The interaction of intense electromagnetic waves with plasma has been of considerable interest due to its significance in number of applications such as plasma based beat-wave accelerators,^[1] plasma-based accelerators,^[2] inertial confinement fusion,^[3–4] ionospheric modification,^[5–7] laser charge particle accelerator,^[8–9] and X-ray lasers.^[10] The success of these applications depends on propagation of laser over several Rayleigh lengths while maintaining the efficient interaction with plasma. Pulse focusing and pulse compression have been proved to be an effective way of guiding a laser pulse in the medium over many Rayleigh lengths and to increase radiation power and intensity while retaining its energy. Therefore the dynamics of self-focusing and self-compression of laser beam in plasma have been studied both theoretically and experimentally by many investigators extensively since long and have been reported in a number of investigations.^[11–16] The effect of self-compression mechanism on the self-focusing of a Gaussian laser beam in an unmagnetized plasma has already been investigated by Bokaei and Niknam.^[17] They have shown in their work that in the presence of ponderomotive nonlinearity the self-focusing mechanism is obstructed by the self-compression one. Shibu et al.^[14] have investigated the self-compression of Gaussian laser beam due to the relativistic mass nonlinearity and have observed that the self-focusing mechanism interferes strongly with the non-linear self-compression process. Recently Bokaei et al.^[18] have studied the effect of external magnetic field and plasma inhomogeneity on simultaneous self-focusing and self-compression of a Gaussian laser beam through the plasma. Their results showed that the simultaneous use of both external magnetic field and density ramp leads to generate highly focused and compressed pulses.

Most of the theoretical study on self-focusing and self-compression is devoted to cylindrically symmetric Gaussian laser beams.^[19–20] Few studies of self-focusing have been reported on elliptical Gaussian beam,^[21–23] cosh Gaussian beams,^[24] Hermite Gaussian beams,^[25] Hermite-Cosh-Gaussian beams^[26] and super Gaussian beams.^[27] From our studies, we have observed that, presently, there is an increase in interest in exploring a new technique of combining multiple beams to achieve high power densities at the target.^[28–31] However combining identical four beams is mathematically simpler than combining two beams. This is because the intensity distribution of the beam formed by combining four identical beams is symmetrical in x- and y-directions, hence only one beam width parameter is required to describe the whole beam dynamics.

In the present investigation, we have focused on the self-focusing of a quadruple Gaussian laser beam comprising four coherent Gaussian laser beams propagating along the z-direction, but having intensity maxima in the x-y plane at (−x_o,0), (x_o,0), (0,−x_o), (0,x_o), in a collisionless magnetized plasma. However, in most plasma fusion experiments, the externally applied or self generated magnetic fields play an important role in laser beam propagation.^[32–33] Therefore, it is justified to investigate the propagation of the quadruple Gaussian laser beam through plasma in the presence of a static magnetic field. The self-focusing mechanism in the presence of self-compression mechanism will be analyzed in contrast to the case in the absence of it. On a short time scale (t < τ_e) where τ_e is the energy relaxation time in a collisionless plasma, the nonlinearity due to ponderomotive force is dominant. This ponderomotive force of the laser causes perturbations in the electron density of plasma, which consequently leads to more refractive index gradient. Therefore, the electron density perturbation due to ponderomotive force is the main source of nonlinearity which modifies the plasma refractive index and makes it intensity dependent. The longitudinal refractive index gradient causes self-compression and transverse refractive index gradient leads to self-focusing of a laser beam. When the magnetic field is applied along the direction of propagation of beam, there are two modes of propagation, namely extraordinary and ordinary mode. Therefore, the redistribution of carrier is affected by the change in the strength of the static magnetic field. In weak coupling limit, we can consider the propagation to be in either of the two modes. In Sec. 2, authors have given brief description of the intensity profile of quadruple Gaussian laser beam in paraxial ray approximation. In Sec. 3, we have set up and solved the wave equation for the quadruple Gaussian beam. The effect of self-compression on self-focusing of the beam has also been studied in the same section. In Sec. 4, the authors have studied the self trapped mode and in the last section, discussion of the present investigation is presented.

