## Abstract

The electron-neutral collisions in the plasma become crucial with regard to the generation of THz radiation when thermal motion of the electrons is considerable. If we look at the mechanism of THz emission, this is only the movement/oscillations of the electrons which is responsible for the excitation of nonlinear current that generated the THz radiation. The present work aims to disclose the role of thermal motion of the plasma electrons to the resonance condition and the THz emission when two co-propagating super-Gaussian laser beams beat in the plasma. The dynamics of the plasma electrons and subsequent generation of nonlinear current are discussed in greater detail for the emission of THz radiation.

## Introduction

Several applications such as characterization of materials, imaging, communication, medical diagnostics make use of THz radiations for their better efficiency. The frequency of these radiations lies between 100 GHz and 10 THz. These radiations can be achieved by various mechanisms which include Cherenkov wakes [1,2,3] and inverse Compton scattering based on X-ray radiation [4]; even soft X-rays have been found to emit from TIN plasmas [5]. The interaction of laser pulses with magnetized plasma has led to the emission of THz radiation [6,7,8]. Some instabilities can be responsible for the emission of such radiation [9]. Laser beating is one of the mechanisms of producing the THz radiation [10]. Some researchers have also established a relationship between the signals corresponding to the THz radiation and X-rays [11]. Magnetic field has been found to enhance the intensity of the emitted THz radiation in laser-plasma interaction [12]. The tunnel ionized plasma can also radiate when a femtosecond laser pulse is launched to this and where dipole oscillations take place due to the residual momentum imparted by the lasers to the plasma electrons [13]. Extremely powerful THz radiations have also been achieved by interaction of chirped [14] and few cycle laser pulses [15] with plasmas.

The above literature proves that the spatial profile of the laser pulses is very important for achieving the THz radiation of specific characteristics during the interaction of laser beams with plasmas. The temporal profile of the lasers has also played an important role in other mechanisms [16, 17]. For laser-plasma interaction, Ostermayr et al. [18] and Devi and Malik [19] have employed super-Gaussian pulses of laser beam for achieving the particle acceleration and to get the self-focusing effect in plasmas, respectively. In almost all the work, people have neglected thermal motion of the plasma electrons during laser-plasma interaction, when talked about the THz radiation generation in a collisional plasma. However, this is vital to understand the impact of finite temperature of the electrons for the emission of THz radiation in a collisional plasma, as these two effects are correlated and should be analyzed together. Hence, the present work is aimed at examining the combined effect of electron-neutral collisions and electron temperature on the emitted THz radiation when two co-propagating super-Gaussian laser beams beat in the plasma.

### Dynamics of plasma electrons

We consider a propagation of two linearly polarized lasers of frequencies *ω*_{1} and *ω*_{2}, and wave numbers *k*_{1} and *k*_{2}, respectively, in the *z* direction of preformed plasma of density N_{0}. The field of the lasers is assumed as

where *E*_{0} has super-Gaussian (sG) field profile, and *j* = 1 and 2. The motion of plasma electrons is described by the momentum equation, whereas plasma ions remain unaffected being heavier than the electrons; hence, these are considered to be immobile. Consistently, the plasma is taken to be weakly ionized where the electron-neutral collisions (frequency *ν*_{en} = *ν*) are present. T_{0} is taken to be the equilibrium temperature of the plasma electrons. For the matching of wave number of ponderomotive force and the nonlinear current, we shall require space-periodically modulated density *N* = *N*_{0} + *N*_{α}*e*^{iαz} together with *N*_{α} as the amplitude and *α* as the wave number of the density ripples along the direction of propagation of the lasers. On passage of the lasers along the *z*-axis, the plasma electrons are under the action of lasers fields (polarized in *y*-direction). The force acting on them is expressed through the equation of motion, i.e.,

in the presence of collisions (frequency \(\nu\)).

