# Faraday-Fresnel rotation and splitting of orbital angular momentum carrying
waves in a rotating plasma

J.-M. Rax1  R. Gueroult2  1IJCLab - Université de Paris-Saclay & Département
de Physique - Ecole Polytechnique, 91405 Orsay, France 2LAPLACE, Université de
Toulouse, CNRS, INPT, UPS, 31062 Toulouse, France

###### Abstract

Rotational Fresnel drag - or orbital Faraday rotation - in a rotating
magnetised plasma is uncovered and studied analytically for Trivelpiece-Gould
and Whistler-Helicon waves carrying orbital angular momentum (OAM). Plasma
rotation is shown to introduce a non-zero phase shift between OAM-carrying
eigenmodes with opposite helicities, similarly to the phase-shift between spin
angular momentum eigenmodes associated with the classical Faraday effect in a
magnetised plasma at rest. By examining the dispersion relation for these two
low-frequency modes in a Brillouin rotating plasma, this Faraday-Fresnel
rotation effect is traced back to the combined effects of Doppler shift,
centrifugal forces and Coriolis forces. In addition, rotation is further shown
to lead to rotation- and azimuthal mode-dependent longitudinal group velocity,
therefore prediciting the Faraday-Fresnel splitting of the enveloppe of a wave
packet containing a superposition of OAM-carrying eigenmodes with opposite
helicities.

##  1 Introduction

The dynamics of magnetised plasmas and beams confined by an axial magnetic
field and put in rotation through a radial electric field has been extensively
investigated, both from a basic plasma physics standpoint and to enable
applications. Rotating plasmas have for instance been investigated - and
continue to - for the purpose of magnetic confinement fusion (Wilcox, 1959),
including in homopolar configurations (Anderson et al., 1958), rotating
mirrors (Lehnert, 1971; Bekhtenev et al., 1980; Ellis et al., 2005; Fetterman
& Fisch, 2008, 2010) and rotating tokamaks (Stix, 1971; Rax et al., 2017; Ochs
& Fisch, 2017). Rotating plasmas have also been the object of continued
attention for mass separation applications (Dolgolenko & Muromkin, 2009, 2017;
Zweben et al., 2018), notably in the form of plasma centrifuges (Bonnevier,
1966; Krishnan et al., 1981; Prasad & Krishnan, 1987), crossed-field rotating
plasmas (Ohkawa & Miller, 2002; Shinohara & Horii, 2007; Gueroult et al.,
2019b; Liziakin et al., 2021) and rotating magnetic field configurations (Rax
& Gueroult, 2016). In parallel to these applications driven studies, the
complementary problem of angular momentum coupling between electromagnetic
waves and particles has been examined in the study of particle acceleration
(Rax & Robiche, 2010; Thaury et al., 2013), magnetic field generation in
plasma channels and plasma bubbles (Shvets et al., 2002; Kostyukov et al.,
2002) and isotope separation (Rax et al., 2007).

A basic question of importance to these studies, but also to shed light on the
intricate effects of rotation in astrophysics (Balbus & Hawley, 1998; Benomar
et al., 2018), is how to diagnose plasma rotation. Asking this question is
fundamentally asking for physical probes sensitive to fluid vorticity.
Interestingly, it has recently been shown that rotation could imprint its
signature on the polarisation of a wave propagating through a rotating plasma
(Gueroult et al., 2019a), offering in turn unique opportunities to infer the
rotation direction in pulsars. A challenge here, however, is that in a
magnetised plasma both this polarisation drag effect (Player, 1976; Jones,
1976) (also referred to as mechanical Faraday effect) and the classic Faraday
rotation (Faraday, 1846; Stix, 1992) are at play (Gueroult et al., 2020).
Measuring plasma rotation through polarisation drag hence requires
disambiguating this effect from Faraday rotation. Similarly, harnessing
polarisation drag in laboratory experiments likely requires conditions such
that polarisation drag dominates over Faraday rotation (Gueroult et al.,
2020). Polarisation however corresponds only to the spin component of the wave
angular momentum. Indeed, waves can also carry orbital angular momentum
through helical phase fronts of the type $\exp j\left(l\theta-\beta z\right)$
with $l\in\mathbb{Z}$ and $\beta\in\mathbb{R}$, and where $\theta$ and $z$ are
the transverse azimuthal angle and the propagation axis in a cylindrical
coordinate system (Gough, 1986; Allen et al., 1992; van Enk & Nienhuis, 1994;
Barnett & Allen, 1994; Stenzel, 2016; Barnett et al., 2017).

In isotropic media mechanical rotation has been postulated (Padgett et al.,
2006; Götte et al., 2007; Leach et al., 2008) and then demonstrated
experimentally (Franke-Arnold et al., 2011) to lead to image rotation. This
phenomena, analogously to the polarisation rotation that stems from a phase
shift between right- and left-circularly polarised modes, results from a
mechanically induced phase shift between the positive and negative OAM
carrying modes (Wisniewski-Barker et al., 2014). Image rotation has hence been
coined as mechanical Faraday effect for OAM-carrying beams. Yet, owing to its
similarities with Fresnel drag for a uniform motion (Fresnel, 1818), image
rotation is equally referred to as rotatory photon drag (Franke-Arnold et al.,
2011). Reflecting these two roots, we refer to it here as Faraday-Fresnel
rotation (FFR).

An interesting property of FFR in a dielectric medium is that, in contrast to
polarisation rotation, it is insensitive to a magnetic field aligned with the
propagation direction. Measuring FFR could hence lift the challenges faced by
polarisation drag to measure rotation in a rotating magnetised plasma, as
suggested for ultracold atomic gases (Ruseckas et al., 2007). Yet, while both
linear and angular Fresnel drag have been studied in isotropic, non-gyrotropic
rotating media (Götte et al., 2007; Leach et al., 2008), the case of rotating
gyrotropic media such as a rotating magnetised plasma has not yet been
investigated. Conceptually, anisotropy and gyrotropy bring out additional
complexity in that the polarisation state and the wave vector direction can be
set independently for a given frequency in isotropic media, whereas these two
parameters are no longer independent in anisotropic or gyrotropic media.

Two approches can be used to examine the effect of rotation on waves carrying
OAM in a rotating magnetised plasma. One possibility is to consider the
transformation of the various parameters from the rotating plasma rest frame
to the lab frame. This is the approach we previously used to uncover the
effect of plasma rotation on the wave’s spin angular momentum (Gueroult et
al., 2019a, 2020). Another possibility is to carry out the calculations in the
lab frame starting from first principles. This is the approach we will use in
this work as it proves more straightforward and also provides more directly
physical insights into this new plasma effect. The approach in this study is
therefore to solve the particles dynamics to calculate the rotating plasma
linear response. We specifically consider here the low frequency electronic
branches where the coupling between the fields and the particles is strong,
that is Trivelpiece-Gould (TG) and Whistler-Helicon (WH) modes (Stix, 1992;
Davidson, 2001).

This paper is organised as follows. In the next section, Section 2, we
introduce the effect of rotation on plasma waves carrying OAM through an
analogy with SAM-carrying waves in the presence of a magnetic field, and
briefly recall the characteritics of Trivelpiece-Gould (TG) and Whistler-
Helicon (WH) plasma waves as they will be the focus of our work. Section 3
then summarises the properties of Brillouin rotation in a cold magnetised
plasma column used later on as our rotating plasma equilibrium. With these
tools in hand, the dispersion relation for TG and WH waves in a rotating
plasma are derived and analysed in Sections 4 and 5, respectively. In both
instances Faraday-Fresnel rotation and its parametric dependencies are
identified and antennae designs that could excite these rotating waves are
proposed. Building on this finding, Section 6 examines the effect on a wave
packet containing a combination of counter rotating OAM-carrying eigenmodes,
and shows how rotation lead to the Faraday-Fresnel splitting of this wave
packet. Finally Section 7 summarises the main findings of this study.

##  2 Phenomenology of Faraday-Fresnel rotation in a rotating plasma

Throughout this study we write $\left(r,\theta,z\right)$ a set of cylindrical
coordinates on the cylindrical coordinate system
$\left(\mathbf{e}_{r},\mathbf{e}_{\theta},\mathbf{e}_{z}\right)$, and define
$z$ as the direction of both the background static magnetic field
$\mathbf{B}_{0}=B_{0}\mathbf{e}_{z}$ and the plasma angular velocity
$\bm{\Omega}=\Omega\mathbf{e}_{z}$.

###  2.1 Polarisation rotation from phase shift between circularly polarised
eigenmodes

To draw an analogy between the classical Faraday effect and Faraday-Fresnel
rotation, let us first recall the basic picture relating polarisation rotation
to circularly polarised eigenmodes. For this consider a plane wave with a
phase factor $\exp j\left(\omega t-\beta z\right)$ propagating along the
magnetic field in an homogeneous plasma at rest. The study of the cold plasma
dispersion relation between the axial wave vector $\beta$ and the frequency
$\omega$ leads to the identification of the two left (L) and right (R)
circularly polarised modes such that

$\beta_{L/R}=\frac{\omega}{c}\sqrt{\frac{\left(\omega\pm\omega_{L}\right)\left(\omega\mp\omega_{R}\right)}{\left(\omega\mp\omega_{ce}\right)\left(\omega\pm\omega_{ci}\right)}}.$
(1)

Here $\omega_{c\alpha}=|q_{\alpha}|B_{0}/m_{\alpha}$, $\alpha=i,e$ is the
(positive) cyclotron frequency and
$\omega_{L/R}=({\omega_{pe}}^{2}+{\omega_{ce}}^{2}/4)^{1/2}\pm\omega_{ce}/2$
are the left and right cut-off with $\omega_{pe}=ne^{2}/(m_{e}\epsilon_{0})$
the plasma frequency. At high frequency $\omega>\omega_{ce}$ the electrons
inertia response dominates the dispersion relation and Eq. (1) can be expanded
to give the phase difference $\delta\varphi$ between the $L$ and $R$ modes
accumulated when propagating one wavelength
$\lambda=(\lambda_{R}+\lambda_{L})/2$ along $z$

$\left.\frac{\delta\varphi}{4\pi}\right|_{SAM}=\frac{\beta_{L}-\beta_{R}}{\beta_{L}+\beta_{R}}\approx\frac{\omega_{ce}}{\omega}\frac{\omega_{pe}^{2}}{\omega^{2}}.$
(2)

The phase difference derived in Eq. (2) is the classical formula describing
the high-frequency Faraday SAM rotation in a magnetised plasma at rest. A very
similar result, though not with the same frequency scaling, can be obtained by
considering the phase velocity difference between right- and left-circularly
polarised wave as a result of the medium’s rotation (Player, 1976; Jones,
1976; Götte et al., 2007; Gueroult et al., 2020).

