2. PROPAGATION OF CGLBS IN PLASMA

Let us consider the propagation of a linearly polarized CGLB of frequency ω₀ along the z-direction through collisionless plasma. The field distribution of the beam is given by (Casperson et al., Reference Casperson, Hall and Tovar1997; Lu et al., Reference Lu, Ma and Zhang1999; Lu & Luo, Reference Lu and Luo2000)

(1) $$\eqalign{E(r,z) & = \displaystyle{{E_0} \over {2f}}\exp \left( {\displaystyle{{b^2} \over 4}} \right) \cr & \quad \times \left[{\exp \left\{{ -{\left({\displaystyle{r \over {r_0f}} + \displaystyle{b \over 2}}\right)}^2}\right\} + \exp \left\{{- {\left({\displaystyle{r \over {r_0f}} - \displaystyle{b \over 2}} \right)}^2} \right\}}\right]}$$

where r is the radial coordinate of the cylindrical coordinate system, r ₀ is the initial beam width, b is the decentered parameter of the beam, f is the dimensionless beam-width parameter of the laser beam in plasma, which measures axial intensity and width of the beam and E ₀ is the amplitude of the electric field at the central position of r = z = 0.

The propagation of the CGLB in a collisionless plasma is governed by the wave equation

(2) $$\nabla ^2E - \nabla (\nabla \cdot E) + \displaystyle{{{\rm \omega} _0^2} \over {c^2}}{\rm \varepsilon} E = 0$$

where ε is the effective dielectric function of the plasma and c is the speed of light in free space.

The effective dielectric constant of the plasma at frequency ω₀ is given by

(3) $${\rm \varepsilon} = {\rm \varepsilon} _0 + {\rm \varphi} (E.E^{\ast})$$

where ε ₀ and ϕ represent the linear and nonlinear parts of dielectric constant respectively. The liner part of dielectric constant of the plasma can be expressed as

(4) $${\rm \varepsilon} _0 = 1 - \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {{\rm \omega} _0^2}} $$

where ω_p0 is the plasma frequency given by ω_p0 = 4πn ₀ e ²/m ₀, with e and m ₀ (in the absence of external field) are the electronic charge and rest mass respectively, n ₀ is the equilibrium electron density in the absence of laser beam. In the presence of an intense laser field, the RP force on the electrons modifies the electron density. The modified electron density profile of plasma due to RP force can be written as (Brandi et al., Reference Brandi, Manus and Mainfray1993a, Reference Brandi, Manus and Mainfrayb; Kumar et al., Reference Kumar, Gupta and Sharma2006)

(5) $$F_{\rm P} = - m_0c^2\nabla ({\rm \gamma} _0 - 1)$$

where γ₀ is called the relativistic factor defined as follows

$${\rm \gamma} _0 = \left[ {1 + \displaystyle{{e^2} \over {c^2m_0^2 {\rm \omega} _0^2}} E.E^{\ast}} \right]^{\textstyle{1 \over 2}}$$

The modified electron density (n _T) due to RP force is given by (Brandi et al., Reference Brandi, Manus and Mainfray1993a; Kumar et al., Reference Kumar, Gupta and Sharma2006)

(6) $$n_{\rm T} = n_0 + n_2 = n_0 + \displaystyle{{c^2n_0} \over {{\rm \omega} _{{\rm p}0}^2}} \left( {\nabla ^2{\rm \gamma} _0 - \displaystyle{{{(\nabla {\rm \gamma} _0)}^2} \over {{\rm \gamma} _0}}} \right)$$

where n ₂ is the nonlinear variation of the density together with the relativistic correction that is, the second order correction in the electron density equation and

