1. Introduction

A self-focusing phenomenon is considered as one of the major self-action effects in laserplasma interaction [1, 2]. During this process, the laser beams are able to modify their front medium by means of a nonlinear response of plasma so as to make it more suitable for propagation. In this situation, the refractive index of plasma could be expressed as n ¼ n<sup>0</sup> þ n2I rð Þ in which n<sup>0</sup> presents a linear term, n<sup>2</sup> is considered as an optical constant characterizing the strength of optical nonlinearity, and I rð Þ determines the beam intensity distribution along the radial coordinate r. In other words, when an intense laser beam with an intensity gradient along its cross-section propagates through plasma, the self-focusing and the

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

self-defocusing of laser beam occur frequently. In this stage, if the electric field is strong enough, the laser beam will create a dielectric waveguide in the path ahead. This typical waveguide results in reducing or entirely eliminating the divergence of the beam. From an optical perspective, the refractive index of the medium in such situations acts as a convex lens; consequently, the central part of the laser beam would move slower than the edge parts. Therefore, while the beam is propagated through the nonlinear medium, its wave front becomes increasingly distorted, as depicted in Figure 1.

According to the self-focusing of laser beam through plasma, the irradiance of a focused pulse laser with a range of 1015 1016W=cm2, for instance, can reach approximately 1018W=cm2. Under this high intensity, the relativistic motion of the electrons may be expected. This mechanism is of great importance because it is able to produce ultra-high laser irradiances (10<sup>19</sup> 1020W=cm2) over distances much greater than the Rayleigh length [5]. Sun et al. [6], Barnes et al. [7], Kurki-Suonio et al. [8], and Borisov et al. [9] investigated both relativistic mass variation and charge displacement due to the ponderomotive force by solving the propagation within the envelope approximation and fluid equations. With advancement in laser technology using the chirped-pulse amplification technique [10, 11], the propagation of laser beam in inhomogeneous plasmas has been also studied [12, 13]. Furthermore, guiding nonlinear relativistic laser pulses in preformed plasma with density

Self-Focusing of High-Power Laser Beam through Plasma

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The simplest wave equation describing self-focusing is a prototype of an important class of nonlinear partial differential equations in physics, such as the Landau Ginsberg equation for the macroscopic wave function of type II superconductors or the Schrodinger's equation for a particle with self-interactions [18]. In this situation, there are several approximate analytical approaches to analyse the effect of self-focusing, namely paraxial ray approximation, moment theory approach, variation approach, and source-dependent expansion method. However, each of these theories has limitations in describing completely the experimental and computer

Our discussion begins with a review of the equations which are normally utilized to describe self-focusing in Wentzel-Kramers-Brillouin (WKB) and paraxial ray approximation (Section 2), followed by an outline of the ramp-density profile and its impact on self-focusing of highintense laser beams. The next sections deal with consideration of relativistic self-focusing in

It is reasonable to assume that the paraxial wave equation presents an accurate description for laser beams propagating near the axis throughout the propagation. Akhmanove et al. [19] illustrated that in a limit when the eikonal term is expanded only up to the second power in r, the shape of the radial intensity profile remains unchanged. However, in the experimental situation with high-intense laser beams, one needs to go beyond the paraxial approximation for which the predictions of such an approximation are often not sufficiently accurate [20]. Thus, it would be interesting, on high-intense laser-plasma interaction, to investigate propagation of laser beams using the extended paraxial approximation. In this case, Liu [21] and Tripathi [22] reported a useful theoretical framework that accounts for the combined several effects of interaction of an intense short pulse laser with plasma, the laser frequency blue shift, self-defocusing, ring formation and self-phase modulation. The expansion of the eikonal term

classical and quantum plasma for Gaussian and Cosh-Gaussian laser beams.

2. Self-focusing equation for high-intense laser-plasma interaction

transition has recently become a subject of great interests [14–17].

simulation results.

2.1. High-order paraxial theory

Overall, the generation of self-focusing phenomenon could be connected with various physical causes. The basic physical mechanism which is responsible for self-focusing of laser beam is the nonlinearity of the medium which originates in its interaction with the laser field. Therefore, the self-focusing of laser beam through plasma is categorized into three options according to nonlinear mechanisms that they are listed here:

1. Thermal self-focusing (TS)

This effect is due to collisional heating of plasma exposed to electromagnetic radiation. In fact, the rise in temperature induces the hydrodynamic expansion, which leads to an increase in refractive index and further heating [3].

2. Ponderomotive self-focusing (PS)

A nonlinear radial ponderomotive force of the focused laser beam pushes electrons out of the propagation axis. It expels the plasma from the beam centre, high-intensity region, and increases the plasma dielectric function, leading to self-focusing of the laser in plasmas.

3. Relativistic self-focusing (RS)

The increase of electrons' mass traveling by velocity approaching the speed of light modifies the plasma refractive index. This phenomenon has been observed in several experiments and has been proved to be an efficient way to guide a laser pulse over distances much longer than the Rayleigh length.

R.W. Boyd et al. [4] reviewed the self-focusing methods, which are recommended by the authors for more details on the topic.

Figure 1. A schematic showing distortion of the wave-front and self-focusing of a laser beam in plasma.

According to the self-focusing of laser beam through plasma, the irradiance of a focused pulse laser with a range of 1015 1016W=cm2, for instance, can reach approximately 1018W=cm2. Under this high intensity, the relativistic motion of the electrons may be expected. This mechanism is of great importance because it is able to produce ultra-high laser irradiances (10<sup>19</sup> 1020W=cm2) over distances much greater than the Rayleigh length [5]. Sun et al. [6], Barnes et al. [7], Kurki-Suonio et al. [8], and Borisov et al. [9] investigated both relativistic mass variation and charge displacement due to the ponderomotive force by solving the propagation within the envelope approximation and fluid equations. With advancement in laser technology using the chirped-pulse amplification technique [10, 11], the propagation of laser beam in inhomogeneous plasmas has been also studied [12, 13]. Furthermore, guiding nonlinear relativistic laser pulses in preformed plasma with density transition has recently become a subject of great interests [14–17].

The simplest wave equation describing self-focusing is a prototype of an important class of nonlinear partial differential equations in physics, such as the Landau Ginsberg equation for the macroscopic wave function of type II superconductors or the Schrodinger's equation for a particle with self-interactions [18]. In this situation, there are several approximate analytical approaches to analyse the effect of self-focusing, namely paraxial ray approximation, moment theory approach, variation approach, and source-dependent expansion method. However, each of these theories has limitations in describing completely the experimental and computer simulation results.

Our discussion begins with a review of the equations which are normally utilized to describe self-focusing in Wentzel-Kramers-Brillouin (WKB) and paraxial ray approximation (Section 2), followed by an outline of the ramp-density profile and its impact on self-focusing of highintense laser beams. The next sections deal with consideration of relativistic self-focusing in classical and quantum plasma for Gaussian and Cosh-Gaussian laser beams.
