The propagation of electromagnetic beams carrying orbital angular momentum $l$ is investigated in a cold collisionless plasma where a static magnetic field is applied in the axial direction. The relativistic and ponderomotive nonlinearities are taken into consideration simultaneously. A stationary nonlinear Schrödinger equation is derived using the Wentzel–Kramers–Brillouin method and the slowly varying envelope approximation. The critical condition for the self-trapped mode is achieved as a function of orbital angular momentum (OAM), magnetic field, and initial laser intensity of the beam. The response of the medium to the two types of polarizations, i.e., left circular polarization (LCP) and right circular polarization (RCP), is compared, and it is observed that the RCP laser shows better focusing than the LCP laser and also requires a smaller beam radius for achieving the self-trapped mode. The effect of applied magnetic field and OAM of the laser is also studied on the beam width evolution. The laser is found to be focused earlier in the cases of a larger applied magnetic field. A Laguerre–Gaussian laser with higher OAM is observed to show efficient self-focusing. This study enables exploration in the fields of particle acceleration, electron bunch generation, x-ray sources, and more.

## I. INTRODUCTION AND MOTIVATION

Orbital angular momentum (OAM), which is a form of angular momentum carried by a monochromatic electromagnetic beam, is found to be of central interest to recent researchers because of its growing applications in generation of ultra-relativistic monoenergetic electron bunches, high harmonic generation, laser wakefield acceleration, generation of high energetic gamma photons, optical control of plasma accelerators, and so on.^{1–11} It was Allen^{12} who showed that the electromagnetic beams can have OAM along with the spin angular momentum (SAM) if the phase fronts are helical in nature. There are some experimental techniques found in the literature, which show the generation of such electromagnetic waves.^{13–15} Laguerre–Gaussian beams $LGpl$, which have helical phase fronts $exp\u2009il\varphi $, consist of two indices: $l$ and p. Here, $l$ is the topological charge or OAM of the beam (which is also called as vortex index), and $p$ is the radial index of the beam. In the literature, the radial index is mentioned as “the forgotten quantum number.”^{16,17} $\varphi $ is the azimuthal angle that shows self-healing properties as a Bessel–Gaussian beam does under similar condition but propagates further than them. Growing applications of the laser technologies demand higher intensity of the lasers, and this has been made possible with the utilization of the chirped pulse amplification (CPA) technique^{18} involving the stretching of temporal and spectral profiles of the laser, its amplification, and then its compression. Another important phenomenon that shows enhancement of laser intensity is the self-focusing during the laser plasma interaction process. In recent years, much research has been done on this laser self-focusing technique both theoretically and experimentally.^{19–26} However, most of them considered the Gaussian lasers, super-Gaussian lasers, Hermite–Gaussian lasers, and q-Gaussian beams. Consideration of the orbital angular momentum $l$ in the self-focusing process gives another degree of freedom to the study. Given the importance of OAM modes of laser, there are very few works done on the self-focusing phenomenon of Laguerre–Gaussian (LG) laser beams. For example, Gupta^{27} considered the moment theory to investigate the propagation properties of a LG beam with ponderomotive force as optical nonlinearity, and he observed oscillatory behavior of the laser beam width. He also observed that the focusing becomes less efficient for larger value of the topological charge $l$. Assuming ponderomotive and collisional nonlinearity in the medium, Suo *et al.*^{28} have considered Wentzel–Kramers–Brillouin (WKB) and paraxial-like approximations to study the phenomenon and observed similar behavior. The literature also shows that for higher value of the topological charge $l$, the focusing gets enhanced when either only relativistic nonlinearity^{29} or only ponderomotive nonlinearity^{30} is taken into account. Kad *et al.*^{31} studied the variation of a spatiotemporal LG pulse in magnetized plasma and observed that $LG02$ shows better spatial focusing than $LG00$ and $LG01$ modes. The observations direct that the self-focusing of the Laguerre–Gaussian phenomenon needs further investigations. In addition, to the best of our knowledge, there are no studies that examine the propagation characteristics of laser carrying OAM modes in a plasma medium while taking into account the combined effect of ponderomotive and relativistic nonlinearities. It has been shown that the ponderomotive effect aids in relativistic self-focusing of the laser beams provided the intensity is below a certain intensity level,^{32} showing the importance of ponderomotive nonlinearity.

