2. FORMALISM

In this section, we develop the formalism to investigate the propagation of laser beam of Gaussian profile in weakly relativistic magnetized cold quantum plasma in cylindrical coordinate system. We have assumed initial electron density n _o in the absence of the laser beam. The linearly polarized laser pulse is assumed to be propagating along the z-axis with angular frequency ω _o . A static magnetic field $B_o \hat z$ is taken along the direction of propagation of the beam. The electric field amplitude of laser beam can be expressed as

(1) $$E(r,z) = A(r,z)\exp [ - \iota ({\rm \omega} _o t - kz)],$$

where $A^2 \vert_{z = 0} = A_{oo}^2 \exp ( - r^2 /r_o^2 )$ is the initial intensity distribution of the beam propagating along the z-axis. Here, r _o is the initial beam width and A _oo is the axial amplitude at the central position z = 0. $k = ({\rm \omega} _o /c)\sqrt {{\rm \epsilon} _o} $ is the wave propagation constant and c is the speed of the light in vacuum. The field distribution A(r, z) for z > 0 is given by

(2) $$A^2 (r,z) = \displaystyle{{A_{oo}^2} \over {\,f^2}} \exp \left[ { - \displaystyle{{r^2} \over {r_o^2 f^2}}} \right].$$

In the present case, pulse duration is assumed to be shorter than the electron response time. Therefore, the non-linear contribution due to ponderomotive force can be neglected. Thus, the relativistic non-linearity and longitudinal magnetic field are the only factors responsible for the occurrence of self-focusing. The effective electron density “n _e” arising from these factors is given by

(3) $$n_{\rm e} = \displaystyle{{n_o} \over {{\rm \gamma} (1 - ({\rm \omega} _{\rm c} /{\rm \gamma \omega} _o ))}}.$$

Here, ω_c is the cyclotron frequency and γ is the modified relativistic factor, which can be obtained iteratively for (ω_c/γω _o ) <1 as (Pandey & Tripathi, Reference Pandey and Tripathi2009; Gill et al., Reference Gill, Kaur and Mahajan2010)

(4) $${\rm \gamma} = \left[ {1 + A^2 + 2A^2 \left( {\displaystyle{{{\rm \omega} _{\rm c}} \over {{\rm \omega} _o}}} \right)\left( {\displaystyle{1 \over {\sqrt {1 + A^2}}}} \right) + 3A^2 \left( {\displaystyle{{{\rm \omega} _{\rm c}} \over {{\rm \omega} _o}}} \right)^2 \left( {\displaystyle{1 \over {1 + A^2}}} \right)} \right]^{1/2}. $$

The general form of effective dielectric constant of the plasma medium can be written as

(5) $${\rm \epsilon} = {\rm \epsilon} _o + {\rm \epsilon} _2 (EE^* ),$$

where ε _o is the linear part of the dielectric constant and can be written as

(6) $${\rm \epsilon} _o = 1 - \displaystyle{{{\rm \omega} _{\rm p}^{\rm 2}} \over {{\rm \omega} _o^2}}, $$

ω_p = (4πn _o e ²/m)^1/2 is the plasma frequency, e and m are the electronic charge and mass, respectively. ε₂ (EE*) represents the non-linear part of the plasma permittivity comprising of magnetized cold quantum plasma and is given by (Jung & Murakami, Reference Jung and Murakami2009; Kumar et al., Reference Kumar, Aggarwal, Richa and Gill2016).

(7) $${\rm \epsilon} _2 (EE^* ) = \displaystyle{{{\rm \omega} _{\rm p}^{\rm 2}} \over {{\rm \omega} _o^2}} \left( {1 - \displaystyle{{n_{\rm e}} \over {n_o}} (1 - \displaystyle{{\rm \delta} \over {\rm \gamma}} )^{ - 1}} \right).$$

here ${\rm \delta} = 4{\rm \pi} ^4 h^2 /m^2 {\rm \omega} _o^2 {\rm \lambda} ^4 $ is the parameter which is a function of quantum wavelength, h is the planck's constant and λ is the wavelength of the laser beam. Equation (7) represents the nonlinear part of the dielectric constant which is function of intensity. It is noticed that the linear part of the dielectric constant is intensity-independent, as appears in the equation for dynamics of self-focusing as multiplicative constant in Eq. (16). On the other hand, intensity-dependent non-linear part takes part in self-focusing of laser beam in a dynamic way during its transit through the plasma. In the paraxial ray approximation $(r^2 \ll r_0^2 f^2 )$ , ε₂ (EE*) is Taylor expanded around r = 0 as follows