###  2.2 Faraday-Fresnel rotation from phase shift between OAM-carrying
eigenmodes

Plasma waves carrying orbital angular momentum (see Appendix A for a
discussion of spin and orbital angular momentum in waves and particles) have
been the object of growing attention over the last decade. In unmagnetised
plasmas, OAM-carrying plasma waves have been studied theoretically mostly
under the scalar paraxial approximation (see, _e. g._ , Mendonca et al.
(2009); Mendonça (2012b, a)), although exact solutions of the vector Maxwell
equations have recently been derived (Chen et al., 2017). Meanwhile, in cold
magnetised plasmas, waves carrying orbital angular momentum have been reported
in the form of Trivelpiece-Gould (TG) and Whistler-Helicon (WH) waves (Stenzel
& Urrutia, 2015a; Urrutia & Stenzel, 2016; Stenzel, 2016), as well as
predicted for twisted shear Alfvén waves (Shukla, 2012). Because our interest
in this study, as it will be detailed in the next section, is in Brillouin
rotating plasmas, we focus our attention on magnetised plasmas and consider
more specifically TG and WH modes.

Whistler-Helicon (WH) waves are intermediate frequencies
$\omega_{lh}<\omega<\omega_{ce}$ waves propagating along the magnetic field,
with $\omega_{lh}$ the lower hybrid frequency. The plasma response in this
regime is dominated by the electron Hall current, which allows simplifying the
dispersion relation Eq. (1) to obtain the classical Whistler dispersion

${\beta_{W}}^{2}\approx\frac{\omega}{c^{2}}\frac{\omega_{L}\omega_{R}}{\omega_{ce}}\approx\frac{{\omega_{pe}}^{2}}{c^{2}}\frac{\omega}{\omega_{ce}}.$
(3)

On the other hand non-rotating Trivelpiece-Gould (TG) are electrostatic modes
that propagate near the electron cyclotron frequency $\omega_{ce}$ and the
electron plasma frequency $\omega_{pe}$. These waves are characterised by an
electric potential proportional to $J_{l}\left(\beta_{\bot}r\right)\exp
j\left(\omega t-\beta_{TG}z\right)$ where $J_{l}$ is the ordinary Bessel
function of order $l\in\mathbb{Z}$, and the dispersion relation in this case
is (Trivelpiece & Gould, 1959)

${\beta_{TG}}^{2}=\beta_{\bot}^{2}\frac{\omega^{2}\left({\omega_{uh}}^{2}-\omega^{2}\right)}{\left({\omega_{ce}}^{2}-\omega^{2}\right)\left({\omega_{pe}}^{2}-\omega^{2}\right)}$
(4)

with $\omega_{uh}\sim({\omega_{pe}}^{2}+{\omega_{ce}}^{2})^{1/2}$ the upper
hybrid frequency.

As mentioned above, both TG waves (Urrutia & Stenzel, 2016) and WH waves
(Stenzel & Urrutia, 2015a) with helical phase front of the form $\exp
j\left(\omega t\pm l\theta-\beta_{\pm l}z\right)$ can propagate in a cold
magnetised plasma at rest (Stenzel, 2016). Both positive and negative
azimuthal mode numbers $l$, that is to say both positive and negative orbital
helicities for the same absolute value of OAM content, can propagate and be
generated through an appropriate phasing of the generating antenna array. It
is important to underline here though that the wave rotation should not be
construed as the rotation of the field vectors (_i. e._ the waves
polarisation), and only indicates a rotation of the transverse wave pattern.
Whistler modes are for instance right circularly polarised modes. The
superposition of azimuthal modes of equal amplitudes but opposite azimuthal
mode number (OAM of $\pm l\hbar$) then leads to an azimuthally standing wave
(zero OAM), very much like the superposition of right- and left-circularly
polarised modes (SAM of $\pm\hbar$) leads to a linearly polarised wave (zero
SAM).

By analogy with the phase shift between opposite SAM eigenmodes given in Eq.
(2), we can define the phase shift between opposite OAM-carrying eigenmodes

$\left.\frac{\delta\varphi}{4\pi}\right|_{OAM}=\left.\frac{\beta_{+l}-\beta_{-l}}{\beta_{+l}+\beta_{-l}}\right|_{TG/W}\approx
l\frac{\Omega}{\omega}F_{TG/W}\left(\omega_{pe},\omega_{ce},\omega\right).$
(5)

As we will show, this OAM phase difference is zero absent rotation (_i.e._ for
$\Omega=0$) for both WH and TG modes. This is because in this case
$\beta_{+l}=\beta_{-}$, so that the transverse wave pattern produced by the
superposition of opposite OAM-carrying eigenmodes does not depend on $z$. This
is analog to the constant linear polarisation observed if the phase velocity
of right- and left-circularly polarised modes is the same. Building on this
analogy, we will show that plasma rotation $\Omega\neq 0$ leads to
$\beta_{+l}\neq\beta_{-}$, and therefore now to a rotation of the transverse
wave pattern along $z$, which we refer to as Faraday-Fresnel rotation. The
dispersive form factor $F$ introduced in Eq. (5) and derived in Eq. (33) and
Eq. (62) for TG and WH waves, respectively, quantifies the strength of this
effect.

Finally, note that this study is limited to the analysis of the OAM dynamics
through the reactive (or adiabatic) part of the plasma response. The active
(or resonant) part of the plasma response, which leads to SAM and OAM
transfer, is assumed negligible as we consider a cold collisionless plasma
model. This active or resonant part of the interaction is only briefly
discussed in Appendix A to provide a fuller picture of the SAM and OAM
dynamics.

##  3 Brillouin rotation of a magnetised plasma column

Before embarking on the derivation of Faraday-Fresnel rotation for the
Trivelpiece-Gould and Whistler-Helicon modes, the properties of the Brillouin
rotating plasmas considered in the rest of this manuscript are briefly
recalled here.

Brillouin rotation refers to the rotation of a magnetised plasma column which
results from the combination of an axial confining magnetic field
$\mathbf{B}_{0}=B_{0}\mathbf{e}_{z}$ and a radial electric field
$\mathbf{E}_{0}=B_{0}\Omega r\mathbf{e}_{r}$. As shown next the plasma angular
velocity $\Omega=d\theta/dt$ is then the $E\times B$ drift rigid body rotation
corrected by inertial effects (Brillouin, 1945), as well as possibly by
collisional effects (Rax et al., 2015, 2019). In a non-neutral plasma the
electric field $\mathbf{E}_{0}$ is the space charge field (Davidson, 2001). On
the other hand, the electric field $\mathbf{E}_{0}$ in a quasi-neutral plasma
requires in steady-state maintaining a small uniform space charge
$\varepsilon_{0}\bm{\nabla}\cdot\mathbf{E}_{0}$ = $2\varepsilon_{0}\Omega
B_{0}$, and a DC or RF power input is thus needed to maintain this small
deviation from quasi-neutrality against short circuiting radial currents (Rax
et al., 2019; Kolmes et al., 2019).

Neglecting for simplicity collisions in this study and writing
$\mathbf{v}_{0}$ the particle fluid velocity, the steady state momentum
balance between inertia, electric and magnetic forces writes

$m\left(\mathbf{v}_{0}\cdot\bm{\nabla}\right)\mathbf{v}_{0}=q\left(\mathbf{E}_{0}+\mathbf{v}_{0}\times\mathbf{B}_{0}\right)$
(6)

with $q$ the particle charge. The left hand side describes inertia while the
right hand side accounts for electric and magnetic forces. Equation (6) has
two solutions,

$\mathbf{v}_{0}=\Omega_{\pm}r\mathbf{e}_{\theta},$ (7)

associated with the two roots $\Omega_{\pm}$ of the characteristic equation

$2\frac{\Omega_{\pm}}{\omega_{c}}=-\frac{q}{|q|}\left[1\pm\sqrt{1-4\frac{q}{|q|}\frac{\Omega}{\omega_{c}}}\right]$
(8)

obtained by substituting Eq. (7) into Eq. (6) (Brillouin, 1945; Davidson,
2001). Consistent with earlier notations, we use here $\omega_{c}=|q|B_{0}/m$
as the unsigned cyclotron frequency.

In the remaining of this study we consider only the small root associated with
the slow $E\times B$ rotation, that is $\Omega_{-}\approx-\Omega$, as it is
the rotation mode spontaneously observed quasi-neutral plasmas experiments. It
also leads to $\Omega_{e}=\Omega_{i}$, or in other words solid body rotation
at equal angular frequency for ions and electrons. However, since both slow
$\Omega_{-}\approx-\Omega$ and fast $\Omega_{+}\approx-q\omega_{c}/|q|$
rotation modes have been observed and analysed in non-neutral plasmas
(Davidson, 2001), it is worth noting here that our findings could easily be
extended and applied to higher angular velocities.

In the next two sections we will perturb this
$\left[\mathbf{v}_{0},\mathbf{E}_{0},\mathbf{B}_{0}\right]$ Brillouin
equilibrium given by Eqs. (6, 7) with helical electrostatic and
electromagnetic waves
$\left[\mathbf{E}\left(r,\theta,z,t\right),\mathbf{B}\left(r,\theta,z,t\right)\right]$
with a space and time structure of the form

$\mathbf{A}=\mathbf{a}\left(r\right)\exp j\left(\omega t-l\theta-\beta
z\right).$ (9)

According to Appendix A, $\widehat{L}_{z}\mathbf{A}=-l\mathbf{A}$ and
$\widehat{P}_{z}\mathbf{A}=-\beta\mathbf{A}$, so that one quantum of these
waves carry an orbital angular momentum $l\hbar$ and a linear angular momentum
$\beta\hbar$. We will then derive the expressions describing the rotating
plasma response to this helical electromagnetic wave to obtain the dispersion
relation. The dispersion relation will finally be used to identify Faraday-
Fresnel rotation and Faraday-Fresnel splitting, that is to say the effect of
plasma rotation on OAM-carrying waves. The following set of relations

$\displaystyle\bm{\nabla}\times\left(r\mathbf{e}_{\theta}\times\mathbf{A}\right)$
$\displaystyle=$ $\displaystyle
jl\mathbf{A}+r\left(\bm{\nabla}\cdot\mathbf{A}\right)\mathbf{e}_{\theta},$
(10)
$\displaystyle\bm{\nabla}\times\left(\mathbf{e}_{z}\times\mathbf{A}\right)$
$\displaystyle=$ $\displaystyle
j\beta\mathbf{A}+\left(\bm{\nabla}\cdot\mathbf{A}\right)\mathbf{e}_{z},$ (11)
$\displaystyle\bm{\nabla}\cdot\left(r\mathbf{e}_{\theta}\times\mathbf{A}\right)$
$\displaystyle=$ $\displaystyle 2\mathbf{A\cdot
e}_{z}-r\mathbf{e}_{\theta}\cdot\mathbf{\nabla\times A},$ (12)
$\displaystyle\bm{\nabla}\cdot\left(\mathbf{e}_{z}\times\mathbf{A}\right)$
$\displaystyle=$ $\displaystyle-\mathbf{e}_{z}\cdot\mathbf{\nabla\times A},$
(13)
$\displaystyle\left(\mathbf{A}\cdot\bm{\nabla}\right)r\mathbf{e}_{\theta}$
$\displaystyle=$ $\displaystyle\mathbf{e}_{z}\times\mathbf{A},$ (14)
$\displaystyle\left(r\mathbf{e}_{\theta}\cdot\bm{\nabla}\right)\mathbf{A}$
$\displaystyle=$ $\displaystyle-jl\mathbf{A}+\mathbf{e}_{z}\times\mathbf{A},$
(15)

will be particularly useful to solve Maxwell’s equation for the TG and WH
modes.