(7) $$\displaystyle{{n_{\rm T}} \over {n_0}} = 1 + \displaystyle{{c^2a} \over {{\rm \omega} _{{\rm p}0}^2 4f^2}}\left[ {\displaystyle{1 \over {{\rm \gamma} _0}}AC + \displaystyle{1 \over {{\rm \gamma} _0}}r^2B_{}^2 - \displaystyle{{2a} \over {4f^2{\rm \gamma} _0^3}} A^2r^2B^2} \right]$$

where

$$A = \exp \left[ { - \left( {\displaystyle{{r^2} \over {\,f^2r_0^2}} + \displaystyle{{br} \over {\,fr_0}}} \right)} \right] + \exp \left[ { - \left( {\displaystyle{{r^2} \over {\,f^2r_0^2}} - \displaystyle{{br} \over {\,fr_0}}} \right)} \right]$$

$$\eqalign{B & = \exp \left[ { - \left( {\displaystyle{{r^2} \over {\,f^2r_0^2}} + \displaystyle{{br} \over {\,fr_0}}} \right)} \right]\left( { - \displaystyle{2 \over {\,f^2r_0^2}} - \displaystyle{b \over {\,frr_0}}} \right) \cr & \quad + \exp \left[ { - \left( {\displaystyle{{r^2} \over {\,f^2r_0^2}} - \displaystyle{{br} \over {\,fr_0}}} \right)} \right]\left( { - \displaystyle{2 \over {\,f^2r_0^2}} + \displaystyle{b \over {\,frr_0}}} \right)} $$

$$\eqalign{C & = \exp \left[ { - \left( {\displaystyle{{r^2} \over {\,f^2r_0^2}} + \displaystyle{{br} \over {\,fr_0}}} \right)} \right]\, \cr & \quad\times \left( { - \displaystyle{4 \over {\,f^2r_0^2}} - \displaystyle{b \over {\,frr_0}} + \displaystyle{{4r^2} \over {\,f^4r_0^4}} + \displaystyle{{b^2} \over {\,f^2r_0^2}} + \displaystyle{{4br} \over {\,f^3r_0^3}}} \right) \cr & \quad + \exp \left[ { - \left( {\displaystyle{{r^2} \over {\,f^2r_0^2}} - \displaystyle{{br} \over {\,fr_0}}} \right)} \right] \cr & \quad\times \,\left( { - \displaystyle{4 \over {\,f^2r_0^2}} + \displaystyle{b \over {\,frr_0}} + \displaystyle{{4r^2} \over {\,f^4r_0^4}} + \displaystyle{{b^2} \over {\,f^2r_0^2}} - \displaystyle{{4br} \over {\,f^3r_0^3}}} \right)} $$

The nonlinear dielectric constant of the plasma (in the presence of RP nonlinearity) is given by

(8) $${\rm \varphi} (E.E^{\ast}) = \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {{\rm \omega} _0^2}} \left( {1 - \displaystyle{{n_T} \over {n_0{\rm \gamma}}}} \right)$$

Expanding the dielectric constant around r = 0 in Eq. (3) by Taylor expansion, one can write

$${\rm \varepsilon} = {\rm \varepsilon} _{\rm f} + {\rm \gamma} _1r^2$$

where,

(9) $${\rm \varepsilon} _{\rm f} = {\rm \varepsilon} _0 + {\rm \Omega} ^2\left[ {1 - \displaystyle{1 \over {{\rm \gamma} _0}} - \displaystyle{{c^2a} \over {{\rm \omega} _0^2 \,f^4r_0^2}} (b^2 - 4)} \right]$$

(10) $${\rm \gamma} _1 = - \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {{\rm \omega} _0^2}} \,\left[ {\displaystyle{a \over {{\rm \gamma} _0^3 f^4r_0^2}} + \displaystyle{{c^2a} \over {{\rm \omega} _{{\rm p}0}^2 f^2}}\left( {\displaystyle{{(16 - 4b^2)} \over {{\rm \gamma} _{_{} 0}^2 f^4r_0^4}} + \displaystyle{{a(2b^2 - 16)} \over {{\rm \gamma} _0^4 r_0^4 f^6}}} \right)} \right]$$

Here, Ω = ω_p0/ω₀, and a = αE ₀ ² is the intensity parameter.

