Introduction
Self-focusing generally refers to a nonlinear phenomenon which is caused by the incident high-intensity laser under the action of an electric field, as the intensity of the light increases, and the refractive index of the plasma increases at the same time, thus causing the outer profile of the laser beam to deflect toward the center. Previous researches have shown many effects in self-focusing (Gao and Shim, Reference Gao and Shim2019; Kovalev and Bychenkov, Reference Kovalev and Bychenkov2019; Shao et al., Reference Shao, Zeng, Lin, Wu and Zhang2019) such as the coupling efficiency (Kodama et al., Reference Kodama, Norreys, Mima, Dangor, Evans, Fujita, Kitagawa, Krushelnick, Miyakoshi, Miyanaga, Norimatsu, Rose, Shozaki, Shigemori, Sunahara, Tampo, Tanaka, Toyama, Yamanaka and Zepf2001), fluid instability (Srinivasan et al., Reference Srinivasan, Cagas, Masti, Rathod, Shetty and Song2019), and other nonlinear processes. Self-focusing (Simmons and Godwin, Reference Simmons and Godwin1983) is considered as one of the most important issues in the total designing, engineering advancement such as direct-drive laser fusion (Watkins and Kingham, Reference Watkins and Kingham2018; Gopalaswamy et al., Reference Gopalaswamy, Betti, Knauer, Luciani, Patel, Woo, Bose, Igumenshchev, Campbell, Anderson, Bauer, Bonino, Cao, Christopherson, Collins, Collins, Davies, Delettrez, Edgell, Epstein, Forrest, Froula, Glebov, Goncharov, Harding, Hu, Jacobs-Perkins, Janezic, Kelly, Mannion, Maximov, Marshall, Michel, Miller, Morse, Palastro, Peebles, Radha, Regan, Sampat, Sangster, Sefkow, Seka, Shah, Shmyada, Shvydky, Stoeckl, Solodov, Theobald, Zuegel, Johnson, Petrasso, Li and Frenje2019), the research of which is of vital significance both in theory and practical applications.
To date, there have been four types of formation mechanisms (Sharma et al., Reference Sharma, Prakash and Verma2003) of self-focusing from discovery and which have been confirmed: ponderomotive self-focusing (Brandi et al., Reference Brandi, Manus and Mainfray1993; Aggarwal et al., Reference Aggarwal, Vij and Kant2015b; Patil et al., Reference Patil, Chikode and Takale2018; Rawat and Purohit, Reference Rawat and Purohit2019), thermal self-focusing (Craxton and McCrory, Reference Craxton and McCrory1984), resonance self-focusing (Joshi et al., Reference Joshi, Clayton and Chen1982; Gill and Saini, Reference Gill and Saini2007; Zare et al., Reference Zare, Rezaee, Yazdani, Anvari and Sadighi-Bonabi2015), and relativistic self-focusing (Sprangle et al., Reference Sprangle, Tang and Esarey1987; Hora et al., Reference Hora, Hoelss, Scheid, Wang, Ho, Osman and Castillo2000). Different formation mechanisms correspond to different laser–plasma interaction (LPI) systems, and a large number of experimental studies have shown that the mechanism of self-focusing often appears two or more and does not appear alone (Lu et al., Reference Lu, Huang, Zhou, Mori and Katsouleas2006). The different approaches to analyze the contributions of self-focusing have been reported such as paraxial-ray theory (Lam et al., Reference Lam, Lippmann and Tappert1977), variational approach (Wilson et al., Reference Wilson, Li, Weng, Chen, McKenna and Sheng2019), and source-dependent expansion method (Malekshahi et al., Reference Malekshahi, Dorranian and Askari2014). Alexopoulos and Uslenghi (Reference Alexopoulos and Uslenghi1981) obtained the second-order Wentzel–Kramers–Brillouin (WKB) solutions for the TE and TM waves, the method of which then has been put forward in the research developments from Bud'ko and Liberman (Reference Bud'ko and Liberman1992). Based on the expansion of the eikonal and nonlinear constant, the ray theory could expand the distance from the axis to the beam up to square term. Then, Gill et al. (Reference Gill, Kaur and Mahajan2010) developed the ray theory to the expansion of the distance up to the higher-order term.