##  4 Faraday-Fresnel rotation for Trivelpiece-Gould waves

In a magnetised plasma column close to the plasma and cyclotron frequencies
both cyclotron and Langmuir oscillations are expected to propagate along the
magnetic field. The analysis of these electrostatic elementary excitations
suggests considering a potential electric field $\mathbf{E}$ such that
$\bm{\nabla}\times\mathbf{E}=\mathbf{0}$ (Davidson, 2001; Stenzel & Urrutia,
2016). This electric field derives from a potential $\phi$ whose dynamics is
associated with an oscillating electron space charge $-ne$. We therefore
consider the propagation of potential and density perturbations

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\phi\left(r\right)\exp
j\left(\omega t-l\theta-\beta z\right),$ (16) $\displaystyle n$
$\displaystyle=$ $\displaystyle n\left(r\right)\exp j\left(\omega
t-l\theta-\beta z\right),$ (17)

over the equilibrium static radial electric potential $\phi_{0}=-B_{0}\Omega
r^{2}/2$ from which derives the electric field $\mathbf{E}_{0}$ driving the
equilibrium Brillouin rotation and uniform densities $n_{0}$. Equation (6)
provides the unperturbed momentum balance for a Brillouin flow while the
relation $\bm{\nabla}\cdot n_{0}\mathbf{v}_{0}=0$ yields the unperturbed
charge conservation relation.

###  4.1 Disperson relation

For a weak electronic perturbation of this Brillouin flow the linearised
momentum balance, Eq. (18), and the linearised charge conservation relation,
Eq. (19),

$\displaystyle
j\omega\mathbf{v}+\left(\mathbf{v}_{0}\cdot\bm{\nabla}\right)\mathbf{v}+\left(\mathbf{v}\cdot\bm{\nabla}\right)\mathbf{v}_{0}$
$\displaystyle=$
$\displaystyle-\frac{e}{m_{e}}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}_{0}\right),$
(18) $\displaystyle j\omega
n+n_{0}\bm{\nabla}\cdot\mathbf{v}+\mathbf{v}_{0}\cdot\mathbf{\nabla}n$
$\displaystyle=$ $\displaystyle 0,$ (19)

yield a set of coupled equations relating $n$ to $\phi$. Making use of Eqs.
(14, 15), Eq. (18) can be rearranged as

$j\left(\omega-l\Omega\right)\mathbf{v}+\left(2\Omega-\omega_{ce}\right)\mathbf{e}_{z}\times\mathbf{v}=\frac{e}{m_{e}}\bm{\nabla}\phi,$
(20)

while Eq. (19) can be rewritten as

$n_{0}\bm{\nabla}\cdot\mathbf{v}=-j\left(\omega-l\Omega\right)n.$ (21)

We then apply the three linear operators $\bm{\nabla}\times$,
$\bm{\nabla}\cdot$ and $\mathbf{e}_{z}\cdot$ to Eq. (20) to obtain three
relations relating the divergence, the curl and the longitudinal component of
the velocity $\mathbf{v}$. With the help of Eqs (11, 13) this yields

$\displaystyle
j\left(\omega-l\Omega\right)\bm{\nabla}\times\mathbf{v}+\left(2\Omega-\omega_{ce}\right)\left[\left(\bm{\nabla}\cdot\mathbf{v}\right)\mathbf{e}_{z}+j\beta\mathbf{v}\right]$
$\displaystyle=$ $\displaystyle\mathbf{0},$ (22) $\displaystyle
j\left(\omega-l\Omega\right)\bm{\nabla}\cdot\mathbf{v}-\left(2\Omega-\omega_{ce}\right)\mathbf{e}_{z}\cdot\bm{\nabla}\times\mathbf{v}$
$\displaystyle=$ $\displaystyle{\omega_{pe}}^{2}\frac{n}{n_{0}},$ (23)
$\displaystyle\left(\omega-l\Omega\right)\mathbf{e}_{z}\cdot\mathbf{v}$
$\displaystyle=$ $\displaystyle-\frac{e}{m_{e}}\beta\phi,$ (24)

where we have used Poisson equation to eliminate $\Delta\phi$ in Eq. (23).
Velocity dependent terms $\bm{\nabla}\times\mathbf{v}$,
$\bm{\nabla}\cdot\mathbf{v}$ and $\mathbf{v}$ are then eliminated between Eqs.
(21, 22, 23, 24) to write the oscillating space charge $-ne$ as a function of
the oscillating potential $\phi$

$-\frac{ne}{\varepsilon_{0}}=\beta^{2}\frac{{\omega_{pe}}^{2}}{\left(\omega-l\Omega\right)^{2}}\frac{\left(\omega_{ce}-2\Omega\right)^{2}}{\omega_{pe}^{2}+\left(\omega_{ce}-2\Omega\right)^{2}-\left(\omega-l\Omega\right)^{2}}\phi.$
(25)

Poisson equation $\Delta\phi$ = $ne/\varepsilon_{0}$ for the perturbation
potential $\phi\left(r\right)$ given in Eq. (16) finally leads to the final
equation describing TG modes propagation

$\frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial\phi}{\partial
r}-\frac{l^{2}}{r^{2}}\phi+\beta^{2}T_{l}^{2}\phi=0$ (26)

where we have defined the parameter

$T_{l}^{2}=\frac{{\omega_{pe}}^{2}\left(\omega_{ce}-2\Omega\right)^{2}}{\left(\omega-l\Omega\right)^{2}\left[{\omega_{pe}}^{2}+\left(\omega_{ce}-2\Omega\right)^{2}-\left(\omega-l\Omega\right)^{2}\right]}-1.$
(27)

Looking at Eqs. (26) and (27), one first verifies that the classical TG
relation Eq. (4) first derived by Trivelpiece & Gould (1959) is recovered in
the limit of zero rotation $\Omega=0$. One additionally notes that
$T_{+l}=T_{-l}$ in this same limit of zero rotation, but that $T_{+l}\neq
T_{-l}$ for $\Omega\neq 0$. Finally, a closer examination of Eq. (27) shows
that the $\Omega$ dependent terms $\left(\omega_{ce}-2\Omega\right)$ can be
traced back to the Coriolis effect in the plasma rest frame (Lehnert, 1962;
Uberoi & Das, 1970), while the $\left(\omega-l\Omega\right)$ terms are to be
attributed to the Doppler effect in the plasma rest frame.

###  4.2 Solutions as superpositions of eigenmodes with opposite OAM contents

A general non-singular solution of Poisson equation Eq. (26) describing TG
modes with an OAM content such that $\left|L_{z}/P_{z}\right|=l/\beta$
consists in a linear combination of two counter-rotating Bessel modes

$\phi=\phi_{+}J_{l}\left(\beta T_{l}r\right)\exp j\left(\omega t-l\theta-\beta
z\right)+\phi_{-}J_{-l}\left(\beta T_{-l}r\right)\exp j\left(\omega
t+l\theta-\beta z\right),$ (28)

where $l\in\mathbb{N}$ is a positive number and the mode amplitudes
$\phi_{\pm}$ are assumed to be set by the antenna. Classically TG modes are
studied as confined modes in a conducting cylindrical cavity of length $L$ and
radius $a$, which leads to the boundary conditions
$E_{\theta}\left(a\right)=E_{z}\left(a\right)=0$, and from there to
$J_{l}\left(\beta T_{l}a\right)=0$. The axial wave vector $\beta$ is then
determined through the usual requirement of standing wave along $z$, that is
$\beta L=p\pi$ with $p\in\mathbb{Z}$. This set of conditions fully determines
the resonant mode frequency in the cylindrical cavity through $J_{l}\left(p\pi
T_{l}a/l\right)=0$.

Instead of following this approach, we look here for free TG modes propagating
along $z$. Specifically we consider free TG modes displaying the same radial
structure $J_{l}\left(\alpha r\right)$. In other words, instead of the
solution in Eq. (28) consisting in the superposition of modes with opposite
azimuthal mode numbers $\pm l$, different radial structure and identical axial
wave vectors $\beta$, we consider non-singular solutions

$\phi=\phi_{0}J_{l}\left(\alpha r\right)\exp j\left(\omega
t-l\theta-\beta_{l}z\right)+\phi_{0}J_{-l}\left(\alpha r\right)\exp
j\left(\omega t+l\theta-\beta_{-l}z\right).$ (29)

that are the superposition of modes with opposite azimuthal mode numbers $\pm
l$, identical radial structure but different axial wave vectors
$\beta_{+l}\neq\beta_{-l}$. Both the frequency $\omega$ and the transverse
parameter $\alpha$ are set by the antenna, whereas the wave vectors
$\beta_{+l}$ and $\beta_{-l}$ depend on the rotating plasma properties through
the condition

$\beta_{\pm l}T_{\pm l}(\omega,\omega_{ce},\omega_{pe},\Omega)=\alpha$ (30)

which guarantees that $\phi$ verifies the radial Poisson equation Eq. (26).
Following the analysis given in Appendix A, the longitudinal electric field
associated with the $\pm l$ modes in Eq. (29) are eigenvectors of $S_{z}$ with
$0$ eigenvalue and of $L_{z}$ with eigenvalues $\pm l$, and thus also of the
sum $L_{z}+S_{z}$.

###  4.3 Faraday-Fresnel rotation

Going back to Eq. (5), the normalised difference between these wave vectors
$\beta_{+l}$ and $\beta_{-l}$ is

$\frac{\beta_{+l}-\beta_{-l}}{\beta_{+l}+\beta_{-l}}=\frac{T_{-l}-T_{+l}}{T_{-l}+T_{+l}}=\frac{\Omega}{2\left.T_{l}\right|_{\Omega=0}}\left.\frac{\partial\left(T_{-l}-T_{+l}\right)}{\partial\Omega}\right|_{\Omega=0}+\mathcal{O}\left(\frac{\Omega}{\omega}\right)^{2}.$
(31)

From Eq. (27), the lowest order in $\Omega/\omega$ estimate for this phase
shift then writes

$\left.\frac{\beta_{+l}-\beta_{-l}}{\beta_{+l}+\beta_{-l}}\right|_{TG}=\frac{\Omega}{4T_{l}^{2}}\left(\left.\frac{\partial
T_{-l}^{2}}{\partial\Omega}\right|_{\Omega=0}-\left.\frac{\partial
T_{+l}^{2}}{\partial\Omega}\right|_{\Omega=0}\right)=-l\frac{\Omega}{\omega}F_{TG}\left(\omega_{pe},\omega_{ce},\omega\right),$
(32)

where we have defined the form factor

$F_{TG}=\frac{{\omega_{pe}}^{2}}{\left({\omega_{pe}}^{2}-\omega^{2}\right)}\frac{{\omega_{ce}}^{2}}{\left({\omega_{ce}}^{2}-\omega^{2}\right)}\frac{\left({\omega_{uh}}^{2}-2\omega^{2}\right)}{\left({\omega_{uh}}^{2}-\omega^{2}\right)}$
(33)

which is $\mathcal{O}(1)$ at low frequency.