### Ponderomotive force by super-Gaussian laser profile

The field of the super-Gaussian lasers is assumed as

Here *p* represents the index of the laser beams and *p* > 2 (even numbers) represents the case of super-Gaussian lasers, whereas *p* = 2 would represent the Gaussian laser only. The super-Gaussian beams have highest field/intensity (peak) for larger region on the axis, whereas the field/intensity of the Gaussian beams is within a smaller region. Because of this, the gradient in the intensity of the super-Gaussian beams and hence the ponderomotive force is larger than the Gaussian beams. The ponderomotive force is realized at the frequency *ω* = *ω*_{1} − *ω*_{2} and wave number *k* = *k*_{1} − *k*_{2}, and the oscillatory velocity of the electrons is obtained from Eq. (2) as

The corresponding ponderomotive potential is calculated from the expression \(\varphi_{p} = - \frac{m}{2e}(\vec{\upsilon }_{1} \cdot \vec{\upsilon }_{2} *).\) Hence, the ponderomotive force is obtained as

### Generation of plasma currents

The motion of plasma electrons is modified due to the ponderomotive force and the pressure gradient force which appears due to the finite temperature of the electrons. Let us find out nonlinear velocity of these electrons \(\vec{\upsilon }^{NL}\) by using the following equation of motion and equation of continuity

Here *k*_{B} represents the Boltzmann constant and its value in CGS units is *k*_{B} = 1.38 × 10^{−16} cm^{2}g/s^{2}K. The above equations yield

Based on this we evaluate the nonlinear current density developed in the plasma. This is given by

The nonlinear density may also be expressed in terms of the electrical susceptibility of the plasma as \(N{}^{NL} = - \frac{{\chi_{e} N_{0} \vec{\nabla } \cdot \vec{F}_{p}^{NL} }}{{m\omega_{p}^{2} }},\) where \(\chi_{e} = - \frac{{\omega_{p}^{2} }}{{\left[ {(\omega^{2} - k^{2} \upsilon_{th}^{2} ) + i\nu \omega } \right]}},\) and \(\upsilon_{th}^{2} = \frac{{k_{B} T_{e} }}{m}.\) This is clear that the electrical susceptibility of the plasma has been modified due to the finite temperature of the plasma. For the cold plasma, where *T*_{e} = 0 (\(\upsilon_{th} = 0\)), however, this expression takes the form of susceptibility derived by other researchers in unmagnetized cold plasma [20] and magnetized cold plasma with the neglect of external magnetic field [12].

The linear density perturbations caused by \(N{}^{NL}\) are given as below

Clearly these perturbations are also affected by the finite temperature of plasma electrons. Now we calculate the force \(\vec{F}^{L}\), which takes the contribution of \(N{}^{NL}\) and \(N{}^{L}\). This is obtained as

The resultant transverse nonlinear electron velocity under the combined action of forces \(\vec{F}^{L}\) and \(\vec{F}_{p}^{NL}\) is obtained as

Based on \(\vec{J}^{NL} = - \frac{1}{2}N^{\prime } e\vec{\upsilon }_{y}^{\prime }\), where \(N^{\prime } = N_{\alpha } e^{i\alpha z}\) yields the following nonlinear oscillatory current density

This current is mainly responsible for the resonant generation of THz radiation provided phase matching condition is satisfied. Here it can be seen that \(\vec{J}^{NL}\) is modified due to the term \(\upsilon_{th}^{2}\) or the finite temperature of the electrons. In order to check the authenticity of the calculations, we discuss the limiting case for the cold plasma (\(\upsilon_{th} = 0\) = *T*_{e}) which was discussed earlier. This can be seen that \(\vec{J}^{NL}\) takes the same form as obtained by other investigators for *T*_{e} = 0. For example, it reduces to the expression obtained in Ref. [20] when *T*_{e} = 0. On the other hand, this expression is the same when one neglects the cyclotron motion of the electrons in the investigation carried out in Ref. [12].

