2.1. Polarisation rotation from phase shift between circularly polarised eigenmodes

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

(2.1)\begin{equation} \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) }}. \end{equation}

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 cutoff 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 (2.1) 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 $z$:

(2.2)\begin{equation} \left. \frac{\delta \varphi }{4{\rm \pi} }\right| _{SAM}= \frac{\beta _{L}-\beta _{R}}{\beta _{L}+\beta _{R}}\approx \frac{\omega _{ce}}{\omega }\frac{\omega _{pe}^{2}}{\omega ^{2}}. \end{equation}

The phase difference derived in (2.2) 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 (Jones Reference Jones1976; Player Reference Player1976; Götte et al. Reference Götte, Barnett and Padgett2007; Gueroult et al. Reference Gueroult, Rax and Fisch2020; Steinitz & Averbukh Reference Steinitz and Averbukh2020; Milner et al. Reference Milner, Steinitz, Averbukh and Milner2021).

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

Plasma waves carrying OAM (see Appendix A for a discussion of spin and OAM 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, Ali & Thidé Reference Mendonca, Ali and Thidé2009; Mendonça Reference Mendonça2012a,Reference Mendonçab), although exact solutions of the vector Maxwell equations have recently been derived (Chen, Qin & Liu Reference Chen, Qin and Liu2017). Meanwhile, in cold magnetised plasmas, waves carrying OAM have been reported in the form of TG and WH waves (Stenzel & Urrutia Reference Stenzel and Urrutia2015a; Stenzel Reference Stenzel2016; Urrutia & Stenzel Reference Urrutia and Stenzel2016), as well as predicted for twisted shear Alfvén waves (Shukla Reference Shukla2012). Because our interest in this study is in Brillouin rotating plasmas, that is as it will be detailed in the next section in a plasma rotating as a result of an $\boldsymbol {E}\times \boldsymbol {B}$ configuration in cylindrical geometry, we focus our attention on magnetised plasmas and consider more specifically TG and WH modes.

Whistler–helicon waves are intermediate frequency $\omega _{lh}<\omega <\omega _{ce}$ 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 simplification of the dispersion relation (2.1) to obtain the classical Whistler dispersion

(2.3)\begin{equation} {\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}}. \end{equation}

On the other hand non-rotating 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 $J_{l}( \beta _{\bot }r) \exp j( \omega t-\beta _{TG}z)$ where $J_{l}$ is the ordinary Bessel function of order $l\in \Bbb {Z}$, and the dispersion relation in this case is (Trivelpiece & Gould Reference Trivelpiece and Gould1959)

(2.4)\begin{equation} {\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), } \end{equation}

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

As mentioned above, both TG waves (Urrutia & Stenzel Reference Urrutia and Stenzel2016) and WH waves (Stenzel & Urrutia Reference Stenzel and Urrutia2015a) with helical phase front of the form $\exp j(\omega t\pm l\theta -\beta _{\pm l}z)$ can propagate in a cold magnetised plasma at rest (Stenzel Reference Stenzel2016). 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 (left- and right-circularly polarised modes) given in (2.2), we can define the phase shift between opposite OAM-carrying eigenmodes

(2.5)\begin{equation} \left. \frac{\delta \varphi }{4{\rm \pi} }\right| _{OAM}=\left. \frac{\beta _{{+}l}-\beta _{{-}l}}{\beta _{{+}l}+\beta _{{-}l}}\right| _{TG/W}\approx l \frac{ \varOmega }{\omega }F_{TG/W}\left( \omega _{pe},\omega _{ce},\omega \right), \end{equation}

with $\beta _{\pm l}$ the longitudinal wavevector of the wave with azimuthal mode number $\pm l$. As we will show, this OAM phase difference is zero absent rotation (i.e. for $\varOmega =0$) 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 analogue 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 $\varOmega \neq 0$ leads to $\beta _{+l} \neq \beta _{-}$, and therefore now to a rotation of the transverse wave pattern along $z$, which we refer to as FFR. The dispersive form factor $F$ introduced in (2.5) and derived in (4.18) and (5.26) 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.

