2. THEORETICAL CONSIDERATIONS

We start by considering the propagation of a Gaussian laser beam along the z-direction through underdense, unmagnetized, and collisionless plasma. Initial intensity distribution of the beam in this situation is expressed as

(1) $$EE^{\ast} = E_0^2 \;\exp \left( { - \displaystyle{{r^2} \over {r_0^2}}} \right),$$

where r is the radial coordinate of cylindrical coordinate system and r ₀ is the initial beam width. The Gaussian laser beam exerts a ponderomotive force on plasma electrons due to the transverse intensity gradient of the beam. In a steady-state condition, the modified plasma density can be written as (Gill et al., Reference Gill, Mahajan and Kaur2011b ), $n_{\rm e} = n_0\exp [ - {\rm \beta} _0({\rm \gamma} - 1)]$ , where n ₀ is the unperturbed plasma electron density, β₀ = m ₀ c ²/T _e and ${\rm \gamma} = (1 + {\rm \alpha} EE^{\ast})^{{1 / 2}}$ is the Lorentz relativistic factor with ${\rm \alpha} = e^2/m_0^2 {\rm \omega} ^2c^2$ . Here e and m ₀ are the charge and rest mass of electron; respectively, ω is the frequency of laser beam and c is the speed of light in free space. This perturbed plasma electron density leads to a modified dielectric function of the plasma as

(2) $${\rm \varepsilon} = 1 - \displaystyle{{n_{\rm e}} \over {n_0{\rm \gamma}}}. $$

One can formally express the dielectric function of plasma as (Sodha et al., Reference Sodha, Ghatak and Tripathi1976)

(3) $${\rm \varepsilon} = {\rm \varepsilon} _0 + \Phi (EE^{\ast}) - i{\rm \varepsilon} _i,$$

where ε₀ and Φ are the linear and non-linear parts of the dielectric function respectively, ε _i takes care of absorption.

3. SELF-FOCUSING

The wave equation governing the electric field E of the beam in plasmas with the dielectric function given by Eq. (3) can be written as

(4) $$\nabla ^2E + \displaystyle{{{\rm \omega} ^2} \over {c^2}}{\rm \varepsilon} E = 0.$$

Keeping in mind that $(c^2/{\rm \omega} ^2)\vert(1/{\rm \varepsilon} )\nabla ^2\ln {\rm \varepsilon} \vert \ll 1$ , within the WKB approximation, we neglect the term $\nabla (\nabla. E)$ while writing Eq. (4). Now, $E = A(r,z)\exp ( - ik_0z)$ is introduced, where A(r, z) is the complex function of its argument and can be described by the parabolic equation in the WKB approximation as

(5) $$2ik_0\displaystyle{{\partial A} \over {\partial z}} + \nabla _ \bot ^2 A + \displaystyle{{{\rm \omega}^2} \over {c^2}}\left[ {\Phi \left( {EE^{\ast}} \right) - i{\rm \varepsilon}_i} \right]A = 0. $$

Expressing $A(r,z) = A_0(r,z)\exp ( - ik_0S)$ , where A ₀ and S are real functions of r and z (S being the eikonal of the beam) and substituting in Eq. (5), one can obtain

(6) $${2\;\left( {\displaystyle{{\partial S} \over {\partial z}}} \right) + \left( {\displaystyle{{\partial S} \over {\partial r}}} \right)^2 = \displaystyle{1 \over {k_0^2 A_0}}\nabla _ \bot ^2 A_0 + \displaystyle{{{\rm \omega}_{\rm p}^2 } \over {{\rm \omega}^2{\rm \varepsilon}_0}}\left\{ {1 - \displaystyle{1 \over {\rm \gamma}}\exp \left[ { - {\rm \beta}_0\left( {{\rm \gamma} - 1} \right)} \right]} \right\},} $$

(7) $$\displaystyle{{\partial A_0^2} \over {\partial z}} + \displaystyle{{\partial A_0^2} \over {\partial r}}\displaystyle{{\partial S} \over {\partial r}} + A_0^2 \left( {\displaystyle{{\partial ^2S} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial S} \over {\partial r}} - k_0\displaystyle{{{\rm \varepsilon} _i} \over {{\rm \varepsilon} _0}}} \right) = 0.$$

