Introduction
Theoretical and experimental studies on the propagation characteristics of high-intensity electromagnetic beam in a nonlinear medium have been an interesting and fascinating field of research since decades. The nonlinear interaction of laser beam with plasma gives rise to many applications like ionospheric modification (Gondarenko, Reference Gondarenko2005; Sodha and Sharma, Reference Sodha and Sharma2008), particle accelerator (Leemans et al., Reference Leemans, Nagler, Gonsalves, Tóth, Nakamura, Geddes, Esarey, Schroeder and Hooker2006; Gonsalves et al., Reference Gonsalves, Nakamura, Lin, Panasenko, Shiraishi, Sokollik, Benedetti, Schroeder, Geddes, van Tilborg, Osterhoff, Esarey, Toth and Leemans2011), inertial confinement fusion (Tabak et al., Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell, Perry and Mason1994; Roth et al., Reference Roth, Cowan, Key, Hatchett, Brown, Fountain, Johnson, Pennington, Snavely, Wilks, Yasuike, Ruhl, Pegoraro, Bulanov, Campbell, Perry and Powell2001; Betti and Hurricane, Reference Betti and Hurricane2016), and high-harmonic generation (Kant et al., Reference Kant, Gupta and Suk2011; Zhang and Thomas, Reference Zhang and Thomas2015). The success of these applications is constrained to longer propagation distance of the electromagnetic beam through the nonlinear media with minimal loss in its intensity. When a laser pulse propagates through the plasma, it tends to change the refractive index of the plasma medium and results in many nonlinear effects and instabilities such as self-focusing and self-phase modulation of the laser beam, high-harmonic generation, filamentation instability, and group-velocity dispersion. Self-focusing is an important nonlinear process among others and is widely studied and discussed. In the self-focusing process, the refractive index of the medium changes due to the difference in intensity of the incoming electromagnetic beam between its on-axis and off-axis of the beam. The nonlinearities responsible for the self-focusing of the beam are relativistic, ponderomotive, and collisional type.
Beside these nonlinearities, the dielectric constant of the nonlinear medium is also affected by electron plasma temperature (T e) which is independent of electric field of the incoming electromagnetic wave. Various studies can be found based upon plasma temperature effects on studying self-focusing properties of laser beam in the light of above-said nonlinearity (Gupta et al., Reference Gupta, Islam, Jang, Suk and Jaroszynsk2013; Niknam et al., Reference Niknam, Barzegar and Hashemzadeh2013; Varaki and Jafari, Reference Varaki and Jafari2017; Walia et al., Reference Walia, Tripathi and Tyagi2017; Kumar and Aggarwal, Reference Kumar and Aggarwal2018). Malik and Aria (Reference Malik and Aria2010) investigated the interaction of microwave radiations with the plasma medium while taking into account the ponderomotive nonlinearity and plasma temperature variations. They observed that plasma temperature helps in increasing ponderomotive nonlinearity through density perturbation and hence self-focusing of the beam can be enhanced. Wang and Zhou (Reference Wang and Zhou2011) have rigorously studied the plasma temperature effects on the propagation characteristics of Gaussian laser beams in collisionless cold plasma with ponderomotive nonlinearity. In their investigations, they obtained four plasma temperature regions, namely oscillatory divergence, self-trapping, self-focusing and steady-divergence, which determine the propagation characteristics of Gaussian laser beam. Ping and Lin (Reference Ping and Lin2012) studied the effect of higher-order axial electron temperature on self-focusing behavior of the electromagnetic beam in collisional plasma. They observed that higher-order axial electron temperature tends to decrease the effect of collisional nonlinearity unlike lower-order terms of plasma temperature which assist the nonlinearity. Hence, different trends in self-focusing of the beam are observed. Milani et al. (Reference Milani, Niknam and Bokaei2014) took different temperature ranges to study self-focusing of Gaussian beam through plasma and observed that at higher temperatures the beam divergence increases. This occurs because at higher temperature, the amplitude of beam-width increases to larger extent while frequency kept on decreasing and hence beam divergence increases. Patil et al. (Reference Patil, Chikode and Takale2018) discussed self-focusing under the combined effects of light absorption and plasma temperature and observed that absorption coefficient helps to improve the self-focusing process with in a given range of temperatures. Recently, Ouahid et al. (Reference Ouahid, Dalil-Essakali and Belafhal2018) studied the light absorption parameter and plasma temperature effects on the self-focusing property of finite Airy–Gaussian beams in a relativistic–ponderomotive plasma regime. They studied the effects of different controlling parameters like absorption coefficient, plasma intensity and electron temperature.