Eq. (32) shows that, indeed, plasma rotation introduces a phase shift between
OAM-carrying TG eigenmodes with opposite azimuthal mode numbers, or in other
words to Faraday-Fresnel rotation. The wave transverse phase structure is
hence rotated as the wave propagates along the rotation and magnetic axis
$\mathbf{e}_{z}$. Comparing now Eq. (32) with the phase shift introduced by
rotation between circularly polarised eigenmodes (Gueroult et al., 2019a,
2020), that is to say to the SAM mechanical Faraday effect, one finds that the
phase shifts are in both case proportional to the ratio of plasma angular
frequency to wave angular frequency $\Omega/\omega$, but that in addition Eq.
(32) depends linearly on the azimuthal wave number $l$. This $\Omega/\omega$
scaling is also consistent with image rotation in an isotropic dielectric
(Götte et al., 2007).

###  4.4 Possible antenna design

To conclude this section we propose here a coupling structure designed to
launch the mode Eq. (29), _i. e_ two TG modes with identical radial structure
but opposite azimuthal mode numbers. For this we draw on the inductive antenna
design reviewed by Stenzel (2019).

The proposed configuration is shown in Fig. 1(b). It consists in a capacitive
antenna made of $2l$ phased conductive plates positioned perpendicularly to
the magnetic field $\mathbf{B}_{0}$ in the plane $z=0$ and arranged along the
polar angle $\theta$ at

$\theta_{n}=n\frac{\pi}{l},\qquad n\in\left[0,1,2,3...2l-1\right],$ (34)

on a ring of radius $a$. The radius of the antenna ring $a$ is chosen to match
the first extremum of $J_{l}\left(\alpha r\right)$. To both have a good
impedance matching to the $\pm l$ TG modes components of Eq. (29) and minimise
the excitation of $\pm l^{\prime}$ satellites, the size and shape of the $2l$
plates are chosen similar to the size and shape of the pattern
$\left|J_{l}\left(\alpha r\right)\cos\left(l\theta\right)\right|$ illustrated
in Fig. 1(a) for $l=4$. The applied oscillating bias $\phi_{n}(t)$ on each of
these azimuthally distributed $2l$ plates must then verify

$\phi_{n}\left(t\right)=\phi_{0}J_{l}\left(\alpha a\right)\left[\exp-
jn\pi+\left(-1\right)^{l}\exp jn\pi\right]\exp j\omega t,\qquad
n\in\left[0,1,2,3...2l-1\right],$ (35)

to preferentially excite the targeted modes

$\phi\left(r,\theta,z,t\right)=\phi_{0}\left[J_{+l}\left(\alpha r\right)\exp
j\left(-l\theta-\beta_{+l}z\right)+J_{-l}\left(\alpha r\right)\exp
j\left(l\theta-\beta_{-l}z\right)\right]\exp j\omega t.$ (36)

Going further, an improved antenna design could be constructed by adding a
second ring of $2l$ phased conductive plates near the second extremum of
$J_{\left|l\right|}\left(\alpha r\right)$, as illustrated in Fig. 1(c).

Figure 1: (a) Bessel TG mode $l=4$ at $z=0$ and $t=0$. The radius $a$ of
extrema on the first ring verifies $\alpha a=5.317$. (b) Corresponding
capacitive antenna with one ring of phased polarised plates located at $z=0$
and $r=a$. (c) Advanced antenna using two rings of polarised plates.

##  5 Faraday-Fresnel rotation for Whistler-Helicon waves

Whistler waves in a cold magnetised plasma at rest are right circularly
polarised waves whose propagation below $\omega_{pe}$ is sustained by the
electron Hall current. Because of the low-frequency nature of these waves, the
displacement current in Maxwell-Ampère equation can be neglected, and
propagation is characterised by the dispersion relation given in Eq. (3). The
antennae design to excite this WH branch (Stenzel & Urrutia, 2014; Stenzel,
2019; Urrutia & Stenzel, 2015b), as well as the properties associated with
their angular momentum dynamics, have been extensively investigated in Refs.
(Urrutia & Stenzel, 2015a, 2016; Stenzel & Urrutia, 2015b, a, 2018).

Consider Whistler-Helicon (WH) electric $\left(\mathbf{E}\right)$ and magnetic
$\left(\mathbf{B}\right)$ fields with helical phase fronts propagating along
the $z$ axis and rotating around this axis

$\displaystyle\mathbf{E}$ $\displaystyle=$
$\displaystyle\mathbf{E}\left(r\right)\exp j\left(\omega t-l\theta-\beta
z\right),$ (37) $\displaystyle\mathbf{B}$ $\displaystyle=$
$\displaystyle\mathbf{B}\left(r\right)\exp j\left(\omega t-l\theta-\beta
z\right).$ (38)

This electromagnetic perturbation is solution to Maxwell-Faraday and Maxwell-
Ampère equations

$\displaystyle\bm{\nabla}\times\mathbf{E}$ $\displaystyle=$
$\displaystyle-j\omega\mathbf{B},$ (39)
$\displaystyle\bm{\nabla}\times\mathbf{B}$ $\displaystyle=$
$\displaystyle\mu_{0}\mathbf{J},$ (40)

and is associated with an electron velocity

$\mathbf{v}=\mathbf{v}\left(r\right)\exp j\left(\omega t-l\theta-\beta
z\right)$ (41)

and an electron current

$\mathbf{J}=-n_{0}e\mathbf{v}-ne\mathbf{v}_{0},$ (42)

with $\mathbf{v}_{0}$ the equilibrium Brillouin velocity given in Eq. (7).
However, Maxwell-Ampère equation Eq. (40) can be used to get
$\bm{\nabla}\cdot\mathbf{J}=0$ and from there $j\omega ne=0$, so that the
current simply writes $\mathbf{J}=-n_{0}e\mathbf{v}$.

###  5.1 Dispersion relation

The linearised momentum balance

$j\omega\mathbf{v}+\left(\mathbf{v}_{0}\cdot\bm{\nabla}\right)\mathbf{v}+\left(\mathbf{v}\cdot\bm{\nabla}\right)\mathbf{v}_{0}=-\frac{e}{m_{e}}\left(\mathbf{E}+\mathbf{v}_{0}\times\mathbf{B}+\mathbf{v}\times\mathbf{B}_{0}\right)$
(43)

allows, with the help of Eqs. (14, 15), to write the electron current as a
function of the fields

$j\left(\omega-l\Omega\right)\mathbf{J}+\left(2\Omega-\omega_{ce}\right)\mathbf{e}_{z}\times\mathbf{J}={\omega_{pe}}^{2}\varepsilon_{0}\left(\mathbf{E}+\Omega
r\mathbf{e}_{\theta}\times\mathbf{B}\right).$ (44)

The operator $\bm{\nabla}\times$ is then applied to Eq. (44) to eliminate both
the electric field $\mathbf{E}$ and the current $\mathbf{J}$. Using Eqs. (10,
11, 39, 40) finally yields the wave equation

$\bm{\nabla}\times\bm{\nabla}\times\mathbf{B}-\beta\frac{\omega_{ce}-2\Omega}{\omega-l\Omega}\bm{\nabla}\times\mathbf{B}+\frac{{\omega_{pe}}^{2}}{c^{2}}\mathbf{B=0}.$
(45)

One verifies that Eq. (45) reduces to the classical WH equation (Klozenberg et
al., 1965) for $\Omega=0$. In addition, Eq. (45) can be rewritten as

$\left(\beta_{1}-\bm{\nabla}\times\right)\left(\beta_{2}-\bm{\nabla}\times\right)\mathbf{B}=\mathbf{0}$
(46)

with

$\displaystyle\beta_{1}\beta_{2}$ $\displaystyle=$
$\displaystyle\frac{{\omega_{pe}}^{2}}{c^{2}},$ (47)
$\displaystyle\beta_{1}+\beta_{2}$ $\displaystyle=$
$\displaystyle\beta\frac{\omega_{ce}-2\Omega}{\omega-l\Omega}.$ (48)

The associated second order algebraic equation
$x^{2}-(\beta_{1}+\beta_{2})x+\beta_{1}\beta_{2}=0$ has two real roots. The
large root is related to TG modes whereas the small one describes the
dispersive properties of the WH branch

$\beta_{W}=\frac{\beta}{2}\frac{\omega_{ce}-2\Omega}{\omega-l\Omega}\left[1-\sqrt{1-4\frac{{\omega_{pe}}^{2}}{{\beta_{W}}^{2}c^{2}}\left(\frac{\omega-l\Omega}{\omega_{ce}-2\Omega}\right)^{2}}\right]\approx\frac{{\omega_{pe}}^{2}}{\beta_{W}c^{2}}\frac{\omega-l\Omega}{\omega_{ce}-2\Omega},$
(49)

where we have used the WH ordering $\omega_{ce}\gg\omega$ to obtain an
estimate for $\beta_{W}$ in the last equation. Note that we recover Eq. (3)
for $\Omega=0$, and also that
$\beta_{W}\left(+l\right)\neq\beta_{W}\left(-l\right)$ for $\Omega\neq 0$.
Similarly to the analysis done above for TG modes, further examination shows
that the $\left(\omega_{ce}-2\Omega\right)$ terms are associated with the
Coriolis effect in the plasma rest frame (Lehnert, 1962; Uberoi & Das, 1970)
while the $\left(\omega-l\Omega\right)$ terms come from the Doppler effect in
the plasma rest frame.

