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

### 2.1. High-order paraxial theory

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

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

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

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.

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

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

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

becomes increasingly distorted, as depicted in Figure 1.

to nonlinear mechanisms that they are listed here:

increase in refractive index and further heating [3].

1. Thermal self-focusing (TS)

198 High Power Laser Systems

2. Ponderomotive self-focusing (PS)

3. Relativistic self-focusing (RS)

for more details on the topic.

much longer than the Rayleigh length.

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 to the fourth power in r makes significant difference in studying laser beam propagating through plasma and even other nonlinear materials.

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 z:

$$\frac{\partial^2}{\partial z^2}E(r,z,t) + \nabla\_\perp^2 E(r,z,t) - \frac{\varepsilon}{c^2} \frac{\partial^2}{\partial t^2} E(r,z,t) = 0\tag{1}$$

b is called the decentred parameter as well as r<sup>0</sup> introduced the initial spot size of the laser beam. Two functions <sup>a</sup>2ð Þ<sup>z</sup> and <sup>a</sup>4ð Þ<sup>z</sup> , the coefficients of <sup>r</sup><sup>2</sup> and <sup>r</sup>4, respectively, are considered as

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

ε2ð Þz , and ε4ð Þz corresponding to the nonlinearity play an important role in investigating the selffocusing of laser beam through a nonlinear medium. By substituting both Eqs. (5) and (6) into Eq. (3) and equating the coefficients of r0, r2, and r<sup>4</sup> on both sides of the resulting equation, the

differential equations governing the parameters a2, a4, f zð Þ, and S4ð Þz can be given by:

0 f 2

<sup>2</sup> <sup>þ</sup> ð Þ <sup>4</sup> � <sup>8</sup>a<sup>2</sup> <sup>=</sup><sup>3</sup> <sup>þ</sup> <sup>2</sup>b<sup>6</sup> =15 b<sup>2</sup> ε0f

0

propagation distance and an original beam-width parameter, respectively. The first term on the right-hand side of Eq. (8) represents the diffraction effect, while the second term plays a vital role in self-focusing of laser beam. Both of them are nonlinear and responsible for the defocusing and focusing of the ChG laser beam through plasma. The behaviour of laser beam propagating through plasma can be investigated by solving these equations, Eqs. (7)–(9), with the initial

derivative is removed from the wave equation using the independent variable transformation (Z ¼ z, T ¼ t � z=vg) where vg is group velocity. We have presented non-stationary self-focusing of high-intense Gaussian laser beams for different portions of a pulse in classical and quantum plasmas in the weakly relativistic as well as ponderomotive regimes [23, 24]. We have recorded a

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

> p ω<sup>2</sup> þ

ω2 p ω2 αEE<sup>∗</sup> =2Te

<sup>1</sup> <sup>þ</sup> <sup>α</sup>EE<sup>∗</sup>

=2Te

, (10)

very significant focusing near the peak of the pulse and the rear portion of the pulse.

<sup>ε</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup>

 ε0f

0

, a<sup>4</sup> <sup>¼</sup> <sup>3</sup>a<sup>2</sup>

=3 � 2a<sup>2</sup>

<sup>3</sup> � <sup>ε</sup><sup>2</sup>

<sup>ε</sup>2ð Þ� <sup>z</sup> <sup>r</sup><sup>4</sup>=r<sup>4</sup>

<sup>2</sup> � 4a<sup>2</sup>

<sup>6</sup> � <sup>ε</sup><sup>4</sup>

ε0

2ε<sup>0</sup>

<sup>z</sup> and <sup>r</sup> <sup>¼</sup> <sup>r</sup>0ω=<sup>c</sup> are considered as a dimensionless

<sup>0</sup> ¼ 0 and a<sup>2</sup> ¼ 0 at ξ ¼ 0. In non-stationary situations, the time

<sup>r</sup><sup>2</sup><sup>f</sup> � <sup>1</sup> 2ε<sup>0</sup> dε<sup>0</sup> dξ df