In order to solve Eq. (2), the second term [i. e. ∇ (∇. E)] on left hand side can be neglected provided that (c ²/ω₀ ²)|(1/ε)∇²Inε|≪1 (Sodha et al., Reference Sodha, Ghatak and Tripathi1974). Thus

(11) $$\nabla ^2E + \displaystyle{{{\rm \omega} _0^2} \over {c^2}}{\rm \varepsilon} E = 0$$

Using (WKB) approximation and following Akhmanov et al. (Reference Akhmanov, Sukhorukov and Khokhlov1968) and Sodha et al. (Reference Sodha, Ghatak and Tripathi1974), the solution of Eq. (11) can be written as

(12) $$E = A(r,z)\exp ( - ik_0z)$$

where A(r, z) is the complex amplitude of the electric field E, $k_0 = ({\rm \omega} {}_0/c)({\rm \varepsilon} _0)^{1/2}$ is the wave number. Substituting the value of E into Eq. (11), we obtain the following equation

(13) $$\eqalign{& - 2ik_0\displaystyle{{\partial A} \over {\partial z}} + \left( {\displaystyle{{\partial ^2} \over {\partial r^2}} + \displaystyle{{\partial ^2} \over {\partial z^2}} + \displaystyle{1 \over r}\displaystyle{\partial \over {\partial r}}} \right) A + \displaystyle{{{\rm \omega} _0^2} \over {c^2}}{\rm \varphi} (E.E^{\ast})A = 0} $$

Further, the variation of A can be expressed as

(14) $$A = A_0(r,{\rm} z)exp( - ik_0S_0)$$

where A ₀ and S ₀ are the real function of r and z (S ₀ being the eikonal of the beam). Substituting the expression for A into Eq. (13) and separating real and imaginary parts, the following set of equations is obtained

(15) $$\eqalign{& 2\displaystyle{{\partial S_0} \over {\partial z}} + \left( {\displaystyle{{\partial S_0} \over {\partial z}}} \right)^2 = \displaystyle{1 \over {k_0^2 A_0}}\left( {\displaystyle{{\partial ^2} \over {\partial r^2}} + \displaystyle{{\partial ^2} \over {\partial z^2}} + \displaystyle{1 \over r}\displaystyle{\partial \over {\partial r}}} \right) A_0 + \displaystyle{1 \over {{\rm \varepsilon} _0}}{\rm \varphi} (E.E^{\ast})} $$

(16) $$\displaystyle{{\partial A_0^2} \over {\partial z}} + \displaystyle{{\partial A_0^2} \over {\partial r}}\displaystyle{{\partial S_0} \over {\partial r}} + A_0^2 \left( {\displaystyle{1 \over r}\displaystyle{{\partial S_0} \over {\partial r}} + \displaystyle{{\partial ^2S_0} \over {\partial r^2}}} \right) = 0$$

The solution of the above coupled equations can be written as (Akhmanov et al., Reference Akhmanov, Sukhorukov and Khokhlov1968; Sodha et al., Reference Sodha, Ghatak and Tripathi1974)

(17) $$S_0 = {\rm \varphi} _0(z) + \displaystyle{{r^2} \over 2}{\rm \beta} (z)$$

where

$${\rm \beta} (z) = \displaystyle{{{\rm \omega} _0} \over c}\displaystyle{{{\rm \varepsilon} _{\,f0}^{1/2}} \over f}\displaystyle{{df} \over {dz}},\,\,\,\,{\rm and}\,{\rm \varphi} _{\rm 0}{\rm (}z{\rm )}\,{\rm is}\,{\rm the}\,{\rm axial}\,{\rm phase}{\rm.} $$

and the intensity of the beam as

(18) $$\eqalign{A_0^2 & = \displaystyle{{E_0^2} \over {4f^2}}\exp \left( {\displaystyle{{b^2} \over 2}} \right) \left[ {\exp \left\{ { - {\left( {\displaystyle{r \over {r_0f}} + \displaystyle{b \over 2}} \right)}^2} \right\} + \exp \left\{ { - {\left( {\displaystyle{r \over {r_0f}} - \displaystyle{b \over 2}} \right)}^2} \right\}} \right]^2} $$