## Discussion on modified resonance and phase matching condition

Equation (13) shows that the nonlinear current \(\vec{J}^{NL}\) oscillates at the frequency *ω*, but its wave number is *k* + *α*. This is clear that the matching of wave number has been done with the application of periodicity of the density ripples. Further, in order to obtain the amplitude of THz radiation, we have made use of the following condition in Eq. (16), later

The real part of above condition is represented as

This is the required condition for the resonant excitation of THz radiation. Clearly, the resonance condition has been modified due to the finite temperature *T*_{e} of the electrons via the term \(\upsilon_{th}^{2}\) in Eq. (15). If we look at the resonance condition, we find that the condition *ω* = *ω*_{p} obtained in the collision-less plasma is departed due to the electron-neutral collisions and thermal motion of plasma electrons in the present case, and hence, the resonance condition modifies. The modified resonance condition in the present case of finite temperature plasma reads \(\omega = \sqrt {\omega_{p}^{2} + \nu^{2} + k_{B}^{2} \upsilon_{th}^{2} }\).

Figure 1 shows the variation of normalized ripple wave number *αc*/*ω*_{p} with the temperature *T*_{e}. This is evident that *α* increases with the increase of temperature. The nature of variation, i.e., the increase of wave number with the temperature, remains the same for different values of collision frequency *ν*. It means the density ripples should be constructed at smaller distances in the case of finite temperature plasma for the resonant excitation of the THz radiation. However, the opposite behavior of *α* with beat wave frequency *ω* (Fig. 2) and collision frequency *ν* (Fig. 3) is observed. This variation suggests that the ripples should be constructed at larger distances if the collision frequency or the beat wave frequency is enhanced in the plasma.

A comparative study of Figs. 1, 2 and 3 reveals that the impact of collisions is much significant on the ripples wave number in comparison with the effect of thermal motion of the electrons. Keeping in mind the variation of *α*c/*ω*_{p} with *T*_{e}, *ν* and *ω*, one can optimize the separation of density ripples in the plasma for the resonant excitation of the THz radiation. The optimization can be done based on Eq. (15). The separation and amplitude of the density ripples in plasma can be achieved experimentally. For example, the use of laser machining with a liquid–crystal spatial light modulator can create the pattern mask [21]. Periodic tunnel ionization and strong ripples in plasma can be achieved by focusing of two counter propagating laser beams through axicon lens [22]. There are other techniques also which can create the periodic density ripples [23,24,25] in plasma and where the periodicity and size can also be controlled by using standing waves created by microwave radiation [25].

### Calculation of field of THz radiation

We make use of the following wave equation for obtaining the field of the THz radiation (say *E*_{THz})

However, we shall use the modified expression of ε due to finite temperature *T*_{e}, given as

The use of Eqs. (15) and (17) in Eq. (16) gives rise to the following expression for \(E_{\rm THz}\)

The real part of the above relation shall be the magnitude of the THz radiation field \(E_{0THz}\). The normalized form of \(E_{0THz}\) is obtained as

where *H*_{R} reads

This is clear that the amplitude of \(E_{0THz}\) is affected by the finite temperature *T*_{e} of the electrons due to the appearance of the term \(\upsilon_{th}\).

### Effect of electron-neutral collisions on THz field

Figure 4 shows the variation of normalized THz amplitude with electron temperature in collision-less plasma (*ν* = 0), depicting the enhancement of THz field with temperature. Figure 5 shows the variation of normalized THz field amplitude with electron temperature for two values of collision frequency as *ν* = 0.05 *ω*_{p} and *ν* = 0.5 *ω*_{p}. Figure 6 depicts the situation of collision frequency *ν* = 0.45 *ω*_{p} and *ν* = 0.6 *ω*_{p}.

It is evident from Figs. 4 and 5 that the thermal motion of the electrons supports the THz emission as the amplitude of THz field is enhanced with increasing *T*_{e}. This is true for the collision-less plasma or the plasma where *ν* ≪ *ω*_{p}. However, for the case of larger collision frequency, negative impact of electron temperature *T*_{e} is realized and we get diffused THz emission for the higher electron temperature. This reveals that one should include the electron temperature when the electron-neutral collisions sustain in the plasmas. When we focus on Fig. 6, we realize that there is an optimum value of collision frequency below which the electron temperature helps achieving the stronger THz emission. On the other hand, if the collision frequency is larger than this optimum value of *ν*, the electron temperature reduces the emission of THz radiation.