With the passage of time, various authors studied the effects of magnetic field on the self-focusing property of Gaussian laser beam. When the magnetic field is applied parallel to the direction of the propagation of laser beam, the electrons get rotated in the direction of magnetic field lines, thereby improving the natural oscillating frequency of the electrons. This phenomena also changes the density distribution of electrons in comparison with the unmagnetized collisionless plasma as studied by Darian et al. (Reference Darian, Abari and Sedaghat2016). Sharma et al. (Reference Sharma, Koueakis and Sodha2008) investigated the self-focusing and defocusing regions for Gaussian electromagnetic beam using kinetic theory approach in magnetized plasma.
Recently, quantum effects are gaining interest in the studies of self-focusing properties of the electromagnetic beam, as they have several applications in astrophysical system (Opher et al., Reference Opher, Silva, Dauger, Decky and Dawson2001; Benvenuto and De Vito, Reference Benvenuto and De Vito2005; Harding and Lai, Reference Harding and Lai2006), biophotonics (Barens et al., Reference Barens, Dereux and Ebbensen2003), neutron star (Chabrier et al., Reference Chabrier, Douchin and Potekhin2002), ultra-cold plasmas (Killian, Reference Killian2006), ultra-small electronic devices (Markowich et al., Reference Markowich, Ringhofer and Schmeiser1990), quantum well (Ang et al., Reference Ang, Koh, Lau and Kwan2006), and quantum dots (Ang and Zhang, Reference Ang and Zhang2007). Quantum plasmas are characterized as low temperature and high-density plasma medium and can be differentiated from classical plasma by the parameter χ = T f/T e, where T f and T e are Fermi and plasma temperatures, respectively. We can define Fermi temperature and Fermi energy by the following relation (Landau and Lifshitz, Reference Landau and Lifshitz1980):
$$k_{\rm B}T_{\rm f}=E_{\rm f}={\hbar^2 \over 2m_{\rm e}}(3{\rm \pi}^2n_{\rm e})^{2/3}$$
$${\rm \chi}={T_{\rm f} \over T_{\rm e}}={1 \over 2}(3{\rm \pi}^2)^{2/3}(n_{\rm e}{\rm \lambda}_{\rm B}^3)^{2/3}$$where λB is the de-Broglie wavelength. When (T e ≤ T f), i.e. when plasma temperature is less than or equal to the Fermi temperature, then the system shifts from Maxwell–Boltzmann to Fermi–Dirac statistics and quantum effects starts dominating over the classical case. Alternatively, when de-Broglie wavelength becomes greater than or equal to interparticle distance (
$n_{\rm e}\lambda _{\rm B}^3\geq 1\rpar$, then quantum effects are more effective over the classical case. Several authors have studied the self-focusing properties of laser beam in quantum plasma (Patil et al., Reference Patil, Takale, Navare, Dongare and Fulari2013; Zare et al., Reference Zare, Yazdani, Rezaee, Anvari and Sadighi-Bonabi2015; Kumar et al., Reference Kumar, Aggarwal, Richa and Gill2016; Aggarwal et al., Reference Aggarwal, Goyal, Kumar, Richa and Gill2017a, Reference Aggarwal, Kumar, Richa and Gill2017b; Mahajan et al., Reference Mahajan, Gill, Kaur and Aggarwal2018). Aggarwal et al. (Reference Aggarwal, Goyal, Kumar, Richa and Gill2017a) have studied the self-focusing property of Gaussian laser beam propagating in weakly relativistic magnetized cold quantum plasma (WRMCQ) and observed the enhanced self-focusing under the combined effects of quantum conditions and magnetic field as compared with their individual cases.
In the present paper, we have studied the role of plasma electron temperature in the presence of applied magnetic field on the self-focusing properties of the Gaussian laser beam propagating in cold quantum plasma. To the best of our knowledge, the effect of plasma electron temperature on the normalized intensity in the presence of cold quantum plasma have not been studied so far. The paper is organized in four sections: The theoretical calculations for obtaining nonlinear plasma permittivity are carried out in the Formalism and Self-focusing sections. Numerical simulations and results are obtained in the Numerical results and discussion section and discussed thereafter. Further, the conclusions are presented in the last section.