The solution of Eqs (6) and (7) satisfying the initial condition for intensity distribution of a Gaussian beam can be expressed as

(8) $$S = \displaystyle{{r^2} \over {2f}}\displaystyle{{df} \over {dz}} + {\rm \varphi} (z),$$

(9) $$A_0^2 = \displaystyle{{E_0^2} \over {\,f^2}}\exp \left( { - \displaystyle{{r^2} \over {r_0^2 f^2}} - 2K_iz} \right),$$

where K _i is the absorption coefficient and ϕ(z) is the axial phase.

Following the paraxial approach given by Sodha et al. (Reference Sodha, Ghatak and Tripathi1976) the dimensionless beam-width parameter f is obtained as

(10) $${\displaystyle{{d^2f} \over {d{\rm \eta}^2}} = \displaystyle{1 \over {\,f^3}} - {\rm \rho}_0^2 \displaystyle{{P_{\rm \eta}} \over {2Q_{\rm \eta}^3 f^3}} \left[ {1 + \left( {{\rm \beta}_0Q_{\rm \eta}} \right)} \right] \times \exp \left\{ {{\rm \beta}_0\left[ {1 - Q_{\rm \eta}} \right]} \right\},} $$

where K _i ′ = K _i R _d is the normalized absorption coefficient, $R_d = k_0r_0^2 $ is the Rayleigh length, $P_{\rm \eta} = P_0\exp ( - 2K_i^{\prime} {\rm \eta} )$ with $P_0 = {\rm \alpha} E_0^2 $ ^, $Q_{\rm \eta} = [1 + (P_{\rm \eta} /f^2)]^{{1 / 2}}$ , η = z/R _d is the dimensionless distance of propagation and ρ₀ = r ₀ω_p/c is the normalized equilibrium beam radius. Equation (10) can be solved numerically with appropriate boundary conditions. For an initially plane wavefront at η = 0, f = 1 and df / dη = 0, the condition d ² f/dη² = 0 leads to the propagation of Gaussian laser beam in the self-trapped mode.

4. NUMERICAL RESULTS AND DISCUSSION

Equation (10) is the second-order non-linear differential equation governing beam-width parameter f as a function of dimensionless distance of propagation η to study the effect of light absorption on temperature-based self-focusing and defocusing of Gaussian laser beam in a plasma. In order to present the behavior of beam-width parameter, Eq. (10) is solved by using fourth-order Runge–Kutta method for an initially plane wavefront (f = 1, df/dη = 0 at η = 0). The dependence of beam-width parameter and electron density distribution is studied for laser–plasma parameters; λ = 800 nm, n ₀ = 2.6 × 10¹⁷ cm⁻³, T _e = 20 − 100 KeV, r ₀ = 20 µm, P ₀ = 0.15. When d ² f/dη ² = 0, the initial beam-width does not change along the propagation in the plasma and the so-called waveguide/self-trapped mode. Herein, when f < 1 or f > 1, the laser beam will be focused (converged) or defocused (diverged) during the propagation of laser beam in the plasma. It is to be noted that in absence of light absorption effect (K _i ′ = 0), Eq. (10) reduces to Eq. (12) of Milani et al. (Reference Milani, Niknam and Bokaei2014a ) Also, on setting β₀ = 0 in Eq. (10), one can cast Eq. (10) of our earlier result (Patil et al., Reference Patil, Takale and Gill2015) for influence of light absorption under relativistic case only. Further by choosing suitable temperature and absorption level, we can investigate the self-focusing of Gaussian laser beam in a plasma under weakly relativistic and ponderomotive regime.

