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.

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 Allen12 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 expilϕ, 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 ϕ 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) technique18 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, Gupta27 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 nonlinearity29 or only ponderomotive nonlinearity30 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 l0,p=0 and investigate the quasi-steady situation where space charge field Es, generated when the electrons are expelled out from the higher intensity region to the lower one, is equal and opposite to the ponderomotive field Ep, i.e., ϕs=ϕp. This situation occurs in the timescale of ωpe1<t<ωpi1, where the electrons have enough time to respond but the ions do not. In the timescale of t>ωpi1, 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 B0ẑ, as the magnetic field is found to enhance the self-focusing process.33 The relativistic factor34 in the presence of static magnetic field reads γ=1+ωωαωc2a212=1+Ωa212, where α=+1 for the right circular polarization (RCP), α=1 for the left circular polarization (LCP), and a=eEmωc is the normalized field amplitude of the laser, ωc=eBmc is the cyclotron frequency, and Ω is given as Ω=ωωαωc2. The effective dielectric constant33 of the magnetized plasma can be written as ϵ=1ωp02ω2γαωcωnen0, which for weakly relativistic case reads as ϵ1ωp02ω21αωcω1γnen0.

The electric field profile of circularly polarized Laguerre–Gaussian laser is taken as
E=E00rr0lLplr2r02expr22r02expilϕx̂+iαŷ,
(1)
where E00 is the electric field amplitude of the fundamental mode. The transverse intensity profiles of these beams for different values of OAM (l) are shown in Fig. 1.
FIG. 1.

The transverse intensity profile of the Laguerre–Gaussian laser for different values of orbital angular momentum l.

FIG. 1.

The transverse intensity profile of the Laguerre–Gaussian laser for different values of orbital angular momentum l.

Close modal
The wave equation governing the propagation is as follows:
2E·E+ω2c2ϵE=0,
(2)
where E=Ar,zeiωtkzx̂+iαŷ is the electric field of the laser, ϵ is the dielectric constant of the medium, ω is the laser frequency, and c is the speed of light.

The term ·E of the earlier equation is neglected provided E·ϵϵk2E, as mentioned by Sodha et al.35 

In cylindrical coordinates, the wave equation can be written as
2z2E+2E+ω2c2ϵE=0,
(3)
where 2=2r2+1rr is the transverse Laplacian.

Using Wentzel–Kramers–Brillouin (WKB) approximation, we can write the solution as E=Ar,zexpi0zkzdz together with Ar,z as the complex amplitude of the electric field.

Now, assuming Ar,z as the slowly varying electric vector, we can neglect 2Az2as,2Az2k2A from Eq. (3). Then, we have
2ikAz+iAkz=2A+ω2c2ϵϵ0A,
(4)
which is known as the stationary form of nonlinear Schrödinger equation (NSE).
To obtain the solution of Eq. (4), we introduce the eikonal representation, given as
Ar,z=A0r,zexpikSr,zilϕ,
(5)
where both A0 and the eikonal S are functions of r and z.
Substituting this in Eq. (4), we find the real part as
2Sz+Sr2+2Skkz=1k2A02A0r2+1rA0rl2r2A0ω2c2k2r2r02ϵ2
(6)
and the imaginary part as
A02z+A022Sr2+1rSr+A02rSr+A02kkz=0.
(7)
The eikonal function can be written in paraxial approximation as S=r221fdfdz+θz, 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 η and z, where η=r/r0fl,
2Szη+lfdfdzSη+1r02f2Sη2+2Skkz=1k2A0r02f22A0η2+1η+lA0ηl2η+l2A0ω2c2k2η2ϵ2,
(8)
A02zη+lfdfdzA02η+A02r02f22Sη2+1η+lSη+1r02f2A02ηSη+A02kkz=0.
(9)
Here, we have used the following transformation rules:
r=1r0fη,
(10a)
z=zη+lfdfdzη,
(10b)
2=2r2+1rr1r02f22η2+1η+lη,
(10c)
S=η+l22βz+ϕz,
(11)
β=r02fdfdz.
(12)
We consider the following ansatz in terms of the new variables, which is required for the solution of Eq. (4) as
a2=a0f2η+l2lexpη+l2.
(13)
Using this in Eq. (8), we obtain
ϵ0d2fdζ2=1ρ04f31ρ02fϵ212dfdζdϵ0dζ,
(14)
where ζ=ωz/c is the normalized distance of propagation.