> <sup>r</sup><sup>2</sup> � <sup>S</sup><sup>4</sup> 0 2ε<sup>0</sup> dε<sup>0</sup> <sup>d</sup><sup>ξ</sup> � <sup>4</sup>S<sup>4</sup>

0

<sup>ε</sup>4ð Þ<sup>z</sup> . The parameters <sup>ε</sup>0ð Þ<sup>z</sup> ,

<sup>d</sup><sup>ξ</sup> (8)

0 f df <sup>d</sup><sup>ξ</sup> (9)

=4 (7)

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an indicative of the departure of the beam from the Gaussian nature.

da<sup>2</sup>

<sup>d</sup><sup>ξ</sup> ¼ �16S<sup>4</sup>

<sup>2</sup> � <sup>8</sup>a<sup>2</sup> <sup>þ</sup> <sup>4</sup> � <sup>b</sup><sup>2</sup> <sup>4</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup>

a2

higher power in <sup>r</sup><sup>2</sup> to obtain <sup>ε</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup> <sup>ε</sup>0ð Þ� <sup>z</sup> <sup>r</sup><sup>2</sup>=r<sup>2</sup>

d2 f <sup>d</sup>ξ<sup>2</sup> <sup>¼</sup> <sup>3</sup>a<sup>2</sup>

conditions f ¼ 1, df =dξ ¼ 0, S<sup>4</sup>

<sup>2</sup> � <sup>52</sup>a<sup>3</sup>

8ε0f

<sup>2</sup> þ 32a<sup>2</sup>

<sup>6</sup> þ

<sup>0</sup> <sup>¼</sup> <sup>S</sup>4ω=c. Additionally, <sup>ξ</sup> <sup>¼</sup> <sup>c</sup>=ωr<sup>2</sup>

2.2. Importance of nonlinearity in self-focusing equation

dS<sup>4</sup> 0 <sup>d</sup><sup>ξ</sup> <sup>¼</sup> <sup>40</sup>a<sup>2</sup>

where S<sup>4</sup>

condition:

where <sup>α</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup>=3δmω<sup>2</sup>kB

where E shows amplitude of the electric field, c is the speed of light in vacuum and ε is the dielectric constant of plasma. In this stage, we consider the solution of the Eq. (1):

$$E(r,z,t) = A(r,z,t) \exp\left[\mathrm{i}\left(\omega t - \int\_0^z k(z)dz\right)\right] \tag{2}$$

E rð Þ ; <sup>z</sup>; <sup>t</sup> can be substituted with Eq. (2) in Eq. (1). This substitution leads to neglecting <sup>∂</sup><sup>2</sup> A=∂z<sup>2</sup> and ∂<sup>2</sup> A=∂t <sup>2</sup> on the assumption that A is a slowly varying function of z and t is compared with ω:

$$-2ik\frac{\partial A}{\partial z} - iA\frac{\partial k}{\partial z} - k^2 A + \frac{\partial^2 A}{\partial r^2} + \frac{1}{r}\frac{\partial A}{\partial r} + \frac{\omega^2}{c^2}\varepsilon A = 0. \tag{3}$$

The complex amplitude of the electric vector A rð Þ ; z; t is expressed as,

$$A(r,z,t) = A\_0(r,z,t) \exp\left[-ik(z)S(r,z)\right] \tag{4}$$

From Eq. (4), it is noticed that the envelope A rð Þ ; z; t has been separated into real amplitude and complex phase terms in which the eikonal function S rð Þ ; z is.

