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

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

ω2 p

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

> k 4 =4γm<sup>0</sup> 2

2 <sup>ω</sup><sup>2</sup> � <sup>ℏ</sup><sup>2</sup> k4 =4γm<sup>0</sup> <sup>2</sup>ω<sup>2</sup> �<sup>1</sup> (14)

kBTFe=m<sup>0</sup>

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

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

charge density profile for modelling inhomogeneous plasma can be considered as:

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>ω</sup><sup>2</sup> �<sup>1</sup> (13)

neð Þ¼ ξ n0Fð Þ ξ=d (15)

<sup>0</sup>=c is the Rayleigh length, r<sup>0</sup>

<sup>ω</sup><sup>2</sup> <sup>1</sup> � exp � <sup>β</sup>EE<sup>∗</sup>

<sup>∘</sup> <sup>=</sup><sup>2</sup> <sup>1</sup>=<sup>2</sup> Circular Polarization, a <sup>∘</sup> <sup>¼</sup> eE<sup>∘</sup>

Te

, (11)

<sup>ω</sup>mc (12)

p ω<sup>2</sup> þ

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

<sup>ω</sup><sup>2</sup> , <sup>γ</sup><sup>≃</sup> <sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

constant in relativistic cold quantum plasma (CQP) is expressed by:

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

γ ω2 p <sup>ω</sup><sup>2</sup> <sup>1</sup> � <sup>2</sup>k<sup>2</sup>

γ ω2 p <sup>ω</sup><sup>2</sup> <sup>1</sup> � <sup>ℏ</sup><sup>2</sup>

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

where <sup>β</sup> <sup>¼</sup> <sup>e</sup><sup>2</sup>

202 High Power Laser Systems

8mω<sup>2</sup>kB

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

and in the relativistic warm quantum plasma:

2.3. Ramp density profile

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

γ ω2 p

> 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 plasma, including higher-order paraxial theory.

> 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 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> 0 � �ε2ð Þ� <sup>z</sup> <sup>r</sup><sup>4</sup>=r<sup>4</sup> 0 � �ε4ð Þ<sup>z</sup> . The parameters <sup>ε</sup>0ð Þ<sup>z</sup> , <sup>ε</sup>2ð Þ<sup>z</sup> , and ε4ð Þz corresponding to the relativistic nonlinearity are:

$$\varepsilon\_0(z) = 1 - \frac{\omega\_p^2}{\omega^2} \left( 1 + \frac{\Gamma}{f^2(z)} \right)^{-\frac{1}{2}} \left[ 1 - \hbar^2 k^4 \left( 1 + \frac{\Gamma}{f^2(z)} \right)^{-1/2} / 4m\_0^{-2} \omega^2 \right]^{-1} \tag{16}$$

$$\varepsilon\_{2}(\mathbf{z}) = -\frac{\Gamma\left(-2+b^{2}+a\_{2}(\mathbf{z})\right)a\_{p}^{2}}{2f^{4}(\mathbf{z})} \frac{\alpha\_{p}^{2}}{\alpha^{2}} \left(1+\frac{\Gamma}{f^{2}(\mathbf{z})}\right)^{-\frac{3}{7}} \left[1-\hbar^{2}k^{4}\left(1+\frac{\Gamma}{f^{2}(\mathbf{z})}\right)^{-1/2}/4m\_{0}^{2}\alpha^{2}\right]^{-2} \tag{17}$$

$$\begin{split} \varepsilon\_{4}(\mathbf{z}) &= -\frac{\Gamma\left(\beta\_{1}f^{2}(\mathbf{z}) + \Gamma\beta\_{2}/4\right)a\_{p}^{2}}{6f^{8}(\mathbf{z})} \frac{a\_{p}^{2}}{\alpha^{2}} \left(1 + \frac{\Gamma}{f^{2}(\mathbf{z})}\right)^{-\frac{1}{2}} \\ &\times \left[-2\beta\_{1}f^{2}(\mathbf{z}) + 3\Gamma\left(a\_{2}^{2}(\mathbf{z}) - 2a\_{4}(\mathbf{z})\right) + 1 + 4b^{4} + \hbar^{2}k^{4} \left(1 + \frac{\Gamma}{f^{2}(\mathbf{z})}\right)^{-1/2}/4m\_{0}c^{2}\alpha^{2}\right] \end{split} \tag{18}$$

