Generation of E cient Terahertz Radiation by Relativistic Self-Focusing of A Cosh-Gaussian Laser Beam in A Magnetized Plasma


 This paper presents a scheme for the generation of terahertz (THz) radiation by self-focusing of a cosh-Gaussian laser beam in the magnetized and rippled density plasma, when relativistic nonlinearity is operative. The strong coupling between self-focused laser beam and pre-existing density ripple produces nonlinear current that originates THz radiation. THz radiation is produced by the interaction of the cosh-Gaussian laser beam with electron plasma wave under the appropriate phase matching conditions. Expressions for the beamwidth parameter of cosh-Gaussian laser beam and the electric vector of the THz radiation have been obtained using higher-order paraxial theory and solved numerically. The self-focusing of the cosh-Gaussian laser beam and its effect on the generated THz amplitude have been studied for specific laser and plasma parameters. Numerical study has been performed on various values of the decentered parameter, incident laser intensity, magnetic field, and relative density. The results have also been compared with the paraxial region as well as the Gaussian profile of laser beam. Numerical results suggest that the self-focusing of the cosh-Gaussian laser beam and the amplitude of THz radiation increase in the extended paraxial region compared to the paraxial region. It is also observed that the focusing of the cosh-Gaussian laser beam in the magnetized plasma and the amplitude of the THz radiation increases at higher values of the decentered parameter.


Introduction
Terahertz (THz) radiation generation has been the most interesting topic of research in recent years. THz radiation sources have wide applications in the elds of THz time-domain spectroscopy, material characterization, explosives science, imaging, topography, medical diagnostics, security identi cation, etc. ( Mittleman et al. 1996). Various schemes such as photoconductive antennas, optical recti cation, quantum cascade lasers, and the semiconductors and electro-optic crystals (e.g., ZnSe, ZnTe, GaSe and LiNbO 3 ) have been used to generate THz radiation (Tani et al. 2002;Chen et al. 2011;Vodopyanov 2008; Lee et al. 2000). But due to some limitations of these schemes such as low conversion e ciency, narrow bandwidth of emitted THz radiation, and material breakdown in intense laser pulses, high energy THz radiation is not su ciently generated. These limitations can be overcome by the use of plasma. Plasma is an attractive medium for THz radiation production because it is able to handle very high laser powers without any constraint of medium breakdown.
The production of THz radiation by laser plasma interaction has been the subject of intense study for the past two decades (Liao and Li 2019; Tonouchi 2007). Several schemes have been proposed to produce e cient THz radiation through laser-plasma interactions. Coherent Thz radiation has been measured by laser accelerated electron bunches moving from plasma to vacuum (Leemans et al. 2003; Leemans et al. 2004;Tilborg et al. 2006). Coherent radiation in the range of 0.3-3 THz generated from femtosecond electron bunches at the plasma-vacuum boundary via transition radiation has been experimentally observed (Leemans et al. 2003). Terahertz radiation can also be generated from radiation generated by a two-color laser pulse (fundamental and second harmonic eld) during gas ionization (Kim et al. 2007; Kim et al. 2008;Kim 2009). It has been reported that THz energy of more than 5 µJ is generated through this mechanism (Kim et al. 2008). THz radiation has also been generated by laser pulses propagating in short corrugated plasma channels (Antonsen et al. 2007; Pearson et al. 2011;Miao et al. 2017). It is found that a pulse energy of 0.5 J generates a terahertz energy of 6 mJ through the corrugated plasma channel (Pearson et al. 2011). These types of channels can also be used to generate THz radiation by bunched electron beams. In addition, high power THz radiation has been generated from short laser pulses in the plasma through various mechanisms (Chen 2013; Sharma and Singh 2014; Malik et al. 2020; Koulouklidis et al. 2020).
In recent years THz radiation by self-focusing/ lamentation of an intense laser beam in plasma has been studied.
