2. DIELECTRIC FUNCTION OF PLASMA

Consider the propagation of two coaxial, linearly polarized laser beams having electric field vectors

(1)$${{\bf E}_{\bf j}}(r,z,t) = {A_j}(r,z){\kern 1pt} {e^{ - \iota ({{\rm \omega} _j}t - {k_j}z)}}{{\bf e}_{\bf x}}$$

through an under dense collisional plasma having dielectric function

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

where, j = 1, 2 and ω_j and k _j, respectively are angular frequencies and vacuum wave numbers of the laser beams, A _j(r,z) are the slowly varying complex amplitudes of the laser beams and

(3)$${\rm \omega} _{\rm p}^2 = \displaystyle{{4{\rm \pi} {e^2}} \over m}{n_{\rm e}}$$

is the plasma frequency in the presence of laser beams. The initial intensity distribution of the laser beams along their wavefronts is assumed to be ChG and is given by Konar et al. (Reference Konar, Mishra and Jana2007); Gill et al. (Reference Gill, Kaur and Mahajan2011); Patil et al. (Reference Patil, Takale, Navare, Fulari and Dongare2012)

(4)$${A_j}A_j^{\rm \star} {\vert_{{\rm z} \;= \;0}} = E_{\,j0}^2 \,{e^{ - ({r^2}/r_j^{\rm 2})}}{\rm cos}{{\rm h}^2}\left( {\displaystyle{{{b_j}} \over {{r_j}}}r} \right)$$

where r _j are the radii of the laser beams at the plane of incidence, that is, at z = 0, E _j0 are the axial amplitudes of the electric fields of the laser beams, b _j/r _j are the parameters associated with cosh function and are known as cosh factors. Such ChG laser beams can be produced in the laboratory by superposition of two decentered Gaussian laser beams having same spot size and that are in phase with each other (Lu et al., Reference Lu, Ma and Zhang1999). Hence, the parameters b _j are also known as decentered parameters.

For z > 0, energy conserving ansatz for the intensity distribution of the ChG laser beams propagating along z-axis are given by

(5)$${A_j}A_j^{\rm \star} = \displaystyle{{E_{j0}^2} \over {f_j^2}} e^{- {r^2}/r_j^2 f_j^2} \cos {\rm h}^2 \left( {\displaystyle{{b_j} \over {r_j\,f_j}}r} \right)$$

where r _jf_j are the instantaneous radii of the laser beams. Hence, the parameters f _j are termed as dimensionless beam width parameters that are measure of both axial intensity and spot size of the laser beams. For b _j = 0, the ChG distribution gradually gets converted into usual Gaussian distribution that is,

$$\mathop {\lim} \limits_{{b_j} \to 0} {A_j}A_j^{\rm \star} = \displaystyle{{E_{\,j0}^2} \over {\,f_j^2}} {e^{ - {r^2}/r_j^2 f_j^2}}$$

The nonuniform intensity distribution along the wavefronts of the laser beams produce nonuniform heating of plasma electrons as a result of which redistribution of electrons takes place. The resultant distribution of electrons is given by Sodha et al., (Reference Sodha, Ghatak, Tripathi and Wolf1976)

(6)$${n_{\rm e}} = {n_0}{\left( {\displaystyle{{2{T_0}} \over {{T_0} + T}}} \right)^{1 - (s/2)}}$$

where n ₀ is the equilibrium electron density of plasma, T ₀ is the equilibrium plasma temperature and T is the temperature of plasma in the presence of laser beams which is related to laser electric fields by

(7)$$\displaystyle{T \over {{T_0}}} = 1 + \sum\limits_j {{{\rm \beta} _j}} {E_j}E_j^{\rm \star} $$

where β_j = (e ²M / 6K ₀T ₀m ²ω_j²) are the coefficients of collisional nonlinearity; e, m, respectively are the electronic charge and mass, M is the mass of ions, and K ₀ is the Boltzmann constant. Using Eqs (3)–(7) in Eq. (2) we get

