Orbital angular momentum of Lienard Wiechert fields

particle with arbitrary periodic orbit, in the form of multi-pole expansion of the Liénard–Wiechert potential,whichexplicitlyincludesthechargedparticlemotion.Usingthisexpression,wediscusstheorbitalangularmomentumradiatedfromarelativisticchargedparticle.Ithasrecentlybeenindicatedthattheradiationemittedbycircularlyorbitingchargedparticlescarrieswell-deﬁnedorbitalangularmomentum.Weshowthat,evenforthegeneralcasesofarbitraryperiodicorbits,theradiationﬁeldpossesseswell-deﬁnedorbitalangularmomentum....................................................................................................................


Introduction
Electromagnetic waves carry not only energy and linear momentum but also angular momentum. In particular, it is well known that electromagnetic waves carry spin angular momentum, which is associated with circular polarization. However, it is less well known that they also carry orbital angular momentum. In 1992, Allen et al. showed that an electromagnetic wave in Laguerre-Gaussian modes carries well-defined orbital angular momentum, distinct from spin angular momentum [1]. Such light waves have spiral wavefronts and are called optical vortices, photons carrying orbital angular momentum, or twisted photons. Since this pioneering work, optical vortices have been investigated not only from the scientific but also the technological viewpoint [2][3][4]. Some methods for artificially modifying the wavefront structure from a plane to a spiral by using spiral phase plates or numerically designed holograms have been established [2], and various applications of their unique features have been discussed. For example, optical vortices can be used to manipulate micrometer-sized objects by using their torque [5], and can be made useful for future high-capacity communications owing to the orthogonality between their different spiral modes [4]. Another attractive application of optical vortices may be for astronomical observations. It has been indicated that much more information can be obtained from electromagnetic wave signals from the universe by considering their spiral modes rather than conventional plane-wave spectrum channels [4,[6][7][8]. However, there have only been a few discussions on the generation of optical vortices in nature [6,9].
It has recently been indicated that the radiation emitted by circularly orbiting charged particles exhibits spiral wavefronts and carries well-defined orbital angular momentum [10]. This process forms the basis for various important radiation processes in plasma physics and astrophysics, such PTEP 2019, 083A02 H. Kawaguchi and M. Katoh as cyclotron radiation, synchrotron radiation, and Compton scattering of circularly polarized light. Therefore, this radiation has been well investigated in the literature [11,12]; however, its angular momentum has not been discussed explicitly until recently [13][14][15].
In this paper, we discuss the angular momentum of the radiation field emitted by a relativistic charged particle in more general cases, using a multi-pole expansion of the Liénard-Wiechert fields [16,17]. The multi-pole expansion of electromagnetic fields has been discussed in textbooks in the context of angular momentum [18][19][20][21]. However, in these textbooks, the authors consider electromagnetic radiation fields that are solutions of the homogeneous Helmholtz equation. This means that they examine electromagnetic waves propagating in free space without considering their source. In contrast, here, we treat the radiation field by explicitly including the motion of the charged particle.

Multipole expansion of Liénard-Wiechert fields
In this section, we briefly review the multi-pole expansion of Liénard-Wiechert fields [16,17]. The Liénard-Wiechert potentials, which are general expressions for the electromagnetic field produced by a relativistic charged particle, are given as follows: where q is the electric charge of the particle, ε 0 is the dielectric constant, c is the velocity of light in vacuum, R(τ ) = x − s(τ ), R(τ ) = |R(τ )| , s(t) is the charged particle trajectory, and v(t) =ṡ(t) is the particle velocity. The right-hand sides of Eqs. (1) and (2) should be evaluated at the retarded time τ , which is defined by the following recursive causality relation: The Fourier expansion of the potentials are expressed as follows: where ω is the angular frequency of the original electron motion and σ = ωt is the phase of the periodic motion. Then, by employing the well-known formula [20]  and replacing x by s(σ ), Eqs. (4) and (5) can be rewritten as where r, θ , and ϕ are the spherical coordinates, and h (2) l (x), j l (x), and P m l (x) are the spherical Hankel function of the second type, the spherical Bessel function, and the associated Legendre function, respectively. If we introduce the spherical harmonic functions [20] Eqs. (7) and (8) can be expressed as where are definite integrals with respect to σ = ωt over the period of the particle motion; these have constant values in Cartesian coordinates x,ŷ,ẑ .