The main motive is to increase the conversion e ciency of THz radiation. Self-focusing of an intense laser pulse into the plasma increases the e ciency of THz radiation. Hamster et al. (1994) shown experimentally that plasmas produced by high intensity lasers generates coherent Thz radiation. Mun et al. (2007) observed an intense Thz radiation from a relativistic plasma on metal and plastic targets irradiated with 10-TW, 30-fs laser pulses. Hussain et al. (2014; proposed a scheme for the generation of Thz radiation by ponderomotive/relativistic self-focusing of a hollow Gaussian laser beam in a collissionless magnetized rippled density plasma. They found that these schemes increase the power level of the THz wave to the order of Gigawatts. Kumar et al. (2015) studied the effect of selffocusing of an amplitude-modulated Gaussian laser beam on THz generation in rippled density plasma. They observed that when the self-focusing effect is taken into account, the amplitude of the generated THz wave increases signi cantly. Miao et al. (2016) developed a mechanism for ponderomotive driven resonant THz transition radiation generated at plasma boundaries and found that broad-band THz radiation is generated with frequencies up to the maximum plasma frequency. Liao et al. (2016) reported intense THz radiation from relativistic laser-plasma interactions under various experimental conditions. Their results indicate that relativistic laser plasma is a promising source of intense THz radiation. Rawat et al. (2017) studied the self-focusing effect of hollow Gaussian laser beam on the generation of THz radiation in a collisionless magnetized plasma, where relativistic and ponderomotive nonlinearities are operate together. Amouamouha et al. (2020) proposed a scheme to generate THz radiation at the modulation frequency, which is based on the pondermotive self-focusing of an amplitude modulated super Gaussian laser beam in preformed ripple density plasma. This study has been performed in the higher-order paraxial region, where pondermotive nonlinearity is operative. They found that the e ciency of THz radiation reaches 6.5%. Recently, Gupta and Jain (2021) have investigated Thz radiation production by a super-Gaussian laser pulse in a magnetized plasma. They observed that stable THz radiation is generated with a maximum eld strength of 1 GV/cm and a relatively broad spectrum spanning 50 THz, which corresponds to a conversion e ciency for a magnetic eld Various pro les of laser beams such as Gaussian, super Gaussian, hollow-Gaussian and triangular pro les etc. have been used to generate THz radiation in plasma in these studies. Such beams have different types of irradiances across their wavefront, which exhibit different characteristics in plasma. Recently, at-top decentred cosh-Gaussian laser beams have attracted much attention due to their higher e cient power and attractive applications (Konar et al. 2007; Nanda and Kant 2014; Rawat and Purohit 2019). The most important specialty of the cosh-Gaussian laser beam is that it becomes focused before the Gaussian laser beam in the plasma. Furthermore, most studies of THz radiation schemes based on laser-plasma interactions have used the paraxial-ray approximation (Akhmanov et al. 1968). But at high intensities the paraxial-ray theory is not su ciently accurate because this theory does not describe the variation of the radial pro le of the beam from the initial to the ring position. The extended-paraxial ray approximation (Bonnaud et al. 1994; Sodha and Faisal 2008) more accurately describes the propagation of an intense laser beam in plasma than the paraxial ray approximation. At higher laser intensities, the relativistic nonlinearity functional occurs in the plasma. Therefore, it would be interesting to investigate the self-focusing effect of cosh-Gaussian laser beam in plasma with relativistic nonlinearity on the generation of THz radiation under the extended-paraxial approximation theory.