$$\mathcal{S}(r,z) = \mathcal{S}\_0(z) + \left(r^2/r\_0^2\right)\mathcal{S}\_2(z) + \left(r^4/r\_0^4\right)\mathcal{S}\_4(z) \tag{5}$$

where S0ð Þz is the axial phase shift, S2ð Þ¼ z ð Þ df zð Þ=dz =2f zð Þ is indicative of the spherical curvature of the wave front, and S4ð Þz corresponds to its departure from the spherical nature. Moreover, the beam irradiance A∘ð Þ r; z; t of Cosh-Gaussian (ChG) laser beam can be written as:

$$\begin{split} A\_0^2(r, z, t) = EE^\* &= \frac{E\_0^2}{4f^2(z)} \exp\left(\frac{b^2}{2}\right) \left(1 + \frac{r^2}{r\_0^2 f^2(z)} a\_2(z) + \frac{r^4}{r\_0^4 f^4(z)} a\_4(z)\right) \\ &\times \left(\exp\left[-\left(\frac{r}{t\eta(z)} + \frac{b}{2}\right)^2\right] + \exp\left[-\left(\frac{r}{t\eta(z)} - \frac{b}{2}\right)^2\right]\right)^2 \mathbf{g}(t) \end{split} \tag{6}$$

In the near-axis approximation (i.e. a2, a<sup>4</sup> ! 0), Eq. (6) converts to a general solution of ChG laser beam. In Eq. (6), the initial laser intensity at the central position r ¼ z ¼ 0 is presented by E0, and the beam-width parameter characterized by f zð Þ depends on z. Temporal shape of the pulse can be considered as a step function, g tðÞ¼ 1 at ti0 and g tðÞ¼ 0 otherwise. In addition, b is called the decentred parameter as well as r<sup>0</sup> introduced the initial spot size of the laser beam. Two functions <sup>a</sup>2ð Þ<sup>z</sup> and <sup>a</sup>4ð Þ<sup>z</sup> , the coefficients of <sup>r</sup><sup>2</sup> and <sup>r</sup>4, respectively, are considered as an indicative of the departure of the beam from the Gaussian nature.

Furthermore, in the higher-order paraxial theory, the dielectric constant is expanded to the next higher power in <sup>r</sup><sup>2</sup> to obtain <sup>ε</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup> <sup>ε</sup>0ð Þ� <sup>z</sup> <sup>r</sup><sup>2</sup>=r<sup>2</sup> 0 <sup>ε</sup>2ð Þ� <sup>z</sup> <sup>r</sup><sup>4</sup>=r<sup>4</sup> 0 <sup>ε</sup>4ð Þ<sup>z</sup> . The parameters <sup>ε</sup>0ð Þ<sup>z</sup> , ε2ð Þz , and ε4ð Þz corresponding to the nonlinearity play an important role in investigating the selffocusing of laser beam through a nonlinear medium. By substituting both Eqs. (5) and (6) into Eq. (3) and equating the coefficients of r0, r2, and r<sup>4</sup> on both sides of the resulting equation, the differential equations governing the parameters a2, a4, f zð Þ, and S4ð Þz can be given by:

$$\frac{da\_2}{d\xi} = -16S\_4/f^2, \ a\_4 = \left(3a\_2^2 - 4a\_2\right)/4\tag{7}$$

$$\frac{d^2f}{d\xi^2} = \frac{\left\{ \left(3a\_2^2 - 8a\_2 + 4\right) - b^2 \left(4 + b^2/3 - 2a\_2\right) \right\}}{\varepsilon\_0 f^3} - \frac{\varepsilon\_2}{\varepsilon\_0} \rho^2 f - \frac{1}{2\varepsilon\_0} \frac{d\varepsilon\_0}{d\xi} \frac{df}{d\xi} \tag{8}$$

$$\frac{dS\_4^{'}}{d\xi} = \frac{40a\_2^2 - 52a\_2^3 + 32a\_2}{8\varepsilon\_0 f^6} + \frac{\left(a\_2^2 + (4 - 8a\_2)/3 + 2b^6/15\right)b^2}{\varepsilon\_0 f^6} - \frac{\varepsilon\_4}{2\varepsilon\_0}\rho^2 - \frac{S\_4^{'}}{2\varepsilon\_0}\frac{d\varepsilon\_0}{d\xi} - \frac{4S\_4^{'}}{f}\frac{df}{d\xi} \tag{9}$$