Formalism
Consider the linearly polarized high intense Gaussian laser beam propagating with angular frequency “ω0” along the z-axis through the magnetized cold quantum plasma. The electric field amplitude of the electromagnetic beam is given by
$$E\lpar r\comma z\rpar =A\lpar r\comma z\rpar \,{\exp[ {-{\rm \iota}\lpar {\rm \omega}_0 t-k z\rpar }] }$$where 
$A^2\vert _{z=0}=A_{oo}^2 \exp \lpar \!\!-r^2/r_0^2\rpar$ is the intensity of the laser beam, r 0 is the initial spot size, and A oo is the initial amplitude of the beam at z = 0. The wave number is given by 
$k=\lpar {\rm \omega} _0/c\rpar \sqrt {{\rm {\epsilon}} _{o}}$, c is the speed of light in the vacuum, and ε0 is the axial dielectric function along the central position z = 0. The intensity distribution A 2(r,z) for z > 0 is given by the following equation:
$$A^2\lpar r\comma z\rpar ={A_{oo}^2\over f^2}\exp\left[-{r^2\over r_0^2f^2}\right]$$The modified plasma electron density distribution n e, which is the function of plasma electron temperature, is given by Niknam et al. (Reference Niknam, Hashemzadeh and Shokri2009):
$$n_{\rm e}=n_0\exp\left[-{m_{\rm e}c^2\over T_{\rm e}}\lpar {\rm \gamma}-1\rpar \right]$$where n 0 is the unperturbed plasma electron density, m e is the mass of the electron, and T e is the plasma electron temperature in the units of energy. γ is the Lorentz relativistic factor obtained iteratively for (ωc/γω0) < 1 as given by Pandey and Tripathi (Reference Pandey and Tripathi2009) and Gill et al. (Reference Gill, Kaur and Mahajan2010):
$${\rm \gamma}=\left[1+A^2+2 A^2\left({{\rm \omega}_c\over {\rm \omega}_0}\right)\left({1\over \sqrt{1+A^2}}\right)+3A^2\left({{\rm \omega}_c\over {\rm \omega}_0}\right)^{2}\left({1\over 1+A^2}\right)\right]^{1/2}$$In the present model, we are investigating the beam propagation characteristics in weakly relativistic magnetized cold quantum (WRMCQ) plasma. Since the quiver velocity of electrons is much more than their thermal velocities; hence, we can assume that the plasma behavior is cold.
The dielectric function ε here is of a second-rank tensor εij and takes the simple form when the magnetic field is directed along one of the Cartesian co-ordinate axes. Therefore, the magnetic field is along the direction of wave propagation, i.e. along the z-axis. In this case, εij has only three nonvanishing components, namely εxx = εyy, εxy = −εyx, and εzz with εxz = εzx = εyz = εzy = 0 (Sharma et al., Reference Sharma, Koueakis and Sodha2008). The general form of dielectric permittivity can be written as follows:
$${\rm {\epsilon}}={\rm {\epsilon}}_{xx}-i{\rm {\epsilon}}_{xy}={\rm {\epsilon}}_0+{\rm {\epsilon}}_r\lpar {\rm EE}^\ast \rpar $$where ε0 and 
${\rm {\epsilon}} _r\lpar {\rm EE}^\ast \rpar$ are the linear and nonlinear part of the plasma dielectric function.