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 d2fdζ2<0 and the de-focusing occurs when d2fdζ2>0.

In the so-called quasi-steady state, we can use ϕs=ϕp=mc2eγ1 relation in the Poisson's equation to calculate the modified plasma density. Hence,
nen0=1+c2ωp022γ=1+c2ωp02Ωγa2a+1r02f21γ2aη2.
(15)
Since the relativistic factor γ is a function of l, it is obvious that the ponderomotive force is also a function of l. Therefore, the plasma density of the medium will be modified depending on the orbital angular momentum of the laser propagating through it.
To separate the linear and nonlinear parts of the dielectric constant ϵ, we can expand ϵ in Taylor series in terms of η2 as
ϵ=ϵ0η2ϵ2,
(16)
with ϵ2=ϵη2η=0.
ϵ0z=1ωp02ω21αωcω1γ01c2ωp02r02f2Ω2a0l2f2γ0,
(17)
ϵ2z=ωp02ω21αωcωΩa0l2f2γ031+c2ωp02r02f212+4Ωa0l2f2γ0,
(18)
dϵ0dζ=ωp02ω21αωcωΩa0l2γ031+c2ωp02r02f28+4Ωa0l2f2γ01f3dfdζ,
(19)
where a0l2=a02llel.
The critical condition for which the diffraction divergence is canceled out by the nonlinearity leads to the situation of self-trapped mode. This is obtained by equating the right-hand side of Eq. (14) to zero and using the initial conditions f=1 and dfdz=0 in Eqs. (14) and (17)–(19). Therefore, the beam will satisfy the condition for self-trapped mode when
r02ω2c2=ω2ωp02110Ωa0l23Ω2a0l4αωcω1+Ωa0l22Ωa0l21+Ωa0l212.
(20)
Clearly the self-trapping condition depends on the OAM of the laser and the applied magnetic field. The set of parameters of LG laser, which satisfies the critical condition, will lead to the stationary soliton.
In order to solve Eq. (14) numerically, we use the Runge–Kutta fourth-order method considering the initial conditions as mentioned earlier. The initial laser and plasma parameters are considered as ω=3×1014 rad/s, pulse width r0=8μm, a02=0.06, λ=1μm, and ωp0=0.6ω. Similar parameters were used in the theoretical findings of Sharma et al.21 The corresponding intensity 8×1016Wcm2 of the laser is obtained using the relation
I=1.37×1018a0λμm2W/cm2.
(21)
All the numerical calculations are valid17,36 as long as these satisfy the condition nen00, 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 ωc, 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.

FIG. 2.

Variation of equilibrium beam radius (self-trapped mode) with respect to (a) initial intensity for the fixed ωc=0.1ω and (b) to cyclotron frequency for fixed a02=0.06 for LCP and RCP lasers when l=1. ωp0=0.6ω is taken as another parameter.

FIG. 2.

Variation of equilibrium beam radius (self-trapped mode) with respect to (a) initial intensity for the fixed ωc=0.1ω and (b) to cyclotron frequency for fixed a02=0.06 for LCP and RCP lasers when l=1. ωp0=0.6ω is taken as another parameter.

Close modal

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 ωc=0.1ω in a plasma and ωp0=0.6ω. 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 ωc=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.

FIG. 3.

Evolution of the laser beam width parameter with respect to the propagation distance (normalized) for the left and right circularly polarized Laguerre–Gaussian LGpl lasers, when a static magnetic field ωc=0.1ω is applied in the medium when the initial intensity of the laser a02=0.06, plasma frequency ωp0=0.6ω, r0ωc=10, and l=1.