(8)$${{\rm \epsilon} _j} = 1 - \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {{\rm \omega} _j^2}} {\left\{ {1 + \displaystyle{1 \over 2}\displaystyle{{{{\rm \beta} _j}E_{\,j0}^2} \over {\,f_j^2}} {e^{ - {r^2}/r_j^2 f_j^2}} {\rm cos}{{\rm h}^2}\left( {\displaystyle{{{b_j}} \over {{r_j}{\,f_j}}}r} \right)} \right\}^{(s/2) - 1}}$$

where ω_p0² = (4πe ² / m)n ₀ is the plasma frequency in the absence of the laser beams. The parameter s describes the nature of collisions and can be defined through the dependence of collision frequency ν on electron's random velocity v and temperature T as ν ∝ (v ² / T)^s. For velocity-dependent collisions s = 0, for collisions between electrons and diatomic molecules s = 2 and for electron–ion collisions s = −3. Taking

(9)$${\rm \epsilon} = {{\rm \epsilon} _{0j}} + {{\rm \phi} _j}({A_1}A_1^{\rm \star}, {A_2}A_2^{\rm \star})$$

we get,

(10)$${{\rm \epsilon} _{0j}} = 1 - \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {{\rm \omega} _j^2}} $$

and

(11)$$\eqalign{{{\rm \phi} _j}({A_1}A_1^{\rm \star}, {A_2}A_2^{\rm \star}) & = \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {{\rm \omega} _j^2}} \left[ {1 - \left\{ {1 + \displaystyle{1 \over 2}\sum\limits_j {\displaystyle{{{{\rm \beta} _j}E_{\,j0}^2} \over {\,f_j^2}}} {e^{ - ({r^2}/r_j^2 f_j^2)}}} \right.} \right. \cr & \quad \left. {{{\left. {{\rm cos}{{\rm h}^2}\left( {\displaystyle{{{b_j}} \over {{r_j}{\,f_j}}}r} \right)} \right\}}^{(s/2) - 1}}} \right]} $$

where, ε_0j and ϕ_j, respectively, are the linear and nonlinear parts of the dielectric function of the plasma.

3. CROSS-FOCUSING OF LASER BEAMS

Starting from Ampere's and Faraday's laws for an isotropic, non-conducting, and non-absorbing medium (J = 0, ρ = 0, μ = 1), we get

(12)$$\nabla \times {{\bf B}_{\bf j}} = \displaystyle{1 \over c}\displaystyle{{\partial {{\bf D}_{\bf j}}} \over {\partial t}}$$

(13)$$\nabla \times {{\bf E}_{\bf j}} = - \displaystyle{1 \over c}\displaystyle{{\partial {{\bf B}_{\bf j}}} \over {\partial t}}$$

where E_j and B_j are electric and magnetic fields associated with the laser beams and D_j = ε_jE_j are the electric displacement vectors. Eliminating B_j  from Eqs (12) and (13) it can be shown that electric field vectors E_j of the laser beams satisfy the wave equation

(14)$${\nabla ^2}{{\bf E}_{\bf j}} + \displaystyle{{{\rm \omega} _j^2} \over {{c^2}}}{{\rm \epsilon} _j}{{\bf E}_{\bf j}} = 0$$

In deriving Eq. (14) the polarization term ∇(∇.E_j) has been neglected under the assumption that either the transverse dielectric variations are weak or the plasma is significantly under dense.

Using Eq. (1) in Eq. (14) we get

(15)$$\iota \displaystyle{{d{A_j}} \over {dz}} = \displaystyle{1 \over {2{k_j}}}\nabla _ \bot ^2 {A_j} + \displaystyle{{{k_j}} \over {2{{\rm \epsilon} _{0j}}}}{{\rm \phi} _j}({A_1}A_1^{\rm \star}, {A_2}A_2^{\rm \star}){A_j}$$

where

$$\nabla _ \bot ^2 = \displaystyle{{{\partial ^2}} \over {\partial {r^2}}} + \displaystyle{1 \over r}\displaystyle{\partial \over {\partial r}}$$

is the Laplacian along the transverse direction. Equation (15) is the well-known nonlinear Schrödinger wave equation that describes propagation characteristics of electromagnetic beams in nonlinear media under the assumption that wave-amplitude scale length along the z-axis is much larger as compared with characteristic scale in the transverse direction.