$${\rm {\epsilon}}_{zz}={\rm {\epsilon}}_{ozz}+{\rm \phi}$$where 
${\rm {\epsilon}} _{ozz}=1-\lpar {1}/{\rm \gamma }\rpar \lpar {{\rm \omega} _{\rm p}^2}/{{\rm \omega} _0^2}\rpar$ and 
${\rm \phi} =\lpar {1}/{\rm \gamma }\rpar \lpar {{\rm \omega} _{\rm p}^2}/{{\rm \omega} _0^2}\rpar$. Where 
${\rm {\epsilon}} _0=1-{{\rm \omega} _{\rm p}^2}/{{\rm \omega} _0^2}$ and 
${\rm \omega} _{\rm p}=\sqrt {4{\rm \pi} n_0e^2}/{m_{\rm e}}$ is the plasma frequency. The nonlinear plasma dielectric function in WRMCQ plasma can be written as
$${\rm {\epsilon}}_r\lpar {\rm EE}^\ast \rpar ={{\rm \omega}_{\rm p}^2\over {\rm \omega}_0^2}\left[1-{n_{\rm e}\over n_0}{1\over {\rm \gamma}}\left(1-{{\rm \delta}\over {\rm \gamma}}\right)^{-1}\right]$$Expanding εr using paraxial ray approximation and Taylor's series expansion around r = 0 and retaining the terms up to second power of r 2, we get the nonlinear plasma permittivity as given below
$${\rm {\epsilon}}\lpar r\comma z\rpar ={\rm {\epsilon}}_r\lpar r=0\rpar ={\rm {\epsilon}}_0\lpar z\rpar -{r^2\over r_0^2}{\rm \Phi}$$where ε0(z) is the plasma permittivity, which corresponds to the maximum electric field on the axis:
$${\rm {\epsilon}}_0\lpar z\rpar =1-{{\rm \omega}_{\rm p}^2\over {\rm \omega}_0^2}{1\over {\rm \gamma}_0} \exp\left[-{m_{\rm e}c^2\over T_{\rm e}}\lpar {\rm \gamma}_0-1\rpar \right]\left(1-{{\rm \delta}\over {\rm \gamma}_0}\right)^{-1}\comma $$and Φ is the nonlinear plasma permittivity, which is calculated as
$$\eqalign{{\rm \Phi} =&{A_{oo}^2 \over 2f^4}{{\rm \omega}_{\rm p}^2 \over {\rm \omega}_0^2}{1 \over {\rm \gamma}_0^3} \exp\left[-{m_{\rm e}c^2 \over T_{\rm e}}({\rm \gamma}_0-1)\right]\left(1-{{\rm \delta} \over {\rm \gamma}_0}\right)^{-2} \left[1+{m_{\rm e}c^2{\rm \gamma}_0 \over T_{\rm e}}\left(1-{{\rm \delta} \over {\rm \gamma}_0}\right)\right] \cr & \times \left[1+{{\rm \omega}_c \over {\rm \omega}_0}{1 \over \sqrt{1+{A_{oo}^2 \over f^2}}} \left(2-{{A_{oo}^2 \over f^2} \over 1+{A_{oo}^2 \over f^2}}\right) +3 {1 \over 1+ {A_{oo}^2 \over f^2}}\left({{\rm \omega}_c \over {\rm \omega}_0}\right)^{2} \left(1-{{A_{oo}^2 \over f^2} \over 1+ {A_{oo}^2 \over f^2}}\right)\right]}$$here γ0 is the relativistic factor defined at r = 0 and can be written given by
$${\rm \gamma}_0 \! = \!\! \left[ \! \!1 \! \!+ {A_{oo}^2\over f^2}+2{A_{oo}^2\over f^2}\left({{\rm \omega}_c\over {\rm \omega}_0}\right)\left({1\over \sqrt{1+ {A_{oo}^2 \over f^2}}}\right) \! \! + \! 3{A_{oo}^2\over f^2} \left({{\rm \omega}_c\over {\rm \omega}_0}\right)^{2} \!\left({1\over 1+{A_{oo}^2 \over f^2}}\right) \! \right]^{1/2}.$$Self-focusing
To obtain the equation governing self-focusing of the electromagnetic beam, we use the Maxwell's equation and arriving at the nonlinear wave equation which satisfies the amplitude of Gaussian beam as follow
$${\partial^2{\bf E}\over \partial{z^2}}+\alpha\nabla_\bot^2{\bf E}+{{\rm \omega}_0^2\over c^2}{\rm {\epsilon}}{\bf E}=0.$$The quasioptic equation governing the evolution of electric field of laser beam through the plasma medium can be obtained under Wentzel–Kramers–Brilluion (WKB) and slowly varying envelope approximation as
$$-2{\rm \iota} k{\partial A\over \partial z}+ \alpha\nabla_\bot^2A-{r^2\over r_0^2}{{\rm \omega}_0^2\over c^2}{\rm {\epsilon}} A=0\comma$$where α = (1/2)(1 + ε0/εozz) and εozz = 1 − (1/γo)(ωp/ω0) 2.