$$\eqalign{{\rm \epsilon }_2(EE^*) = {\rm \epsilon }_2(r = 0) - \displaystyle{{r^2} \over {r_o^2 }}\Phi (EE^*),} $$

where

(8) $$\eqalign{ \Phi &= \displaystyle{{A_{oo}^2 } \over {2f^4}}\displaystyle{{{\rm \omega }_{\rm p}^{\rm 2} } \over {{\rm \omega }_o^2 }}\displaystyle{1 \over {{\rm \gamma }_o^3 }}\displaystyle{1 \over {(1 - ({\rm \omega }_{\rm c}/{\rm \gamma }_o{\rm \omega }_o))}}\left( {1 - \displaystyle{{\rm \delta } \over {{\rm \gamma }_o}}} \right)^{ - 2}\left( {1 - \displaystyle{{{\rm \omega }_{\rm c}{\rm \delta }} \over {{\rm \gamma }_o^2 {\rm \omega }_o}}} \right) \cr &\times\left[ {1 + \displaystyle{{{\rm \omega }_{\rm c}} \over {{\rm \omega }_o}}\displaystyle{1 \over {\sqrt {1 + (A_{oo}^2 /f^2)} }}\left( {2 - \displaystyle{{A_{oo}^2 /f^2} \over {1 + (A_{oo}^2 /f^2)}}} \right)}\right. \cr &+ \left.3\displaystyle{1 \over {1 + (A_{oo}^2 /f^2)}}{\left( {\displaystyle{{{\rm \omega }_{\rm c}} \over {{\rm \omega }_o}}} \right)}^2\left( {1 - \displaystyle{{A_{oo}^2 /f^2} \over {1 + (A_{oo}^2 /f^2)}}} \right) \right],}$$

where γ _o is the relativistic factor defined at r = 0 and is given by

(9) $$\eqalign{{\rm \gamma} _o &= \left[ {1 + \displaystyle{{A_{oo}^2} \over {\,f^2}} + 2\displaystyle{{A_{oo}^2} \over {\,f^2}} \left( {\displaystyle{{{\rm \omega} _{\rm c}} \over {{\rm \omega} _o}}} \right)\left( {\displaystyle{1 \over {\sqrt {1 + (A_{oo}^2 /f^2 )}}}} \right)}\right. \cr &+ \left. 3\displaystyle{{A_{oo}^2} \over {\,f^2}} \left( {\displaystyle{{{\rm \omega} _{\rm c}} \over {{\rm \omega} _o}}} \right)^2 \left( {\displaystyle{1 \over {1 + (A_{oo}^2 /f^2 )}}} \right) \right]^{1/2}. }$$

3. SPOT SIZE EVOLUTION

For the spot size evolution, we start with the Maxwell's equations governing the non-linear wave equation, which oversees the propagation of Gaussian laser beam in the plasma. The non-linear wave equation is given by

(10) $$\displaystyle{{\partial ^2 {\rm E}} \over {\partial z^2}} + {\rm \alpha} \nabla _ \bot ^2 {\bf E} + \displaystyle{{{\rm \omega} _o^2} \over {c^2}} {\rm \epsilon} {\bf E} = 0,$$

where α = (1/2)(1 + ε _op /ε _ozz ) and ε _ozz  = 1 − (ω_p/ω _o )². Here ε _op  = ε _o  + ε₂ (r = 0), substituting for E in Eq. (10) and by using WKB (Wentzel-Kramers-Brillouin) and slowly varying envelope approximation, one can neglect $\partial ^2 {\bf A}/\partial z^2 $ . The quasioptic equation reduces to

(11) $$- 2\iota k\displaystyle{{\partial A} \over {\partial z}} + {\rm \alpha} \nabla _ \bot ^2 A - \displaystyle{{r^2} \over {r_o^2}} \displaystyle{{{\rm \omega} _o^2} \over {c^2}} \Phi A = 0,$$

where $\nabla _ \bot ^2 A$ represents spatial dispersion of the beam in transverse direction. The last term is a non-linear term and is responsible for the compression of the beam. The relative competition between a spatial dispersion term and a non-linear term will decide the focusing and defocusing of laser beam. Further, following the approach given by Akhmanov et al. (Reference Akhmanov, Sukhorukov and Khokhlov1968) and developed by Sodha et al. (Reference Sodha, Ghatak and Tripathi1976), one may express the complex amplitude of the electric vector A(r, z) as

(12) $$A(r,z) = A_o (r,z)\exp [ - {\rm \iota} k(z)S(r,z)],$$

where A _o (r, z) and S(r, z) are the real functions of space variables z and r. In order to obtain the coupled set of equations for S and $A_0^2 $ , we substitute for A(r, z) from Eq. (12) into Eq. (11). Separating the real and imaginary parts of the resulting equation, we get