This paper presents a model for the production of THz radiation by relativistic self-focusing of an intense cosh-Gaussian laser beam in a magnetized plasma. This study is carried out under the extended-paraxial ray approximation, which is based on the expansion of the eikonal and nonlinear dielectric constant up to the fourth power of r, where r is the distance from the axis of the beam. Due to the nonlinear coupling between the cosh-Gaussian laser beam and the electron plasma wave (EPW) in a rippled density plasma, THz radiation is emitted. The ripple in plasma density plays a major role for the production of THz radiation as it provides the necessary conditions for phase matching. The paper is structured as follows: In section 2, the self-focusing equation of an intense cosh-Gaussian laser beam through a magnetized plasma in the presence of relativistic nonlinearity is obtained using a higher-order paraxial ray approximation. In the same section, the expression for the nonlinear effective dielectric constant of plasma in the presence of relativistic nonlinearity is obtained. Section 3 presents the analysis for the nonlinear current density and the emitted THz radiation amplitude. Section 4 presents a discussion of numerical results for the relevant parameters. Finally, Section 5 contains conclusions. 2 Relativistic Self-focusing Of Cosh-gaussian Laser Beam In Magnetized Plasma Consider the propagation of a cosh Gaussian laser beam (CGB) of angular frequency (ω 0 + ) and wave vector (k 0 + ) along the direction of the static magnetic eld in a collisionless magneto plasma. The external applied static magnetic eld (B 0 ) is perpendicular to the propagation direction (towards z-axis) of laser beam. The electric eld of cosh-Gaussian laser beam can be written as E 0 + = E x + iE y = A 0 + (r, z)exp i ω 0 + t − k 0 + z 1 where+ sign denotes the right circular mode of propagation, A 0+ is the amplitude of the electric eld, and k 0+ is the propagation wave vector of the laser beam.
The initial amplitude of the cosh-Gaussian laser beam at z = 0 is given by (Lü et al. 1999 where E 00+ is the amplitude of cosh-Gaussian laser beam for the central position at r = z = 0, b is the decentred parameter of the beam, r is the radial coordinate of the cylindrical coordinate system, and r 0 is the initial beam width. The dielectric constant (ϵ + ) corresponding to wave propagation vector (k 0+ ) is given by  The relativistic factor (γ) is given by The propagation of cosh-Gaussian laser beam in magnetized plasma is governed by the general wave equation (Sodha et al. 1974) where ε + is the effective dielectric constant of the plasma.
The effective dielectric constant (ε + ) of the plasma in the presence of relativistic nonlinearity for the right circularly polarized laser beam is given by where ε xx and ε xy are the components of plasma dielectric constant tensor. Putting the value of relativistic factor (γ) in above equation, one may get By using Eq. (1), Eq. (7) can be written in terms of A 0+ is as where ϵ 0 + , ϵ 2 + , and ϵ 4 + are the expansion coe cients. Substituting Eqs. (11) and (12) The solution of Eqs. (13) and (14)  Substituting the value of A 2 00 + and S + from Eqs. (15) and (16) in Eq. (14), and equating the coe cients of r 0 , r 2 , and r 4 on both sides of the resulting equation, one can obtain S 2 (z) = r 2 3 Generation Of Terahertz Radiation The self-focused cosh-Gaussian laser beam excites terahertz radiation in magnetized plasma. Now we consider propagation of a cosh-Gaussian laser beam (whose electric eld is given by Eq. (1)) through rippled density plasma (which is presumed to exist across the magnetic eld). Due to strong nonlinear coupling between cosh-Gaussian laser beam (E 0+ , ω 0+ ,\overrightarrow{k} 0+ ) and electron plasma wave (\overrightarrow{E} p , ω p , \overrightarrow{k} p ), terahertz radiation (\overrightarrow{E} t+ , ω t , \overrightarrow{k} t ) is generated at the difference frequency of cosh-Gaussian laser beam and electron plasma wave. The ripple density plasma ful ls the requirement of momentum phase matching that paves the way for e cient terahertz radiation generation. where {E}_{t0+} is amplitude of the right circular polarized THz eld, and \widehat{r}=\widehat{x}+i\widehat{y} (\widehat{x} and \widehat{y} are the unit vectors along the xand y-axis respectively).