The solution of above the equation can be expressed in the form of complex amplitude A(r,z) and following the approach given by Akhmanov et al. (Reference Akhmanov, Sukhorukov and Khokhlov1968) and Sodha et al. (Reference Sodha, Ghatak and Tripathi1976) as follows:
$$A\lpar r\comma z\rpar =A_0\lpar r\comma z\rpar \exp[\!\! -{\rm \iota} k\lpar z\rpar S\lpar r\comma z\rpar ]$$The envelope A(r,z) includes real amplitude and complex phase terms which are functions of z and r. Substituting the value of A(r,z) from Eq. (16) into Eq. (15) to obtain two separate coupled equations of real and imaginary parts given below
$$2{\partial S\over \partial z}+\left({{\partial S\over \partial r}}\right)^2 -{\alpha\over k^2 A_0} \left({{\partial^2 A_0\over \partial r^2}}+{1\over r}{{\partial A_0\over \partial r}}\right)+ {r^2\over r_0^2}{{\rm {\epsilon}}_r\over {\rm {\epsilon}}_0}=0$$and
$${\partial A_0^2\over \partial z}+{\partial S\over \partial r}{\partial A_0^2\over \partial r}+ {\rm \alpha} A_0^2\left({{\partial^2 S\over \partial r^2}}+{1\over r}{{\partial S\over \partial r}}\right)= 0$$Expanding eikonal S(r,z) as
$$S\lpar r\comma z\rpar ={r^2\over 2}{\rm \beta}_0\lpar z\rpar +S_0\lpar z\rpar$$where β0(z) = (1/f)(1/(1 + ε0/εozz))(df/dz) is the radius of curvature of the beam, and S 0(z) is a constant related to the phase. Further, in order to obtain the equation of the dimensionless beam-width parameter f, we substitute 
$A_0^2(=A^2)$ and S(r,z) from Eqs (4) and (19), respectively, in Eq. (17). Equating the coefficients of r 2 on both sides of the resulting equation and normalizing by using ξ = z/R d, where 
$R_d= kr_0^2$. The dimensionless beam-width parameter equation is derived and given by the following expression
$$\eqalign{d^2f \over d{\rm \xi}^2 & = {{\rm \alpha}^2\over{\,f^3}}-{{\rm \alpha} R_d^2 \over {\rm {\epsilon}}_0}f^{\ast} {A_{oo}^2 \over 2f^4} {{\rm \omega}_{\rm p}^2 \over {\rm \omega}_0^2} {1 \over {\rm \gamma}_0^3} \exp\left[-{m_{\rm e}c^2 \over T_{\rm e}}({\rm \gamma}_0-1)\right]\left(1-{{\rm \delta} \over {\rm \gamma}_0}\right)^{-2} \left[1+{m_{\rm e}c^2{\rm \gamma}_0 \over T_{\rm e}}\left(1-{{\rm \delta} \over {\rm \gamma}_0}\right)\right] \cr & \quad \times \left[1+{{\rm \omega}_c \over {\rm \omega}_0}{1 \over \sqrt{1+ {A_{oo}^2 \over f^2}}} \left(2-{{A_{oo}^2 \over f^2} \over 1+{A_{oo}^2 \over f^2}}\right) +3{1 \over 1+ {A_{oo}^2 \over f^2}}\left({{\rm \omega}_c \over {\rm \omega}_0}\right)^{2} \left(1-{{A_{oo}^2 \over f^2} \over 1+{A_{oo}^2 \over f^2}}\right)\right]}$$Numerical results and discussion
The second-order nonlinear differential equation (20) is solved numerically by the fourth-order Runge–Kutta method to study the behavior of the beam-width parameter “f” with the dimensionless distance of propagation ξ. The boundary conditions applied for numerical solutions of Eq. (20) are f = 1, f′ = 0 at ξ = 0. The other parameters of laser and plasma are as follows: ω0 = 1.778 × 1020 rad/s, r 0 = 0.003 cm, λ = 0.0106 nm, n 0 = 3.14 × 1018 cm−3, ωc/ω0 = 0.1 − 0.3, and plasma electron temperature T e are taken as 50, 75, 100, and 125 keV. The right-hand side of Eq. (20) is comprised of two terms which are responsible for convergence and divergence of the beam. The beam undergoes self-focusing when the second term dominates over the first term, while diffraction of the beam takes place when the first term dominates over the second term.