Since the present mechanism talks about the resonant excitation of THz emission via the nonlinear current density \(\vec{J}^{NL}\), the reason for an optimum combination of the collision frequency *ν* and electron temperature *T*_{e} for the maximum THz amplitude may be attributed to the nonlinearity of the plasma. This is plausible that the plasma becomes highly nonlinear at this optimum combination of *ν* and *T*_{e}. It means the collisions and thermal motions of electrons play a vital role in setting up the nonlinearity in the plasma via the modified electrical susceptibility or the dielectric constant of the plasma. Under this situation, an enhanced (maximum) current is excited that leads to the maximum radiation through the oscillations of plasma electrons.

### Effect of electron temperature on THz field

Through Fig. 7 we uncover the role of electron temperature *T*_{e} on the resonant excitation of the THz radiation. This is evident from this figure that there is a peak of THz amplitude at a particular value of *ω*/*ω*_{p}, which is not at *ω*/*ω*_{p} = 1, i.e., at *ω* = *ω*_{p.}

In the absence of collisions, the resonance is achieved when beat wave frequency matches with the plasma frequency (*ω* = *ω*_{p}), leading to maximum transfer of laser energy to THz radiation through nonlinear current. However, collisions bring a slight departure in the resonance condition, i.e., resonance occurs at \(\omega \approx \sqrt {\omega_{p}^{2} + \nu^{2} }\). On the other hand, in the present case of collisional plasma where electron thermal motion is also considerable, the resonance condition has been drastically modified, being around \(\omega \approx \sqrt {\omega_{p}^{2} + \nu^{2} + k_{B}^{2} \upsilon_{th}^{2} }\). Because of this we realize a shift in the peak of THz amplitude in Fig. 7. The peak refers to the resonance where maximum energy and momentum from lasers to plasma electrons are transferred for the generation of nonlinear current and then the THz radiation. Hence, the role of electron temperature is to shift the frequency and amplitude of the emitted THz radiation.

### Efficiency of THz radiation mechanism

We employ the similar approach, as used in Refs. [10, 12, 20], for obtaining the efficiency of THz radiation. Hence, the efficiency (say *η*) is obtained using the following formula

For the present case of super-Gaussian lasers and collisional plasma, we obtain

and

where \(\varGamma\) is the Gamma function. Finally, the efficiency *η* reads

where all the symbols have already been defined and have their usual meanings.

Figure 8 depicts the variation of the efficiency of the mechanism of THz radiation generation with electrons temperature for different values of collisional rates. Consistent to the observations of THz field amplitude, the efficiency increases straight with temperature in the presence of low collisions *ν* = 0.05 *ω*_{p}, whereas the efficiency decays for the larger collisions in the plasma (*ν* = 0.55*ω*_{p} and 0.6*ω*_{p}).

On the other hand, the efficiency attains a maximum value at a particular value of beating frequency (Fig. 9). However, lower efficiency is attained for the frequency larger or lower than this critical frequency. This critical frequency is actually the resonance frequency where the maximum transfer of momentum and energy takes place. Interestingly this peak or the resonance frequency is shifted when the electron temperature is changed.

## Conclusions

It is concluded that the resonance condition for maximum amplitude of emitted THz radiation is altered in the presence of thermal motion of the electrons in collisional plasma, but the frequency of emitted radiation can be tuned by changing electron temperature *T*_{e} owing to the thermal effects involved in the process. The amplitude of the emitted THz radiation varies differently with collisional effects. In the presence of low collision frequency *ν* < 0.5*ω*_{p}, thermal motions support the THz emission. However, in the presence of high-frequency collision *ν* ˃ 0.5*ω*_{p}, the THz amplitude is found to decrease with the increased electron temperature.

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Malik, H.K., Singh, D. Terahertz emission during laser-plasma interaction: effect of electron temperature and collisions.
*J Theor Appl Phys* **14, **359–365 (2020). https://doi.org/10.1007/s40094-020-00392-3

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### Keywords

- Laser-plasma interaction
- Ponderomotive force
- Terahertz radiation
- Electron temperature
- Super-Gaussian laser
- Efficiency