(13) $$2\displaystyle{{\partial S} \over {\partial z}} + \left( {\displaystyle{{\partial S} \over {\partial r}}} \right)^2 - \displaystyle{\alpha \over {k^2 A_o}} \left( {\displaystyle{{\partial ^2 A_o} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial A_o} \over {\partial r}}} \right) + \displaystyle{{r^2} \over {r_o^2}} \displaystyle{\Phi \over {{\rm \epsilon} _o}} = 0$$

and

(14) $$\displaystyle{{\partial A_o^2} \over {\partial z}} + \displaystyle{{\partial S} \over {\partial r}}\displaystyle{{\partial A_o^2} \over {\partial r}} + {\rm \alpha} A_o^2 \left( {\displaystyle{{\partial ^2 S} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial S} \over {\partial r}}} \right) = 0.$$

Expanding eikonal S(r, z), in paraxial ray approximation as

(15) $$S(r,z) = \displaystyle{{r^2} \over 2}{\rm \beta} _o (z) + S_o (z),$$

where β _o (z) = (2/f)(1/(1 + ε _op /ε _ozz ))(df/dz) and ${\rm \beta} _0^{ - 1} $ is elucidated as the radius of the curvature of the beam. S _o (z) is a constant related to the phase and is not required in further analysis.

Substituting $A_o^2 (r,z)$ and S(r, z) from Eqs. (2) and (15), respectively, in Eq. (13) and equating the coefficients of r ² on both sides and using ξ = z/R _d , where $R_d = kr_o^2 $ , one can obtain the equation of dimensionless beam width parameter f as

(16) $$\eqalign{ {{d^2 f} \over {d\xi ^2}} &= {{{\rm \alpha} ^2} \over {\,f^3}} - {{{\rm \alpha} R_d^2} \over {{\rm \epsilon} _o}} f*{{A_{oo}^2} \over {2f^4}} {{{\rm \omega} _{\rm p}^{\rm 2}} \over {{\rm \omega} _o^2}} {1 \over {{\rm \gamma} _o^3}} {1 \over {(1 - ({\rm \omega} _{\rm c} /{\rm \gamma} _o {\rm \omega} _o ))}}\left( {1 - {{\rm \delta} \over {{\rm \gamma} _o}}} \right)^{ - 2} \left( {1 - {{{\rm \omega} _{\rm c} {\rm \delta}} \over {{\rm \gamma} _o^2 {\rm \omega} _o}}} \right) \cr &\times\, \left[ {1 + {{{\rm \omega} _{\rm c}} \over {{\rm \omega} _o}} {1 \over {\sqrt {1 + (A_{oo}^2 /f^2 )}}} \left( {2 - {{A_{oo}^2 /f^2} \over {1 + (A_{oo}^2 /f^2 )}}} \right)}\right. \cr &\left. +\, 3{1 \over 1 + (A_{oo}^2 /f^2 )} \left( {{\rm \omega}_{\rm c} \over {\rm \omega} _o} \right)^2 \left( 1 - {{A_{oo}^2 /f^2} \over {1 + (A_{oo}^2 /f^2 )}} \right) \right].}$$

4. NUMERICAL RESULTS AND DISCUSSION

Equation (16) is the second-order non-linear differential equation that governs the dynamics of the beam width parameter f as a function of the normalized distance of propagation ξ. The first term on its righ-hand side is the diffractional term, which has its origin coming from Eq. (11) and is responsible for the divergence of the laser beam. The second term being non-linear in character results in self-focusing. There is a relative competition between the diffractional and non-linear terms. The fate of the laser beam further depends upon the several parameters such as plasma density (ω_p), relativistic factor (γ), contribution of cold quantum plasma (δ), and relative value of magnetic field (ω_c/ω₀). All these parameters determines the beam dynamics of laser beam in the presence of externally applied magnetic field. As the analytical solution of Eq. (16) is not possible, we seek help of numerical techniques to find the solution. The following set of laser and plasma parameters is considered: ω _o  = 1.778 × 10²⁰ rad/s, r _o  = 0.002 cm, λ = 0.0106 nm, n _o  = 3.14 × 10¹⁸ cm⁻³, and ω_c/ω _o  = 0.1 − 0.5.

It is to be understood that the non-linear term increases as the plasma permittivity increases. Obviously, we explore the combined effect of static magnetic field and cold quantum plasma when Gaussian laser beam propagates in such a non-linear medium. Numerical computation is performed and results are plotted in the form of graphs. Critical curves are obtained when laser beam travels in self-trapped mode. This can be done by setting f = 1 and vanishing of left-hand side of the Eq. (16).