Figure 1a shows the plot for the nonlinear plasma dielectric function Φ against normalized laser intensity 
$A^2/A_{oo}^2$ at different values of plasma electron temperature T e = 50, 75, 100, and 125 keV with the fixed value of magnetic field ωc/ω0 = 0.3. From the curves, we can see that the nonlinearity responsible for self-focusing saturates earlier at temperature T e = 50 keV as compared with T e = 75, 100, and 125 keV. The nonlinearity thereafter falls rapidly with normalized intensity. Thus, plasma temperature plays a vital role in increasing the nonlinearity of the medium. It is further observed that with the increase in plasma electron temperature, the nonlinearity achieves lesser saturation. This is due to fact that as the temperature increases, the electron oscillations increases to larger extent and electrons are expelled away from the laser beam axis. This leads to the decrease in oscillatory frequency of the beam. Hence, the dielectric permittivity decreases with the plasma electron temperature. Thus, self-focusing is achieved with delayed focusing length at higher temperature due to redistribution in electron density which changes the effective dielectric constant of the medium. Figure 1b shows the plot between Φ and 
$A/A_{oo}^2$ but for different nonlinearities, namely WRMCQ plasma, weakly relativistic cold quantum (WRCQ), weakly relativistic magnetized (WRM), and classical relativistic (CR) plasma. The plasma electron temperature T e is fixed at 50 keV. The nonlinearity saturates earlier for WRMCQ plasma than for WRCQ, WRM, and CR cases of references, respectively. This proves that nonlinearity comprising of magnetic field and quantum conditions together helps in enhancing self-focusing of the beam as compared with the case when quantum parameters or magnetic field is taken alone as obvious from Fig. 2.
(a) The plot of nonlinearity (Φ) versus normalized laser intensity 
$\lpar {A^2}/{A_{oo}^2}\rpar$ for different plasma electron temperature T e values in WRMCQ plasma for the fixed value of magnetic field ωc/ω0 = 0.3. The other parameters are ω0 = 1.778 × 1020 rad/s, r 0 = 0.003 cm, λ = 0.0106 nm, and n 0 = 3.14 × 1018 cm−3. (b) The plot between (Φ) and 
$\lpar {A^2}/{A_{oo}^2}\rpar$ for WRMCQ, WRCQ, WRM, and CR cases of references at T e = 50 keV. The other parameters are same as mentioned in (a).
Variation of the beam-width parameter “f” with the normalized propagation distance “ξ” for WRMCQ, WRCQ, WRM, and CR cases of references for the fixed value of plasma electron temperature T e = 50 keV. The other parameters are same as mentioned in the caption of Fig. 1a.
Figure 2 shows the plot between the dimensionless beam-width parameter “f” and the dimensionless distance of propagation “ξ” for different nonlinearities, namely the beam propagation in plasma with (1) magnetic field (WRMCQ), (2) without magnetic field (WRCQ), (3) when quantum effects are neglected, δ = 0 (WRM), and the last curve is for classical relativistic case (CR). The values of magnetic field and electron temperature are ωc/ω0 = 0.3 and T e = 50 keV, respectively. From the figure, it is clear that self-focusing is maximum for the WRMCQ case in which both magnetic field and quantum effects are taken as compared to their individual effects i.e. WRCQ and WRM plasma cases, respectively. The beam is least focused for the CR case of reference. Hence, we can conclude that plasma electron temperature (T e = 50 keV) for which quantum effects are dominant assists the magnetic field and the pinching effect of quantum plasma further helps in increasing the self-focusing of the beam.