In this article, we mainly focus on the LG beams having purely orbital angular momentum $l\u22600,p=0$ and investigate the quasi-steady situation where space charge field $E\u2192s$, generated when the electrons are expelled out from the higher intensity region to the lower one, is equal and opposite to the ponderomotive field $E\u2192p$, i.e., $\varphi s=\u2212\varphi p$. This situation occurs in the timescale of $\omega pe\u22121<t<\omega pi\u22121$, where the electrons have enough time to respond but the ions do not. In the timescale of $t>\omega pi\u22121$, the ions are accelerated because of the space-charge potential and causing deviation from the quasi-steady state. We also consider a static magnetic field $B0z\u0302$, as the magnetic field is found to enhance the self-focusing process.^{33} The relativistic factor^{34} in the presence of static magnetic field reads $\gamma =1+\omega \omega \u2212\alpha \omega c2a212=1+\Omega a212$, where $\alpha =+1$ for the right circular polarization (RCP), $\alpha =\u22121$ for the left circular polarization (LCP), and $a=eEm\omega c$ is the normalized field amplitude of the laser, $\omega c=eB\u2192mc$ is the cyclotron frequency, and $\Omega $ is given as $\Omega =\omega \omega \u2212\alpha \omega c2$. The effective dielectric constant^{33} of the magnetized plasma can be written as $\u03f5=1\u2212\omega p02\omega 2\gamma \u2212\alpha \omega c\omega nen0$, which for weakly relativistic case reads as $\u03f5\u22481\u2212\omega p02\omega 21\u2212\alpha \omega c\omega 1\gamma nen0$.

## II. ANALYTICAL SCHEME

The term $\u2207\u2192\u2207\u2192\xb7E\u2192$ of the earlier equation is neglected provided $\u2207\u2192E\u2192\xb7\u2207\u2192\u03f5\u03f5\u226ak2E\u2192$, as mentioned by Sodha *et al.*^{35}

Using Wentzel–Kramers–Brillouin (WKB) approximation, we can write the solution as $E=Ar,z\u2009exp\u2009\u2212i\u222b0zkzdz$ together with $Ar,z$ as the complex amplitude of the electric field.

*eikonal*representation, given as

*eikonal S*are functions of

*r*and

*z*.

*eikonal*function can be written in paraxial approximation as $S=r221fdfdz+\theta z$, where $f(z)$ is the laser beam width parameter (dimensionless). Since $r0$ is the initial radius of the laser, the actual radius of the laser after a distance of propagation can be written as $r0f(z)$. This shows how the spot size changes as the laser keeps on propagating in the plasma. Since, the peak intensity of the laser with topological charge is not in the center $r=0$, we would rewrite the above-mentioned equations in terms of new variables $\eta $ and z, where $\eta =r/r0f\u2212l$,

The final equation (14) of the beam width parameter consists of three terms in the right-hand side: the first one describes the diffraction divergence of the beam, and the remaining two terms correspond to the nonlinear response of the medium. The nonlinear response of the medium can be investigated by finding the nonlinear part of the dielectric constant, which is a function of the OAM of the laser. The spot size evolution will be such that the self-focusing occurs when $d2fd\zeta 2<0$ and the de-focusing occurs when $d2fd\zeta 2>0$.

## III. NUMERICAL ANALYSIS AND RESULTS

*et al.*

^{21}The corresponding intensity $\u20098\xd71016Wcm2$ of the laser is obtained using the relation

^{17,36}as long as these satisfy the condition $nen0\u22650$, and the dielectric constant given by Eq. (16) is positive.

Figures 2(a) and 2(b) show the variation of the required beam radius (normalized) for the self-trapped mode with the initial intensity $a02$ and cyclotron frequency $\omega c$, respectively, for lasers $LG01$ with left and right circular polarizations. For both types of LCP and RCP lasers, the initial beam radius decreases as the initial intensity increases. However, the RCP laser requires a smaller beam radius for the self-trapped mode. It is also observed [Fig. 2(b)] that the beam radius decreases with the cyclotron frequency in the case of RCP laser but it increases monotonically for the LCP laser. Initial beam radius greater than that required for self-trapped mode corresponds to the self-focusing phenomenon, while the beam radius less than that required for self-trapped mode corresponds to the defocusing phenomenon. Our study aims to determine the suitable regime for self-focusing and investigate the impact of plasma medium and LG laser parameters on the laser's mode of propagation.

Figure 3 shows evolution of the beam width parameter (f) for both the left and right circularly polarized lasers when a static magnetic field is applied to the medium such that $\omega c=0.1\omega $ in a plasma and $\omega p0=0.6\omega $. For the RCP laser $LG01$, the beam width parameter is found to decrease with propagating distance as the relativistic and ponderomotive nonlinearities dominate over the diffraction, leading to the beam self-focusing during propagation. However, for similar plasma and laser parameters, the LCP laser is observed to show diverging behavior, with the spot size progressively increasing during laser propagation. Therefore, further investigations of the spot size evolution are performed for right circularly polarized LG lasers. Figure 2(b) depicts that when we increase the magnetic field (for RCP laser), the required beam radius for the self-trapped mode decreases as the applied magnetic field makes the nonlinearity stronger over the diffraction. Therefore, the self-focusing occurs earlier (enhanced focusing) when a stronger static axial magnetic field is applied in the medium (as shown in Fig. 4). For the chosen laser and plasma parameters, the $LG01$ laser is observed to show diverging behavior when there is no applied magnetic field $\omega c=0$. Such a behavior of the laser was observed by Devi *et al.*^{33} who only considered relativistic nonlinearity and neglected the ponderomotive nonlinearity to study the self-focusing of super-Gaussian beam in magnetized plasma. In the case of our study, the beam self-focuses strongly in the presence of larger magnetic field.