Due to its symmetrical properties, the wave Eq. (15) possesses a number of conserved quantities. In view of self-focusing of the laser beams, the two most important invariants are

(16)$${I_{0j}} = \int_0^{2{\rm \pi}} {\int_0^\infty {{A_j}}} A_j^{\rm \star} r\,dr\,d{\rm \theta} $$

(17)$${H_j} = \int_0^{2\pi} {\int_0^\infty {\displaystyle{1 \over {2k_j^2}}}} (\left \vert{\nabla _ \bot} {A_j}\right\vert{^2} - {F_j})r\,dr\,d{\rm \theta} $$

where

(18)$${F_{\rm j}} = \displaystyle{1 \over {2{{\rm \epsilon} _{{\rm 0j}}}}}\int_0^{{A_{\rm j}}A_{\rm j}^{\rm \star}} {{{\rm \phi} _{\rm j}}} ({A_1}A_1^{\rm \star}, {A_2}A_2^{\rm \star})d({A_{\rm j}}A_{\rm j}^{\rm \star})$$

The first invariant I _0j is merely a statement of the conservation of energy of the laser beams and second invariant H _j relates the wavefront curvature of the laser beams to the plasma nonlinearity. In moment theory approach the average value of a physical quantity M _j is defined as

(19)$$\left\langle {M_j^2} \right\rangle = \displaystyle{1 \over {{I_{0j}}}}\int_0^{2{\rm \pi}} {\int_0^\infty {{A_j}}} A_j^{\rm \star} {M_j}r{\kern 1pt} dr{\kern 1pt} d{\rm \theta} $$

The quantity of particular importance from the standpoint of self-focusing is mean square radius of the laser beams,

(20)$$\left\langle {R_j^2} \right\rangle = \displaystyle{1 \over {{I_{0j}}}}\int_0^{2{\rm \pi}} {\int_0^\infty {{r^2}}} {A_j}A_j^{\rm \star} r{\kern 1pt} dr{\kern 1pt} d{\rm \theta} $$

Following the procedure of Lam et al. (Reference Lam, Lippmann and Tappert1977), we get the following quasi-optic equation governing the evolution of mean-square radius of laser beams with distance of propagation

(21)$$\eqalign{\displaystyle{{{d^2}} \over {d{z^2}}}\left\langle {a_j^2} \right\rangle & = 4\displaystyle{{{H_j}} \over {{I_{0j}}}} - \displaystyle{4 \over {{I_{0j}}}} \int_0^{2{\rm \pi}} {\int_0^\infty {\left( {2{F_j} - \displaystyle{1 \over {2{{\rm \epsilon} _{0j}}}}{A_j}A_j^{\rm \star} {{\rm \phi} _j}} \right)}} r{\kern 1pt} dr{\kern 1pt} d{\rm \theta}} $$

Using Eqs (5), (11), (16)–(19), (20) in (21), we get the following coupled differential equations governing the cross-focusing of the laser beams.

(22)$$\eqalign{\displaystyle{{{d^2}{\,f_1}} \over {d{{\rm \xi} ^2}}} + \displaystyle{1 \over {{\,f_1}}}{\left( {\displaystyle{{d{\,f_1}} \over {d{\rm \xi}}}} \right)^2} & = \left( {\displaystyle{{1 + {e^{ - b_1^2}} (1 - b_1^2)} \over {2(1 + b_1^2)}}} \right) \cr & \quad \displaystyle{1 \over {\,f_1^3}} + \left( {\displaystyle{s \over 2} - 1} \right){\left( {\displaystyle{{{{\rm \omega} _{{\rm p0}}}{r_1}} \over c}} \right)^2}\left( {\displaystyle{{{e^{ - b_1^2}}} \over {1 + b_1^2}}} \right){J_1}({\,f_1},{\,f_2})} $$

(23)$$\eqalign{\displaystyle{{{d^2}{\,f_2}} \over {d{{\rm \xi} ^2}}} + \displaystyle{1 \over {{\,f_2}}}{\left( {\displaystyle{{d{\,f_2}} \over {d{\rm \xi}}}} \right)^2} & = {\left( {\displaystyle{{{r_1}} \over {{r_2}}}} \right)^4}{\left( {\displaystyle{{{{\rm \omega} _1}} \over {{{\rm \omega} _2}}}} \right)^2}\left( {\displaystyle{{{{\rm \epsilon} _{01}}} \over {{{\rm \epsilon} _{02}}}}} \right)\left[ {\left( {\displaystyle{{1 + {e^{ - b_2^2}} (1 - b_2^2)} \over {2(1 + b_2^2)}}} \right)\displaystyle{1 \over {\,f_2^3}}} \right. \cr & \left. {\quad + \left( {\displaystyle{s \over 2} - 1} \right){{\left( {\displaystyle{{{{\rm \omega} _{{\rm p0}}}{r_1}} \over c}} \right)}^2}\left( {\displaystyle{{{e^{ - b_2^2}}} \over {1 + b_2^2}}} \right){J_2}({\,f_1},{\,f_2})} \right]} $$