4.1. Dispersion relation

For a weak electronic perturbation of this Brillouin flow, the linearised momentum balance (4.3), and the linearised charge conservation relation (4.4),

(4.3)\begin{gather} j\omega \boldsymbol{v}+\left( \boldsymbol{v}_{0}\boldsymbol{\cdot} \boldsymbol{\nabla}\right) \boldsymbol{v}+\left( \boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{\nabla}\right) \boldsymbol{v}_{0} ={-}\frac{e}{m_e}\left( \boldsymbol{E}+\boldsymbol{v}\times \boldsymbol{B}_{0}\right), \end{gather}

(4.4)\begin{gather}j\omega n+n_{0}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v}+\boldsymbol{v}_{0}\boldsymbol{\cdot} \boldsymbol{ \nabla }n = 0, \end{gather}

yield a set of coupled equations relating $n$ to $\phi$. Making use of (3.9), (3.10) and (4.3) can be rearranged as

(4.5)\begin{equation} j\left( \omega -l\varOmega \right) \boldsymbol{v}+\left( 2\varOmega -\omega_{ce}\right) \boldsymbol{e}_{z}\times \boldsymbol{v}=\frac{e}{m_e}\boldsymbol{\nabla }\phi, \end{equation}

while (4.4) can be rewritten as

(4.6)\begin{equation} n_{0}\boldsymbol{\nabla }\boldsymbol{\cdot} \boldsymbol{v}={-}j\left( \omega -l\varOmega \right) n. \end{equation}

We then apply the three linear operators $\boldsymbol {\nabla }\times$, $\boldsymbol {\nabla }\boldsymbol {\cdot }$ and $\boldsymbol {e}_z\boldsymbol {\cdot }$ to (4.5) to obtain three relations relating the divergence, the curl and the longitudinal component of the velocity $\boldsymbol {v}$. With the help of (3.6) and (3.8) this yields

(4.7)\begin{gather} j\left( \omega -l\varOmega \right) \boldsymbol{\nabla }\times \boldsymbol{v}+ \left(2\varOmega -\omega _{ce}\right) \left[ \left( \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{v}\right) \boldsymbol{e}_{z}+j\beta \boldsymbol{v}\right] = \boldsymbol{0}, \end{gather}

(4.8)\begin{gather}j\left( \omega -l\varOmega \right) \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{v}- \left(2\varOmega -\omega _{ce}\right) \boldsymbol{e}_{z}\boldsymbol{\cdot} \boldsymbol{\nabla}\times \boldsymbol{v} = {\omega_{pe}}^{2}\frac{n}{n_{0}}, \end{gather}

(4.9)\begin{gather}\left( \omega -l\varOmega \right) \boldsymbol{e}_{z}\boldsymbol{\cdot} \boldsymbol{v} ={-}\frac{e}{m_e} \beta \phi, \end{gather}

where we have used the Poisson equation to eliminate $\Delta \phi$ in (4.8). Velocity-dependent terms $\boldsymbol {\nabla }\times \boldsymbol {v}$, $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {v}$ and $\boldsymbol {v}$ are then eliminated between (4.6), (4.7), (4.8) and (4.9) to write the oscillating space charge $-ne$ as a function of the oscillating potential $\phi$:

(4.10)\begin{equation} -\frac{ne}{\varepsilon _{0}}=\beta ^{2}\frac{{\omega_{pe}}^{2}}{\left( \omega -l\varOmega \right) ^{2}} \frac{\left(\omega _{ce}-2\varOmega\right) ^{2}}{\omega _{pe}^{2}+ \left(\omega _{ce}-2\varOmega\right) ^{2}-\left( \omega -l\varOmega \right) ^{2}}\phi. \end{equation}

Poisson equation $\Delta \phi = ne/\varepsilon _{0}$ for the perturbation potential $\phi ( r)$ given in (4.1) finally leads to the final equation describing TG modes propagation,

(4.11)\begin{equation} \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 ,\end{equation}

where we have defined the parameter

(4.12)\begin{equation} T_{l}^{2}=\frac{{\omega_{pe}}^{2}\left(\omega _{ce}-2\varOmega\right) ^{2}}{\left( \omega -l\varOmega \right) ^{2} \left[ {\omega_{pe}}^{2}+\left(\omega _{ce}-2\varOmega\right) ^{2}-\left( \omega -l\varOmega \right) ^{2}\right] }-1. \end{equation}