where S<sup>4</sup> <sup>0</sup> <sup>¼</sup> <sup>S</sup>4ω=c. Additionally, <sup>ξ</sup> <sup>¼</sup> <sup>c</sup>=ωr<sup>2</sup> 0 <sup>z</sup> and <sup>r</sup> <sup>¼</sup> <sup>r</sup>0ω=<sup>c</sup> are considered as a dimensionless propagation distance and an original beam-width parameter, respectively. The first term on the right-hand side of Eq. (8) represents the diffraction effect, while the second term plays a vital role in self-focusing of laser beam. Both of them are nonlinear and responsible for the defocusing and focusing of the ChG laser beam through plasma. The behaviour of laser beam propagating through plasma can be investigated by solving these equations, Eqs. (7)–(9), with the initial conditions f ¼ 1, df =dξ ¼ 0, S<sup>4</sup> <sup>0</sup> ¼ 0 and a<sup>2</sup> ¼ 0 at ξ ¼ 0. In non-stationary situations, the time derivative is removed from the wave equation using the independent variable transformation (Z ¼ z, T ¼ t � z=vg) where vg is group velocity. We have presented non-stationary self-focusing of high-intense Gaussian laser beams for different portions of a pulse in classical and quantum plasmas in the weakly relativistic as well as ponderomotive regimes [23, 24]. We have recorded a very significant focusing near the peak of the pulse and the rear portion of the pulse.

#### 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:

$$\varepsilon(r,z) = 1 - \frac{\omega\_p^2}{\omega^2} + \frac{\omega\_p^2}{\omega^2} \frac{a \to E^\*/2T\_e}{1 + a \to E^\*/2T\_e} \tag{10}$$

where <sup>α</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup>=3δmω<sup>2</sup>kB

to the fourth power in r makes significant difference in studying laser beam propagating

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 z:

<sup>⊥</sup>E rð Þ� ; z; t

dielectric constant of plasma. In this stage, we consider the solution of the Eq. (1):

E rð Þ¼ ; z; t A rð Þ ; z; t exp i ωt �

E rð Þ ; <sup>z</sup>; <sup>t</sup> can be substituted with Eq. (2) in Eq. (1). This substitution leads to neglecting <sup>∂</sup><sup>2</sup>

From Eq. (4), it is noticed that the envelope A rð Þ ; z; t has been separated into real amplitude

where S0ð Þz is the axial phase shift, S2ð Þ¼ z ð Þ df zð Þ=dz =2f zð Þ is indicative of the spherical curvature of the wave front, and S4ð Þz corresponds to its departure from the spherical nature. Moreover, the beam irradiance A∘ð Þ r; z; t of Cosh-Gaussian (ChG) laser beam can be written as:

1 þ

<sup>r</sup>0f zð Þ <sup>þ</sup> <sup>b</sup> 2 � �<sup>2</sup> � �

In the near-axis approximation (i.e. a2, a<sup>4</sup> ! 0), Eq. (6) converts to a general solution of ChG laser beam. In Eq. (6), the initial laser intensity at the central position r ¼ z ¼ 0 is presented by E0, and the beam-width parameter characterized by f zð Þ depends on z. Temporal shape of the pulse can be considered as a step function, g tðÞ¼ 1 at ti0 and g tðÞ¼ 0 otherwise. In addition,

r2 r2 0 f 2 ð Þz

� �<sup>2</sup> � � � � <sup>2</sup>

2 =r 2 0 � �S2ð Þþ <sup>z</sup> <sup>r</sup>

b2 2 � �

where E shows amplitude of the electric field, c is the speed of light in vacuum and ε is the

ε c2 ∂2 ∂t

0 @

<sup>2</sup> on the assumption that A is a slowly varying function of z and t is compared with ω:

2 4 ðz

k zð Þdz

A rð Þ¼ ; z; t A0ð Þ r; z; t exp ½ � �ik zð ÞS rð Þ ; z (4)

4 =r 4 0

a2ð Þþ z

<sup>þ</sup> exp � <sup>r</sup>

!

r4 r4 0 f 4 ð Þz a4ð Þz

<sup>r</sup>0f zð Þ � <sup>b</sup> 2

1 A

3

0

<sup>2</sup> E rð Þ¼ ; z; t 0 (1)

5 (2)

<sup>c</sup><sup>2</sup> <sup>ε</sup><sup>A</sup> <sup>¼</sup> <sup>0</sup>: (3)

� �S4ð Þ<sup>z</sup> (5)

gð Þt

(6)

A=∂z<sup>2</sup>

through plasma and even other nonlinear materials.