where ξ = z / k ₁r ₁² is the dimensionless distance of propagation and,

$$\eqalign{{J_1}({\,f_1},{\,f_2}) & = \left\{ {\displaystyle{{{{\rm \beta} _1}E_{10}^2} \over {\,f_1^3}} ({T_1} - {b_1}{T_2})} \right. \cr & \quad \left. { + \displaystyle{{{{\rm \beta} _2}E_{20}^2} \over {\,f_1^3}} \left( {\displaystyle{{r_1^2} \over {r_2^2}} \displaystyle{{\,f_1^4} \over {\,f_2^4}} {T_3} - {b_2}\displaystyle{{{r_1}} \over {{r_2}}}\displaystyle{{\,f_1^3} \over {\,f_2^3}} {T_4}} \right)} \right\} \cr {J_2}({\,f_1},{\,f_2}) & = \left\{ {\displaystyle{{{{\rm \beta} _1}E_{10}^2} \over {\,f_2^3}} ({T_3} - {b_1}{T_5})} \right. \cr & \left. {\quad + \displaystyle{{{{\rm \beta} _2}E_{20}^2} \over {\,f_2^3}} \left( {\displaystyle{{r_1^2} \over {r_2^2}} \displaystyle{{\,f_1^4} \over {\,f_2^4}} {T_6} - {b_2}\displaystyle{{{r_1}} \over {{r_2}}}\displaystyle{{\,f_1^3} \over {\,f_2^3}} {T_7}} \right)} \right\} \cr {T_1} & = \int_0^\infty {{x^3}} {e^{ - 2{x^2}}}{\rm cos}{{\rm h}^4}({b_1}x)G(x)dx \cr {T_2} & = \int_0^\infty {{x^2}} {e^{ - 2{x^2}}}{\rm cos}{{\rm h}^3}({b_1}x){\rm sinh}({b_1}x)G(x)dx \cr {T_3} & = \int_0^\infty {{x^3}} {e^{ - {x^2}}}{e^{ - {{({r_1}{\,f_1}/{r_2}{\,f_2})}^2}{x^2}}} \cr & \quad {\rm cos}{{\rm h}^2}({b_1}x){\rm cos}{{\rm h}^2}\left( {{b_2}\displaystyle{{{r_1}{\,f_1}} \over {{r_2}{\,f_2}}}x} \right)G(x)dx \cr {T_4} & = \int_0^\infty {{x^2}} {e^{ - {x^2}}}{e^{ - {{({r_1}{\,f_1}/{r_2}{\,f_2})}^2}{x^2}}}{\rm cos}{{\rm h}^2}({b_1}x) \cr & \quad {\rm cosh}\left( {{b_2}\displaystyle{{{r_1}{\,f_1}} \over {{r_2}{\,f_2}}}x} \right){\rm sinh}\left( {{b_2}\displaystyle{{{r_1}{\,f_1}} \over {{r_2}{\,f_2}}}x} \right)G(x)dx }$$

$$\eqalign{\cr {T_5} & = \int_0^\infty {{x^2}} {e^{ - {x^2}}}{e^{ - {{({r_1}{\,f_1}/{r_2}{\,f_2})}^2}{x^2}}}{\rm cosh}({b_1}x){\rm sinh}({b_1}x) \cr & \quad {\rm cos}{{\rm h}^2}\left( {{b_2}\displaystyle{{{r_1}{\,f_1}} \over {{r_2}{\,f_2}}}x} \right){\rm sinh}\left( {{b_2}\displaystyle{{{r_1}{\,f_1}} \over {{r_2}{\,f_2}}}x} \right)G(x)dx \cr {T_6} & = \int_0^\infty {{x^3}} {e^{ - 2{{({r_1}{\,f_1}/{r_2}{\,f_2})}^2}{x^2}}}{\rm cos}{{\rm h}^4}\left( {{b_2}\displaystyle{{{r_1}{\,f_1}} \over {{r_2}{\,f_2}}}x} \right)G(x)dx \cr {T_7} & = \int_0^\infty {{x^3}} {e^{ - 2{{({r_1}{\,f_1}/{r_2}{\,f_2})}^2}{x^2}}}{\rm cos}{{\rm h}^3}\left( {{b_2}\displaystyle{{{r_1}{\,f_1}} \over {{r_2}{\,f_2}}}x} \right) \cr & \quad {\rm sinh}\left( {{b_2}\displaystyle{{{r_1}{\,f_1}} \over {{r_2}{\,f_2}}}x} \right)G(x)dx \cr G(x) & = \left\{ {1 + \displaystyle{1 \over 2}\displaystyle{{{{\rm \beta} _1}E_{10}^2} \over {\,f_1^2}} {e^{ - {x^2}}}{\rm cos}{{\rm h}^2}({b_1}x) } \right. \cr & \quad{\left. +\;{\displaystyle{1 \over 2}\displaystyle{{{{\rm \beta} _2}E_{20}^2} \over {\,f_2^2}} {e^{ - {{({r_1}{\,f_1}/{r_2}{\,f_2})}^2}{x^2}}}{\rm cos}{{\rm h}^2}\left( {{b_2}\displaystyle{{{r_1}{\,f_1}} \over {{r_2}{\,f_2}}}x} \right)} \right\}^{^{(s/2) - 2}}} \cr x & = \displaystyle{r \over {{r_1}{\,f_1}}}} $$