Figure 3 shows the plot between “f” and “ξ” at different values of plasma electron temperature, namely T e = 50, 75, 100, and 125 keV for WRMCQ plasma. From the curve, it is clear that maximum self-focusing is obtained at the lowest electron temperature value, that is, at T e = 50 keV. This is because at T e = 50 keV, de-Broglie wavelength is found to be λB = 1.06 × 10−6 and 
$n_{\rm e}{\rm \lambda} _{\rm B}^3=3.74\gt 1$. Thus quantum effects are dominant at T e = 50 keV. Further, with the increase in T e values i.e. at T e = 75, 100, and 125 keV, weaker self-focusing is observed. This is because at T e = 125 keV, λB = 6.71 × 10−7, and 
$n_{\rm e}{\rm \lambda} _{\rm B}^3=0.94\lt 1$, quantum effects start diminishing with the rise in plasma electron temperature. Our results are in good agreement with Bokaei et al. (Reference Bokaei, Niknam and Jafari Milani2013).
Variation of the beam-width parameter “f” with the normalized propagation distance “ξ” at different values of plasma electron temperature T e = 50, 75, 100, and 125 keV, the other parameters are same as mentioned in the caption of Fig. 1a.
Figure 4 shows the variations of “f” versus “ξ” at different values of ωc/ω0 = 0, 0.1, 0.2, and 0.3 for the fixed electron temperature T e = 50 keV. From the plot, we observe that with the increase in cyclotron frequency, the spot size of the beam decreases, and hence self-focusing of the beam increases. The reason is quite vivid that cyclotron frequency provided by the magnetic field adds to the natural oscillating frequency of the electrons and results in maximum self-focusing of the beam. This further increases the normalized laser intensity. So we have plotted the normalized laser intensity with beam radius in Fig. 5d at different values of ωc/ω0 = 0, 0.1, 0.2, and 0.3.
Variation of the beam-width parameter “f” with the normalized propagation distance “ξ” at different values of ωc/ω0 = 0, 0.1, 0.2, and 0.3. The other parameters are same as mentioned in the caption of Fig. 1a.
Variation of the normalized beam intensity (
$A^2/A_{oo}^2$) against normalized beam radius (
$r^2/r_{oo}^2$) at different plasma electron temperature T e = 50, 75, 100, and 125 keV (a) for ωc/ω0 = 0.1; (b) for ωc/ω0 = 0.2; (c) for ωc/ω0 = 0.3; and (d) at different values of magnetic field ωc/ω0 = 0, 0.1, 0.2, and 0.3 at T e = 50 keV. The other parameters are same as mentioned in the caption of Fig. 1a.
Figure 5a is plotted for normalized intensity against normalized beam radius at ωc/ω0 = 0.1 for different values of plasma electron temperature T e = 50, 75, 100, and 125 keV. In the similar case, Fig. 5b is plotted at the fixed value of ωc/ω0 = 0.2, and Fig. 5c is plotted at ωc/ω0 = 0.3, respectively. From these plots, it is observed that the beam intensity achieves maximum at T e = 50 keV and decreases in the order of increasing electron temperature values i.e. T e = 75, 100, and 125 keV. Further, we have observed that the normalized intensity increases with the increase in cyclotron frequency. The normalized intensity achieved is about tenfolds in case of T e = 50 keV as compared with T e = 125 keV (Fig. 5c). Figure 5d is sketched between normalized beam intensity and normalized beam radius at different values of magnetic field at T e = 50 keV. The increase in intensity is about twelverfolds in case of ωc/ω0 = 0.3 as compared with ωc/ω0 = 0.0 (when no magnetic field is applied) at the fixed value of plasma electron temperature T e = 50 keV. Thus, we can conclude that studying the plasma electron temperature along with magnetic field significantly assists in improving the beam propagation characteristics, as well as normalized intensity of the beam.
Conclusions
(1) In the present investigation, we have studied the effect of electron plasma temperature and its vital role in improving the self-focusing in WRMCQ plasma for different plasma electron temperatures. For T e = 50 keV, λB = 1.06 × 10−6 and
$n_{\rm e}{\rm \lambda} _{\rm B}^3=3.74\gt 1$, the normalized intensity increases to many folds where quantum effects are dominant.(2) The normalized intensity gains a boost of twelvefold in intensity at ωc/ω0 = 0.3 as compared tothe case when ωc/ω0 = 0.0 at a temperature T e = 50 keV. Thus, we conclude that magnetic field assists quantum effects in improving the self-focusing of Gaussian laser beam.
Acknowledgments
The authors Vimmy Goyal, Harish Kumar, and Richa acknowledge the guidance and support provided by the research and innovation committee (RIC) of I.K. Gujral Punjab Technical University, Jalandhar (India).