Consider the propagation of a linearly polarized quadruple Gaussian laser beam of angular frequency ω_o along the z-direction in a collisionless magnetoplasma. The external magnetic field B_o is applied along the direction of propagation of wave. In linear approximation, the propagation of laser beam can be assumed in either of the two modes of propagation namely ordinary and extraordinary. The electric field vector of the laser beam may be expressed as

The nonlinear wave equation governing the evolution of the electric field in magnetoplasma is

The initial condition d²f_±/dζ² = 0, for a plane wave front at ζ = 0, f_± = 1, df_±/dζ = 0, and g_± = 1, dg_±/dζ = 0 leads to the propagation of the quadruple Gaussian beam in the uniform waveguide/self trapped mode. By putting d²f_±/dζ² = 0, in Eq. (15), we obtain a relation between equilibrium beam radius ρ_o = r_oω_p/c and critical intensity of the beam Π_o = . After simplification the expression can be written as

It is observed from Fig. 1 (for extraordinary mode), that there is a decrease in equilibrium beam radius with increase in . After attaining minimum, beam radius stabilizes at lower values of magnetic field i.e. at ω_c/ω_o = 0.2. For every point (Π_o, ρ_o) lying below the curve (d²f_±/dζ² > 0), corresponds to defocusing while for every point (Π_o, ρ_o) lying above the curve (d²f_±/dζ² < 0), corresponds to self-focusing of the laser beam. The beam radius is lesser at higher x_o and thus results in better stabilization. Increasing magnetic field disrupts this stabilization and there is monotonic increase in beam radius at lesser x_o.

Fig. 1

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	Fig. 1 (Color online) Square of equilibrium beam radius plotted against axial beam irradiance = for different combinations of xo/ro and ωc/ωo with ωp/ωo = 0.1 for extraordinary mode.

Equations (15) and (16) are nonlinear ordinary differential equations governing the behavior of the dimensionless beam width parameters f_± and g_± respectively as a function of normalized distance of propagation ζ. The first term on the right hand side of Eq. (15) has its origin in the Laplacian () appearing in Eq. (7). The other term arises due to ponderomotive self-focusing and depends on an intensity parameter (), relative separation x_o/r_o and magnetic field ω_c/ω_o. The diffractional term leads to diffractional divergence of the beam, while the nonlinear term is responsible for self-focusing of the laser beam. The divergence and convergence of the beam depend on whether first term of Eq. (15) (self-focusing equation) dominates the second or vice versa. Equation (16) is known as the self-compression equation. The first term on the right hand side of this equation represents the dispersion broadening; while the second term is responsible for the pulse compression due to ponderomotive effects. Since analytical solutions of these equations are not possible, so we have solved these coupled equations numerically by using Runge-Kutta method with following set of parameters; initial intensity I = 1.22 × 10¹⁵ W/cm², initial beam width r_o = 1 μm, initial pulse duration τ_o = 20 fs, laser frequency ω_o = 2 × 10¹⁵ rad/sec, plasma frequency ω_p = 2 × 10¹⁴ rad/sec, electron temperature T = 1 keV, and external magnetic field B_o = 56.8 MG. By choosing suitable laser and plasma parameters, one can study the focusing/defocusing of laser beam in plasma.

Figures 2 plot the beam width parameters f_± and g_± with normalized distance of propagation ζ at different values of normalized intensity parameter a = = 0.06, 0.07, 0.08, and 0.09. It can be observed from the figure that by increasing the initial laser intensity the self-focusing and self-compression lengths decrease. Also the strength of the self-focusing and self-compression processes improve as the intensity parameter increases up to 0.08.