where Γ ¼ αE<sup>0</sup> 2 , <sup>β</sup><sup>1</sup> <sup>¼</sup> <sup>6</sup> � <sup>6</sup>b<sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>4</sup> <sup>þ</sup> <sup>3</sup>a<sup>4</sup> � <sup>6</sup>a<sup>2</sup> <sup>þ</sup> <sup>3</sup>a2b<sup>2</sup> , and <sup>β</sup><sup>2</sup> ¼ �3a<sup>2</sup> <sup>2</sup> <sup>þ</sup> <sup>6</sup>b<sup>2</sup> � <sup>12</sup> <sup>a</sup><sup>2</sup> � <sup>12</sup>b<sup>2</sup> <sup>þ</sup>b<sup>4</sup> <sup>þ</sup> <sup>12</sup>a<sup>4</sup> <sup>þ</sup> 12. By substituting these expressions in Eq. (8), the relativistic self-focusing of ChG laser beam through cold quantum plasma could be investigated. Eqs. (7)–(9) are numerically solved simultaneously using the fourth-order Runge-Kutta method with the initial conditions fð Þ¼ 0 1, f 0 ð Þ¼ 0 0, S<sup>4</sup> <sup>0</sup> ¼ 0, a<sup>2</sup> ¼ 0 at ξ ¼ 0 as well as following the set of parameters: <sup>Γ</sup> <sup>¼</sup> <sup>0</sup>:08, <sup>0</sup>:12, <sup>0</sup>:14, <sup>r</sup><sup>∘</sup> <sup>¼</sup> <sup>20</sup>μm, and <sup>ω</sup><sup>p</sup> <sup>¼</sup> <sup>0</sup>:59ω, <sup>ω</sup> <sup>¼</sup> <sup>1</sup>:<sup>77</sup> � 1016s�<sup>1</sup> which correspond to the gold metallic plasma at room temperature. It is noted that the case of a Gaussian beam b ¼ 0:0 is similar to a dark ring, maximum irradiance on the axis. Therefore, no parts of the beam penetrate beyond the determined depth of penetration. While in the case of a ChG beam, where b 6¼ 0:0 like a bright ring, the irradiance is maximum on a ring and hence the portion of the beam around the bright ring is able to penetrate farther than that of on the axis. As a result, the decentred parameter plays an effective role in improving relativistic self-focusing of highpower Cosh-Gaussian laser beams in quantum plasma.

beam as it is moving through plasma. For more investigation and better comparison, the variation of beam-width parameter with propagation distance for various powers of pump laser beam with b ¼ 0:8 and through plasma density ω<sup>p</sup> ¼ 0:75ω is shown in Figure 3. It is obvious that the laser beam focused substantially when the initial power of the laser beam grew from 0:08 to 0:14. As a result, the intensity of laser also plays an important role in enhancing the focusing of ChG laser beam through the cold dense plasmas along with the

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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 r<sup>4</sup> term in the eikonal function modifies the radial profile. According to equation of RSF in CQP and using cost-effective decentred parameters, decreasing beam-width parameter is observed so that it could produce ultra-high laser irradi-

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

Figure 3. Variation of beam-width parameter f with propagation distance ð Þ ξ for different powers of pump laser beam

ance over distances much greater than the Rayleigh length.

Γ ¼ 0:08, 0:12, 0:14 at plasma density ω<sup>p</sup> ¼ 0:75ω, b ¼ 0:8, and r<sup>0</sup> ¼ 20μm.

decentred parameter.

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 f on ξ in both Gaussian and Cosh-Gaussian of intensity distribution through the Q-plasma for three decentred parameters is shown in Figure 2. As seen from this figure, the focusing term becomes dominant with an increasing value of the decentred parameter b. The self-focusing effect is stronger for a higher decentred parameter at b ¼ 0:8 than Gaussian laser beam b ¼ 0. Consequently, increase of decentred parameters in the ChG laser beams will result in better reduction in the focusing length and more enhancements in localization of non-Gaussian laser

Figure 2. Variation of beam-width parameter f with the normalized propagation distance ð Þ ξ for different values of decentred parameters b ¼ 0:0, 0:4, 0:8 and Γ ¼ 0:12, r<sup>0</sup> ¼ 20μm, ω<sup>p</sup> ¼ 0:59ω.