For initially plane wavefronts Eqs (22) and (23) are subjected to boundary conditions f _j = 1 and df _j / dξ = 0 at ξ = 0.

Equations (22) and (23) are the coupled nonlinear differential equations governing the relativistic cross-focusing of two coaxial ChG laser beams in an under dense plasma. Analytical solutions to these equations are not possible. We therefore seek numerical computational techniques to investigate the beam dynamics. Before that, it is worth noting to understand the physical mechanisms that originate various terms on the right-hand sides (RHS) of Eqs (22) and (23). The first terms on RHS of Eqs (22) and (23) are responsible for diffraction divergence of the laser beams and have their origin in the Laplacian $\nabla_{\bot}^2 $, appearing in nonlinear wave Eq. (15). The second terms on RHS of these equations arise under the combined effect of relativistic mass nonlinearity and nonlinear coupling between the two laser beams. These terms are responsible for nonlinear refraction of the laser beams. It is the relative competition between the two terms that determine the focusing/defocusing of the laser beams in plasma.

Equations (22) and (23) have been solved for the following set of laser–plasma parameters

$$\eqalign{{{\rm \omega} _1} = & 1.78 \times {10^{15}}\;{\rm rad}/{\rm s};\quad {{\rm \omega} _2} = 1.98 \times {10^{15}}\;{\rm rad}/{\rm s} \cr {r_1} = & 15\,{\rm \mu m};\quad {r_2} = 16.67\,{\rm \mu m} \cr {T_0} = & {10^6}\;{\rm K}}$$

to analyze the effect of decentered parameters and plasma density on cross focusing of the laser beams and beat wave excitation of EPW.

Figures 1 and 2 illustrate the effect of decentered parameter b ₁ on focusing/defocusing of first laser beam. The plots in Fig. 1 depict that as the value of decentered parameter of first laser beam increases for 0 ≤ b ₁ < 1, there is increase in the extent of its self-focusing. This is due to the fact that for 0 ≤ b ₁ < 1 the intensity profile of ChG laser beam resembles to that of flat topped beam and extent of flatness increases with increase in the value of b ₁. As a result of this nonlinear refraction of most of the transverse part of its wavefront opposes the diffraction divergence. Hence, increase in the value of b ₁ from 0 to 1 leads to stronger focusing of the first laser beam. The plots in Fig. 2 depict that as the value of b ₁ increases for b ₁ ≥ 1 there is decrease in extent of self-focusing of first laser beam. This is due to the fact that as the value of decentered parameter b ₁ increases from 1 onwards up to 1.50 the diffractive term relatively dominates the refractive term but still maintaining the focusing character of the first laser beam. As the value of b ₁ becomes more than 1.50 (or b ₁ > 1.50) the diffractive term completely dominates over refractive term leading to complete defocusing of the first laser beam.

Fig. 1. Variation of normalized intensity of first beam with normalized distance of propagation ξ and radial distance r / r ₁, keeping (ω_p0r ₁ / c)² = 12, βE ₁₀² = 1.0, βE ₂₀² = 1.5, b ₂ = 0.25, collisional parameter s = −3 fixed and at different values of decentered parameter, (a) b ₁ = 0, (b) b ₁ = 0.25, (c) b ₁ = 0.50, (d) b ₁ = 0.75, respectively.