Figure 1 depicts the plot of critical curve ( $a_o^2, {\rm \rho} _o $ ) for weakly relativistic magnetized cold quantum (WRMCQ) plasma, weakly relativistic magnetized (WRM) plasma, and weakly relativistic cold quantum (WRCQ) plasma as well as for classical relativistic (CR) case of reference. By neglecting the quantum effects (δ → 0), the equilibrium beam radius can be plotted for weakly relativistic magnetized (WRM) plasma. Similarly, ω_c/ω₀ → 0 leads to WRCQ case and by putting δ → 0 and ω_c/ω₀ → 0 for CR case of reference. As apparent from the graph, with increase of $a_o^2 $ , there is a monotonic fall in the equilibrium beam radius initially. The equilibrium beam radius falls faster for WRMCQ plasma as compared with WRM, WRCQ, and CR cases. The magnetic field enhances the plasma permittivity and there is thus a wide range of laser intensities which can be focused in the case of WRMCQ plasma. Obviously, the non-linearity is strongest for WRMCQ plasma, weaker for WRM and WRCQ plasma model, respectively, and weakest for CR case. Therefore, self-focusing region is varied accordingly, that is, maximum for WRMCQ plasma and minimum for CR case.

Fig. 1.

Critical curves in ( $a_o^2, \rho _o $ ) plane for weakly relativistic magnetized cold quantum (WRMCQ) plasma, weakly relativistic magnetized (WRM), weakly relativistic cold quantum (WRCQ), and classical relativistic (CR) cases. The parameters are, λ = 0.0106 nm, ω_c/ω _o  = 0.5, r _o  = 0.002 cm, and n _o  = 3.14 × 10¹⁸ cm⁻³.

Figure 2 exhibits the graph of non-linear plasma permittivity Φ as a function of intensity for four cases of plasma model. The inclusion of magnetic field in cold quantum plasma acts as a catalyst, thereby improving the possibility of self-focusing. Thus, it leads to suitable conditions required for self-focusing in a WRM plasma. However, saturation achieved in case of WRCQ is quite delayed, therefore, enhanced self-focusing length is observed. Obviously, the combined role of cold quantum plasma and magnetic field in non-linear plasma permittivity is to assist laser beam in setting earlier self-focusing as evident from Figure 4. From Figure 2, it is observed that the magnetic field enhances the non-linear term, thus increasing the focussed region of WRMCQ and WRM plasma model as compared to WRCQ and CR case of reference.

Fig. 2.

Variation of non-linear plasma permittivity Φ against intensity parameter for WRMCQ, WRM, WRCQ, and CR cases, and other parameters being same as mentioned in the caption of Figure 1.

Figure 3 represents the variation of beam width parameter f with normalized distance of propagation ξ for different values of ω_c/ω _o . It has been found that self-focusing starts earlier with increase in magnetic field. For ω_c/ω _o  = 0.5, the self-focusing is strongest. This is due to the fact that non-linear term increases with increasing magnetic field by accelerating the electrons in the relativistic factor γ. Our results are thus in agreement with the results obtained by Vij et al. (Reference Vij, Gill and Aggarwal2016) wherein the role of magnetic field is seen to enhance the self-focusing and self-compression mechanism.

Fig. 3.

Variation of beam width parameter f with normalized propagation distance ξ for different values of ω_c/ω _o and other parameters being same as mentioned in the caption of Figure 1.

Figure 4 represents the variation of beam width parameter f with normalized distance of propagation ξ for different cases of plasma models, that is, WRMCQ, WRM, WRCQ, and CR cases of references. The self-focusing observed is stronger in the case of WRMCQ plasma due to enhancement in non-linear plasma permittivity. The results are consistent with Figures 1 and 2. Periodic focusing is observed due to the dominance of non-linear refraction term over diffraction term. The self-focusing length is considerably decreased with the combined effect of magnetic field and cold quantum plasma. The results are improved over the previous model studied by Kumar et al. (Reference Kumar, Aggarwal, Richa and Gill2016) in their (Fig. 2). The beam width parameter also decreases in WRMCQ plasma and WRM (with magnetic field) in comparison to WRCQ and CR cases (without magnetic field) as apparent from the figure. The present results and analysis may be useful in understanding the mechanism of laser beam propagation in non-linear medium where the role of quantum plasma is envisaged.

Fig. 4.

Variation of beam width parameter f with normalized propagation distance ξ for WRMCQ, WRM, WRCQ, and CR cases of references, and other parameters being same as mentioned in the caption of Figure 1.