beam as it is moving through plasma. For more investigation and better comparison, the variation of beam-width parameter with propagation distance for various powers of pump laser beam with b ¼ 0:8 and through plasma density ω<sup>p</sup> ¼ 0:75ω is shown in Figure 3. It is obvious that the laser beam focused substantially when the initial power of the laser beam grew from 0:08 to 0:14. As a result, the intensity of laser also plays an important role in enhancing the focusing of ChG laser beam through the cold dense plasmas along with the decentred parameter.

where Γ ¼ αE<sup>0</sup>

204 High Power Laser Systems

conditions fð Þ¼ 0 1, f

2

0

ð Þ¼ 0 0, S<sup>4</sup>

power Cosh-Gaussian laser beams in quantum plasma.

, <sup>β</sup><sup>1</sup> <sup>¼</sup> <sup>6</sup> � <sup>6</sup>b<sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>4</sup> <sup>þ</sup> <sup>3</sup>a<sup>4</sup> � <sup>6</sup>a<sup>2</sup> <sup>þ</sup> <sup>3</sup>a2b<sup>2</sup>

<sup>þ</sup>b<sup>4</sup> <sup>þ</sup> <sup>12</sup>a<sup>4</sup> <sup>þ</sup> 12. By substituting these expressions in Eq. (8), the relativistic self-focusing of ChG laser beam through cold quantum plasma could be investigated. Eqs. (7)–(9) are numerically solved simultaneously using the fourth-order Runge-Kutta method with the initial

<sup>Γ</sup> <sup>¼</sup> <sup>0</sup>:08, <sup>0</sup>:12, <sup>0</sup>:14, <sup>r</sup><sup>∘</sup> <sup>¼</sup> <sup>20</sup>μm, and <sup>ω</sup><sup>p</sup> <sup>¼</sup> <sup>0</sup>:59ω, <sup>ω</sup> <sup>¼</sup> <sup>1</sup>:<sup>77</sup> � 1016s�<sup>1</sup> which correspond to the gold metallic plasma at room temperature. It is noted that the case of a Gaussian beam b ¼ 0:0 is similar to a dark ring, maximum irradiance on the axis. Therefore, no parts of the beam penetrate beyond the determined depth of penetration. While in the case of a ChG beam, where b 6¼ 0:0 like a bright ring, the irradiance is maximum on a ring and hence the portion of the beam around the bright ring is able to penetrate farther than that of on the axis. As a result, the decentred parameter plays an effective role in improving relativistic self-focusing of high-

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 f on ξ in both Gaussian and Cosh-Gaussian of intensity distribution through the Q-plasma for three decentred parameters is shown in Figure 2. As seen from this figure, the focusing term becomes dominant with an increasing value of the decentred parameter b. The self-focusing effect is stronger for a higher decentred parameter at b ¼ 0:8 than Gaussian laser beam b ¼ 0. Consequently, increase of decentred parameters in the ChG laser beams will result in better reduction in the focusing length and more enhancements in localization of non-Gaussian laser

Figure 2. Variation of beam-width parameter f with the normalized propagation distance ð Þ ξ for different values of

decentred parameters b ¼ 0:0, 0:4, 0:8 and Γ ¼ 0:12, r<sup>0</sup> ¼ 20μm, ω<sup>p</sup> ¼ 0:59ω.

, and <sup>β</sup><sup>2</sup> ¼ �3a<sup>2</sup>

<sup>0</sup> ¼ 0, a<sup>2</sup> ¼ 0 at ξ ¼ 0 as well as following the set of parameters:

<sup>2</sup> <sup>þ</sup> <sup>6</sup>b<sup>2</sup> � <sup>12</sup> <sup>a</sup><sup>2</sup> � <sup>12</sup>b<sup>2</sup>

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 r<sup>4</sup> term in the eikonal function modifies the radial profile. According to equation of RSF in CQP and using cost-effective decentred parameters, decreasing beam-width parameter is observed so that it could produce ultra-high laser irradiance over distances much greater than the Rayleigh length.