Fig. 2. Variation of normalized intensity of first beam with normalized distance of propagation ξ and radial distance r / r ₁, keeping (ω_p0r ₁ / c)² = 12, βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₂ = 0.25, collisional parameter s = −3 fixed and at different values of decentered parameter, (a) b ₁ = 1.0, (b) b ₁ = 1.25, (c) b ₁ = 1.50, (d) b ₁ = 1.75, respectively.

Figures 3 and 4 illustrate the effect of decentered parameter b ₁ of first laser beam on focusing/defocusing of second laser beam. It is observed that with increase in the decentered parameter of first laser beam there is decrease in the extent of self-focusing of the second laser beam. This is due to the fact that with an increase in the value of b ₁ the magnitude of refractive term in Eq. (23) decreases which leads to reduced focusing of second laser beam.

Fig. 3. Variation of normalized intensity of second beam with normalized distance of propagation ξ and radial distance r / r ₂, keeping (ω_p0r ₁ / c)² = 12, βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₂ = 0.25, collisional parameter s = −3 fixed and at different values of decentered parameter, (a) b ₁ = 0, (b) b ₁ = 0.25, (c) b ₁ = 0.50, (d) b ₁ = 0.75, respectively.

Fig. 4. Variation of normalized intensity of second beam with normalized distance of propagation ξ and radial distance r / r ₂, keeping (ω_p0r ₁ / c)² = 12, βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₂ = 0.25, collisional parameter s = −3 fixed and at different values of decentered parameter, (a) b ₁ = 1.0, (b) b ₁ = 1.25, (c) b ₁ = 1.50, (d) b ₁ = 1.75, respectively.

Figures 5 and 6 illustrate the effect of plasma density (ω_p0²r ₁² / c ²) on focusing/defocusing of laser beams. It is observed that with increase in plasma density there is increase in the extent of self-focusing of both the laser beams. This is due to the fact that with increase in plasma density the number of electrons contributing to the collisional nonlinearity also increases.

Fig. 5. Variation of normalized intensity of first beam with normalized distance of propagation ξ and radial distance r / r ₁, keeping βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₁ = 0.25, b ₂ = 0.25, collisional parameter s = −3 fixed and at different values of normalized density, (a) (ω_p0r ₁ / c)² = 12, (b) (ω_p0r ₁ / c)² = 14, (c) (ω_p0r ₁ / c)² = 16, (d) (ω_p0r ₁ / c)² = 18, respectively.

Fig. 6. Variation of normalized intensity of second beam with normalized distance of propagation ξ and radial distance r / r ₂, keeping βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₁ = 0.25, b ₂ = 0.25, collisional parameter s = −3 fixed and at different values of normalized density, (a) (ω_p0r ₁ / c)² = 12, (b) (ω_p0r ₁ / c)² = 14, (c) (ω_p0r ₁ / c)² = 16, (d) (ω_p0r ₁ / c)² = 18, respectively.

Figures 7 and 8 illustrate the effect of nature of collisions on focusing/defocusing of laser beams. It is observed that in plasmas dominant with collisions between electrons and diatomic molecules, no self-focusing of both the laser beams takes place. This is due to the fact that s = 2 corresponds to the situation where n _e = n ₀ = n _0i and hence no redistribution of electrons takes place. Thus, nonlinear terms in Eqs (22) and (23) are zero. Diffraction is dominant mechanism in this case.

Fig. 7. Variation of normalized intensity of first beam with normalized distance of propagation ξ and radial distance r / r ₁, keeping βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₁ = 0.25, b ₂ = 0.25, (ω_p0r ₁ / c)² = 12 fixed and at different values of collisional parameter (a) s = −3,, (b) s = 0,, (c) s = 2, respectively.

Fig. 8. Variation of normalized intensity of second beam with normalized distance of propagation ξ and radial distance r / r ₂, keeping βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₁ = 0.25, b ₂ = 0.25, (ω_p0r ₁ / c)² = 12 fixed and at different values of collisional parameter (a) s = −3, (b) s = 0, (c) s = 2, respectively.