The formation plasma of intense laser and matters interaction, the density of plasmas mainly exhibits several forms such as the homogeneous manners (Fuchs et al., Reference Fuchs, Labaune, Depierreux and Tikhonchuk2000; Sen et al., Reference Sen, Rathore, Varshney and Varshney2010; Kant et al., Reference Kant, Wani and Kumar2012), inhomogeneous manners (Gahn et al., Reference Gahn, Tsakiris, Pukhov, Meyer-ter-Vehn, Pretzler, Thirolf, Habs and Witte1999), and even exponential manners (Kemp et al., Reference Kemp, Sentoku and Tabak2008). Applied paraxial-ray theory, Wang et al. (Reference Wang, Liang, Yao, Yuan and Zhou2019) studied the nonlinear propagation characteristics of a Gaussian beam in collisionless plasma by the high-order paraxial-ray theory. Pathak et al. (Reference Pathak, Vieira, Silva and Nam2018) applied the paraxial axis approximation to study the laser dynamics in a laterally inhomogeneous plasma and its correlation with the wakefield acceleration and it was found that the nonuniformity of the plasma can lead to a stronger self-focusing. Kaur et al. (Reference Kaur, Kaur, Kaur and Gill2017) studied the self-focusing and defocusing of the Hamiltonian hyperbolic cosine-type Gaussian laser beam (HChG) in an inhomogeneity corrugated density plasma. Tikhonchuk et al. (Reference Tikhonchuk, Hüller and Mounaix1997) studied the effect of laser light self-focusing in speckles on stimulated Brillouin scattering (SBS) in an inhomogeneous plasma. Bornatici and Maj (Reference Bornatici and Maj2003) put forward the various methods for the description of paraxial wave beams propagating in weakly inhomogeneous media. Varshney et al. (Reference Varshney, Qureshi and Varshney2006) investigated the relativistic self-focusing of intense laser radiation in an axially inhomogeneous plasma. In the above studies, as the significant nonlinear effect, self-focusing must be taken into consideration.
According to the research results by Zhang et al. (Reference Zhang, Xiao, Wang, Feng, Liu and Zheng2017) and Trtica and Gaković (Reference Trtica and Gaković2003), they have found that there are such plasmas similar to the plasma with periodical density ripple. Ripples could be produced by different mechanisms, the following references could be declared, Liu and Tripathi (Reference Liu and Tripathi1995) proposed a mechanism for short-wavelength electromagnetic wave generation in a periodic dielectric material. For a moderately energetic electron beam, when passing through a periodic dielectric, then electromagnetic wave which will get amplified along the propagation, then the periodic density ripple would be formed. Another mechanism has been noted by Liu and Tripathi (Reference Liu and Tripathi2008) which noted that impinged on a gas jet target, the intense machining laser beam causes space periodic ionization of the gas and heats the electrons, the inhomogeneous plasma pressure leads to the redistribution of atomic density, when the intense leaser pulse propagates after the certain time delay, there could produce the plasma density ripple. Thakur and Kant (Reference Thakur and Kant2018) have presented stronger self-focusing of chirped pulse laser with exponential plasma density ramp profile in cold quantum magnetoplasma, and they observed the low value of the beam width parameter in the focal region. What is more, the significant contribution of the cold quantum magnetoplasma with the optimized value of magnetic field lead to enhanced self-focusing. Also, the intensity of the laser beam imparts a chief role in achieving the stronger self-focusing. Thakur et al. (Reference Thakur, Wani and Kant2019) have revealed an exploration of self-focusing of Hermite–cosine–Gaussian laser beam in a collisionless plasma under relativistic nonlinearity. Wani and Kant (Reference Wani and Kant2016) have studied nonlinear propagation of Gaussian laser beam in an inhomogeneous plasma under plasma density ramp and derived the differential equation for beam width parameter by the parabolic wave equation approach under paraxial approximation. Aggarwal et al. (Reference Aggarwal, Vij and Kant2015a) have investigated the propagation of circularly polarized quadruple Gaussian laser beam in underdense magnetoplasma. Their results revealed that the propagation of the quadruple laser beam can be studied in three different regimes, that is, steady divergence, oscillatory divergence, and self-focusing regime depending on the initial point. For an initial point not lying on the critical curve, the beam width parameter will either increase or decrease. The beam is more focused at lower intensity in both cases, namely extraordinary and ordinary mode.