Looking at (4.11) and (4.12), one first verifies that the classical TG relation (2.4) first derived by Trivelpiece & Gould (Reference Trivelpiece and Gould1959) is recovered in the limit of zero rotation $\varOmega =0$. One additionally notes that $T_{+l}=T_{-l}$ in this same limit of zero rotation, but that $T_{+l}\neq T_{-l}$ for $\varOmega \neq 0$. Finally, a closer examination of (4.12) shows that the $\varOmega$-dependent terms $(\omega _{ce}-2\varOmega )$ can be traced back to the Coriolis effect in the plasma rest frame (Lehnert Reference Lehnert1962; Uberoi & Das Reference Uberoi and Das1970), while the $(\omega -l\varOmega )$ 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 (4.11) describing TG modes with an OAM content such that $| L_{z}/P_{z}| =l/\beta$ consists of a linear combination of two counter-rotating Bessel modes

(4.13)\begin{equation} \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), \end{equation}

where $l\in \Bbb {N}$ 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 $E_{\theta }( a) = E_{z}( a) =0$, and from there to $J_{l}( \beta T_{l}a) =0$. The axial wave vector $\beta$ is then determined through the usual requirement of a standing wave along $z$, that is $\beta L=p{\rm \pi}$ with $p\in \Bbb {Z}$. This set of conditions fully determines the resonant mode frequency in the cylindrical cavity through $J_{l}( p{\rm \pi} T_{l}a/l) =0$.

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 $J_{l}( \alpha r)$. In other words, instead of the solution in (4.13) consisting of the superposition of modes with opposite azimuthal mode numbers $\pm l$, different radial structure and identical axial wavevectors $\beta$, we consider non-singular solutions

(4.14)\begin{equation} \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), \end{equation}

that are the superposition of modes with opposite azimuthal mode numbers $\pm l$, identical radial structure, but different axial wavevectors $\beta _{+l}\neq \beta _{-l}$. 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

(4.15)\begin{equation} \beta _{{\pm} l}T_{{\pm} l}(\omega ,\omega _{ce},\omega _{pe},\varOmega )=\alpha , \end{equation}

which guarantees that $\phi$ verifies the radial Poisson equation (4.11). Following the analysis given in Appendix A, the longitudinal electric field associated with the $\pm l$ modes in (4.14) 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 (2.5), the normalised difference between these wave vectors $\beta _{+l}$ and $\beta _{-l}$ is

(4.16)\begin{equation} \frac{\beta _{{+}l}-\beta _{{-}l}}{\beta _{{+}l}+\beta _{{-}l}}=\frac{T_{{-}l}-T_{{+}l}}{T_{{-}l}+T_{{+}l}} = \frac{\varOmega}{2\left. T_{l}\right| _{\varOmega =0}} \left.\frac{\partial \left(T_{{-}l}-T_{{+}l}\right) }{\partial \varOmega}\right| _{\varOmega =0} + O\left(\frac{\varOmega}{\omega} \right) ^{2}. \end{equation}

From (4.12), the lowest order in $\varOmega /\omega$ estimate for this phase shift then writes as

(4.17)\begin{equation} \left. \frac{\beta _{{+}l}-\beta _{{-}l}}{\beta _{{+}l}+\beta _{{-}l}}\right|_{TG}= \frac{\varOmega}{4T_{l}^{2}}\left( \left. \frac{\partial T_{{-}l}^{2}}{\partial \varOmega}\right| _{\varOmega =0} -\left. \frac{\partial T_{{+}l}^{2}}{\partial \varOmega}\right| _{\varOmega =0}\right) ={-}l \frac{\varOmega }{\omega }F_{TG}( \omega _{pe},\omega _{ce},\omega ), \end{equation}

where we have defined the form factor

(4.18)\begin{equation} F_{TG}=\frac{{\omega_{pe}}^{2}}{( {\omega_{pe}}^{2}-\omega ^{2}) } \frac{{\omega_{ce}}^{2}}{( {\omega_{ce}}^{2}-\omega ^{2}) } \frac{ ( {\omega_{uh}}^{2}-2\omega ^{2}) }{( {\omega_{uh}}^{2}-\omega ^{2}), } \end{equation}

which is $O(1)$ at low frequency.

Equation (4.17) shows that, indeed, plasma rotation introduces a phase shift between OAM-carrying TG eigenmodes with opposite azimuthal mode numbers, or in other words to FFR. The wave transverse phase structure is hence rotated as the wave propagates along the rotation and magnetic axis $\boldsymbol {e}_z$. Comparing now (4.17) with the phase shift introduced by rotation between circularly polarised eigenmodes (Gueroult et al. Reference Gueroult, Shi, Rax and Fisch2019a,  Reference Gueroult, Rax and Fisch2020), that is to say to the SAM mechanical Faraday effect, one finds that the phase shifts are in both cases proportional to the ratio of plasma angular frequency to wave angular frequency $\varOmega /\omega$, but that in addition, (4.17) depends linearly on the azimuthal wavenumber $l$. This $\varOmega /\omega$ scaling is also consistent with image rotation in an isotropic dielectric (Götte et al. Reference Götte, Barnett and Padgett2007).