Figure 1 presents the variation of f with η under weakly relativistic and ponderomotive regime of interaction and only relativistic case of reference (solid line). In the numerical parameters as given above, we have chosen the normalized absorption coefficient K _i ′ = 0.005. It is obvious from Figure 1 that by increasing the electron temperature from T _e = 20 KeV (dashed curve) to T _e = 100 KeV (dotted curve), self-focusing loses its power and beam is diverged for T _e = 100 KeV in relativistic and ponderomotive regime as reported earlier by Bokaei et al. (Reference Bokaei, Niknam and Milani2013) in the absence of light absorption (K _i ′ = 0). It is important to note from Figure 1 that light absorption causes further to improve/obstruct the behavior of self-focusing/defocusing under weakly relativistic and ponderomotive regime of interaction. As obvious, occurrence of self-focusing/defocusing depends not only on the range/limit of requisite numerical parameters such as initial plasma density n ₀, electron temperature T _e, intensity parameter P ₀ etc., but also on the light absorption.

Fig. 1. Variation of beam-width parameter f with dimensionless distance of propagation η in plasma. (a) Relativistic case only, (b) T _e = 20 KeV, and (c) T _e = 100 KeV. The other laser–plasma parameters are: λ = 800 nm r ₀ = 20 µm, n ₀ = 2.6 × 10¹⁷ cm⁻³, P ₀ = 0.15, and K _i ′ = 0.005.

The variation of f with η for different values of normalized absorption coefficients K _i ′ is displayed in Figure 2. It is observed from this figure that in absence of the absorption effect, for T _e = 20 KeV, laser beam is self-focused in stationary oscillatory mode during propagation through plasma under weakly relativistic and ponderomotive regime. Such stationary mode has been reported by Bokaei et al. (Reference Bokaei, Niknam and Milani2013) for same laser–plasma parameters and their regime of interaction. However, such stationary oscillatory character of f gets destroyed due to absorption effect. It is interesting to note that by increasing the absorption level, self-focusing of beam is improved and takes place for earlier values of η. As obvious, penetration decreases as beam propagates through plasma on account of absorption.

Fig. 2. Variation of beam-width parameter f with dimensionless distance of propagation η in plasma under weakly relativistic and ponderomotive regime with T _e = 20 KeV for different absorption levels, (a) K _i ′ = 0.00, (b) K _i ′ = 0.003, (c) K _i ′ = 0.005, (d) K _i ′ = 0.007. The other parameters are same as in Figure 1.

In Figure 3, we have presented the effect of same absorption levels on dependence of f with η for T _e = 100 KeV. In contrast to Figure 2, it is observed from Figure 3 that by increasing the absorption for T _e = 100 KeV, result is reversed. This is due to the existence of temperature interval in which self-focusing can occur, while the beam diverges outside this region as reported (Bokaei et al., Reference Bokaei, Niknam and Milani2013) earlier.

Fig. 3. Variation of beam-width parameter f with dimensionless distance of propagation η in plasma under weakly relativistic and ponderomotive regime with T _e = 100 KeV for different absorption levels, (a) K _i ′ = 0.00, (b) K _i ′ = 0.003, (c) K _i ′ = 0.005, (d) K _i ′ = 0.007. The other parameters are same as in Figure 1.

An effect of initial electron temperature T _e on profile of normalized electron density has been shown in Figures 4 and 5 for K _i ′ = 0.005. Figure 4 depicts the normalized electron density at T _e = 20 KeV for self-focusing regime. It is observed from this figure that the slight depressed oscillatory trend of normalized electron density along the distance of propagation occurs due to absorption. This is in accordance with behavior of beam-width parameter f with dimensionless distance of propagation η (Fig. 2). Figure 5 presents the normalized electron density at T _e = 100 KeV  for defocusing regime. From Figure 5 it is observed that the normalized electron density at T _e = 100 KeV shows slight increased oscillatory behavior due to absorption in accord with Figure 3.

Fig. 4. Dependence of normalized plasma electron density n _e/n ₀ on r/r ₀ and η at T _e = 20 KeV. The other parameters are same as in Figure 1.

Fig. 5. Dependence of normalized plasma electron density n _e/n ₀ on r/r ₀ and η at T _e = 100 KeV. The other parameters are same as in Figure 1.