It can be seen from Fig. 5 that during the $LG01$ beam propagation through the medium, the relativistic and ponderomotive nonlinearities change the dielectric constants (both $\u03f50$ and $\u03f52$), leading to the change in the beam spot size (Fig. 4), which, in turn, changes the relativistic nonlinearity [Fig. 5(a)]. If we increase the magnitude of the applied magnetic field, the relativistic factor tends to reach its maximum earlier [Fig. 5(a)], causing similar changes in $\u03f50$ and $\u03f52$. However, in the case of unmagnetized plasma $\omega c=0$, the relativistic nonlinearity decreases with propagation distance, causing $\u03f50$ and $\u03f52$ to decrease, eventually leading to de-focusing of the laser (as shown in Fig. 4).

Now, to investigate the effect of orbital angular momentum on the spot size evolution, we have plotted Fig. 6. Figure 6(a) shows that the normalized beam radius decreases with the initial laser intensity. It further shows that the required beam radius of the laser for self-trapped mode in the case when beam carries OAM $l=1,\u20092$ is greater than that in the case of the laser not carrying any OAM $l=0$, i.e., Gaussian beam. Again, larger the value of $l$, smaller is the required beam radius of the beam. In addition to these observations, Fig. 6(b) shows that the beam gets focused earlier for a larger value of $l$. This can be explained as follows. The relativistic nonlinearity increases with the increase in $l$ and shifts toward the smaller value of $\zeta $, leading to the change in both the dielectric constants, i.e., $\u03f50$ and $\u03f52$ (as shown in Fig. 7). For the chosen parameters, it is observed that the self-focusing is better for the Gaussian mode $LG00$ than for the $LG01$ and $LG02$ modes but weaker than $LG03$.

Figure 8(a) depicts the transverse intensity profile of $LG01$ laser at three different locations, i.e., $z=0,\u2009150$ and 200 $\mu m$. Here, the intensity is observed to enhance with distance of the propagation for the chosen parameters. The modified plasma density given by Eq. (15) is also plotted in Fig. 8(b) for these locations. It is inferred that deeper density profiles are achieved due to the larger number of electrons expelled out from the higher intensity to lower intensity region when the intensity of the laser increases during propagation. A comparison of Figs. 8(a) and 8(b) shows that the plasma density with the use of $LG01$ beam is found to acquire a ring-shaped profile concentering the beam axis $r=0$ under the said medium and laser parameters.

## IV. CONCLUSIONS

This article described the propagation properties of Laguerre–Gaussian $LG0l$ laser in a cold collisionless plasma, where a static axial magnetic field is applied, and the relativistic and ponderomotive nonlinearities act as the optical nonlinearities of the medium. The RCP-type lasers required smaller beam radius for achieving the self-trapped mode compared to the LCP-type lasers, and when the magnetic field was increased, the required beam radius decreased monotonically for the RCP lasers and increased monotonically for the LCP lasers. Better focusing was achieved in the case of RCP laser compared to the LCP laser. In the presence of magnetic field, the nonlinearity became stronger than the diffraction divergence, and self-focusing started. Increment in the magnetic field gave rise to better focusing. Numerical solution of Eq. (20) revealed that the laser with OAM (*l* = 1, 2) requires larger beam radius for self-trapping than the laser with zero OAM, and also higher values of OAM gave rise to better focusing. The overall conclusion is that the OAM gives rise to an extra degree of freedom to the investigation of the laser propagation characteristics under relativistic and ponderomotive nonlinearities that may play a crucial role in the field of laser plasma interaction, high harmonic generation, monoenergetic electron bunch generation, acceleration of energetic particles, and so on.

## ACKNOWLEDGMENTS

Subhajit Bhaskar acknowledges the Ministry of Human Resource and Development (MHRD), Government of India, for the financial support.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Subhajit Bhaskar:** Conceptualization (equal); Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – original draft (equal). **Hitendra Kumar Malik:** Supervision (lead); Validation (equal); Writing – review & editing (lead).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

## REFERENCES

*Laser-Matter Interaction for Radiation and Energy*

*γ*-ray vortex generation in near-critical-density plasma driven by twisted laser pulses

*q*-Gaussian laser beam in relativistic plasma