Using Eq. (18) in Eq. (15), and equating the coefficients of r ² on both sides of the resulting equation, we obtain the equation governing the beam width parameter f as

(19) $$\eqalign{\displaystyle{{d^2f} \over {d{\rm \xi} ^2}} & = \left( {\displaystyle{{12 - 12b^2 - b^4} \over {3f^3}}} \right) \cr & \quad - \left[ {\left( {\displaystyle{{{\rm \omega} _{{\rm p}0}^2 r_0^2} \over {c^2}}} \right)\displaystyle{a \over {{\rm \gamma} _0^3 f^3}} + \displaystyle{a \over f}\left( {\displaystyle{{(16 - 4b^2)} \over {{\rm \gamma} _0^2 f^4}} + \displaystyle{{a(2b^2 - 16)} \over {{\rm \gamma} _0^4 f^6}}} \right)} \right]} $$

where (ξ = z/k ₀ r ₀ ²) is the dimensionless distance of propagation. Equation (12) describes the beam width of CGLB with the distance of propagation in collisionless plasma, when both relativistic and ponderomotive nonlinearities are operative.

2.1. Self-Trapped Mode

The initial conditions for the propagation of CGLB in the uniform waveguide/self-trapped mode are: d ² f/dξ² = 0 with df/dξ = 0, ξ = 0 and f = 1. This condition is known as critical condition under which CGLB propagates in plasma without convergence or divergence. We obtain a relation between the dimensionless initial beam width parameter (ρ_0 = r ₀ω_p0/c) and critical values of power of the beam (a = αE ₀₀ ²) in the presence of RP nonlinearity by substituting d ² f/dξ² = 0 in Eq. (19). The general expression for determination of critical threshold for various values of b can be obtained by using the critical condition. Eq. (19) is simplified for self-trapped mode and is given as follows:

(20) $$\eqalign{{\rm \rho} _0^2 & = \left( {\displaystyle{{12 - 12b^2 - b^4} \over {3a}}} \right){\rm \gamma} _0^3 \cr & - \left[ {(16 - 4b^2){\rm \gamma} _0 + \displaystyle{{a(2b^2 - 16)} \over {{\rm \gamma} _0}}} \right]}$$

The critical condition reflects that the CGLB undergoes self-focusing when the condition d ² f/dz ² < 0 is satisfied, whereas for d ² f/dz ² > 0, the CGLB either oscillatory or steady state self-focusing.

3. GENERATION OF EPW AND PARTICLE ACCELERATION

Nonlinear interaction of the EPW with the self-focused CGLB leads to their excitation. It is clear from the earlier analysis that the CGLB can be self-focused in plasma if the initial laser power is larger than the critical power and the change in background density due to ponderomotive force and the relativistic effects. In this process, the laser beam intensity is very intense and the plasma density is also changed due to the ponderomotive force. Thus, the amplitude of EPW, which depends on the background electron density is modified. The magnitude of EPW can be obtained by using the equation of continuity, the equation of motion and Poisson's equation. Following the standard techniques, one obtains the general equation governing the electron density variation in the EPW (neglecting the contribution of the ions) as (Purohit et al., Reference Purohit, Sharma and Sharma2012):