In this paper, we have used high-order paraxial-ray theory and WKB approximation to study the relativistic laser self-focusing in plasmas with periodic density ripple, revealing some properties of self-focusing.
The theory of laser beam propagation in plasma
In the process of the interaction between laser and plasma, the wave equation that determines the laser propagation in the plasma is:
$$\nabla ^2\vec{E}+ \displaystyle{{{\rm \omega }^ 2} \over {c^2}}{\rm \varepsilon }\vec{E} = 0$$where 
$\vec{E}$ is the electric field of incident laser which may be the function of position and time, c refers to the light speed, ɛ is the effective dielectric constant of the plasma, ω is the relativistic electron plasma frequency in the absence of electromagnetic beam and wave frequency.
For Gaussian laser beam, when the laser propagates along the axis in the z-direction, according to the wave equation, at the initial position 
$z{\rm} = 0$, the laser intensity distribution can be expressed as 
${\rm EE^\ast } = E_ 0^ 2 {\exp}\lpar -r^ 2/r_ 0^ 2 \rpar$, where r 0 is the width of the initial laser beam, r is the radial component in the cylindrical coordinate system, E 0 is the axial amplitude of the beam.
When the laser intensity reaches to 1019 W/cm2, the relativistic nonlinear effect becomes very obvious at this time. When the value is lower than 1019 W/cm2, there also exists the nonlinear effect, such as the electron mass begins to change significantly (Umstadter, Reference Umstadter2003), changes in electronic mass will form the difference between and the rest mass of electron, and which further leads to the form of relativistic nonlinear effect. Under the effect of relativistic effect, the dielectric constant of the plasma (Walia et al., Reference Walia, Tripathi and Tyagi2017) can be expressed as follows:
$${\rm \varepsilon } = {\rm \varepsilon }_0{\rm \varphi }\lpar {\rm EE}^\ast \rpar $$where ɛ0 is the linear part and φ(EE*) is the nonlinear part, the expression of ɛ0 is 
${\rm \varepsilon }_ 0 = 1-\lpar {\rm \omega }_{{\rm p} 0}^ 2 /{\rm \omega }^ 2 \rpar \exp \lpar -{\rm \beta }E_{ 00}^ 2 /f^ 2\rpar$, and 
${\rm \varphi }\lpar {{\rm EE}^\ast} \rpar = \lsqb 1-N_{\rm e}/ \lpar N_{ 0{\rm e}}{\rm \gamma} \rpar \rsqb \lpar {\rm \omega }_{{\rm p} 0}^ 2 /{\rm \omega }^ 2\rpar$, ωp0 and ω are the relativistic electron plasma frequency in the absence of electromagnetic beam and wave frequency. Here, 
${\rm \omega }_{{\rm p} 0}^ 2 = 4{\rm \pi }N_{ 0{\rm e}}e^ 2/ \lpar m_ 0{\rm \gamma }\rpar$ and 
${\rm \beta} = e^ 2 / \lpar {8m{\rm \omega }^ 2k_{\rm B}T} \rpar$, where e is the charge of an electron, m 0 is the rest mass, N 0e is the initial entity of plasma electrons, and k B is the Boltzmann constant, respectively.