∂2

�2ik <sup>∂</sup><sup>A</sup>

<sup>∂</sup><sup>z</sup> � iA <sup>∂</sup><sup>k</sup> ∂z � k 2 A þ ∂2 A ∂r<sup>2</sup> þ 1 r ∂A ∂r þ ω2

The complex amplitude of the electric vector A rð Þ ; z; t is expressed as,

and complex phase terms in which the eikonal function S rð Þ ; z is.

S rð Þ¼ ; z S0ð Þþ z r

0 4f 2 ð Þz exp

� exp � <sup>r</sup>

and ∂<sup>2</sup>

A=∂t

200 High Power Laser Systems

A2

<sup>0</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup>; <sup>t</sup> EE<sup>∗</sup> <sup>¼</sup> <sup>E</sup><sup>2</sup>

<sup>∂</sup>z<sup>2</sup> E rð Þþ ; <sup>z</sup>; <sup>t</sup> <sup>∇</sup><sup>2</sup>

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

$$\varepsilon(r,z) = 1 - \frac{\omega\_p^2}{\omega^2} + \frac{\omega\_p^2}{\omega^2} \left[ 1 - \exp\left(-\frac{\beta E E^\*}{T\_e}\right) \right],\tag{11}$$

plasma density transitions (the so-called ramp density profile), which is usable to investigate several laser-plasma mechanisms such as self-focusing of laser beam, electron acceleration and high harmonic generation. This mathematical model of density transition has been also supported by a wide range of numerical and experimental work [29–35]. For example, Chunyang et al. [36] observed the high harmonics in the reflection spectra from short-intense

laser-pulse interaction with over-dense plasmas the particle-in-cell (PIC) simulations.

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

plasma, including higher-order paraxial theory.

power in <sup>r</sup><sup>4</sup> to obtain <sup>ε</sup>ð Þ¼ <sup>r</sup>; <sup>z</sup> <sup>ε</sup>0ð Þ� <sup>z</sup> <sup>r</sup><sup>2</sup>=r<sup>2</sup>

<sup>ε</sup>0ð Þ¼ <sup>z</sup> <sup>1</sup> � <sup>ω</sup><sup>2</sup>

<sup>Γ</sup> �<sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>a</sup>2ð Þ<sup>z</sup> � � 2f 4 ð Þz

> Γ β1f 2 ð Þþ <sup>z</sup> <sup>Γ</sup>β2=<sup>4</sup> � � 6f 8 ð Þz

> > 2

ð Þþ <sup>z</sup> <sup>3</sup><sup>Γ</sup> <sup>a</sup><sup>2</sup>

� �2β1f

ε2ð Þ¼� z

ε4ð Þ¼� z

and ε4ð Þz corresponding to the relativistic nonlinearity are:

Γ f 2 ð Þz

!�<sup>1</sup>

ω2 p <sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup>

> ω2 p <sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup>

p <sup>ω</sup><sup>2</sup> <sup>1</sup> <sup>þ</sup>

Theoretical investigations of quantum effects on propagation of Gaussian laser beams are carried out within the framework of quantum approach in dense plasmas [37–40]. 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

The figure for a ChG laser beam makes a substantial contribution with an even stronger selffocusing 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

� �ε2ð Þ� <sup>z</sup> <sup>r</sup><sup>4</sup>=r<sup>4</sup>

<sup>1</sup> � <sup>ℏ</sup><sup>2</sup> k <sup>4</sup> <sup>1</sup> <sup>þ</sup>

2

<sup>2</sup>ð Þ� <sup>z</sup> <sup>2</sup>a4ð Þ<sup>z</sup> � � <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>4</sup>b<sup>4</sup> <sup>þ</sup> <sup>ℏ</sup><sup>2</sup>k<sup>4</sup> <sup>1</sup> <sup>þ</sup> <sup>Γ</sup>