(21) $$\displaystyle{{\partial ^2N_{\rm e}} \over {\partial t^2}} + 2{\rm \Gamma} _{\rm e}\displaystyle{{\partial N_{\rm e}} \over {\partial t}} - {\rm \upsilon} _{{\rm th}}^2 \nabla ^2N_{\rm e} + \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {\rm \gamma}} \displaystyle{{n_{\rm T}} \over {n_0}}N_{\rm e} = 0$$

where Γ_e is the Landau damping coefficient for EPW (Krall & Trivelpiece, Reference Krall and Trivelpiece1973), υ_th is the electron thermal velocity. Using the WKB and paraxial approximations (Akhmanov et al., Reference Akhmanov, Sukhorukov and Khokhlov1968; Sodha et al., Reference Sodha, Ghatak and Tripathi1974), the solution of Eq. (21) can be expressed as

(22) $$N_{\rm e} = N_{{\rm e}0}(r,z)\exp [i({\rm \omega} t - kz)]$$

where N _e0 is the slowly varying real function of r and z, ω and k are the frequency and the propagation vector of the EPW related by the following dispersion relation

(23) $${\rm \omega} ^2 = {\rm \omega} _{{\rm p}0}^2 + k^2{\rm \upsilon} _{{\rm th}}^2 $$

Substituting Eqs. (22) and (23) into Eq. (21) one gets

(24) $$\eqalign{& - N_{{\rm e}0}{\rm \omega} ^2 + 2{\rm \Gamma} _{\rm e}N_{{\rm e}0}i{\rm \omega} - {\rm \upsilon} _{{\rm th}}^2 \left( {\displaystyle{{\partial ^2N_{{\rm e}0}} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{\partial \over {\partial r}}} \right) \cr & \qquad \times N_{{\rm e}0} + 2ik{\rm \upsilon} _{{\rm th}}^2 \displaystyle{{\partial N_{{\rm e}0}} \over {\partial z}} + k^2{\rm \upsilon} _{{\rm th}}^2 N_{{\rm e}0} + \displaystyle{{n_{\rm T}} \over {n_0}}\displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {\rm \gamma}} N_{{\rm e0}} = 0} $$

Further, we express N _e0 as

(25) $$N_{{\rm e}0} = N_{{\rm e}00}\left[ { - ikS(r,{\rm} z)} \right]$$

where N _e00 is the real function of its argument and S is the eikonal of the EPW. Substituting N _e0 into Eq. (24) and separating real and imaginary parts of the resulting equation, we obtain

(26) $$\eqalign{& 2\displaystyle{{\partial S} \over {\partial z}} + \left( {\displaystyle{{\partial S} \over {\partial r}}} \right)^2 = \displaystyle{1 \over {k^2N_{{\rm e}00}}}\left( {\displaystyle{{\partial ^2N_{{\rm e}00}} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial N_{{\rm e}00}} \over {\partial r}}} \right) + \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {k^2{\rm \upsilon} _{{\rm th}}^2}} \left( {1 - \displaystyle{1 \over {\rm \gamma}} \displaystyle{{n_{\rm T}} \over {n_0}}} \right)} $$

(27) $$\eqalign{& \displaystyle{{\partial N_{{\rm e}00}^2} \over {\partial z}} + \displaystyle{{\partial S} \over {\partial r}}\displaystyle{{\partial N_{{\rm e}00}^2} \over {\partial r}} + N_{{\rm e}00}^2 \left( {\displaystyle{{\partial ^2S} \over {\partial r^2}} + \displaystyle{1 \over r}\displaystyle{{\partial S} \over {\partial r}}} \right) + \displaystyle{{2{\rm \Gamma} _{\rm e}{\rm \omega} N_{{\rm e}00}^2} \over {k{\rm \upsilon} _{{\rm th}}^2}} = 0} $$

To solve the coupled Eqs. (26) and (27), we assume the initial radial variation of the density perturbation to be

