Direct Electron Acceleration By An Intense Nonparaxial Cosh-Gaussian Laser Beam Driven Electron Plasma Wave in Plasma

Excitation of electron plasma wave by an intense short laser pulse is relevant to electron acceleration process in laser plasma interactions. In this work, the self-focusing of an intense cosh-Gaussian laser beam in collissionless plasma have been studied in the non-paraxial region with relativistic and ponderomotive nonlinearities. Further, the effect of self-focusing of the cosh-Gaussian laser beam on the excitation of electron plasma wave and on subsequent electron acceleration has been investigated. Analytical expressions for the beam width parameter/intensity of cosh-Gaussian laser beam and the electron plasma wave have been established and solved numerically. The energy of the accelerated electrons has also been obtained. The strong self-focusing of the cosh-Gaussian laser beam in plasmas stimulates a large amplitude electron plasma wave, which further accelerates the electrons. The well-established laser and plasma parameters have been used in numerical computation. The results have been compared with paraxial ray approximation, Gaussian profile of laser beam and only with the relativistic nonlinearity. Numerical results suggest that the focusing of the cosh-Gaussian laser beam, the amplitude of electron plasma wave, and energy gain by electrons increases in non-paraxial region, when relativistic and ponderomotive nonlinearities are simultaneously operative. In addition, it has also been observed that the electron plasma wave is driven more efficiently by a cosh-Gaussian laser beam that accelerates plasma electrons to higher energies.


Introduction
The development of short-pulse high intensity lasers has led to research in laser-plasma interactions (Mourou et al. 2006;Norreys et al. 2009). The propagation of intense laser pulses into plasma is the subject of active research that is relevant to many prospective applications such as the acceleration of charged particles, fast ignition in inertial confinement fusion, and new radiation sources (Joshi 2017;Tanaka et al. 2000;Jaroszynski et al. 2006). Laser beams must be propagated over a several Rayleigh lengths in the plasma without divergence or loss of the energy for the realization of these applications. When an ultra-short intense laser beam propagates through the plasma, the beam itself becomes focused and produces ultra-high laser intensity over a large distance, which is used for the excitation of large amplitude electron plasma wave. These laser-driven plasma waves are used for ultra-high energies in the electron acceleration process (Modena et al. 1995;Yadav et al. 2018). Therefore, self-focusing of an intense laser beam in plasma is the most important nonlinear phenomena in laser plasma interaction.
Self-focusing is related to the distortion of the laser beam wavefront, which arises due to the modification of the dielectric constant/refractive index of the plasma during the propagation of an intense laser beam into plasma (Chekalin and Kandidov 2013;Sun et al. 1987). The modification in the dielectric constant/refractive index of the plasma depends on the nonlinearities of the plasma. At high laser intensity, the dielectric constant/refractive index of the plasma is mainly modified by ponderomotive nonlinearity and relativistic nonlinearity.
Ponderomotive nonlinearity arises due to the electron density perturbations by the ponderomotive force of the beam. This nonlinearity is important in the self-focusing of the laser beam due to the expulsion of electrons from the focal spot. The ponderomotive force increases the dielectric constant/refractive index of the plasma, which leads to strong selffocusing of the laser beam. On the other hand, relativistic nonlinearity is set up due to intense electric field of the laser beam. The quiver motion of electrons increases due to high electric field of the laser beam, which further enhances the relativistic mass of electrons and the dielectric constant/refractive index of the plasma.
Self-focusing of laser beam in the plasma and its effect on the excitation of electron plasma wave and on the electron acceleration process have been studied extensively in the past (Tajima and Dawson 1979;Clayton et al. 1993, Everett et al. 1994Nakajima et al. 1995;Singh and Gupta 2003;Liu and Tripathi 2005;Kumar et al. 2006;Esarey et al. 2009;Leemans and Esarey 2009;Priyanka et al. 2013;Gaur et al. 2016;Xia et al. 2017;Vranic et al. 2018;Yadav et al. 2020;Raynaud et al. 2020). The extent of self-focusing of an intense laser beams in plasmas and the amplitude of electron plasma wave depends on the spatial profile of laser beams and the nonlinearities associated with plasma. Most of these studies have been confined to the Gaussian distribution of the laser beams under paraxial-ray approximation, where the effect of ponderomotive nonlinearity and relativistic nonlinearity have been taken separately.
Nevertheless, various profiles of laser beams such as super Gaussian beams (Gill et al. 2015), elliptic Gaussian beams (Gaur et al. 2018), Hermite-Gaussian beams (Wadhwa and Singh 2020), hollow Gaussian beams , Hermite-cosh-Gaussian laser beams (Belafhal and Ibnchaikh 2000), and q-Gaussian beams (Yadav et al. 2020) have been used in a few studies of self-focusing and plasma wave excitation. Such beams having different types of irradiance across their wavefront, which show different features in plasma. Recently, flattop decentred cosh-Gaussian laser beams have attracted much attention because of their higher efficient power and attractive applications (Konar et al. 2007;Aggarwal et al. 2014;Habibi and Ghamari 2015). One of the most important specialty of cosh-Gaussian laser beam is that it becomes focus earlier than Gaussian laser beam. In addition, it has also been observed that when an intense laser beam propagates through the plasma, both relativistic and ponderomotive nonlinearities act simultaneously, which significantly enhance the self-focusing of the laser beam in plasma and the amplitude of electron plasma wave. Furthermore, paraxial-ray theory does not adequately describe the self-focusing of laser beams in plasma at high intensity. The paraxial-ray approximation has failed to describe the variation of radial profile of the beam from the initial to ring position. On the other hand, non-paraxial ray approximation (Sodha and Faisal 2008;Gill et al. 2010) accurately describes the propagation of an intense laser beam in plasma, where the eikonal and nonlinear dielectric constant extends up to the fourth power of the distance from the axis of beam. Therefore, non-paraxial approximation is pertinent for cosh Gaussian laser beam along with relativistic and ponderomotive nonlinearities.
The excitation of electron plasma wave by an intense laser beam is a useful approach to electron acceleration. In the present study, we have investigated the self-focusing of an intense non-paraxial cosh-Gaussian laser beam in collisionless plasma in the presence of relativistic and ponderomotive nonlinearities. The effect of the self-focused cosh-Gaussian laser beam on the excitation of electron plasma wave as well as the electron acceleration has also been studied. The numerical results shows that the focusing of cosh-Gaussian laser beam is strong in non-paraxial region. Due to strong focusing of cosh-Gaussian laser beam in plasma, a dexterous high amplitude electron plasma wave is excited which enhances the energy of accelerated electrons. The structure of the paper is as follows: In section 2, the appropriate expressions for the nonlinear effective dielectric constant of the plasma as well as for the selffocusing of a high-intensity cosh-Gaussian laser beam through collisionless plasma has been derived under the nonparaxial ray approximation, when both relativistic and ponderomotive nonlinearities are operative. The modified equations for the excitation of electron plasma wave and for the electron acceleration process has been acquired in section 3. A brief discussion of the numerical results based on derived equations are presented in section 4. Finally, the main conclusions are summarized in the section 5.