Now by introducing

$K_{l}=\frac{{\omega_{pe}}^{2}}{c^{2}}\frac{\omega-l\Omega}{\omega_{ce}-2\Omega}$
(50)

such that $K_{+l}=K_{-l}$ for $\Omega=0$ but $K_{+l}\neq K_{-l}$ for
$\Omega\neq 0$ and

$K_{0}=\frac{{\omega_{pe}}^{2}}{c^{2}}\frac{\omega}{\omega_{ce}},$ (51)

one finds that Maxwell-Ampere equation for the WH branch reduces to

$\bm{\nabla}\times\mathbf{B}=\left(K_{l}/\beta\right)\mathbf{B}.$ (52)

Applying the operator $\bm{\nabla}\times$ to this last relation leads to the
wave equation

$\Delta\mathbf{B}+\left(K_{l}^{2}/\beta^{2}\right)\mathbf{B}=\mathbf{0}$ (53)

whose $z$ component writes

$\frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial B_{z}}{\partial
r}-\frac{l^{2}}{r^{2}}B_{z}+\left(\frac{K_{l}^{2}}{\beta^{2}}-\beta^{2}\right)B_{z}=0.$
(54)

###  5.2 Solutions as superpositions of eigenmodes with opposite OAM contents

Following a similar approach to that used for TG modes, we look for solutions
of Eq. (54) of the form

$B_{z}=B_{z0}\left[J_{l}\left(\alpha r\right)\exp j\left(\omega
t-l\theta-\beta_{l}z\right)+J_{-l}\left(\alpha r\right)\exp j\left(\omega
t+l\theta-\beta_{-l}z\right)\right],$ (55)

that is to say superpositions of eigenmodes with opposite azimuthal mode
numbers $\pm l$, identical radial structure $J_{l}\left(\alpha r\right)$ but
different axial wave vectors $\beta_{+l}\neq\beta_{-l}$. Both the frequency
$\omega$ and the transverse parameter $\alpha$ are set by the antenna while
the wave vectors $\beta_{\pm l}$ is determined by the properties of the
rotating plasma through the relation

$\beta_{\pm l}^{-2}K_{\pm
l}^{2}(\omega,\omega_{ce},\omega_{pe},\Omega)-\beta_{\pm l}^{2}=\alpha^{2}$
(56)

which guarantees that $B_{z}$ verifies Eq. (54). Separating the $\pm l$
contributions, Eq. (56) can be rewritten as

$\beta_{l}^{-2}K_{l}^{2}-\beta_{l}^{2}=\beta_{-l}^{-2}K_{-l}^{2}-\beta_{-l}^{2}=\alpha^{2}$
(57)

whose solutions are

$\beta_{\pm l}^{2}=\sqrt{\alpha^{4}+4K_{\pm l}^{2}}-\alpha^{2}.$ (58)

###  5.3 Faraday-Fresnel rotation

Expanding $K_{\pm l}^{2}$ given in Eq. (50) for the small parameter
$\Omega/\omega$ and assuming WH ordering $\omega_{ce}\gg\omega$ yields

$K_{\pm l}^{2}=K_{0}^{2}\left(1\mp
2l\frac{\Omega}{\omega}\right)+\mathcal{O}\left[\left(\frac{\Omega}{\omega}\right)^{2}\right].$
(59)

Meanwhile, from Eq. (58),

$\frac{\beta_{+l}^{2}-\beta_{-l}^{2}}{\left(\beta_{+l}+\beta_{-l}\right)^{2}}=\frac{\sqrt{\alpha^{4}+4K_{+l}^{2}}-\sqrt{\alpha^{4}+4K_{-l}^{2}}}{2\sqrt{\alpha^{4}+4K_{0}^{2}}-2\alpha^{2}}.$
(60)

Plugging Eq (59) into this last expression finally leads to the lowest order
in $\Omega/\omega$ estimate for the OAM phase shift

$\left.\frac{\beta_{+l}-\beta_{-l}}{\beta_{+l}+\beta_{-l}}\right|_{WH}=-l\frac{\Omega}{\omega}F_{WH}\left(\omega_{pe},\omega_{ce},\omega\right),$
(61)

where we have defined the form factor

$F_{WH}=\frac{4K_{0}^{2}}{\alpha^{4}+4K_{0}^{2}-\alpha^{2}\sqrt{\alpha^{4}+4K_{0}^{2}}}.$
(62)

Similarly to Eq. (33) which was obtained for TG modes, Eq. (62) shows that
plasma rotation introduces a phase shift between OAM-carrying WH eigenmodes
with opposite azimuthal mode numbers, or in other words to Faraday-Fresnel
rotation. In both instances, the OAM phase shift is proportional to the plasma
angular frequency to wave angular frequency ratio $\Omega/\omega$, and to the
azimuthal wave number $l$. However, closer examination of Eq. (33) and Eq.
(62) shows that the form factor $F$ characterising Faraday-Fresnel rotation
does depend on $\alpha$ (_i. e._ the wave radial structure) for WH modes,
while it did not for TG modes.

###  5.4 Possible antenna design

To conclude this section we point here to a coupling structure designed to
launch WH modes of the type Eq. (55). This type of antenna has been
extensively investigated, both theoretically and experimentally, in Refs.
(Stenzel & Urrutia, 2014; Urrutia & Stenzel, 2015b), and we refer the readers
to the review paper by Stenzel (2019) for an in depth analysis of their
properties.

The envisioned configuration is shown in Fig. 2(b). It consists in an
inductive antenna made of $2l$ phased current loops positioned perpendicularly
to the magnetic field $\mathbf{B}_{0}$ in the plane $z=0$ and arranged along
the polar angle $\theta$ at

$\theta_{n}=n\pi/l,\qquad n\in\left[0,1,2,3...2l-1\right],$ (63)

on a ring of radius $a$. The currents $I_{n}\left(t\right)$ of these $2l$
current loops are phased according to

$I_{n}\left(t\right)=B_{z0}J_{l}\left(\alpha a\right)\left[\exp-
jn\pi+\left(-1\right)^{l}\exp+jn\pi\right]\exp j\omega t,\qquad
n\in\left[0,1,2,3...2l-1\right],$ (64)

while the radius of the antenna ring $a$ is chosen to match the first extremum
of $J_{l}\left(\alpha r\right)$. The set of parameters
$(\alpha,B_{z0},l,\omega)$ are hence all set by the antenna array.

To both have a good impedance matching to the $\pm l$ WH modes components of
Eq. (55) and minimize the excitation of $\pm l^{\prime}$ satellites the size
and shape of the $2l$ loops should be taken as close as possible to the size
and shape of the wave pattern $\left|J_{l}\left(\alpha
r\right)\cos\left(l\theta\right)\right|$ illustrated in Fig. 2(a).
Alternatively, an improved antenna array can be designed with a second ring of
$2l$ current loops near the second extremum of $J_{\left|l\right|}\left(\alpha
r\right)$, as illustrated in Fig. 2(c).

Figure 2: Bessel WH mode $l=4$ at $z=0$ and $t=0$. The radius $a$ of extrema
on the first ring verifies $\alpha a=5.317$. (b) Corresponding inductive
antenna with one ring of phased current loops located at $z=0$ and $r=a$. (c)
Advanced antenna using two rings of current loops.

##  6 Faraday-Fresnel splitting in a rotating plasma

Consider now a wave packet in a plasma exhibiting both a transverse structure
and a longitudinal structure. The rotating phase of the wave packet travels at
the phase velocity $\omega/\beta$ along the $z$ axis, whereas the longitudinal
envelope of the wave packet travels at the group velocity
$\partial\omega/\partial\beta$ along this axis (Stix, 1992; Urrutia & Stenzel,
2015b).

It has been shown in Sections 4 and 5 that plasma rotation introduces a phase
shift between TG and WH OAM-carrying modes with opposite azimuthal mode
numbers, and that for sufficiently small plasma angular frequency $\Omega$ the
longitudinal wave vectors of these rotating modes write

$\beta_{\pm l}=\left.\beta\right|_{\Omega=0}\left[1\mp
l\frac{\Omega}{\omega}F\left(\omega_{pe},\omega_{ce},\omega\right)\right].$
(65)

In these conditions the longitudinal phase velocity of these modes then write

$V_{\varphi,\pm l}=\omega/\beta_{\pm l}=V_{\varphi}\left[1\pm
l\frac{\Omega}{\omega}F\right]$ (66)

with $V_{\varphi}$ = $\left.\omega/\beta\right|_{\Omega=0}$ the phase velocity
along the $z$ axis in a plasma at rest $\Omega=0$. Meanwhile, deriving Eq.
(65) with respect to $\omega$ yields

$V_{g,\pm l}^{-1}=V_{g}^{-1}\left[\left(1\mp
l\frac{\Omega}{\omega}F\right)\mp\frac{V_{g}}{V_{\varphi}}l\Omega\left(\frac{\partial
F}{\partial\omega}-\frac{F}{\omega}\right)\right]$ (67)

with $V_{g,\pm l}=\partial\omega/\partial\beta_{\pm l}$ the longitudinal group
velocity of these $\pm l$ modes, and
$V_{g}=\partial\left.\beta\right|_{\Omega=0}/\partial\omega$ the group
velocity for zero rotation. Assuming that all factors involving the small
parameter $\Omega/\omega$ are small, inverting this last relation yields the
result

$\frac{V_{g,\pm l}}{V_{g}}=1\pm
l\frac{\Omega}{\omega}\left[\left(1-\frac{V_{g}}{V_{\varphi}}\right)F+\frac{V_{g}}{V_{\varphi}}\omega\frac{\partial
F}{\partial\omega}\right].$ (68)

Eq. (68) shows that in the presence of rotation ($\Omega\neq 0$) the
longitudinal group velocity of OAM-carrying eigenmodes with opposite azimuthal
wave numbers - such as the TG and WH modes studied in Section 4 and 5 \- is
different, whereas they are equal absent rotation $\Omega=0$. A single
envelope wave packet consisting of a mix of these OAM-carrying eigenmodes with
opposite azimuthal wave numbers will hence be split into two envelopes. By
analogy with the Faraday splitting of circularly polarised eigenmodes (SAM)
(Weng et al., 2017), we refer to this new plasma effect as Faraday-Fresnel
splitting.

##  7 Conclusions

In this study we have identified and described two new plasma wave effects
which arise as a result of plasma rotation. These effects are traced back to
the adiabatic coupling which takes place between the orbital angular momentum
(OAM) of a rotating wave and the OAM of the rotating plasma.

Through an analysis of the response of a rotating magnetised plasma to
perturbations carrying OAM, it is shown that plasma rotation introduces a
phase shift between OAM-carrying eigenmodes having opposite azimuthal modes
numbers. As a result the transverse wave pattern of a wave obtained by
superposition of these counter-rotating OAM eigenmodes rotates as the wave
propagates. Because of the analogy both with the classical Faraday effect in a
magnetised plasma - which relates to wave’s spin angular momentum (SAM) - and
the rotatory photon drag known in rotating isotropic dielectric - which
relates to the wave’s OAM, this plasma effect is referred to as Faraday-
Fresnel rotation. The dependence of this phase shift on plasma parameters is
derived analytically here for two classical plasma modes, namely, Trivelpiece-
Gould (TG) and Whistler-Helicon (WH) modes, as shown in Eq. (32) and Eq. (61),
respectively.

The difference in longitudinal wave vectors for counter-rotating OAM
eigenmodes is further shown to translate to different longitudinal group
velocity for each of these modes, as given in Eq. (68). A direct consequence
of this result predicted here is the splitting of a wave packet containing a
superposition of these modes, that is to say that the two counter-rotating
eigenmodes will be progressively spatially separated. By analogy with the
Faraday splitting of circularly polarised eigenmodes (SAM), this new plasma
effect is referred to as Faraday-Fresnel splitting.