Fig. 2

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	Fig. 2 (Color online) Beam width parameters f+ and g+ plotted against the dimensionless distance of propagation ζ for different values of intensity parameter a at ωc/ωo = 0.5, ωp/ωo = 0.1, xo/ro = 0.6, ro = 10 μm and τo = 20 fs. (a) Variation of f+ in the absence of self-compression effect; (b) Variation of g+.

The effect of ponderomotive nonlinearity on simultaneous focusing and compression of the pulse is presented in Fig. 3. The variation of beam width parameter f_± with ζ is presented with and without coupling the self-focusing equation with the self-compression equation at different values of intensity parameter a. One can see that with increasing the intensity parameter from 0.07 to 0.09, the self-compression mechanism starts obstructing the pulse self-focusing. At a = 0.07, the self-focusing mechanism is obstructed by the self-compression process between ζ = 2 and 3 where the beam is compressed second time in the self-compression curve.

Fig. 3

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	Fig. 3 (Color online) Beam width parameter f+ plotted against the dimensionless distance of propagation ζ by solving self-focusing equation with and without coupling with self-compression equation. The other parameters are the same as that of Fig. 2. (a) At a = 0.06; (b) At a = 0.07; (c) At a = 0.08; (d) At a = 0.09.

The same trend is observed for a = 0.08. But as the intensity parameter reaches to 0.09, the self-compression mechanism completely obstructs the pulse self-focusing. The physical mechanism behind this phenomenon can be explained as follows: by increasing the initial laser intensity, the ponderomotive force increases and pushes the electrons outward from the central region of the beam. Therefore, after a certain laser intensity value (lower-limit intensity), the refractive index gradient, produced via ponderomotive force becomes strong enough to focus the beam. By further increasing the initial laser intensity, the ponderomotive force becomes stronger and causes the complete expulsion of plasma electrons. As a result, the electron free channel is created in a certain region, which is responsible for defocusing of the laser beam. Hence, after a certain laser intensity threshold (upper-limit), the beam starts to defocus.^[37] By compressing the laser pulse, the laser intensity goes up and reaches the upper-limit, as a result the beam diverges.

However, Fig. 4 shows the effect of external magnetic field on beam width parameters f_± and g_±. It is clear from the figure that the beam width parameters decreases, self-focusing (in Figs. 4(a) and 4(b)) and self-compression (in Fig. 4(c)) becomes stronger with the increase of magnetic field ω_c/ω_o from 0.1 to 0.5 (for extraordinary mode), at a fixed value of a = 0.06, x_o/r_o = 0.6, and ω_p/ω_o = 0.1. This is due to the fact that the magnetic field modifies the refractive index of plasma to intensify the nonlinear effects. Hence the non-linear terms start dominating over the diffractional divergence and dispersion broadening terms, cause the laser beam to get more focussed and compressed.

Fig. 4

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	Fig. 4 (Color online) Beam width parameters f+ and g+ plotted against the dimensionless distance of propagation ζ for different values of magnetic field parameter ωc/ωo, at a = 0.06. The other parameters are the same as that of Fig. 2. (a) Variation of f+ in the absence of self-compression effect; (b) Variation of f+ in the presence of self-compression effect; (c) Variation of g+.

The effect of lateral separation of the beam component is studied in Fig. 5. The variation of beam width parameters f_± and g_± are plotted as a function of normalized distance of propagation ζ, at a fixed value of a = 0.06, ω_c/ω_o = 0.5, ω_p/ω_o = 0.1 and for different values of x_o/r_o = 0.3, 0.6 for extraordinary mode. It is clear from the figures that self-focusing and self-compression get enhanced with an increase in the value of lateral separation parameter x_o/r_o. This is because, the lateral separation parameter x_o/r_o effects the initial intensity distribution of the beam to make it flatten. As a result, both the diffraction divergence of the beam and the ponderomotive force reduce. However, the net effect is to increase the nonlinear term to enhance the focusing of the beam. Further lateral separation of the beam is suitably chosen so that laser intensity be significant enough for ponderomotive force being effective.

