 8. The femtosecond UV pulse is located at the point $z-ct=0$.

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**MDPI and ACS Style**

Bogatskaya, A.V.; Volkova, E.A.; Popov, A.M.
On the Possibility of Intense Unipolar THz Pulses Formation in Nonhomogeneous Nonequilibrium Nitrogen Plasma Channels. *Photonics* **2023**, *10*, 113.
https://doi.org/10.3390/photonics10020113

**AMA Style**

Bogatskaya AV, Volkova EA, Popov AM.
On the Possibility of Intense Unipolar THz Pulses Formation in Nonhomogeneous Nonequilibrium Nitrogen Plasma Channels. *Photonics*. 2023; 10(2):113.
https://doi.org/10.3390/photonics10020113

**Chicago/Turabian Style**

Bogatskaya, Anna V., Ekaterina A. Volkova, and Alexander M. Popov.
2023. "On the Possibility of Intense Unipolar THz Pulses Formation in Nonhomogeneous Nonequilibrium Nitrogen Plasma Channels" *Photonics* 10, no. 2: 113.
https://doi.org/10.3390/photonics10020113