Under the effect of intense laser, the relationship between the relativistic factor and light intensity has the following relationship:
$${\rm \gamma } = \left[{1 + \displaystyle{{e^2} \over {c^2m_0^2 {\rm \omega }^2}}{\rm EE}^\ast } \right]^{1/2}$$ According to the previous research results (Xia and Lin, Reference Xia and Lin2012), under the influence of relativistic ponderomotive force, there is such a relationship between the plasma density and the plasma frequency along the axial direction in the process of laser and plasma interaction: 
$N_{\rm e} / N_{{\rm e} 0}= \lpar {{\rm \omega }_{\rm p} / {\rm \omega }_{{\rm p} 0}} \rpar ^2$, where N e is the density of electron plasma under the influence of intense laser, 
${\rm \omega }_{\rm p} = \sqrt {N_{\rm e}e^ 2 / \lpar {\rm \varepsilon }_ 0m_{\rm e}}\rpar$ is the plasma frequency in the presence of electromagnetic beam, which is related to the initial dielectric constant ɛ0 and the electron mass m e in it, under the influence of the intense laser, the ωp is relativistic. ωp0 is the plasma frequency at 
$z{\rm} = 0$.
According to the discovery by Zhang et al. (Reference Zhang, Xiao, Wang, Feng, Liu and Zheng2017) and Trtica and Gaković (Reference Trtica and Gaković2003) above, the plasmas with periodical density ripple have been found. So, in this paper, based on their results, the plasma density with periodical distribution is assumed as sinusoidal and cosine profile:
$$N_{\rm e}/N_{{\rm e}0} = \lpar {\rm \omega }_{\rm p}/{\rm \omega }_{{\rm p}0}\rpar ^2 = C_1 + D_1\sin \lpar F_1{\rm \zeta }\rpar $$and
$$N_{\rm e}/N_{{\rm e}0} = \lpar \omega _{\rm p}/\omega _{{\rm p}0}\rpar ^2 = C_2 + D_2\cos \lpar F_2 {\rm \zeta} \rpar $$where C 1 and C 2 refer to the initial density without the influence of periodical density ripple, D 1 and D 2 refer to 2, and F 1 and F 2 refer to the frequency of periodic variation in density.
According to the wave equation [see Eq. (1)] in the plasma, the electric field change along the direction is assumed as:
$$\vec{E} = \vec{A}\lpar x\comma \;y\comma \;z\rpar \exp \lsqb i\lpar {\rm \omega }t-k_0z\rpar \rsqb $$ Substituting the value of 
$\overrightarrow {E\;}$ to Eq. (1), one can get the result:
$$-k_0^2 -2ik_0A + \left({\displaystyle{{\partial^2} \over {\partial z^2}} + \displaystyle{1 \over r}\displaystyle{\partial \over {\partial z}}} \right)A = \displaystyle{{{\rm \omega }^2} \over {c^2}}{\rm \varepsilon }A$$where A is a complex function of space and 
$k_ 0 = {\rm \omega }\sqrt {{\rm \varepsilon }_ 0} /c$ is the wave number which is referring to the electromagnetic beam in the plasma. Following Nanda et al. (Reference Nanda, Ghotra and Kant2018), A can be written as follows:
$$A = A_0\lpar r\comma \;z\rpar \exp \lsqb -ik_0S_0\lpar r\comma \;z\rpar \rsqb $$where A 0 and S 0 are the real function of space, 
$S_ 0 = S_ 2r^ 2 / r_ 0^ 2 + S_ 4r^ 4 / r_ 0^ 4$ and 
$S_ 2 = r^ 2{f}^{\prime}\lpar z \rpar /\lpar {2 f} \rpar$, S 2 and S 4 are the components of the eikonal contributions.