Fig. 5

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	Fig. 5 (Color online) Beam width parameters f+ and g+ plotted against ζ for different values of lateral separation parameter xo/ro (a) in the absence of self-compression effect; (b) in the presence of self-compression effect. The other parameters are the same as that of Fig. 2 with a = 0.06.

Comparison of Fig. 5(a) (where the compression parameter is not taken into consideration) and Fig. 5(b) (considering the compression parameter) shows that the self-compression mechanism hinders the self-focusing mechanism of the beam. Also the more compressed beam (corresponding to x_o/r_o = 0.6) defocuses the beam to a greater extent. That is why the defocusing of the beam at x_o/r_o = 0.6 is more than at x_o/r_o = 0.3 in Fig. 5(b) where the compression mechanism due to ponderomotive nonlinearity is taken into account.

Figure 6 presents the spatiotemporal profile of the quadruple Gaussian laser field intensity with dimensionless parameters ρ = rω_p/c and τ₁ = τω_p by considering ponderomotive nonlinearity at positions where the beam is either focussed or compressed (i.e. minima of the self-focusing and self-compression diagrams in Fig. 5(b) i.e. at ζ = 0, 0.35, 0.96, 1.2, and 1.5. Figure 6(a) shows the normalized intensity of the quadruple Gaussian laser pulse at ζ = 0. Figures 6(b), 6(c), and 6(e) depict the spatiotemporal profile of normalized intensity of the pulse after transverse focusing at ζ = 0.35, 0.96, and 1.5 respectively. The figures show how the pulse focuses more and more and its intensity increases as it is focussed second and third time in the self-focusing curve. Figure 6(d) shows the initial normalized intensity of the pulse after compression in a plasma at ζ = 1.2. This figure clearly confirms the occurrence of pulse compression or pulse shortening, in addition to an increase in beam intensity as compared to Fig. 6(a).

Fig. 6

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	Fig. 6 (Color online) The spatiotemporal evolution of the normalized laser intensity at different positions ζ, at a = 0.06 and other parameters are the same as that of Fig. 2. (a) At ζ = 0; (b) At ζ = 0.35; (c) At ζ = 0.96; (d) At ζ = 1.2; (e) At ζ = 1.5.

In the last, Fig. 7 is presented to show the effect of pulse duration on the self-focusing and compression of the beam in the magnetized plasma. It can be clearly observed from Fig. 7(a) that with an increase in initial pulse duration the strength of convergence of beam decreases and there is a significant enhancement in the extent of divergence of the beam. Also, the extent of compression of the beam increases with the increase of pulse duration (Fig. 7(b)).

Fig. 7

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	Fig. 7 (Color online) Beam width parameters f+ and g+ plotted against the dimensionless distance of propagation ζ for different values of initial pulse duration τo at a = 0.06. The other parameters are the same as that of Fig. 2. (a) Variation of f+; (b) Variation of g+.

We have studied the propagation of a quadruple Gaussian laser beam in collisionless magnetized plasma with two normal modes of propagation i.e. extraordinary and ordinary mode. Variation of beam width parameter with normalized distance of propagation has been evaluated and numerically simulated over wide range of initial intensity, magnetic field and lateral separation parameter. The self-focusing mechanism is studied in the presence of self-compression mechanism. It can be observed that, in case of ponderomotive nonlinearity, the pulse self-focusing is obstructed by the self-compression mechanism. From numerical simulation, it is found that the magnetic field is found to have a profound effect on the overall propagation laser beam through plasma and explicit enhancement in self-focusing of extraordinary mode. In the last, we also have plotted the 3-D graphs showing spatiotemporal evolution of a quadruple Gaussian laser pulses in magnetized plasma. Regarding applications, the results of the present investigation may be relevant to the applications where multi-beam mega joule laser is required, especially in laser induced fusion, which further helps in various applications of space like laser propulsion, laser energy network in space, energy supply to the ground energy system etc.