Searching for the basic physic mechanism behind these effects, a closer
examination of Eq. (27) and (50) reveals that rotation manifests itself in the
wave propagation properties through two additional terms besides the usual
$\omega_{pe}/\omega$ and $\omega_{ce}/\omega$ terms associated with
dispersion. The first one is $\left(\omega_{ce}-2\Omega\right)$ and simply
accounts for the modification of the magnetic force by the Coriolis force as
the rest frame of the plasma is rotating at $\Omega$. The second is
$\left(\omega-l\Omega\right)$ and is a consequences of the rotational Doppler
shift as the rest frame of the plasma is rotating at $\Omega$. One is however
typically used to find centrifugal forces affecting the dynamics alongside
Coriolis forces and Doppler shifts in rotating media. One might thus be left
questioning what the role of centrifugal forces is here. The answer is not to
be found in Eq. (27) and (50) but rather in the transverse structure of the
wave. Indeed, since

$J_{l}\left(\alpha r\right)\sim\frac{\left(\alpha r\right)^{l}}{l!}+...,$ (69)

the maxima of the amplitude $J_{l}\left(\alpha r\right)$ are pushed radially
outward for higher angular momentum $l$. This behaviour is reminiscent of the
effect of centrifugal forces. For very high $l$ cylindrical waves, this
pronounced radial outward localisation of the field amplitude is the so called
whispering gallery effect associated with centrifugal forces.

Looking ahead, the Faraday-Fresnel rotation (FFR) and Faraday-Fresnel
splitting (FFS) effects identified in this study open interesting
perspectives. An immediate application would be to use FFR to measure rotation
in laboratory experiments, and notably in tokamaks where rotation provides a
means to control of turbulence and enhance confinement. But these results also
allows for the search of more exotic coupling mechanisms between the plasma
wave’s OAM and plasma rotation. To name a few, SAM cross polarisation
scattering described by Lehner et al. (1989) might be extended to an OAM
scheme where the vorticity fluctuation might now be probed through OAM-
carrying waves. Another promising direction may lie the Zeeman splitting of
the plasmon oscillation identified by Robiche & Rax (2008), for which the
effect of an additional rotation with OAM eigenmodes rather than with simple
SAM eigenmodes should be evaluated. Besides these effects, another important
question that remains to be examined is the complementary active or resonant
part of the coupling between rotating OAM-carrying plasma waves and and the
rotating plasma OAM.

Finally, a strong motivation to focus our analysis on TG and WH modes is that
they are popular low frequency modes for plasma experiments. This should in
turn facilitate experimental efforts to test these findings. Using a pair of
antennae such as those described in Sections 4 and 5 in an emitter-receiver
configuration, the results obtained here predict that the signal on the
receiver should exhibit a double maximum profile as the receiver is rotated
along the azimuthal angle, exposing that two $\pm l$ excited rotating waves
are differentially dragged by the rotating plasma. It should however be
stressed that the new results uncovered in this study for TG and WH modes are
expected to be similarly found for other branches of plasma waves. One example
is compressional rotating Alfvén waves for which a relation akin to that
obtained in Eq. (54) can be derived from ideal MHD for the compressional
magnetic field, suggesting that FFR will be at play. An in-depth systematic
analysis of this broader problem is left for future studies.

## References

  * Allen et al. (1992) Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woerdman, J. P. 1992 Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A 45 (11), 8185–8189.
  * Anderson et al. (1958) Anderson, O. A., Baker, W. R., Bratenahl, A., Ise, J. J., Kunkel, W. B., Stone, J. M. & Furth, H. P. 1958 In Proceedings of the United Nations international conference on the peaceful uses of atomic energy, pp. 155–160.
  * Balbus & Hawley (1998) Balbus, S. A. & Hawley, J. F. 1998 Instability, turbulence, and enhanced transport in accretion disks. Rev. Modern Phys. 70 (1), 1–53.
  * Barnett & Allen (1994) Barnett, S. M. & Allen, L. 1994 Orbital angular momentum and nonparaxial light beams. Opt. Commun. 110 (5-6), 670–678.
  * Barnett et al. (2017) Barnett, S. M., Babiker, M. & Padgett, M. J. 2017 Optical orbital angular momentum. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 375 (2087), 20150444.
  * Bekhtenev et al. (1980) Bekhtenev, A. A., Volosov, V. I., Pal’chikov, V. E., Pekker, M. S. & Yudin, Y. N. 1980 Problems of a thermonuclear reactor with a rotating plasma. Nucl. Fusion 20 (5), 579–598.
  * Benomar et al. (2018) Benomar, O., Bazot, M., Nielsen, M. B., Gizon, L., Sekii, T., Takata, M., Hotta, H., Hanasoge, S., Sreenivasan, K. R. & Christensen-Dalsgaard, J. 2018 Asteroseismic detection of latitudinal differential rotation in 13 sun-like stars. Science 361 (6408), 1231–1234.
  * Bonnevier (1966) Bonnevier, B. 1966 Diffusion due to ion-ion collisions in a multicomponent plasma. Ark. Fys. 33, 255.
  * Brillouin (1945) Brillouin, L. 1945 A theorem of Larmor and its importance for electrons in magnetic fields. Phys. Rev. 67 (7-8), 260–266.
  * Chen et al. (2017) Chen, Q., Qin, H. & Liu, J. 2017 Photons, phonons, and plasmons with orbital angular momentum in plasmas. Sci. Rep. 7 (1).
  * Davidson (2001) Davidson, R. C. 2001 Physics of Nonneutral Plasmas. Imperial College Press.
  * Dolgolenko & Muromkin (2009) Dolgolenko, D. A. & Muromkin, Y. A. 2009 Plasma isotope separation based on ion cyclotron resonance. Physics-Uspekhi 52 (4), 345–357.
  * Dolgolenko & Muromkin (2017) Dolgolenko, D. A. & Muromkin, Y. A. 2017 Separation of mixtures of chemical elements in plasma. Phys. Usp. 60 (10), 994.
  * Ellis et al. (2005) Ellis, R. F., Case, A., Elton, R., Ghosh, J., Griem, H., Hassam, A., Lunsford, R., Messer, S. & Teodorescu, C. 2005 Steady supersonically rotating plasmas in the Maryland centrifugal experiment. Phys. Plasmas 12 (5), 055704\.
  * van Enk & Nienhuis (1994) van Enk, S. J. & Nienhuis, G. 1994 Spin and orbital angular momentum of photons. Europhys. Lett. 25 (7), 497–501.
  * Faraday (1846) Faraday, M. 1846 On the magnetization of light and the illumination of magnetic lines of force. Phil. Trans. Roy. Soc. London 136 (1), 1–20.
  * Fetterman & Fisch (2008) Fetterman, A. J. & Fisch, N. J. 2008 Alpha channeling in a rotating plasma. Phys. Rev. Lett. 101 (20), 205003–.
  * Fetterman & Fisch (2010) Fetterman, A. J. & Fisch, N. J. 2010 Alpha channeling in rotating plasma with stationary waves. Physics of Plasmas 17 (4), 042112.
  * Franke-Arnold et al. (2011) Franke-Arnold, S., Gibson, G., Boyd, R. W. & Padgett, M. J. 2011 Rotary photon drag enhanced by a slow-light medium. Science 333 (6038), 65.
  * Fresnel (1818) Fresnel, A. 1818 Lettre d’ Augustin Fresnel à François Arago sur l’ influence du mouvement terrestre dans quelques phénomènes d’optique. Ann. Chim. Phys 9, 57–66.
  * Gough (1986) Gough, W. 1986 The angular momentum of radiation. European J. Phys. 7 (2), 81–87.
  * Gueroult et al. (2020) Gueroult, R., Rax, J.-M. & Fisch, N. J. 2020 Enhanced tuneable rotatory power in a rotating plasma. Phys. Rev. E 102 (5), 051202(R).
  * Gueroult et al. (2019a) Gueroult, R., Shi, Y., Rax, J.-M. & Fisch, N. J. 2019a Determining the rotation direction in pulsars. Nat. Commun. 10 (1), 3232.
  * Gueroult et al. (2019b) Gueroult, R., Zweben, S. J., Fisch, N. J. & Rax, J.-M. 2019b E x B configurations for high-throughput plasma mass separation: An outlook on possibilities and challenges. Phys. Plasmas 26 (4), 043511.
  * Götte et al. (2007) Götte, J. B., Barnett, S. M. & Padgett, M. 2007 On the dragging of light by a rotating medium. Proc. R. Soc. A 463 (2085), 2185–2194.
  * Jones (1976) Jones, R. V. 1976 Rotary aether drag. Proc. R. Soc. London, Ser. A 349 (1659), 423–439.
  * Klozenberg et al. (1965) Klozenberg, J. P., McNamara, B. & Thonemann, P. C. 1965 The dispersion and attenuation of helicon waves in a uniform cylindrical plasma. J. Fluid Mech. 21 (3), 545–563.
  * Kolmes et al. (2019) Kolmes, E. J., Ochs, I. E., Mlodik, M. E., Rax, J.-M., Gueroult, R. & Fisch, N. J. 2019 Radial current and rotation profile tailoring in highly ionized linear plasma devices. Phys. Plasmas 26 (8), 082309.
  * Kostyukov et al. (2002) Kostyukov, I. Y., Shvets, G., Fisch, N. J. & Rax, J. M. 2002 Magnetic-field generation and electron acceleration in relativistic laser channel. Physics of Plasmas 9 (2), 636–648.
  * Krishnan et al. (1981) Krishnan, M., Geva, M. & Hirshfield, J. L. 1981 Plasma centrifuge. Phys. Rev. Lett. 46 (1), 36–38.
  * Leach et al. (2008) Leach, J., Wright, A. J., Götte, J. B., Girkin, J. M., Allen, L., Franke-Arnold, S., Barnett, S. M. & Padgett, M. J. 2008 “Aether Drag” and moving images. Phys. Rev. Lett. 100 (15), 153902.
  * Lehner et al. (1989) Lehner, T., Rax, J. M. & Zou, X. L. 1989 Linear mode conversion by magnetic fluctuations in inhomogeneous magnetized plasmas. Europhys. Lett. 8 (8), 759–764.
  * Lehnert (1962) Lehnert, B. 1962 Stabilization of flute disturbances by the Coriolis force. Phys. Fluids 5 (6), 740.
  * Lehnert (1971) Lehnert, B. 1971 Rotating plasmas. Nucl. Fusion 11 (5), 485–.
  * Liziakin et al. (2021) Liziakin, G., Antonov, N., Usmanov, R., Melnikov, A., Timirkhanov, R., Vorona, N., Smirnov, V. S., Oiler, A., Kislenko, S., Gavrikov, A. & Smirnov, V. P. 2021 Experimental demonstration of plasma mass separation in a configuration with a potential well and crossed electric and magnetic fields. Plasma Phys. Control. Fusion 63 (3), 032002.
  * Mendonça (2012a) Mendonça, J. T. 2012a Kinetic description of electron plasma waves with orbital angular momentum. Phys. Plasmas 19 (11), 112113.
  * Mendonça (2012b) Mendonça, J. T. 2012b Twisted waves in a plasma. Plasma Phys. Control. Fusion 54 (12), 124031\.
  * Mendonca et al. (2009) Mendonca, J. T., Ali, S. & Thidé, B. 2009 Plasmons with orbital angular momentum. Phys. Plasmas 16 (11), 112103.
  * Ochs & Fisch (2017) Ochs, I. E. & Fisch, N. J. 2017 Particle orbits in a force-balanced, wave-driven, rotating torus. Phys. Plasmas 24, 092513\.
  * Ohkawa & Miller (2002) Ohkawa, T. & Miller, R. L. 2002 Band gap ion mass filter. Phys. Plasmas 9 (12), 5116–5120.
  * Padgett et al. (2006) Padgett, M., Whyte, G., Girkin, J., Wright, A., Allen, L., Öhberg, P. & Barnett, S. M. 2006 Polarization and image rotation induced by a rotating dielectric rod: An optical angular momentum interpretation. Opt. Lett. 31 (14), 2205.
  * Player (1976) Player, M. A. 1976 On the dragging of the plane of polarization of light propagating in a rotating medium. Proc. R. Soc. A 349 (1659), 441.
  * Prasad & Krishnan (1987) Prasad, R. R. & Krishnan, M. 1987 Rigid rotor equilibria of multifluid, neutral plasma columns in crossed electric and magnetic fields. Phys. Fluids 30 (11), 3496.
  * Rax et al. (2015) Rax, J. M., Fruchtman, A., Gueroult, R. & Fisch, N. J. 2015 Breakdown of the Brillouin limit and classical fluxes in rotating collisional plasmas. Phys. Plasmas 22 (9), 092101\.
  * Rax & Gueroult (2016) Rax, J.-M. & Gueroult, R. 2016 Rotation and instabilities for isotope and mass separation. J. Plasma Phys. 82, 595820504.
  * Rax et al. (2017) Rax, J. M., Gueroult, R. & Fisch, N. J. 2017 Efficiency of wave-driven rigid body rotation toroidal confinement. Phys. Plasmas 24 (3), 032504.
  * Rax et al. (2019) Rax, J. M., Kolmes, E. J., Ochs, I. E., Fisch, N. J. & Gueroult, R. 2019 Nonlinear ohmic dissipation in axisymmetric DC and RF driven rotating plasmas. Phys. Plasmas 26 (1), 012303.
  * Rax & Robiche (2010) Rax, J.-M. & Robiche, J. 2010 Theory of unfolded cyclotron accelerator. Physics of Plasmas 17 (10), –.
  * Rax et al. (2007) Rax, J.-M., Robiche, J. & Fisch, N. J. 2007 Autoresonant ion cyclotron isotope separation. Phys. Plasmas 14 (4), 043102.
  * Robiche & Rax (2008) Robiche, J. & Rax, J.-M. 2008 Relativistic wave-induced splitting of the Langmuir mode in a magnetized plasma. Physical Review E 77 (1), 016402.
  * Ruseckas et al. (2007) Ruseckas, J., Juzeliũnas, G., Öhberg, P. & Barnett, S. M. 2007 Polarization rotation of slow light with orbital angular momentum in ultracold atomic gases. Phys. Rev. A 76 (5), 053822.
  * Shinohara & Horii (2007) Shinohara, S. & Horii, S. 2007 Initial trial of plasma mass separation by crossed electric and magnetic fields. Jap. J. App. Phys. 46 (7R), 4276–.
  * Shukla (2012) Shukla, P. K. 2012 Twisted shear Alfvén waves with orbital angular momentum. Phys. Lett. A 376 (44), 2792–2794.
  * Shvets et al. (2002) Shvets, G., Fisch, N. J. & Rax, J.-M. 2002 Magnetic field generation through angular momentum exchange between circularly polarized radiation and charged particles. Physical Review E 65 (4), 046403–.
  * Stenzel (2016) Stenzel, R. L. 2016 Whistler waves with angular momentum in space and laboratory plasmas and their counterparts in free space. Adv. Phys. X 1 (4), 687–710.
  * Stenzel (2019) Stenzel, R. L. 2019 Whistler modes excited by magnetic antennas: A review. Phys. Plasmas 26 (8), 080501.
  * Stenzel & Urrutia (2014) Stenzel, R. L. & Urrutia, J. M. 2014 Magnetic antenna excitation of Whistler modes. II. Antenna arrays. Phys. Plasmas 21 (12), 122108.
  * Stenzel & Urrutia (2015a) Stenzel, R. L. & Urrutia, J. M. 2015a Helicon modes in uniform plasmas. III. Angular momentum. Phys. Plasmas 22 (9), 092113.
  * Stenzel & Urrutia (2015b) Stenzel, R. L. & Urrutia, J. M. 2015b Helicon waves in uniform plasmas. II. High m numbers. Phys. Plasmas 22 (9), 092112.
  * Stenzel & Urrutia (2016) Stenzel, R. L. & Urrutia, J. M. 2016 Trivelpiece-Gould modes in a uniform unbounded plasma. Phys. Plasmas 23 (9), 092103.
  * Stenzel & Urrutia (2018) Stenzel, R. L. & Urrutia, J. M. 2018 Helicons in uniform fields. II. Poynting vector and angular momenta. Phys. Plasmas 25 (3), 032112.
  * Stix (1971) Stix, T. H. 1971 Some toroidal equilibria for plasma under magnetoelectric confinement. Phys. Fluids 14 (3), 692.
  * Stix (1992) Stix, T. H. 1992 Waves in Plasmas. New York: AIP Press.
  * Thaury et al. (2013) Thaury, C., Guillaume, E., Corde, S., Lehe, R., Le Bouteiller, M., Ta Phuoc, K., Davoine, X., Rax, J. M., Rousse, A. & Malka, V. 2013 Angular-momentum evolution in laser-plasma accelerators. Physical Review Letters 111 (13), 135002–.
  * Trivelpiece & Gould (1959) Trivelpiece, A. W. & Gould, R. W. 1959 Space charge waves in cylindrical plasma columns. J. Appl. Phys. 30 (11), 1784–1793.
  * Uberoi & Das (1970) Uberoi, C. & Das, G. C. 1970 Wave propagation in cold plasma in the presence of the Coriolis force. Plasma Phys. 12 (9), 661–684.
  * Urrutia & Stenzel (2015a) Urrutia, J. M. & Stenzel, R. L. 2015a Helicon modes in uniform plasmas. I. Low m modes. Phys. Plasmas 22 (9), 092111.
  * Urrutia & Stenzel (2015b) Urrutia, J. M. & Stenzel, R. L. 2015b Magnetic antenna excitation of whistler modes. III. Group and phase velocities of wave packets. Phys. Plasmas 22 (7), 072109\.
  * Urrutia & Stenzel (2016) Urrutia, J. M. & Stenzel, R. L. 2016 Helicon waves in uniform plasmas. IV. Bessel beams, Gendrin beams, and helicons. Phys. Plasmas 23 (5), 052112.
  * Weng et al. (2017) Weng, S., Zhao, Q., Sheng, Z., Yu, W., Luan, S., Chen, M., Yu, L., Murakami, M., Mori, W. B. & Zhang, J. 2017 Extreme case of Faraday effect: magnetic splitting of ultrashort laser pulses in plasmas. Optica 4 (9), 1086.
  * Wilcox (1959) Wilcox, J. M. 1959 Review of high-temperature rotating-plasma experiments. Rev. Modern Phys. 31 (4), 1045–1051.
  * Wisniewski-Barker et al. (2014) Wisniewski-Barker, E., Gibson, G. M., Franke-Arnold, S., Boyd, R. W. & Padgett, M. J. 2014 Mechanical Faraday effect for orbital angular momentum-carrying beams. Opt. Express 22 (10), 11690–11697.
  * Zweben et al. (2018) Zweben, S. J., Gueroult, R. & Fisch, N. J. 2018 Plasma mass separation. Phys. Plasmas 25 (9), 090901\.

