Self-focusing/Defocusing of Hermite-Sinh-Gaussian Laser Beam in Underdense Inhomogeneous Plasmas

The self-focusing/defocusing of Hermite-sinh-Gaussian (HshG) laser beam in underdense inhomogeneous plasmas is studied by using higher-order approximation theory. It is found that Hermite mode index and the ﬂuctuation of the periodic plasma density have a signiﬁcant eﬀect on the dielectric constant and laser beam self-focusing/self-defocusing. With the increase of mode index, the high-order HshG laser beam is beneﬁcial to suppress self-focusing and enhance self-defocusing. In addition, the eﬀects of decentered parameters, beam intensity, and plasma non-uniformity on self-focusing/self-defocusing are discussed.


Introduction
ere are a lot of non-linear phenomena in the interaction between intense laser and plasma, such as self-focusing/selfdefocusing [1], filamentation [2], stimulated Raman scattering [3] and stimulated Brillouin scattering [4]. ese nonlinearities have important effects on inertial confinement fusion [5] and laser particle accelerator [6]. In particular, plasma self-focusing can improve the yield of these applications by increasing the interaction length between the laser and the medium, but also increase the success of these applications by balancing the natural diffraction of the laser beam to transmit more Rayleigh lengths [7]. erefore, the self-focusing generated by the interaction of laser pulses with plasma is a very worthy topic of study [8].
In interaction intense laser and plasms, the mechanism of self-focusing/defocusing mainly include collisional nonlinearity [9], ponderomotive nonlinearity [10,11], relativistic nonlinearity [12]. Nanda and Kant [13] researched self-focusing of cosh-Gaussian laser beams in collisionless non-uniform magnetic plasmas. By comparing extraordinary mode and ordinary mode, it is found that increasing the plasma density ramp can make the former stronger selffocusing. Asgharnejad et al. [7] investigated self-focusing a Gaussian laser pulse in a weakly relativistic and ponderomotive regime inside a collisionless warm quantum plasma. Zhang et al. [14] analyzed relativistic self-focusing of Gaussian laser beam, the results show that relativistic effect can enhance the oscillation of dielectric function and increase the self-focusing. Abedi-Varaki and Jafari [15] have present that a decrease in plasma temperature and an increase in oblique magnetic field increases defocusing for left-handed polarization. Extending the High-harmonic generation phase matching cutoff by controlling the rapid self-defocusing effect of driving laser is experimentally and theoretically demonstrated by Sun et al. [16].
Lots of research of self-focusing/self-defocusing have focused on interaction of Gaussian beam and homogeneous/ inhomogeneous plasmas. Studies by Gupta et al. [17] and Kant et al. [18] have shown that the existence of plasma density fluctuations can enhance self-focusing. Wani et al. [19] have found that electron temperature can reduce defocusing and plasma density transition increases the range of self-focusing effect. Recently, many researchers have analyzed the effects of different types of laser beams and plasma on self-focusing/self-defocusing. Laser beam extends from Gaussian type to the others type of beam, such as Hermite-Gaussian beam [20], cosh-Gaussian beam [21], super Gaussian beam [22]. Various types plasmas have been researched, such as cold quantum plasma with exponential density variation [23]. Effects of magnetic field on relativistic self-focusing, self-phase modulation and self-trapping of cosh-Gaussian laser beam is exploited by Gill et al. [24]. Kant et al. [25] observed relativistic self-focusing of Hermitecosh-Gaussian laser pulse in plasma with exponential plasma density ramp. Devi and Malik [26] studied the relativistic self-focusing of super-Gaussian laser beams, they have found that the external magnetic field and the cyclotron motion of electrons is important for self-focusing. Based on the moment theory, Malik and Devi [27] developed a formulation to study the self-defocusing of super-Gaussian laser beam in tunnel ionized plasmas, and found that laser beam can get more defocused for higher intensity and smaller spot size. Kaur et al. [28] used paraxial approximation method to study propagation characteristics of Hermite-cosh-Gaussian laser beam in rippled density plasmas and strong oscillatory self-focusing and defocusing are observed in relativistic case.
In this paper, applying higher-order paraxial theory and WKB approximation, the self-focusing/self-defocusing are investigated in the interaction of Hermite-Sinh-Gaussian (HshG) laser beam and the non-uniform underdense plasma with periodic density fluctuations.

