## Abstract

One of the most significant factors in understanding the behaviour of laser beams during the propagation process through plasma is how the spot size of the beam changes with both laser and plasma parameters. Self-focusing and defocusing of the laser beam, therefore, play an important role in this study. This chapter is aimed at presenting a brief investigation into these common phenomena in laser-plasma interaction. In addition to a short overview on the types of self-focusing of laser beams through plasma, a detailed study on the relativistic self-focusing of high-intense laser beam in quantum plasma as a particular nonlinear medium is performed. In this case, the essential equations to model this phenomenon are derived; furthermore, a range of values of laser-plasma parameters which would satisfy these equations is considered.

### Keywords

- self-focusing of laser beam
- laser-plasma interaction

## 1. Introduction

A self-focusing phenomenon is considered as one of the major self-action effects in laser-plasma 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

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:

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].

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.

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.

According to the self-focusing of laser beam through plasma, the irradiance of a focused pulse laser with a range of

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 high-intense laser beams. The next sections deal with consideration of relativistic self-focusing in classical and quantum plasma for Gaussian and Cosh-Gaussian laser beams.

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

### 2.1. High-order paraxial theory

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

For more details, the interaction of an intense laser beam with particular plasma is considered. From Maxwell’s equations, it is noted that the propagation of such a beam can be investigated by solving the scalar wave equation in the cylindrical coordinate system and along the axis

where

The complex amplitude of the electric vector

From Eq. (4), it is noticed that the envelope

where

In the near-axis approximation (i.e.

Furthermore, in the higher-order paraxial theory, the dielectric constant is expanded to the next higher power in

where

### 2.2. Importance of nonlinearity in self-focusing equation

As mentioned in the previous section, the refractive index of plasma, the second term in the right-hand side of the self-focusing equation, Eq.(8), depends on nonlinearity mechanisms. Therefore, this term plays an important role in investigation propagation laser beam through a nonlinear medium with a wide variety of nonlinearities. For example, in collisional nonlinearity condition:

where

or in the ponderomotive regime and in low-intensity laser, it should be expressed like,

where

In addition, in the relativistic regime and high-intense laser-plasma interaction:

where

and in the relativistic warm quantum plasma:

### 2.3. Ramp density profile

Another important parameter in solving the self-focusing equation is plasma density profile. From mathematical and practical perspectives, it can be considered as a uniform or non-uniform function of propagation distance. In inhomogeneous plasmas [12], the propagation of a Gaussian high-intense laser beam in under-dense plasma with an upward increasing density ramp has been investigated. In this chapter, the effect of electron density profile on spot size oscillations of laser beam has been also shown. It leads to further fluctuations in the figure for the spot size of laser beam compared. Therefore, it was confirmed that an improved electron density gradient profile is an important factor in having a good stationary and non-stationary self-focusing in laser-plasma interaction. A mathematical function of non-uniform charge density profile for modelling inhomogeneous plasma can be considered as:

where

## 3. Relativistic self-focusing of ChG laser beam in quantum plasma

Theoretical investigations of quantum effects on propagation of Gaussian laser beams are carried out within the framework of quantum approach in dense plasmas [37, 38, 39, 40]. Habibi et al. have also shown the effective role of Fermi temperature in improving relativistic self-focusing of short wavelength laser beam (X-ray) through warm quantum plasmas [26]. From a theoretical viewpoint, the relativistic effect would be effective as a result of increasing fermions’ number density in dense degenerate plasmas. However, several recent technologies have made it possible to produce plasmas with densities close to solid state. Furthermore, considerable interest has recently been raised in production and propagation of a decentred Gaussian beam on account of their higher efficient power with a flat-top beam shape compared with that of a Gaussian laser beam and their attractive applications in complex optical systems. Generally, focusing of the ChG beam can be analysed like Gaussian beam in plasmas without considering quantum effects. In particular, the present section is devoted to study nonlinear propagation of a ChG laser beam in quantum plasma, including higher-order paraxial theory.

The figure for a ChG laser beam makes a substantial contribution with an even stronger self-focusing effect compared with that of a Gaussian laser beam in cold quantum plasma (CQP). In this chapter, the plasma dielectric function, Eq. (13), which is in the relativistic regime, is considered for unmagnetized and collision-less CQP. Then, it is expanded to the next higher power in

where

Therefore, a comparative analysis can be done among various decentred parameters so as to determine its role in relativistic self-focusing of laser beam. At the first step, the variation of

Therefore, if laser intensity increases, a beam with more relativistic electrons will travel with the laser beam and generate a higher current and consequently a very high quasi-stationery magnetic field. Consequently, while the pinching effect of the magnetic field is becoming stronger, focusing effect will become much more important. Figure 4 illustrates the effects of changing plasma density on the relativistic self-focusing process for a given initial intensity of the laser beam. As seen from the results in the Figure 4, the focusing of laser beam increases while respective focusing length decreases with increasing the slope of ramp density profile.

It is clear that the inclusion of the quadratic

On the other hand, we know that an upward ramp density profile as transition density gives rise more reduction in amplitude of the laser spot size close to the propagation axis [12, 25]. Therefore, we show an analysis of joint relativistic and quantum effects on ChG laser beams in one-species axial inhomogeneous cold quantum plasma (ICQP), using the high-order paraxial approach. For this purpose, Eqs. (7)–(9) should be solved simultaneously again with considering ramp density profile

The linear density profile function

In all figures, it is clear that the self-focusing becomes stronger when the slope of ramp density profile increases. Furthermore, it is noticeable that from these figures, for

In addition, the influence of decentred parameter on the propagation of ChG laser beam in Figure 10 is illustrated for the improved density profile (profile #4). Clearly, the final values of spot size for such a laser with

For more details and better comparison among trends at

## 4. Conclusion

Generally, we aimed to review our results about self-focusing of high-intense laser beams through various plasmas such as cold and warm quantum plasmas. We, in addition, have introduced a ramp density profile for plasma as an effective factor in improving self-focusing and even ion acceleration. The propagation of cosh-Gaussian intensity profile for high-power and short-wavelength laser was also investigated by means of higher-order paraxial approximation. From these investigations, the effects of ramp density profile and off-axial contribution on enhancing the further focusing of laser beam can be drawn. The results show that inclusion of the quadratic