##  Appendix A Wave and particle spin and orbital angular momentum in a
magnetised rotating plasma

In this appendix we briefly review how to separate SAM from OAM components for
plasma waves and particles in light of the complementary dynamical and
geometrical points of view. The up/down degree of freedom of the one half spin
$\left(\hbar/2\right)$ of the electrons is not considered in this study and
SAM refers to the cyclotron motion source of the cyclotron magnetic moment.

The viewpoint adopted here is that separation of both wave’s and classical
particles’ angular momentum into SAM and OAM components does not stem from a
quantum point view, but instead is a simple generalisation of the first
König’s theorem. König’s theorem states that the angular momentum of a system
can be decomposed into an external orbital part and an internal part. This
internal part is nowadays called spin part for waves and magnetised charges,
even within a classical framework. The analysis of this separation for plasma
waves and magnetised particles can be carried both from a geometrical point of
view - through the analysis of the transformations properties of the wave
field - and from a dynamical point of view - through the analysis of SAM and
OAM waves and particles coupling. In this appendix we briefly review (i) the
canonical separation between SAM and OAM for a charged particle, then (ii) the
geometrical characterisation of SAM and OAM eigenvectors for a vector field
and (iii) previous results on the resonant coupling of waves SAM with
particles SAM and waves OAM with particles OAM.

###  A.1 Canonical separation between SAM and OAM for a charged particle

Consider a magnetised plasma column with a background magnetic field
$\mathbf{B}$ directed along the $z$ axis and radially polarized by an electric
field $\mathbf{E}$ sustaining a guiding center slow $E\times B$ rotation at
angular velocity $\Omega$. A particle with charge $e$ and mass $m$ is
described by (i) its position $\mathbf{r}$ = $\mathbf{R}_{gc}$ \+
$\mathbf{\rho}_{L}$, where $\mathbf{R}_{gc}$ is the guiding center position
and $\mathbf{\rho}_{L}$ the rotating Larmor radius, and (ii) its velocity
$\mathbf{v}$ = $\mathbf{V}_{d}$ \+ $\mathbf{v}_{c}$, where $V_{d}$ = $\Omega
R_{gc}$ is the guiding center $E\times B$ velocity and $v_{c}$ =
$\omega_{c}\rho_{L}$ the cyclotron velocity. This configuration is illustrated
on Fig. 3. The instantaneous angular momentum is given by $\mathbf{r}\times
m\mathbf{v}$ and its average over the fast cyclotron motion is given by

$\left\langle\mathbf{r}\times
m\mathbf{v}\right\rangle=\left\langle\left[\mathbf{R}_{gc}+\mathbf{\rho}_{L}\right]\times\left[\mathbf{V}_{d}+\mathbf{v}_{c}\right]\right\rangle=m\Omega
R_{gc}^{2}+m\omega_{c}\rho_{L}^{2}.$ (70)

Introducing the classical magnetic moment of the particle
$\mu=m\omega_{c}^{2}\rho_{L}^{2}/2B$, the angular momentum
$\left\langle\mathbf{r}\times m\mathbf{v}\right\rangle$ is then showed to be
the sum of an orbital component $L_{z}$ plus a spin component $S_{z}$
associated with this magnetic moment,

$\displaystyle L_{z}$ $\displaystyle=$ $\displaystyle m\Omega R_{gc}^{2},$
(71) $\displaystyle S_{z}$ $\displaystyle=$ $\displaystyle 2\frac{m}{e}\mu,$
(72)

where $e/(2m)$ is the classical gyromagnetic factor associated with the ratio
of the magnetic moment to the angular momentum.

Figure 3: Cyclotron (SAM) rotation and guiding center (OAM) $E\times B$
rotations in a magnetised rotating plasma column.