Applied higher-order paraxial-ray theory and WKB approximation, during the calculation, it used the dimensionless propagation parameter 
${\rm \xi} = cz / \lpar{ {\rm \omega }r_ 0^ 2} \rpar$, one can obtain the laser beam intensity, equations for controlling beam width parameters:
$$A_0^2 = \displaystyle{{E_{00}^2 } \over {\,f^2}}\left({1 + \displaystyle{{a_2r^2} \over {r_0^2 f^2}} + \displaystyle{{a_4r^4} \over {r_0^4 f^4}}} \right)\exp \left({-\displaystyle{{r^2} \over {r_0^2 f^2}}} \right)$$
$$\eqalign{\displaystyle{{d^2f} \over {d{\rm \zeta }^2}}& = \displaystyle{{\lpar 1 + 8a_4-3a_2^2 -2a_2\rpar c^2} \over {{\rm \omega }_0^2 {\rm \varepsilon }_0r_0^4 f^3}} + \displaystyle{{{\rm \omega }_p^2 {\rm \alpha }} \over {{\rm \omega }_0^2 {\rm \varepsilon }_0}}\left[{\displaystyle{{a_2-1} \over {2{\rm \gamma }^3r_0^2 f^3}}-\displaystyle{{4c^2} \over {{\rm \omega }_p^2 r_0^2 f^3}}\left({\displaystyle{{2a_4-2a_2 + 1} \over {{\rm \gamma }^2r_0^2 f^2}}-\displaystyle{{{\rm \alpha }a_2^2 -2{\rm \alpha }a_2 + {\rm \alpha }} \over {{\rm \gamma }^4{\rm \gamma }_0^2 f^4}}} \right)} \right]} $$where 
$A_0^2$ refers to the square of the amplitude in optical electric field, which reflects the beam intensity as well, f is the dimensionless beam width parameter of laser beam, and 
${\rm \alpha} = {\rm \alpha }_ 0A_ 0^ 2$. In Eq. (10), the first term in right-hand side is the discrete of the differential equation, which is the linear part related to the initial beam width f along the laser propagation. While the second term is the convergence term which shows the nonlinearity, reflecting the characters of the periodical variations of the plasma, and governed by the term in right hand of Eqs (4) and (5).
Following earlier investigators (Kaur et al., Reference Kaur, Agarwal, Kaur and Gill2018), using WKB approximation and higher-order paraxial theory, and then separating the real and imaginary parts for Eq. (7), the eikonal contributions and the equation for the coefficient a 2 and a 4 have been obtained:
$$\!\!\!\!\!\!\!\!\eqalign{ \displaystyle{{dS_4} \over {d{\rm \zeta }}} & = \displaystyle{{\lpar a_2^3 -a_2^2 -7a_2a_4\rpar c^2} \over {{\rm \omega }_0^2 {\rm \varepsilon }_0r_0^2 f^6}}-\displaystyle{{{\rm \omega }_p^2 r_0^4 } \over {4{\rm \omega }_0^2 {\rm \varepsilon }}}\left\{{\displaystyle{{3{\rm \alpha }^2{\lpar a_2-1\rpar }^2} \over {4{\rm \gamma }^5r_0^4 f^8}}-\displaystyle{{{\rm \alpha }\lpar 2a_4-1\rpar } \over {2{\rm \gamma }^3r_0^2 f^6}}} { + \displaystyle{{2{\rm \alpha }c^2} \over {{\rm \omega }_p^2 r_0^2 f^4}}\left[{\displaystyle{{\lpar 9a_2-18a_4-3\rpar } \over {{\rm \gamma }^2r_0^4 f^4}} + \displaystyle{{{\rm \alpha }\lpar a_2-1\rpar \lpar 17a_2-18a_4-8\rpar } \over {{\rm \gamma }^4r_0^4 f^6}} + \displaystyle{{6{\rm \alpha }^2{\lpar a_2-1\rpar }^3} \over {{\rm \gamma }^6r_0^4 f^8}}} \right]} \right\}} $$
$$\displaystyle{{da_2} \over {d{\rm \zeta }}} = -\displaystyle{{16S_4f^2} \over {r_0^2 }}$$
$$\displaystyle{{da_4} \over {d{\rm \zeta }}} = \displaystyle{{8S_4f^2} \over {r_0^2 }}-\displaystyle{{24a_2S_4f^2} \over {r_0^2 }}$$where a 2 and a 4 are indicative of the departure of the beam from the Gaussian nature.