(28) $$\eqalign{ N_{{\rm e}00}^2 \left \vert {_{z = 0}} \right. &= \left( {\displaystyle{{N_{10}} \over 2}} \right)^2\exp \left( {\displaystyle{{b^2} \over 2}} \right) \cr & \quad\times \left[ {\exp \left\{ { - {\left( {\displaystyle{r \over {a_0}} + \displaystyle{b \over 2}} \right)}^2} \right\} + \exp \left\{ { - {\left( {\displaystyle{r \over {a_0}} - \displaystyle{b \over 2}} \right)}^2} \right\}} \right]^2} $$

where a ₀ is the initial beam width of the plasma wave and N ₁₀ is the initial density associated with the EPW at r = 0. The solution of Eqs. (26) and (27) can be written as (Akhmanov et al., Reference Akhmanov, Sukhorukov and Khokhlov1968)

$$S = \displaystyle{{r^2} \over 2}{\rm \beta} (z) + {\rm \varphi} (z)$$

(29) $$\eqalign{N_{{\rm e}00} & = \displaystyle{{N_{10}} \over {2f_{\rm e}}}\exp \left( {\displaystyle{{b^2} \over 4}} \right) \cr & \quad\times \left[ {\exp \left\{ { \!- {\left( {\displaystyle{r \over {r_0f_{\rm e}}} \!+ \displaystyle{b \over 2}} \right)}^2} \right\} \!+\! \exp \left\{ { \!- {\left( {\displaystyle{r \over {r_0f_{\rm e}}} - \displaystyle{b \over 2}} \right)}^2} \right\}} \right]\exp ( - k_iz)} $$

where k _i = Γ_eω/kυ _th ² and f _e is the dimensionless beam width parameter for EPW. Using Eq. (29) in Eq. (26) and equating the coefficients of r ² on both sides by using the normalized distance (ξ = z/kr ₀ ²), we obtain:

(30) $$\eqalign{ \displaystyle{{d^2f_{\rm e}} \over {d {\rm \xi} ^2}} & = \left( {\displaystyle{{12 - 12b^2 - b^4} \over {3a_0^4 f_{\rm e}^3}}} \right)r_0^4 - f_{\rm e}\left( {\displaystyle{{c^2} \over {{\rm \upsilon} _{th}^2}}} \right) \cr & \quad \times \left[ {\displaystyle{{{\rm \omega} _{{\rm p}0}^2 r_0^2 a} \over {c^2{\rm \gamma} _0^3 f^3}} + \displaystyle{{a(16 - 4b^2)} \over {{\rm \gamma} _0^2 f^6}} + \displaystyle{{a^2(2b^2 - 16)} \over {{\rm \gamma} _0^4 f^8}}} \right]} $$

Equations (29) and (30) respectively give the amplitude of the density perturbation at finite z (intensity of EPW) and the coupling of the EPW with the CGLB when both relativistic and ponderomotive nonlinearities are operative.

The excited EPW by an intense CGLB transfer its energy to electrons and accelerates them [last term on the left hand side of Eq. (21)]. The energy gain (per unit rest mass energy of electron) by the electron is given by

(31) $${\rm \gamma} = \left( {1 - \displaystyle{{{\rm \upsilon} ^2} \over {c^2}}} \right)^{ - 1/2}$$

Differentiating and putting $d(m{\rm \upsilon} )/dt = ikm_0c^2{\rm \varphi} _1$ , we get

(32) $$\displaystyle{{d{\rm \gamma}} \over {dt}} = - ik{\rm \upsilon}. {\rm \varphi} _1$$

where ϕ₁ (=eϕ/m ₀ c ²) is the dimensionless electrostatic potential of EPW. Using the Posisson equation, the expression for ϕ₁ is given by

$${\rm \varphi} _1 = \displaystyle{{i{\rm \omega} _{{\rm p}0}^2} \over {c^2k^2f_{\rm e}}}\exp ( - k_iz)\sin (kz)$$

The first order differential Eq. (32) has been solved numerically, where we have used f _e from Eq. (30).