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

Figure 3. Variation of beam-width parameter f with propagation distance ð Þ ξ for different powers of pump laser beam Γ ¼ 0:08, 0:12, 0:14 at plasma density ω<sup>p</sup> ¼ 0:75ω, b ¼ 0:8, and r<sup>0</sup> ¼ 20μm.

Figure 4. The effect of the change of plasma density on the relativistic self-focusing process for different densities ω<sup>p</sup> ¼ 0:25ω, 0:50ω, 0:75ω, Γ ¼ 0:14, b ¼ 0:8, and r<sup>0</sup> ¼ 20μm.

approach. For this purpose, Eqs. (7)–(9) should be solved simultaneously again with considering ramp density profile ωpð Þ¼ ξ 0:5Fð Þ ξ ω and Γ ¼ 0:10, 0:12, 0:14. Figure 5 provides information about four normalized upward density profiles with various slopes.

Figure 5. The normalized linear density profiles with different slopes: <sup>n</sup>1=n<sup>0</sup> <sup>≈</sup> <sup>1</sup> <sup>þ</sup> <sup>6</sup>:<sup>746</sup> � 102 tan ð Þ <sup>ξ</sup>=<sup>3</sup> (profile#1), <sup>n</sup>1=n<sup>0</sup> <sup>≈</sup> <sup>1</sup> <sup>þ</sup> <sup>33</sup>:<sup>531</sup> � <sup>10</sup><sup>2</sup> tan ð Þ <sup>ξ</sup>=<sup>5</sup> (profile#2), <sup>n</sup>1=n<sup>0</sup> <sup>≈</sup> <sup>1</sup> <sup>þ</sup> <sup>40</sup>:<sup>178</sup> � 102 tan ð Þ <sup>ξ</sup>=<sup>3</sup> (profile#3), <sup>n</sup>1=n<sup>0</sup> <sup>≈</sup> <sup>1</sup> <sup>þ</sup> <sup>20</sup>:<sup>079</sup> � 102 tan ð Þ <sup>ξ</sup> (profile #4).

The linear density profile function Fð Þ¼ ξ 1 þ ð Þ n1=n<sup>0</sup> tan ð Þ ξ=d for axial inhomogeneity has been chosen. Based on this mathematical function, the slope of ramp density profile is adjustable using appropriate d and n<sup>1</sup> so that the range of plasma density has been adjusted from nc=4 to nc=3, nc=2, 3nc=4, and nc. The profile number of each ramp density function has been

Figure 7. Variation of beam-width parameter through ICQP in the presence of ramp and uniform density profiles for

Figure 6. Variation of beam-width parameter through ICQP in the presence of ramp and uniform density profiles for

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b ¼ 0:2.

b ¼ 0:4.

Figure 6. Variation of beam-width parameter through ICQP in the presence of ramp and uniform density profiles for b ¼ 0:2.

approach. For this purpose, Eqs. (7)–(9) should be solved simultaneously again with considering ramp density profile ωpð Þ¼ ξ 0:5Fð Þ ξ ω and Γ ¼ 0:10, 0:12, 0:14. Figure 5 provides informa-

Figure 5. The normalized linear density profiles with different slopes: <sup>n</sup>1=n<sup>0</sup> <sup>≈</sup> <sup>1</sup> <sup>þ</sup> <sup>6</sup>:<sup>746</sup> � 102 tan ð Þ <sup>ξ</sup>=<sup>3</sup> (profile#1), <sup>n</sup>1=n<sup>0</sup> <sup>≈</sup> <sup>1</sup> <sup>þ</sup> <sup>33</sup>:<sup>531</sup> � <sup>10</sup><sup>2</sup> tan ð Þ <sup>ξ</sup>=<sup>5</sup> (profile#2), <sup>n</sup>1=n<sup>0</sup> <sup>≈</sup> <sup>1</sup> <sup>þ</sup> <sup>40</sup>:<sup>178</sup> � 102 tan ð Þ <sup>ξ</sup>=<sup>3</sup> (profile#3), <sup>n</sup>1=n<sup>0</sup> <sup>≈</sup> <sup>1</sup> <sup>þ</sup> <sup>20</sup>:<sup>079</sup> � 102 tan ð Þ <sup>ξ</sup>

Figure 4. The effect of the change of plasma density on the relativistic self-focusing process for different densities

tion about four normalized upward density profiles with various slopes.