Numerical results and analysis
In this paper, applied the fourth order of Runge–Kutta method to solve Eqs (6) and (9)–(13). Some parameters are given as follows: beam intensity I ≈ 1019 W/cm2, frequency ω ≈ 1015 rad/s, wavelength λ = 0.5 μm, and the initial electron density N e0 ≈ 1021 cm−3. The parameters above could further be used to obtain the functions of the spot size which the function 
$I{\rm} = I\lpar r{\rm \comma \;}z\rpar = I_ 0 \lpar {\rm \omega }_ 0^ 2 / {\rm \omega }^ 2\lpar z \rpar \rpar \exp \lsqb - 2r^ 2 / {\rm \omega }^ 2\lpar z\rpar \rsqb$ is to be involved, where r is the radial distance from the center axis of the beam, ω0 is the waist size, and ω(z) is the spot size of the beam. Besides, beam diameter is defined as the distance across the center of the beam for which the 
$I{\rm} = I_{{\rm max}}{\rm /}e^ 2$. The spot size of the beam is the radial distance from the center of 
$I{\rm} = I_{{\rm max}}$ to the 
$I{\rm} = I_{{\rm max}}{\rm /}e^ 2$ points.
The effective dielectric constant in plasmas with cosine density and sinusoidal density ripple in different variations have been shown in Figure 1 that due to the relativistic nonlinearity, variations in plasma frequency and electron energy result in redistribution of electron density and affects the dielectric constant. On the one hand, under the influence of relativistic nonlinear effect, dielectric function has shown the sharp oscillating change with the similar periodical variation. On the other hand, with the increase of initial density (C) and the density ripple amplitude (D), dielectric function presents the fiercer oscillating variation.
Fig. 1. Variation of axial dielectric function ɛ with dimensionless propagation distance ξ in different plasmas. (a) 
$N_{\rm e}/N_{ 0{\rm e}}= 1 + 0.2 \cos \lpar 0.1 {\rm \xi} \rpar$, (b) 
$N_{\rm e}/N_{ 0{\rm e}} = &eqbr;2 + 0.2 \cos \lpar 0.1{\rm \xi} \rpar$, (c) 
$N_{\rm e}/N_{ 0{\rm e}} = 2 + 0.1 \sin \lpar 0.2 {\rm \xi} \rpar$, and (d) 
$N_{\rm e}/N_{ 0{\rm e}}= 2 + 0.3 \sin \lpar 0.2 {\rm \xi} \rpar$.
It can be seen from the enlargement of the details that the dielectric function shows the faster variation within the same distance. In addition, when the plasma density shows cosine distribution (see Fig. 1a and 1b), with the increase of initial density, the envelope of dielectric function oscillation presents the poor periodicity. And for the sinusoidal distribution (see Fig. 1c and 1d), with the increase of the density ripple amplitude, the envelope of dielectric function oscillation presents the better periodicity. And what is more, in such conditions, the frequency increases as well, the variations in Figure 1d is the same as Figure 1a and 1b in comparison.
Figure 2 shows the variation of the beam width along the dimensionless propagation distance. Under the influence of the relativistic nonlinear effect, the beam width shows the obvious oscillating variation with fiercer self-focusing, the envelope of which also presents the similar periodicity. With the increase of initial density and the density ripple amplitude, the variation of beam width becomes fiercer, and the variation frequency increases as well. From the enlargement of the details, the variation of beam width becomes faster, which implies beam self-focusing also becomes faster. Moreover, for the plasma density showing cosine distribution (see Fig. 2a and 2b), when the initial density decreases, the periodical variation of beam self-focusing becomes better. However, for the sinusoidal distribution (see Fig. 2c and 2d), with the density ripple increasing, the envelope of beam width parameter presents the better periodicity, which means the more stable self-focusing.
Fig. 2. Variation of beam width parameter f with dimensionless propagation distance ξ in different plasmas. (a) 
$N_{\rm e} / N_{ 0{\rm e}} = 1 + 0.2 \cos \lpar 0.1 {\rm \xi} \rpar$, (b) 
$N_{\rm e} / N_{ 0{\rm e}} = &eqbr;2 + 0.2 \cos \lpar 0.1 {\rm \xi} \rpar$, (c) 
$N_{\rm e} / N_{ 0{\rm e}} = 2 + 0.1 \sin \lpar 0.2 {\rm \xi} \rpar$, and (d) 
$N_{\rm e} / N_{ 0{\rm e}}= 2 + 0.3 \sin \lpar 0.2 {\rm \xi} \rpar$.