4. BEAT WAVE EXCITATION

In the dynamics of excitation of EPW, it must be mentioned here that the contribution of ions is negligible because they only provide a positive background, that is, only plasma electrons are responsible for excitation of EPW. The background plasma density is modified via non-uniform ohmic heating of plasma. Therefore, amplitude of the EPW, which depends on the background electron density, gets strongly coupled to the laser beams. The generated plasma wave is governed by equation of continuity, equation of motion and Poisson's equation:

(24)$$\displaystyle{{\partial N} \over {\partial t}} + \nabla (N{\bf v}) = 0$$

(25)$$m\displaystyle{{d{\bf v}} \over {dt}} = - e{\bf E} - 3\displaystyle{{{K_0}{T_0}} \over N}\nabla N$$

(26)$$\nabla. {\bf E} = - 4{\rm \pi} eN$$

where N is the total electron density, E is the sum of electric fields of the laser beams and EPW

$$N = {n_0} + {n^{\prime}}$$

$$E = \sum\limits_j {{{\bf E}_{\bf j}}} + {{\bf E}^{\prime}}$$

and

$${\bf v} = {{\bf v}_{\bf e}}$$

is the oscillatory velocity of electrons. Using linear perturbation theory, we get the following wave equation governing the dynamics of EPW

(27)$$\displaystyle{{{\partial ^2}{n^{\prime}}} \over {\partial {t^2}}} - v_{{\rm th}}^2 {\nabla ^2}{n^{\prime}} + {\rm \omega} _{\rm p}^2 {n^{\prime}} = \displaystyle{e \over m}{n_0}\nabla \sum\limits_j {{E_j}} $$

Taking

$${n^{\prime}} = {n_1}{e^{{\rm \iota} ({\rm \omega t} - {\rm kz})}}$$

where k = k₂ − k₁, we get the density perturbation associated with plasma wave

(28)$${n_1} = \displaystyle{{e{n_0}} \over {{m_0}}}\displaystyle{1 \over {({{\rm \omega} ^2} - {k^2}{v}_{{\rm th}}^2 - {\rm \omega} _{\rm p}^2)}}\sum\limits_j {\left( {\displaystyle{{{E_{\,j0}}} \over {{\,f_j}}}{e^{ - {r^2}/2r_j^2 f_j^2}} {F_j}({\,f_j})} \right)} $$

where

$${F_j}({\,f_j}) = \displaystyle{r \over {r_j^2 f_j^2}} \cosh \left( {\displaystyle{{{b_j}} \over {{r_j}{\,f_j}}}r} \right) - \displaystyle{{{b_j}} \over {{r_j}{\,f_j}}}\sinh \left( {\displaystyle{{{b_j}} \over {{r_j}{\,f_j}}}r} \right)$$

Using Poisson's equation

$$\nabla {{\bf E}^{\prime}} = - 4{\rm \pi} e{n^{\prime}}$$

$${E^{\prime}} = {E_{{\rm e}.{\rm p}}}{e^{{\rm \iota} ({\rm \omega t} - {\rm kz})}}$$

we get

(29)$${E_{{\rm e}.{\rm p}}} = \displaystyle{\iota \over k}\displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {({{\rm \omega} ^2} - {k^2}{v}_{{\rm th}}^2 - {\rm \omega} _{\rm p}^2)}}\sum\limits_j {\left( {\displaystyle{{{E_{\,j0}}} \over {{\,f_j}}}{e^{ - {r^2}/2r_j^2 f_j^2}} {F_j}({\,f_j})} \right)} $$

Defining normalized power of EPW as

$${\rm \eta} = \displaystyle{{{P_{{\rm e}.{\rm p}}}} \over {{P_1}}}$$

where

$$\eqalign{{P_{{\rm e}.{\rm p}}} = & \displaystyle{{{{v}_{\rm g}}} \over {8{\rm \pi}}} \int_0^\infty {{E_{{\rm e}.{\rm p}}}} E_{{\rm e}.{\rm p}}^{\rm \star} 2{\rm \pi} rdr \cr {P_1} = & \displaystyle{c \over {8{\rm \pi}}} \int_0^\infty {{A_1}} A_1^{\rm \star} 2{\rm \pi} rdr \cr {{v}_{\rm g}} = & {{v}_{{\rm th}}}{\left( {1 - \displaystyle{{{\rm \omega} _{\,p0}^2} \over {{{\rm \omega} ^2}}}} \right)^{1/2}}}$$

we get

(30)$${\rm \eta} = 2\left( {\displaystyle{{{v_{\rm g}}} \over c}} \right){e^{ - b_1^2}} \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {r_1^2 {k^2}{{\rm \omega} ^4}}}\int_0^\infty {\displaystyle{1 \over {D({{\rm \omega} _1},{{\rm \omega} _2})}}} X({E_{10}},{E_{20}}){\kern 1pt} dx$$