Basic formulation
An intense cosh-Gaussian laser beam is considered to be propagates in z-direction through collisionless unmagnetized plasma. The laser electric field at z = 0 under non-paraxial approximation is given by where r is the radial coordinate of the cylindrical coordinate system, r0 is the initial beam width of the beam, E00 is the amplitude of the electric field at the central position of r = z = 0, b is the decentred parameter of the beam, a20(z) and a40(z) are the coefficient of r 2 and r 4 and are indicative for the departure of the beam from the Gaussian nature.

Effective dielectric constant of the plasma
The dielectric constant of plasma at high laser intensity can be written as In the present study, the electron density and the dielectric constant of plasma are being modified by cumulative effect of relativistic and ponderomotive nonlinearities. These nonlinearities modify the dielectric constant of plasma by relativistic increment in the electron mass and by the ponderomotive expulsion of electrons from the beam path. An intense laser beam exerts the relativistic-ponderomotive force on the electrons, which expels the electrons away from the region of higher electric field and modifies the background electron density.
The relativistic-ponderomotive force is given by (Sun et al. 1987; The modified density of plasma electrons (n) due to relativistic-ponderomotive force is given as ) is the initial intensity of the beam The effective intensity dependent dielectric constant of plasma can be written as In the non-paraxial region, the dielectric function can be expressed as where ε0(z), ε2(z) and ε4(z) are the expansion coefficients.

Self-focusing of non-paraxial cosh-Gaussian laser beam in plasma
The electric field E of the laser beam in the plasma satisfies the following wave equation: where ε the intensity dependent dielectric constant of the plasma, and E is the electric field of the beam. Consider the variation of the electric field can be expressed as where A is the amplitude of the laser field, and is the wave vector of the beam.
By replacing Eq. (10) in Eq. (9) and neglecting the term where A0 is the beam irradiance, and S0is the eikonal i. e. the real function of space, which describes converging/diverging behaviour of the beam in the plasma.
The solution of Eq. (14) where f is the beam width parameter of the cosh-Gaussian laser beam.
The eikonalS0(r, z) can be expressed as where S10(z) is the axial phase shift, S20(z) and S40(z) are indicative of the spherical curvature of the wavefront and its departure from the spherical nature, respectively.
Substituting Eqs. (15) and (16) is the dimensionless propagation distance of laser beam and .
is the original beam width of the beam.
Self-focusing of a cosh-Gaussian laser beam is given by nonlinear differential equation (22), where the first term on right hand side represents the diffractional divergence of the beam, and the second term describes the convergence of the beam, which arises due to combined