FIG. 3.

Evolution of the laser beam width parameter with respect to the propagation distance (normalized) for the left and right circularly polarized Laguerre–Gaussian LGpl lasers, when a static magnetic field ωc=0.1ω is applied in the medium when the initial intensity of the laser a02=0.06, plasma frequency ωp0=0.6ω, r0ωc=10, and l=1.

Close modal
FIG. 4.

Evolution of the laser beam width parameter with respect to the propagation distance (normalized) for different cyclotron frequencies ωc=0,0.1ω,0.15ω,0.2ω when the initial intensity of the laser a02=0.06, r0ωc=8, l=1, and ωp0=0.6ω.

FIG. 4.

Evolution of the laser beam width parameter with respect to the propagation distance (normalized) for different cyclotron frequencies ωc=0,0.1ω,0.15ω,0.2ω when the initial intensity of the laser a02=0.06, r0ωc=8, l=1, and ωp0=0.6ω.

Close modal

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 ϵ0 and ϵ2), 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 ϵ0 and ϵ2. However, in the case of unmagnetized plasma ωc=0, the relativistic nonlinearity decreases with propagation distance, causing ϵ0 and ϵ2 to decrease, eventually leading to de-focusing of the laser (as shown in Fig. 4).

FIG. 5.

Variation of the (a) relativistic factor, (b) linear dielectric constant, and (c) nonlinear dielectric constant with distance of propagation for different cyclotron frequencies ωc=0,0.1ω,0.15ω,0.2ω when a02=0.06, l=1, ωp0=0.6ω, and r0ωc=8.

FIG. 5.

Variation of the (a) relativistic factor, (b) linear dielectric constant, and (c) nonlinear dielectric constant with distance of propagation for different cyclotron frequencies ωc=0,0.1ω,0.15ω,0.2ω when a02=0.06, l=1, ωp0=0.6ω, and r0ωc=8.

Close modal

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=1,2 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 ζ, leading to the change in both the dielectric constants, i.e., ϵ0 and ϵ2 (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.

FIG. 6.

Variation of the (a) equilibrium beam radius with respect to the initial intensity and (b) the laser beam width parameter with respect to the propagation distance (normalized) for different topological charge (OAM): l=0,1,2,3 when a02=0.06, r0ωc=8, ωp0=0.6ω, and ωc=0.05ω.

FIG. 6.

Variation of the (a) equilibrium beam radius with respect to the initial intensity and (b) the laser beam width parameter with respect to the propagation distance (normalized) for different topological charge (OAM): l=0,1,2,3 when a02=0.06, r0ωc=8, ωp0=0.6ω, and ωc=0.05ω.

Close modal
FIG. 7.

Variation of the (a) relativistic factor, (b) linear dielectric constant, and (c) nonlinear dielectric constant with distance of propagation for different topological charge (OAM): l=0,1,2,3 when a02=0.06, ωc=0.05ω, ωp0=0.6ω, and r0ωc=8.

FIG. 7.

Variation of the (a) relativistic factor, (b) linear dielectric constant, and (c) nonlinear dielectric constant with distance of propagation for different topological charge (OAM): l=0,1,2,3 when a02=0.06, ωc=0.05ω, ωp0=0.6ω, and r0ωc=8.

Close modal

Figure 8(a) depicts the transverse intensity profile of LG01 laser at three different locations, i.e., z=0,150 and 200  μm. 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.

FIG. 8.

(a) Laser intensity a2 and (b) modified plasma density nen0 at different distances of propagation for LG01 when a02=0.06, ωc=0.1ω, ωp0=0.6ω, and r0ωc=8.

FIG. 8.

(a) Laser intensity a2 and (b) modified plasma density nen0 at different distances of propagation for LG01 when a02=0.06, ωc=0.1ω, ωp0=0.6ω, and r0ωc=8.

Close modal

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.

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

The authors have no conflicts to disclose.

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

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