where

$$\eqalign{D({{\rm \omega} _1},{{\rm \omega} _2}) & = \left\{ {1 - \displaystyle{{{k^2}v_{{\rm th}}^2} \over {({\rm \omega} _2^2 - {\rm \omega} _1^2)}} - \displaystyle{{{\rm \omega} _{{\rm p}0}^2} \over {{{({{\rm \omega} _2} - {{\rm \omega} _1})}^2}}}} \right. \cr & \qquad\qquad {\left. {{{\left( {1 + \sum\limits_j {\displaystyle{1 \over 2}} \displaystyle{{{{\rm \beta} _j}E_{\,j0}^2} \over {\,f_j^2}} {e^{ - {{({r_1}{\,f_1}/{r_j}{\,f_j})}^2}{x^2}}}{{\cosh} ^2}\left( {{b_j}\displaystyle{{{r_1}{\,f_1}} \over {{r_j}{\,f_j}}}x} \right)} \right)}^{(s/2) - 1}}} \right\}^2} \cr X({E_{10}},{E_{20}}) & = \left[ {\sum\limits_j {\displaystyle{{{E_{\,j0}}} \over {{\,f_j}}}} {e^{ - (1/2){{({r_1}{\,f_1}/{r_j}{\,f_j})}^2}{x^2}}}} \right. \cr & {\left. {\left\{ {{{\left( {\displaystyle{{{r_1}{\,f_1}} \over {{r_{\rm j}}{\,f_{\rm j}}}}} \right)}^2}x{\rm cosh}\left( {{b_j}\displaystyle{{{r_1}{\,f_1}} \over {{r_j}{\,f_j}}}x} \right) - {b_j}\left( {\displaystyle{{{r_1}{\,f_1}} \over {{r_j}{\,f_j}}}} \right)\sinh \left( {{b_j}\displaystyle{{{r_1}{\,f_1}} \over {{r_j}{\,f_j}}}x} \right)} \right\}} \right]^2}} $$

Equation (30) gives the normalized power of EPW generated as a result of beating of the two laser beams and has been solved numerically in conjunction with differential Eqs (22) and (23) for the same set of parameters as in Section 3.

Figures 9 and 10 illustrate the effect of decentered parameter b ₁ of first laser beam on power of generated plasma wave. It is observed that with an increase in value of b ₁ there is decrease in power of generated plasma wave. This is due to the fact that power of plasma wave is decided by the focusing/defocusing behavior of the laser beam with higher intensity and with an increase in decentered parameter of first laser beam there is decrease in extent of self-focusing of second laser beam (i.e., the laser beam with higher initial intensity).

Fig. 9. Variation of normalized power η of beat wave with normalized distance of propagation ξ, keeping (ω_p0r ₁ / c)² = 12, βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₂ = 0.25, collisional parameter s = −3 fixed and at different values of decentered parameter b ₁ = 0, 0.25, 0.50, 0.75.

Fig. 10. Variation of normalized power η of beat wave with normalized distance of propagation ξ, keeping (ω_p0r ₁ / c)² = 12, βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₂ = 0.25, collisional parameter s = −3 fixed and at different values of decentered parameter b ₁ = 1.0, 1.25, 1.50, 1.75.

Figure 11 illustrates the effect of plasma density (ω_p0²r ₁² / c ²) on power of plasma wave. It is observed that with increase in plasma density there is a substantial increase in power of plasma wave. This is due to the fact that increase in plasma density leads to increase in extent of self-focusing of both the laser beams.

Fig. 11. Variation of normalized power η of beat wave with normalized distance of propagation ξ, keeping βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₁ = 0.25, b ₂ = 0.25, collisional parameter s = −3 fixed and at different values of normalized plasma density (ω_p0r ₁ / c)² = 12, 14, 16, 18.

Figure 12 illustrates the effect of nature of collisions on power of plasma wave. It is observed that in case of collisions between electrons and diatomic molecules no plasma wave is generated. This is due to the fact that for s = 2, no self-focusing of the two laser beams takes place and hence there is no density gradient in the plasma.

Fig. 12. Variation of normalized power η of beat wave with normalized distance of propagation ξ, keeping βE ₁₀² = 1.0, βE ₂₀² = 1.50, b ₁ = 0.25, b ₂ = 0.25, (ω_p0r ₁ / c)² = 12 fixed and at different values of collisional parameter s = −3, 0, 2.