The current density can be evaluated by the following equations: Here, the ion contribution to nonlinearity is neglected because of its heavy mass. By replacing the velocity components of electron and ion from Eqs. (39) and (40)  The amplitude of THz radiation can be obtained by solving eq. (45) with appropriate boundary conditions.

Numerical Results And Discussion
The following set of laser and plasma parameters have been used to perform numerical calculations: ω 0 = 1.778×10 15 rad/s, ω p = 2.398×10 14 rad/s, r 0 =15µm, ω ce = 0.1ω 0 , 0.2ω 0 and 0.3ω 0 , ω p0 = 0.03ω 0 , υ th = 0.2c, When an intense cosh-Gaussian laser beam propagates through a collisionless magnetized plasma, the refractive index/dielectric constant of plasma is modi ed due to the relativistic variation of mass and the beam suffers selffocusing. The two terms on the right-hand side of Eq. (22) describe the divergence and convergence of the laser beam, respectively. When the magnitude of the converging term exceeds the diverging term, the beam becomes selffocused in the plasma. Equations (22) and (23) describe the intensity pro le and beamwidth (focusing/defocusing) of a cosh-Gaussian laser beam in plasma, when relativistic nonlinearity operates in the extended-paraxial region. It is clear from Eq. (22) that the intensity pro le of cosh-Gaussian laser beam in extended-paraxial region depends on the beamwidth parameter (f) and the coe cients of r 2 and r 4 respectively. In addition, Eqs. (22) and (23) show the direct proportionality between the A 00 and E 00 . Numerical computation of the Eqs. (19, 20, 23 and 24) have been carried out to understand the self-focusing behavior of the cosh-Gaussian laser beam in plasma. We have solved Eq. (22) numerically with the numerical computation of Eqs. (19), (20), (23), and (24) to obtain the variation in intensity of the cosh-Gaussian laser beam with the normalized distance of propagation. The results are shown in Figures (1-4). Figure 1 shows the intensity pro le of a cosh-Gaussian laser beam in plasma with the normalized distance of propagation(ξ), when relativistic nonlinearity is functional in the paraxial (a 2 = a 4 = 0) and extended-paraxial (a 2 ≠ a 4 ≠ 0) regions. It is clear that the intensity of the cosh-Gaussian laser beam increases in the extended-paraxial region.
The is because that the focusing of the cosh-Gaussian laser beam becomes faster in the extended-paraxial region than in the paraxial region due to the involvement of off-axis parts (a 2 ≠ a 4 ≠ 0). The maximum intensity of the cosh-Gaussian laser beam increases by a factor of about 2.3 in the extended-paraxial region. Figure 2 shows the variation in the intensity of the cosh-Gaussian laser beam in plasma with the normalized propagation distance for different values of b, when relativistic nonlinearity is operated in the extended-paraxial region. Because of the strong selffocusing of the cosh-Gaussian laser beam in the plasma at high values of b, the laser beam intensity increases with increasing values of b. It is also clear from Fig. 2 that the intensity of laser beam is minimum at b = 0. This is because the beam becomes Gaussian at b = 0 like a dark ring. In this case, the beam intensity is maximum on the axis and the propagation length of the beam is reduced. On the other hand, when b ≠ 0, the beam behaves like a bright ring. The intensity of the beam is maximum on a ring and the propagation length of the beam increases. Thus, the decentered parameter (b) plays a very important role in improving the self-focusing of the cosh-Gaussian laser beam in the plasma. Figure 3 shows the variation in the intensity of the cosh-Gaussian laser beam in plasma with the normalized propagation distance for different values of the cyclotron frequency (ω ce ), when the relativistic nonlinearity is operative in the extended-paraxial region. It is observed that the beam intensity increases signi cantly with increase in the value of ω ce . This is due to the fact that the extent of self-focusing of the beam increases with increase in ω ce .  45) is solved numerically for the same set of parameters and the results are presented in Figs. 5-8. By controlling these parameters, a higher amplitude of the THz wave can be obtained which leads to high THz radiation. Figure 5 shows the variation of the amplitude of the emitted THz radiation (E t+ /E 00 ) with the normalized distance of propagation in the paraxial and extended-paraxial regions, respectively, when relativistic nonlinearity is functional. It is clear from Fig. 5 that the amplitude of THz radiation increases signi cantly in the extended-paraxial region. This is because the amplitude of THz radiation depends on the intensity of the laser beam in the plasma and the laser beam intensity further depends on the self-focusing of the laser beam. Figure 6 displays the normalized amplitude of THz radiation with the normalized propagation distance for different values of b in the extended-paraxial region, when relativistic nonlinearity is functional. The amplitude of THz radiation is minimum at b = 0 because the laser beam intensity is minimum for the Gaussian pro le. It is clearly shown from Fig. 6 that the amplitude of the emitted THz radiation increases with increasing b and is maximum at the focal points of the laser beam, where the laser beam intensity is maximum at b. It is found that the amplitude of the THz radiation by the at-topped cosh-Gaussian laser beam increases by a factor of more than two times that of the Gaussian laser beam.