###  A.2 Geometrical characterisation of SAM and OAM eigenvectors for a vector
field

Consider now a wave field $\mathbf{A}(\mathbf{r})\exp j\omega t$. From a
purely geometrical point of view, the identification of (i) linear momentum,
(ii) spin and (iii) orbital momentum wave eigenstates can be guided by the
analysis of the transformation properties of the wave under translations and
rotations.

Let us examine the change of this vector field $\mathbf{A}(\mathbf{r})$ under
an active (change of the object $\mathbf{A}$) or a passive (change of the
reference frame and coordinates used to describe $\mathbf{A}$) infinitesimal
translation $\widehat{T}$ associated with the small vector $\delta\mathbf{r}$

$\widehat{T}\mathbf{A}(\mathbf{r})=\mathbf{A}\left(\mathbf{r}\pm\delta\mathbf{r}\right)=\mathbf{A}(\mathbf{r})\pm\delta\mathbf{r.\nabla
A}(\mathbf{r}),$ (73)

with the minus and plus signs associated with the passive and active point of
view, respectively. Eq. (73) can be rewritten as a near identity
transformation

$\widehat{T}\mathbf{A}(\mathbf{r})=\left[\mathbf{I}\pm
j\delta\mathbf{r.}\widehat{\mathbf{P}}\right]\mathbf{A}(\mathbf{r}),$ (74)

where $\mathbf{I}$ is the identity operator and the linear momentum operator
$\widehat{\mathbf{P}}$ is defined in the usual way as
$\widehat{\mathbf{P}}=-j\bm{\nabla}$. The eigenvectors of this linear momentum
operator $\widehat{\mathbf{P}}$ are the plane waves

$\widehat{\mathbf{P}}\left(\exp
j\bm{\beta}\cdot\mathbf{r}\right)=\bm{\beta}\left(\exp
j\bm{\beta}\cdot\mathbf{r}\right).$ (75)

These waves are also solutions of Maxwell-Ampère and Maxwell-Faraday equations
in an homogeneous linear dispersive plasma provided that
$\bm{\beta}\left(\omega\right)$ verifies the dispersion relation. For a finite
translation : $\mathbf{r}\pm\mathbf{a}$, the limit $N\rightarrow+\infty$ of
the iterated operator

$\left(\mathbf{I}\pm
j\left(\mathbf{a}/N\right)\mathbf{\cdot}\widehat{\mathbf{P}}\right)^{N}$ (76)

provides the expected Taylor expansion identity
$\mathbf{A}\left(\mathbf{r}\pm\mathbf{a}\right)$ =
$\left[\exp\left(\pm\mathbf{a}\cdot\bm{\nabla}\right)\right]\mathbf{A}(\mathbf{r})$.

Let us now examine the change of a vector field $\mathbf{A}(\mathbf{r})$ under
an active or passive infinitesimal rotation $\widehat{R}$ associated with a
small $\delta\theta$ rotation around an axis directed by $\mathbf{n}$
($\mathbf{n}^{2}=1$)

$\widehat{R}\mathbf{A}(\mathbf{r})=\pm\delta\theta\mathbf{n}\times\mathbf{A}\left(\mathbf{r}\pm\delta\theta\mathbf{n}\times\mathbf{r}\right)=\mathbf{A}(\mathbf{r})\pm\delta\theta\left[\mathbf{n}\times+\left(\mathbf{n}\times\mathbf{r}\right)\mathbf{.\nabla}\right]\mathbf{A}(\mathbf{r}).$
(77)

Eq. (77) can be rewritten as a near identity transformation displaying the
separation between OAM and SAM operators

$\widehat{R}\mathbf{A}(\mathbf{r})=\left[\mathbf{I}\pm
j\delta\theta\mathbf{n\cdot}\left(\widehat{\mathbf{L}}+\widehat{\mathbf{S}}\right)\right]\mathbf{A}(\mathbf{r}).$
(78)

The OAM operator $\widehat{\mathbf{L}}$ and the SAM operator
$\widehat{\mathbf{S}}$ are thus defined according to the usual relations

$\displaystyle\widehat{\mathbf{L}}$ $\displaystyle=$
$\displaystyle\mathbf{r}\times\widehat{\mathbf{P}}=-j\left(\begin{array}[]{ccc}y\frac{\partial}{\partial
z}-z\frac{\partial}{\partial y}&0&0\\\ 0&z\frac{\partial}{\partial
x}-x\frac{\partial}{\partial z}&0\\\ 0&0&x\frac{\partial}{\partial
y}-y\frac{\partial}{\partial x}\end{array}\right),$ (82)
$\displaystyle\widehat{\mathbf{S}}$ $\displaystyle=$
$\displaystyle-j\mathbf{n}\times\mathbf{=}-j\left(\begin{array}[]{ccc}0&-n_{z}&n_{y}\\\
n_{z}&0&-n_{x}\\\ -n_{y}&n_{x}&0\end{array}\right),$ (86)

where these matrix representations are given on a cartesian basis
$\left(\mathbf{e}_{x},\mathbf{e}_{y}\mathbf{,e}_{z}\right)$. Note that, since
$\widehat{S}_{x}^{2}+\widehat{S}_{y}^{2}+\widehat{S}_{z}^{2}=2\mathbf{I}$ one
recovers the angular momentum rule for a vector
$\widehat{\mathbf{S}}^{2}=s\left(s+1\right)\mathbf{I}$ with $s=1$.

The eigenvectors of this angular momentum operator
$\widehat{\mathbf{L}}+\widehat{\mathbf{S}}$ are the vector spherical
harmonics. However, since we are considering a magnetised plasma, vector
spherical harmonics cannot be solution of Maxwell-Ampère and Maxwell-Faraday
equations. We have to restrict the transformations to translations along the
magnetic field direction and rotations around the magnetic field direction,
and the usefulness of the angular momentum operator is reduced to its $z$
component $\widehat{L}_{z}+\widehat{S}_{z}$. The eigenvectors of the OAM
operator $\widehat{L}_{z}$ are $\exp\pm jl\theta$ with $\theta$ the polar
angle around the magnetic field, while the eigenvectors of the SAM operator
$\widehat{S}_{z}$ are the L and R circularly polarised wave basis with
eigenvalues $\pm 1$ and $\mathbf{e}_{z}$ with the eigenvalue zero, that is

$\displaystyle\widehat{L}_{z}\left(\exp\pm jl\theta\right)$ $\displaystyle=$
$\displaystyle\pm l\left(\exp\pm jl\theta\right),$ (87)
$\displaystyle\widehat{S}_{z}\left(\frac{\mathbf{e}_{x}\pm
j\mathbf{e}_{y}}{\sqrt{2}}\right)$ $\displaystyle=$
$\displaystyle\mp\left(\frac{\mathbf{e}_{x}\pm
j\mathbf{e}_{y}}{\sqrt{2}}\right).$ (88)

Solutions of Maxwell-Ampère and Maxwell-Faraday equations with a phase factor
$\exp\pm jl\theta$ have a well defined OAM in a rotating magnetised plasma
when the magnetic axis is also the rotation axis. Solutions of Maxwell-Ampère
and Maxwell-Faraday equations with a polarisation $\mathbf{e}_{x}\pm
j\mathbf{e}_{y}$ have a well defined SAM. Solutions of Maxwell-Ampère and
Maxwell-Faraday equations with a phase factor $\exp\pm j\beta z$ have a well
defined linear momentum.

###  A.3 Resonant coupling

The separation of OAM and SAM contents of waves outlined above can also be
analysed through the coupling to the magnetised particles when resonant wave
absorption takes place. To illustrate this point we go back to the magnetised
plasma column depicted in Fig. 3. In this configuration the interplay between
the wave’s SAM and OAM components and particles’ SAM and OAM can be summed up
as follows. The wave’s SAM (L or R polarisation) is coupled to the cyclotron
SAM angular momentum ($2m\mu/e$) while the wave’s OAM (helical phase fronts)
is coupled to the OAM drift rotation of the guiding center around the axis of
the plasma column. These separate coupling, which have previously been
identified and studied within the frameworks of rotating thermonuclear trap
(Rax et al., 2017), dissipation in a fully ionized Brillouin flow (Rax et al.,
2019) and mass separation with rotating fields (Rax & Gueroult, 2016), are
recalled here.

Consider first a magnetised plasma column at rest and a transverse wave with
$L_{z}=0$ and $S_{z}\neq 0$, that is to say with a simple oscillating phase
factor $\exp j\left(\omega t-\beta z\right)$ and a polarisation component co-
rotating with the cyclotron motion. The absorption of this wave by a
population of charged particles with mass $m$ and charge $e$ leads to (i)
energy $\mathcal{E}$, (ii) linear momentum $P_{z}=mv_{z}$ and (iii) cyclotron
angular momentum $S_{z}=mv_{c}^{2}/\omega_{c}$ exchange. Here $v_{c}$ is the
cyclotron velocity around the magnetic field, $v_{z}$ the velocity along the
magnetic field and $\omega_{c}$ the cyclotron frequency. For a collisionless
resonant absorption through the classical Doppler shifted harmonic cyclotron
resonance,

$\omega-\beta v_{z}=n\omega_{c},\qquad n\in\mathbb{Z},$ (89)

the absorption of an amount of energy $\delta\mathcal{E}$ by particles is
associated both with a transfer $\delta P_{z}=m\delta v_{z}$ of linear
momentum,

$\delta P_{z}=\frac{\beta}{\omega}\delta\mathcal{E},$ (90)

along the magnetic field lines and with a transfer $\delta S_{z}=mv_{c}\delta
v_{c}/\omega_{c}$ of SAM defined by Eq. (72),

$\delta S_{z}=2\frac{n}{\omega}\delta\mathcal{E},$ (91)

around the magnetic field lines.

Consider now wave absorption in a magnetised rotating plasma column, and
assume this time a wave with a helical phase factor $\exp j\left(\omega
t-l\theta-\beta z\right)$ verifying the eigenvalue relations Eqs. (87, 75).
This wave has an OAM content such that $L_{z}/P_{z}=l/\beta\neq 0$. In
addition to the previous SAM coupling leading to $\delta S_{z}$, an OAM
coupling takes place with the drift $E\times B$ rotation of the guiding
center. The transfer of energy $\delta\mathcal{E}$ from this wave to a
particle is associated with a transfer of OAM around the $z$ axis

$\delta L_{z}=\frac{l}{\omega}\delta\mathcal{E}.$ (92)

Eqs. (90, 91, 92) are simple consequences of energy, linear momentum and
angular momentum conservations

$d\widehat{P}_{z}[\textit{waves}+\textit{particles}]\Big{/}dt=0$ (93)

and

$d\left(\widehat{L}_{z}+\widehat{S}_{z}\right)[\textit{waves}+\textit{particles}]\Big{/}dt=0.$
(94)

Rather than examining these resonant exchange of linear and angular momentum,
we have chosen in this study to consider the adiabatic interaction of waves
and particles, that is to say the coupling between wave OAM and plasma
$E\times B$ rotation on cold dispersion, as an analog to the coupling between
wave SAM and plasma magnetisation on cold dispersion which yields the
classical Faraday effect.