Propagation Theory of Laser Beam in Plasmas
In laser-plasma interaction, according to Maxwell's equation theory, the wave equation of laser propagation in plasma can be expressed as, For k − 2 ∇ 2 ln(ε) << 1 and k 2 � εω 2 /c 2 , the wave equation can be converted to When the laser propagates in the plasma along the zdirection, the specific form of the field can be expressed as, In this paper, the interaction between HshG laser beam and inhomogeneous plasma is studied. For equation (3), the general solution of the field amplitude is HshG beam.
E 0 is the amplitude of HshG laser beam at r � z � 0, Hermite polynomial r is the radial component in the cylindrical coordinate system, r 0 is the initial beam width, b represents decentred parameter, the dimensionless beam width parameter of laser beam is f(z).
Non-uniform plasmas showing periodic variations have been found experimentally and theoretically by researchers in the process of laser-plasma interaction.
e plasma system studied in this paper is based on the experimental results of Ouahid et al. [29]; electron density N e (z) is sinusoidal function. e expression is as follows, where N e (z), N 0e , α, q represents electron density in equilibrium steady state, initial electron density, parameter of plasma fluctuation, wave number of plasma density fluctuation.
In laser-plasma interaction, when plasma density fluctuation exists, the dielectric function of plasma composed of linear part ε 0 and non-linear part ϕ(EE * ) can be expressed as ε(r, z) � ε 0 − ϕ(EE * ). In the limit c 2 /(r 2 0 ω 2 p0 ) < 1, the linear and non-linear part can be presented as follows 4πn 0 e 2 /m is the plasma frequency, e is the electronic charge and m shows the rest mass. Here the relativistic factor can be written as c � ��������� � 1 + βEE * /2, where β � (e/mωc) 2 . By applying higherorder paraxial theory and extending the dielectric function up to the fourth higher-order term.
where ε 0 (z) is the linear part, ε 2 (z) and ε 4 (z) are expansion coefficients of ε(r, z). Based on the research results of Pathak et al. [30]; using WKB approximation and higher-order paraxial theory, dielectric constant components of different Hermite mode indexes can be obtained in interaction of HshG beams and plasma.For mode index n � 0 For mode index n � 1 where in equations (7)-(9), t � ω 2 p0 /ω 2 (1 + α sin(qξ)) denotes the parameters related to plasma density, B � βE 2 0 , D � 1 − b 2 /2 + b 4 /8 denote intensity dependence and intensity distribution dependence of dielectric constant. For wave equation (2), substitute (3) into equation (2), z 2 A/zz 2 be ignored, can get, Following Nanda et al. [31]; complex amplitude A can be written as follows, A 0 (r, z) and S(r, z) are the real functions of space, according to Habibi and Ghamari [32]; they can be expressed this way

Laser and Particle Beams
a 2 , a 4 , S 0 , S 2 , S 4 are the functions of z, S 0 , S 2 , S 4 are the components of the eikonal. S 2 � (r 2 0 /2f)(df/dξ) corresponds to the spherical curvature of the wave front, S 4 indicates deviation from spherical curvature. a 2 , a 4 are indicative of the departure of the beam from the HshG nature. Can get a complicated differential equation by bringing in equation (10) with equation (11), the following equations can be obtained if the real and imaginary parts are equal.
By substituting equations (12) and (13) in equation (15) for different Hermite mode indexes, and equating the coefficients of r 2 and r 4 on both sides of equation, the equation relation of parameter a 2 and a 4 is represented as.
For mode index n � 0 For mode index n � 1 where S 4 ′ � S 4 (ω/c). Same method, equations (12) and (13) in equation (14) to get the following equations governing the beam width f and S 4 ′ . For mode index n � 0 For mode index n � 1 For mode index n � 2 Normalize the above equations, ξ � cz/ωr 2 0 � z/R d is the dimensionless propagation distance, ρ � ωr 0 /c represents the dimensionless original beam width.