ω<sup>p</sup> ¼ 0:25ω, 0:50ω, 0:75ω, Γ ¼ 0:14, b ¼ 0:8, and r<sup>0</sup> ¼ 20μm.

206 High Power Laser Systems

(profile #4).

Figure 7. Variation of beam-width parameter through ICQP in the presence of ramp and uniform density profiles for b ¼ 0:4.

The linear density profile function Fð Þ¼ ξ 1 þ ð Þ n1=n<sup>0</sup> tan ð Þ ξ=d for axial inhomogeneity has been chosen. Based on this mathematical function, the slope of ramp density profile is adjustable using appropriate d and n<sup>1</sup> so that the range of plasma density has been adjusted from nc=4 to nc=3, nc=2, 3nc=4, and nc. The profile number of each ramp density function has been located in the top left-hand corner as a legend in Figure 5. Figures 6–9 illustrate the results of a numerical solution carried out on the self-focusing equation to assess what behaves like a highintense ChG laser beam through inhomogeneous cold quantum plasma (ICQP).

Figure 8. Variation of beam-width parameter through ICQP in the presence of ramp and uniform density profiles for b ¼ 0:6.

Figure 9. Variation of beam-width parameter through ICQP in the presence of ramp and uniform density profiles for b ¼ 0:8.

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 <sup>ξ</sup> <sup>¼</sup> <sup>1</sup>:<sup>5</sup> � <sup>10</sup>�<sup>3</sup>

Figure 11. Variation of beam-width parameter through ICQP in the presence of ramp density profile #4, b ¼ 0:8, and

Figure 10. Variation of beam-width parameter through ICQP in the presence of ramp density profile #4 and different

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decentred parameters: b ¼ 0:0 for Gaussian beam and b ¼ 0:2, 0:4, 0:6, 0:8 for ChG beam.

different intensities: Γ ¼ 0:10, 0:12, 0:14.

beam-width parameter decreased significantly when the higher-slope ramp density profile was used. Generally, a comparison among Figures 6–9 reveals that there was a substantial change from homogeneous to inhomogeneous quantum plasma by using various decentred parameters.

, the

located in the top left-hand corner as a legend in Figure 5. Figures 6–9 illustrate the results of a numerical solution carried out on the self-focusing equation to assess what behaves like a high-

Figure 8. Variation of beam-width parameter through ICQP in the presence of ramp and uniform density profiles for

Figure 9. Variation of beam-width parameter through ICQP in the presence of ramp and uniform density profiles for

b ¼ 0:6.

208 High Power Laser Systems

b ¼ 0:8.

intense ChG laser beam through inhomogeneous cold quantum plasma (ICQP).

Figure 10. Variation of beam-width parameter through ICQP in the presence of ramp density profile #4 and different decentred parameters: b ¼ 0:0 for Gaussian beam and b ¼ 0:2, 0:4, 0:6, 0:8 for ChG beam.

Figure 11. Variation of beam-width parameter through ICQP in the presence of ramp density profile #4, b ¼ 0:8, and different intensities: Γ ¼ 0:10, 0:12, 0:14.

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 <sup>ξ</sup> <sup>¼</sup> <sup>1</sup>:<sup>5</sup> � <sup>10</sup>�<sup>3</sup> , the beam-width parameter decreased significantly when the higher-slope ramp density profile was used. Generally, a comparison among Figures 6–9 reveals that there was a substantial change from homogeneous to inhomogeneous quantum plasma by using various decentred parameters. 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 <sup>b</sup> from zero (Gaussian profile) to 0:8 in the <sup>ξ</sup> <sup>¼</sup> <sup>1</sup>:<sup>5</sup> � <sup>10</sup>�<sup>3</sup> dropped significantly.

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For more details and better comparison among trends at b ¼ 0:8, Figure 11 shows different intensities. It is obvious that the laser beam becomes self-focused when the initial power of the laser beam increases from 0:10 to 0:14. A significant enhancement in laser self-focusing in underdense plasma with a localized plasma density ramp is observed. It is clear from this figure and the earlier ones that in addition to the type of density profile and decentred parameter, the intensity of laser also plays an effective role in enhancing the focusing of ChG laser beam through the inhomogeneous cold dense plasmas.