Linear and angular momenta of Liénard-Wiechert multipole fields
In this section, we evaluate the linear and angular momenta of the Liénard-Wiechert fields. For this purpose, it is convenient to express the multi-pole expansion of the Liénard-Wiechert potentials in The vector M n,l,m in Eq. (12) has a constant value in Cartesian coordinates, but depends on θ , φ in spherical coordinates. From Eq. (17), we find the following relations between the components of M n,l,m : In spherical coordinates, the multi-pole expansion of the Liénard-Wiechert potentials (11) and (12) is Accordingly, the electromagnetic field components in the far field corresponding to Eqs. (22)-(25) can be expressed approximately as follows (Appendix A): Now, we shall calculate the radiation power emitted by the ultra-relativistic charged particle: where T = 2π ω is the period of the particle motion, n is a unit vector normal to the surface S, and the surface integral dS = r 2 sin θ dθ dφ is calculated for a spherical surface with a sufficiently large radius compared to the charged particle motion region. Considering that only the terms of order 1 r in Eqs. (26)-(31) contribute to Eq. (32), we only need to consider the second terms of Eqs. (27), (28), and (30) and the first term of Eq. (31). Substituting these terms into Eq. (32), we obtain and carrying out the integral with respect to t over the period T , we arrive at Note that the orthogonal formulas relating to spherical harmonics, π 0 P m l (cos θ)P m l (cos θ) sin θdθ = 2 2l + 1 Next we shall calculate the angular momentum of the radiated fields. We consider its projection onto an arbitrary axis. As a matter of convenience, we shall call it the z-axis. The z-component of 6 can be expressed in spherical coordinate as follows: This means that we only need to consider terms of order 1 r 3 in the integrands of Eq. (38). Since the r-components of the electric (26) and magnetic (29) fields are at least of order 1 r 2 , we only need to consider the second terms of Eqs. (27) and (30). Similarly to Eq. (33), substituting these terms into Eq. (38) and carrying out the integral with respect to t over the period T , we obtain the following: The first, second, and third terms of Eq. If we assume, e.g., that the charged particle motion is axis-symmetric with respect to the z-axis on average over the period of the particle motion, meaning that M r n,l,m (θ, φ), M θ n,l,m (θ, φ), M φ n,l,m (φ) are independent of φ, then for the (n-l-m) multi-pole components. We can infer from Eq. (44) that the radiation fields emitted by the ultra-relativistic charged particle carry angular momenta of m per (nω) photons for each nth time harmonic and (l-m)th multi-pole if the charged particle motion is axis-symmetric with respect to the z-axis on average over the period of the particle motion, as in Eq. (41). Another interesting case is when the charged particle motion has the following rotational symmetry: H. Kawaguchi and M. Katoh In this case, the integrand of Eq. (40) becomes In a similar way, using the orthogonal relation (36), the relation (44) can be extended to for the (n-l-m) multi-pole components. For the general case, after tedious calculations (Appendix B), the relation between the momentum and angular momentum of the multi-pole components of the radiation fields can be expressed in the following form, similar to Eq. (47): where α n,l,m is a constant value, which must be determined for each (n-l-m) multi-pole component (see Eq. (B7)). The relation (44) between the momentum and angular momentum of the multi-pole components of the radiation fields emitted by ultra-relativistic charged particles is in agreement with discussions of general multi-pole fields in Refs. [18][19][20][21]. These discussions assume that the radiation fields

Synchrotron radiation
As a typical example of radiating charged particle motion, we apply here the above discussion to circular motion at constant velocity, as shown in Fig. 1. The charged particle's motion can be described as where v = aω is the velocity of the particle. This motion can be expressed in spherical coordinates as In this case, only two harmonics n = m ± 1 exist in the summation which agree with the result in Ref. [10].
Most of α n=20,l,m has an approximate value of −1, and some much bigger values appear depending on the original charged particle motion.

Conclusions
In this paper, we have generalized the discussion on the orbital angular momentum carried by the radiation field from a charged particle in circular motion [10] to arbitrary trajectories, by using a multipole expansion of the Liénard-Wiechert fields. The expression that we have derived is applicable to arbitrary charged particle motion with periodic orbit. We have shown that when the particle motion has an axis of symmetry, the field carries a well-defined angular momentum along the symmetry axis and that this expression for the angular momentum can be extended to the general case.
All other electromagnetic field components can now be calculated straightforwardly.