Discussion
In the paper, intensity of incident laser beam is 1.23 × 10 17 W/cm 2 , βE 2 0 � 0.01, 0.02, 0.1 and 0.2 (corresponding intensities are 1.23 × 10 15 W/cm 2 , 2.46× 10 15 W/cm 2 , 1.23 × 10 16 W/cm 2 and 2.46 × 10 16 W/cm 2 respectively), the frequency and spot size of incident laser beam as ω � 1.78 × 10 15 rad/s, r 0 � 1.68 × 10 − 5 m in interaction system. Appling fourth-order Runge-Kutta method to solve equations (16a)-(21b), the phenomena of self-focusing/self-defocusing in the interaction of intense laser and the inhomogeneous plasma are discussed. e initial conditions are f � 1, df/dξ � 0, a 2 � 1, as well as da 2 /dξ � 0 at ξ � 0, plasmas density is N e (z) � N 0e (1 + α sin qz). Figures 1(a)-1(c) shows the linear component of the dielectric function of the HshG laser beam interaction of plasma with periodic density fluctuation at mode index n � 0, 1 and 2. e linear part of dielectric function shows sinusoidal periodic variations similar to oscillation in plasma of periodic density fluctuation. e peak value of the linear part decreases slightly with the increase of mode index, and the oscillation has intensified the tendency. For the nonlinear part ε 2 , ε 4 at different mode indexes, the value is much smaller compared to the linear part, and mode index has a more obvious influence on the non-linear part. In figure (d), (e) and (f ), it can be seen that mode index obvious affect the peak value of non-linear part, the peak position drifts to the left as the mode increases, and finally all tend to stable. From the above linear and non-linear parts of the dielectric function, it shows an approximate periodic oscillatory variation due to the plasma periodicity, and its oscillatory variation trend to increase with the increase of mode index of Hermite polynomial. In addition, the mode index have more obvious effect on the non-linear part comparison linear part. Figure 2 presents beam width variation for different Hermite mode indexes of HshG laser beams. In the periodic underdense inhomogeneous plasma, the beam width all show oscillatory changes, firstly, beam width becomes small and then increases. During the propagation of the laser beam, ponderomotive force influence on laser beam is obvious at the start, and beam width reduces and appears to convergence, resulting in self-focusing. en, the influence ponderomotive force of laser beam decrease and beam width increase continuously with distance, and beam width presents diverge obviously, resulting in beam self-defocusing. However, at different mode indexes, the size and position of the minimum value of beam width are different. Increase of mode index, the size slightly increase and position appear early, and the beam diverges more quickly, which means the self-defocusing is more intense. For n � 0, reaches the minimum value 0.55 at ξ ≈ 0.14. For n � 1, reaches the minimum value 0.6 at ξ ≈ 0.1. For n � 2, reaches the minimum value 0.998 at ξ ≈ 0.02.
is is due to the combined effects of HshG laser beam and periodic underdense non-uniform plasma, causing a similar periodic oscillatory variation of dielectric function, which leads to self-focusing and self-defocusing. In addition, with the increase of mode index, the oscillation of the dielectric function is enhanced, indicating that the instability is enhanced, which will affect the coupling of the laser plasma, so that the quality of Gaussian beam decreases, self-focusing reduce and self-defocusing enhance. Figure 3 presents the effect of decentered parameters b on beam width f in different modes indexes. It is shown that the decentered parameter has obvious influence on the beam width. For n � 0, the beam width slowly increases in b � 0, and in b ≠ 0, it first decreases, then increases, and with the  Laser and Particle Beams increase of b, the variation of beam width is strengthened. n � 1 is similar to n � 0. For (e) n � 2, beam width increases slightly at first, and then almost linearly increases, moreover, beam width decreases slightly for the same propagation distance with increase of b. (a) (c) correspond to (b) (d), the rules are consistent, the purpose is to study the effect of smaller changes in b on the self-focusing, and optimize the value of the decentered parameter. (f ) depicts the variation of the beam width when βE 2 0 � 0.3, the beam width shows an oscillation trend. Compared with (e), it not only shows the influence of decentered parameters on self-focusing, but also reflects the importance of laser intensity to self-focusing. In summary, (b) (d) (f ) show that the decentered parameter is very sensitive to the self-focusing, which is consistent with that described in the paper of Nanda et al. [33]. For the zero-order and 1-order HshG beams, under the influence of the decentered parameter, there is a significant beam self-focusing at b ≠ 0, however, for the 2-order HshG beam, it presents obvious beam self-defocusing, in this case, the effect of decentered parameter on self-defocusing is relatively slight. It can be clearly seen that the effect of decentered parameters on self-focusing of the HshG beam is significantly stronger than self-defocusing. Figure 4 depicts the variation of the beam width for different light intensities. e figure is shown that, beam width increases at the same propagation distance with the increase of light intensity, it implies that self-focusing decrease and self-defocusing increase. In addition, it is easier to form self-focusing with the decrease of laser intensity, while self-focusing becomes weak when the light intensity  Laser and Particle Beams increases to 0.2, it may be because laser intensity is close or exceed the threshold value of forming self-focusing, leading to self-focusing reduce or disappear. In Figure 4(c), for n � 2 , when laser intensity increases, beam width continues to increase, self-focusing is very weak and produces obvious self-defocusing. Hence, it is shown that the high-order HshG beam and high beam intensity are benefit to restrain selffocusing and enhance self-defocusing. Figure 5 depicts the variation of the beam width for different plasma density parameter. With the increase of plasma inhomogeneity, the beam width increases at n � 0 and n � 1. At n � 2, as the plasma inhomogeneity increases, beam width slowly decreases first and then increases. It implies that the inhomogeneity of plasma has a significant effect on the beam width because the plasma density fluctuation affects the dielectric function, which directly affects laser propagation process in plasma, lead to the change of beam width, and increase non-uniformity of plasmas, it can strength self-focusing/ self-defocusing.

Conclusion
In the paper, apply high-order paraxial theory to analyze the self-focusing/self-defocusing of HshG laser beam in underdense inhomogeneous plasma. Self-focusing and selfdefocusing laws are explored by analyzing the variation of the dielectric constant and the influence of the decentered parameters, laser intensity and plasma density on beam width. e results show that, due to the combined effects of HshG laser beam and periodic underdense non-uniform plasma, lead to dielectric function present a similar periodic oscillation variation. Under the influence of ponderomotive force and diffraction effect, lead to beam self-focusing/self-defocusing. With increase of Hermite mode index, causing the position of self-focusing/selfdefocusing is bring forward, and beam defocusing is strengthened, it implies that the higher mode index HshG beam is benefit to reduce self-focusing and enhance defocusing. In addition, the decentered parameter, beam intensity and non-uniformity of the plasma has a significant influence on the beam width. Influence of the decentered parameter on beam self-focusing is stronger than self-defocusing. In the high-order HshG beam, high beam intensity is benefit to restrain self-focusing and enhance self-defocusing, and increase non-uniformity of plasmas, it can strength self-focusing/self-defocusing.

Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Conflicts of Interest
Authors have no conflict of interest to declare.