The normalized amplitude of THz radiation with normalized propagation distance for different values of ω ce is shown in Fig. 7, when relativistic nonlinearity operates in the extended-paraxial region. It is observed that the amplitude of THz radiation increases signi cantly with increase in the value of ω ce . Due to the strong self-focusing of the cosh-Gaussian laser beam in the plasma at different values of ω ce , the laser beam intensity increases which further increases the amplitude of THz wave. Figure 8 shows the amplitude of THz radiation with the normalized distance of propagation for different values of incident laser beam intensity (\alpha {E}_{00}^{2}), when relativistic nonlinearity operates in the extended-paraxial region. It is clear from the Fig. 8 that the amplitude of the THz wave decreases when the intensity of the laser beam increases. This is due to the fact that the extent of self-focusing and the laser beam intensity decreases at higher values of \alpha {E}_{00}^{2}, which also reduces the nonlinear current density and the amplitude of the THz radiation. These results describe the effect of self-focusing of the cosh-Gaussian laser beam on the production of THz radiation in a magnetized plasma.

Conclusions
In the present investigation, we have studied the self-focusing of a cosh-Gaussian laser beam in a ripple density magnetized plasma and its effect on THz radiation generation, when the relativistic nonlinearity in the extendedparaxial region is functional. The THz wave frequency has been obtained at the difference in the laser beam frequency and the density ripple frequency. Phase matching is an essential requirement for high power THz radiation generation. The necessary condition for phase matching is ful lled by the density ripple. Analytical equations have been derived for the beam width/intensity of the cosh-Gaussian laser beam in the plasma, the nonlinear dielectric constant of plasma, the nonlinear current density, and the amplitude of the THz wave. The effects of various laser and plasma parameters such as the decentred parameter (b), the electron cyclotron frequency (ω ce ), and the incident laser intensity (\alpha {E}_{00}^{2}) have been analysed on the self-focusing of cosh-Gaussian laser beam and the amplitude of the generated THz radiation. The results have been compared to the paraxial-ray approximation and Gaussian pro le of the laser beam. The results show that the self-focusing/intensity of the cosh Gaussian laser beam in the magnetized plasma increases in the extended-paraxial region. Furthermore, the intensity of the cosh-Gaussian laser beam increases at higher values of b and ω ce and decreases with increasing \alpha {E}_{00}^{2} due to the self-focusing behaviour of the beam. The intensity is minimum for Gaussian laser beam (b = 0). It is observed that the strong THz radiation is generated by the cosh-Gaussian laser beam in the extended-paraxial region. The amplitude of THz radiation which depends on the self-focusing/intensity of the laser beam in the plasma signi cantly increases at higher values of b and ω ce and decreases with increasing \alpha {E}_{00}^{2}. Hence this scheme can be considered suitable for generating high-power THz radiation.

Declarations
Disclosures. The authors declare no con icts 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.