� �

2 4

2

<sup>1</sup> � <sup>ℏ</sup><sup>2</sup> k <sup>4</sup> <sup>1</sup> <sup>þ</sup>

0

Γ f 2 ð Þz

!�1=<sup>2</sup>

� �ε4ð Þ<sup>z</sup> . The parameters <sup>ε</sup>0ð Þ<sup>z</sup> , <sup>ε</sup>2ð Þ<sup>z</sup> ,

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3 5 �1

=4m<sup>0</sup> 2 ω2

=4m<sup>0</sup> <sup>2</sup>ω<sup>2</sup> 3 5 �2 (16)

(17)

(18)

=4m<sup>0</sup> 2 ω2

Γ f 2 ð Þz

f <sup>2</sup>ð Þ<sup>z</sup> � ��1=<sup>2</sup>

!�1=<sup>2</sup>

0

2

2 4

Γ f 2 ð Þz

> Γ f 2 ð Þz

!�<sup>5</sup>

!�<sup>3</sup>

where <sup>β</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup> 8mω<sup>2</sup>kB

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

$$\varepsilon(r,z) = 1 - \frac{1}{\mathcal{Y}} \frac{a\_p^2}{a^2}, \mathcal{Y} \simeq \left(1 + a\_\*^2/2\right)^{1/2} \\ \text{Circular Polarization, } a\_\* = \frac{\varepsilon E\_\*}{amc} \tag{12}$$

where γ is the Lorentz factor which arises from the quiver motion of the electron in the laser field. The expressions earlier are just three forms of nonlinearities in plasma. In this case, we have investigated relativistic self-focusing of high-intense laser beam in cold and warm quantum plasma [25–28]. From a quantum-mechanical viewpoint, the de Broglie wavelength of the charge particle is comparable to the inter-particle distance. In this situation, the dielectric constant in relativistic cold quantum plasma (CQP) is expressed by:

$$\varepsilon(r,z) = 1 - \frac{1}{\gamma} \frac{\omega\_p^2}{\alpha^2} \left(1 - \hbar^2 k^4 / 4\gamma m\_0^{-2} \alpha^2\right)^{-1} \tag{13}$$

and in the relativistic warm quantum plasma:

$$\varepsilon(r,z) = 1 - \frac{1}{\gamma} \frac{a\_p^2}{a^2} \left( 1 - 2k^2 k\_B T\_{\rm Fe} / m\_0^2 \omega^2 - \hbar^2 k^4 / 4 \gamma m\_0^2 \omega^2 \right)^{-1} \tag{14}$$

#### 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 nonuniform 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 nonstationary self-focusing in laser-plasma interaction. A mathematical function of non-uniform charge density profile for modelling inhomogeneous plasma can be considered as:

$$n\_e(\xi) = n\_0 F(\xi/d) \tag{15}$$

where <sup>ξ</sup> <sup>¼</sup> <sup>z</sup>=Rd is a dimensionless propagation distance, Rd <sup>¼</sup> <sup>ω</sup>r<sup>2</sup> <sup>0</sup>=c is the Rayleigh length, r<sup>0</sup> is the focused laser beam radius, n<sup>0</sup> is the density of the plasma at ξ ¼ 0, and Fð Þ¼ ξ 1þ ð Þ n1=n<sup>0</sup> tan ð Þ ξ=d is the linear density profile function. The slope of ramp density profile can be determined by changing d and n<sup>1</sup> parameters. This model of density is just one kind of plasma density transitions (the so-called ramp density profile), which is usable to investigate several laser-plasma mechanisms such as self-focusing of laser beam, electron acceleration and high harmonic generation. This mathematical model of density transition has been also supported by a wide range of numerical and experimental work [29–35]. For example, Chunyang et al. [36] observed the high harmonics in the reflection spectra from short-intense laser-pulse interaction with over-dense plasmas the particle-in-cell (PIC) simulations.