4. NUMERICAL RESULTS AND DISCUSSION

The present study is mainly divided into two phases: (1) Propagation of CGLB in collisionless plasma in the presence of RP nonlinearity (RPNL)/only relativistic nonlinearity (RNL), and (2) the effect of the self-focused CGLB on the propagation of EPW and particle acceleration process. Strong self-focusing of CGLB in plasma has been observed when ponderomotive nonlinearity is included in comparison with only relativistic nonlinearity, and Gaussian profile of laser beam, which significantly affected the generation of EPW and particle acceleration. The following set of laser and plasma parameters have been used for carrying out the numerical solution of the problem:

ω₀ = 1.778 × 10¹⁵rad/s, r ₀ = 20 µm, ω_p0 = 0.05ω₀, a ₀ = 10 µm, ν_th = 0.1c, b = 0, 0.5, 0.9 and 1.5, and a = 0.1, 0.5 and 1.

Equation (19) describes the variation in beam width parameter with normalized distance of propagation ξ (=z/kr ₀ ²) and it is solved by the following boundary conditions: f ₀|z = 0 = 1 and df ₀/dz = 0. The right hand side of Eq. (19) contains two terms, where the first one is the diffractive term responsible for the divergence of the laser beam, and the second is a nonlinear term arising due to RP nonlinearity. The self-focusing/de-focusing of the CGLB is determined by the relative magnitude of nonlinear and diffraction terms. The nonlinear term depends on the intensity parameter (a), decentered parameter (b) and plasma density. It is interesting to note that on setting b = 0 in Eq. (19), self-focusing of Gaussian laser beam is observed. Furthermore, if (n _T/n₀ = 1) is inserting in Eq. (8), one can get the equation for relativistic self-focusing of CGLB in plasma. In order to understand the dynamics of the propagation of CGLB in plasma, a comparison with only relativistic regime and Gaussian beam is essentially important. Figure 1 depicts the variation of the beam width parameter (f) of CGLB with the dimensionless distance of propagation (ξ) with the RP nonlinearity and the only relativistic nonlinearity of self-focusing for the constant value of a = b = 0.5. It is obvious that self-focusing is strong and occurs earlier when both relativistic and ponderomotive nonlinearities are taken into account in comparison with taking only relativistic nonlinearity. Figure 2a, 2b respectively illustrate the variation of the beam width parameter (f) of CGLB with the dimensionless distance of propagation (ξ) for different values of b and a, when both relativistic and ponderomotive nonlinearities are operative. It is clear from Figure 2a that the rate of self-focusing increases when we increase the value of decentered parameter ‘b’, while for b = 1.5; the beam width parameter (f) decreases steeply and thus supports the results of Gill et al. (Reference Gill, Mahajanr and Kaur2011). However, when b = 0, strong oscillatory self-focusing effect is observed at certain higher value of normalized distance. The rate of self-focusing is faster for CGLB with RP nonlinearity because the power and intensity of CGLB is relatively higher than Gaussian laser beam. Moreover, it is also observed from Figure 2b that with increase in the intensity of laser beam, there is an increase in self-focusing. This is because, as we increase the intensity of the laser beam, nonlinear refractive term in Eq. (19) dominate the diffractive term and hence there is an increase in focusing of the beam at higher intensity. The tendency of CGLB that is, converge earlier than Gaussian beam, leads us to choose the profile of CGLB to various applications.

Fig. 1.

Variation of the beamwidth parameter (f) of cosh Gaussian laser beam with normalized propagation distance (ξ) for a = b = 0.5.

Fig. 2.

(a)Variation of beam width parameter (f) with normalized propagation distance (ξ) for cosh-Gaussian (different values of decentered parameter b) and Gaussian laser beam (b = 0) with a = 0.5, when both relativistic and ponderomotive nonlinearities are taken into account. (b) Variation of beam width parameter (f) with normalized propagation distance (ξ) for different values of intensity parameter (a) with constant value of decentered parameter (b), when relativistic-ponderomotive nonlinearity is operative.