Figure 3 depicts the spatial plots of the normalized pulse intensity profile in two kinds of plasmas. For the plasma density with cosine and sinusoidal distribution, with the increase of the initial density and the density ripple, some of the top values of laser intense increase, which shows the further compression of beam and stronger self-focusing. For the cosine distribution, when the initial density increase, the number of light spot increases in the same propagation distance. What is more, the number of light spot also increases for the plasma density with the sinusoidal distribution.
Fig. 3. The spatial plots of the normalized pulse intensity, the X- and Y-axes present the transverse pulse width and radial width, respectively, the bar shows the variation in the normalized intensity. (a) 
$N_{\rm e} / N_{ 0{\rm e}}= &eqbr;1 + 0.2 \cos \lpar 0.1 {\rm \xi} \rpar$, (b) 
$N_{\rm e} / N_{ 0{\rm e}} = 2 + 0.2 \cos \lpar 0.1 {\rm \xi} \rpar$, (c) 
$N_{\rm e} / N_{ 0{\rm e}}= &eqbr;2 + 0.1 \sin \lpar 0.2 {\rm \xi} \rpar$, and (d) 
$N_{\rm e} / N_{ 0{\rm e}} =&eqbr; 2 + 0.3 \sin \lpar 0.2 {\rm \xi} \rpar .$
Figure 4 shows the variation of the dimensionless axial beam intensity for different plasmas assumed above. Along with the propagation distance, the beam presents obvious self-focusing and filamentation due to the relativistic effect. The maindifferences are the amplitude of beam intensity and numbers of beam filamentation. Comparing with Figure 4a and 4b, with the decrease of the initial density, the intensity of the light is greatly increased, which implies the enhanced self-focusing, and the number of filaments is also slightly decreased. Obviously, the self-focusing effect is enhanced over a longer distance. In Figure 4c and 4d, with the increasing density ripple amplitude, the number of filaments of the beam is increased, and the stability of self-focusing is enhanced.
Fig. 4. Variation of normalized beam intensity with dimensionless propagation distance ξ and radial distance r. (a) 
$N_{\rm e} / N_{ 0{\rm e}} = 1 + 0.2 \cos \lpar 0.1 {\rm \xi} \rpar$, (b) 
$N_{\rm e} / N_{ 0{\rm e}} = 2 + 0.2 \cos \lpar 0.1 {\rm \xi} \rpar$, (c) 
${\rm N}_{\rm e}{\rm /}{\rm N}_{{\rm 0e}}{\rm} = 2 + 0{\rm.1sin\lpar 0}{\rm.2\xi \rpar }$, and (d) 
$N_{\rm e} / N_{ 0{\rm e}}= 2 + 0.3 \sin \lpar 0.2 {\rm \xi} \rpar$.
Conclusion
In the paper, the self-focusing in LPI system has been studied, where the plasma with periodical density ripple and laser beam with relativistic characters have been chosen. Applied WKB approximation and higher-order paraxial theory, dielectric function and relativistic laser wave equations have been attained. Under the influence of relativistic nonlinear effect, the laser beam presents the intense relativistic self-focusing with a similar periodic variation. During the formation of the relativistic self-focusing, it is shown that the plasma density profile has the important effect on self-focusing. In the plasma with periodic density ripple, both the increase of the initial density and the amplitude of density ripple could lead to the fiercer variation of the dielectric function oscillation, the beam width and its variation frequency as well, which further leads to the strength of self-focusing. The former is more likely to destroy the current self-focusing, the latter is more likely to form the stable self-focusing. In theory and practical applications, if one needs to apply the self-focusing effect, then plasmas with property initial density and the density ripple amplitude value would be selected, so that the better periodicity and more stable self-focusing could be attained.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (Grant No. 11447169) and the Natural Science Foundation of Guangxi province (2018GXNSFAA138180, 2016GXNSFAA380071, and 2016GXNSFBA380204).