Excitation of electron plasma wave
The cosh-Gaussian laser beam becomes self-focused in the plasma when the initial power of laser beam is greater than the critical power. The intensity of laser beam becomes very high at the focused positions. This is due to the modification in electron density of plasma by ponderomotive force and the relativistic effects. Such intense laser beams further excite the electron plasma wave. The amplitude of electron plasma wave depends on the background electron density of plasma. Thus, the electron plasma wave coupled to high intensity laser beam via modified background electron density. In order to analyse the effect of this coupling on the excitation of electron plasma wave, we start with the following set of equations:

(a)
Equation of continuity: where re0 is the initial beam width of the plasma wave and N10 is the initial density associated with the electron plasma wave at r = 0, coefficients in non-paraxial region and are functions of (r, z). (36) in Eq. (33) and equating the coefficients of r 2 and r 4 on both sides, we obtain the following equations:

Electron acceleration
The excited electron plasma wave by an intense cosh-Gaussian laser beam transfers its energy to electrons and accelerates them. The energy gain by the electron is given by Eq. (44) has been solved numerically by using Eq. (37) to obtain energy gain by electrons.

Figure (1) depicts the variation in normalized intensity of cosh-Gaussian laser beam
with normalized distance of propagation, when relativistic and ponderomotive nonlinearities are operative in paraxial and non-paraxial regions. The laser beam becomes self-focused when propagates through the plasma. It is obvious that the intensity of cosh-Gaussian laser beam enhances in non-paraxial region in comparison to the paraxial region due to the influence of the off-axial coefficients. This is due to the fact that the focusing of laser beam becomes strong and fast in nonparaxial region in comparison to the paraxial region. Figure (2) presents the variation of normalized intensity of cosh-Gaussian laser beam in the plasma along the propagation distance in nonparaxial region, when only relativistic and relativisticponderomotive nonlinearities are operative. It is observed that the self-focusing of cosh-Gaussian laser beam becomes significantly enhanced by the inclusion of ponderomotive nonlinearity. Due to the strong focusing of the cosh-Gaussian laser beam by relativisticponderomotive nonlinearity, the intensity of the beam is remarkably increases as compared to only relativistic nonlinearity. Figure (3 and intensity parameter (a). It is clear from Fig. (9) that the energy gain is significantly enhances in non-paraxial region, when the relativistic and ponderomotive nonlinearities are perused together. This is because the amplitude of the laser beam and electron plasma wave is be higher in the non-paraxial region. Figure (10) shows that the amplitude of the electron plasma wave gets enhances by a factor of about 9 by inclusion of ponderomotive nonlinearity with relativistic nonlinearity. Figure (11) represents the variation of energy gain with the normalized propagation distance in the non-paraxial region for different values of b in the presence of relativistic and ponderomotive nonlinearities. It is found that with the increase in the value of b, energy gain by electrons is remarkably increases. This is due to the fact that cosh-Gaussian laser beam excites larger amplitude electron plasma wave at higher values of b (Fig. 7). It is obvious from Fig. 11 that at the optimized value of b = 0.8, the maximum electron energy gain is achieved. Figure (12) shows the effect of intensity parameter (a) on the energy gain by electrons in the nonparaxial region, when both the relativistic and the ponderomotive nonlinearities are operative. Due to the increase in the amplitude of electron plasma wave at higher values of a (Fig. 8), the energy gain also increases along with the increase in the value of a.
In summary, we have studied the self-focusing of cosh-Gaussian laser beam in a collisionless unmagnetized plasma under non-paraxial ray approximation along with the combined effect of relativistic and ponderomotive nonlinearities. Further, the effect of the self-focusing of cosh-Gaussian laser beam on the excitation of electron plasma wave and the electron acceleration have been examined. The results have been compared with the paraxial-ray approximation and with the relativistic nonlinearity. It has been found that the extent of self-focusing of cosh-Gaussian laser beam becomes stronger and faster in the non-paraxial region and in the relativistic-ponderomotive regime. The amplitude/intensity of the electron plasma wave and the energy gain by the electrons are also increases in the non-paraxial region and in the relativistic-ponderomotive regime. Moreover, self-focusing/intensity of cosh-Gaussian laser beam in plasma becomes enhances at higher values of b and a. Consequently, the amplitude/intensity and the energy gain by electrons also increases with increase in the values of b and a respectively. Due to earlier and strong self-focusing of cosh-Gaussian laser beam at higher values of b, the intensity of laser beam, the amplitude/intensity of electron plasma wave and the energy gain by electrons becomes remarkably enhanced. Therefore, on the basis of these results it is suggested that the Gaussian laser beam might be replaced by the cosh-Gaussian laser beam, which can be further used for electron plasma wave excitation and more effectively for electron acceleration.

Conflicts of interest:
The authors declare that they have no conflict of interest.

Data availability:
The data can be obtained on demand.
normalized propagation distance (ξ). Keeping a = 1.4, b = 0.6, andp0 =0.060, when relativistic and ponderomotive nonlinearities are operative. Red curve for paraxial region and blue curve for non-paraxial region.       Red curve for paraxial and blue curve for non-paraxial region.