In order to study the propagation of CGLB in a plasma, we numerically analyze the dependence of dimensionless initial beam width parameter (ρ₀) as a function of critical values of power of the beam (a = αE ₀ ²) for various values of b when both relativistic and ponderomotive nonlinearities are taken into account. We have solved Eq. (20) numerically and the results are presented in the form of Figure 3, which shows the variation of ρ₀ with a for different values of b. It is observed that for uniform waveguide propagation of CGLB, ρ₀ exponentially decreases with a. It is also observed from the figure that with increase in the value of decentered parameter the rate of initial decreases in ρ₀ with a is faster. It shows that highest laser power is required for the Gaussian laser beam (b = 0) to propagate in uniform wave guide mode. Thus, CGLBs are more suitable for propagation in uniform waveguide mode at lowest laser power.

Fig. 3.

Variation of the dimensionless initial beam width (ρ₀) as a function of intensity parameter (a) for different values of decentered parameter (b).

The EPW excited due to nonlinear coupling between the CGLB and plasma with RP nonlinearity. This coupling arises on account of the change in the background density that is, the motion of the charge carrier will be modified due to the RP force in the plasma. Thus, the amplitude of EPW, which depends upon the background electron density, gets strongly coupled to the laser beam. Eq. (29) describes the modified density profile of plasma, when the paraxial approximation is taken into consideration. The intensity profile of the EPW depends on the beam widths (f _e and f) of EPW and the CGLB. We have solved Eq. (29) with the help of Eq. (30) numerically to obtain the amplitude of the density perturbation at finite z for typical laser and plasma parameters. The results are displayed in Figure 4a–4c, which show that the EPW gets excited due to nonlinear coupling with high power laser beam in the presence of RP and only relativistic nonlinearities. In Figure 4a, we have comparatively studied the variation in the intensity of EPW with normalized distance, when RP/only relativistic nonlinearities are operative for a = b = 0.5. It is obvious that the intensity of EPW get enhanced when both nonlinearities are present. It is also clear from Figure 4b, 4c, the intensity of EPW increases with increases the valves of decentered parameter (b) and the intensity of CGLB (a). This is because the intensity of EPW depends directly on the magnitude of the self-focusing of main CGLB and EPW.

Fig. 4.

(a) Variation of electron plasma wave intensity with normalized distance of propagation (ξ) for a = b = 0.5. (b) Variation of electron plasma wave intensity with normalized distance of propagation (ξ) for cosh-Gaussian (different values of decentered parameter b) and Gaussian laser beam (b = 0) with a = 0.5, when both relativistic and ponderomotive nonlinearities are taken into account. (c) Variation of electron plasma wave intensity with normalized distance of propagation (ξ) for different values of intensity parameter (a) with constant value of decentered parameter (b), when relativistic-ponderomotive nonlinearity is operative.

To study the effect of self-focused CGLB and the intensity of EPW on the energy gain by electrons at finite z, we have solved Eq. (32) numerically with the help of Eq. (30). The energy gain by electrons depends on the self-focusing of CGLB in plasma as well as the amplitude of EPW. Figure 5a represents the variation of energy gain with the normalized distance of propagation (ξ), when both RP and only relativistic nonlinearities are operatives. It is evident from Figure 5a that the maximum energy gain by electrons, gets enhanced when both nonlinearities are present. This is due to strong self-focusing of CGLB and the enhancement of EPW intensity in comparison with only relativistic nonlinearity. Figure 5b reflects the effect of decentered parameter (b) on the energy gain with the normalized distance of propagation (ξ). It is observed that maximum energy gain by electrons is significantly increased by increasing the value of b. This is due to the fact that, with an increase in the value of b, self-focusing of CGLB in plasma and the amplitude of EPW enhanced significantly.

Fig. 5.

(a) Variation of energy gain (γ) by the electron with normalized distance of propagation (ξ) for a = b = 0.5. (b) Variation of energy gain (γ) by the electron with normalized distance of propagation (ξ) for different values of decentered parameter (b) with constant value of intensity parameter (a), when relativistic-ponderomotive nonlinearity is operative.