4.4. Possible antenna design

To conclude this section we propose here a coupling structure designed to launch the mode (4.14), 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 (Reference Stenzel2019).

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

(4.19)\begin{equation} \theta _{n}=n\frac{\rm \pi}{l},\quad n\in \left[ 0,1,2,3\cdots2l-1\right], \end{equation}

on a ring of radius $a$. The radius of the antenna ring $a$ is chosen to match the first extremum of $J_{l}( \alpha r)$. Both to have a good impedance matching to the $\pm l$ TG modes’ components of (4.14) and to minimise the excitation of $\pm l^{\prime }$ satellites, the size and shape of the $2l$ plates should be chosen similar to the size and shape of the pattern $| J_{l}( \alpha r) \cos ( l\theta ) |$ illustrated in figure 1(a) for $l=4$. The applied oscillating bias $\phi _n (t)$ on each of these azimuthally distributed $2l$ plates must then verify

(4.20)\begin{align} \phi_{n}\left( t\right) =\phi_{0}J_{l}\left( \alpha a\right) \left[ \exp -jn{\rm \pi} +\left({-}1\right)^{l}\exp jn{\rm \pi} \right] \exp j\omega t,\quad n\in \left[ 0,1,2,3\cdots2l-1\right], \end{align}

to preferentially excite the targeted modes

(4.21)\begin{align} \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. \end{align}

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

5.1. Dispersion relation

The linearised momentum balance

(5.7)\begin{equation} j\omega \boldsymbol{v}+\left( \boldsymbol{v}_{0}\boldsymbol{\cdot} \boldsymbol{\nabla }\right) \boldsymbol{v}+\left( \boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{\nabla }\right) \boldsymbol{v}_{0}={-}\frac{e}{m_e} \left( \boldsymbol{E}+\boldsymbol{v}_{0}\times \boldsymbol{B}+\boldsymbol{v}\times \boldsymbol{B}_{0}\right) \end{equation}

allows us, with the help of (3.9) and (3.10), to write the electron current as a function of the fields

(5.8)\begin{equation} j\left( \omega -l\varOmega \right) \boldsymbol{J}+\left( 2\varOmega -\omega_{ce}\right) \boldsymbol{e}_{z}\times \boldsymbol{J}={\omega_{pe}}^{2}\varepsilon_{0} \left( \boldsymbol{E}+\varOmega r\boldsymbol{e}_{\theta }\times \boldsymbol{B}\right). \end{equation}

The operator $\boldsymbol {\nabla }\times$ is then applied to (5.8) to eliminate both the electric field $\boldsymbol {E}$ and the current $\boldsymbol {J}$. Using (3.5), (3.6), (5.3) and (5.4) finally yields the wave equation

(5.9)\begin{equation} \boldsymbol{\nabla }\times \boldsymbol{\nabla }\times \boldsymbol{B}-\beta \frac{\omega _{ce}-2\varOmega }{\omega -l\varOmega }\boldsymbol{\nabla }\times \boldsymbol{B}+\frac{ {\omega_{pe}}^{2}}{c^{2}}\boldsymbol{B}=\boldsymbol{0}. \end{equation}

One verifies that (5.9) reduces to the classical WH equation (Klozenberg, McNamara & Thonemann Reference Klozenberg, McNamara and Thonemann1965) for $\varOmega =0$. In addition, (5.9) can be rewritten as

(5.10)\begin{equation} \left( \beta _{1}-\boldsymbol{\nabla }\times \right) \left( \beta _{2}- \boldsymbol{\nabla} \times \right) \boldsymbol{B}=\boldsymbol{0} \end{equation}

with

(5.11)\begin{gather} \beta_{1}\beta_{2} =\frac{{\omega_{pe}}^{2}}{c^{2}}, \end{gather}

(5.12)\begin{gather}\beta_{1}+\beta_{2} =\beta \frac{\omega_{ce}-2\varOmega}{\omega -l\varOmega }. \end{gather}

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

(5.13)\begin{equation} \beta _{W}=\frac{\beta }{2}\frac{\omega _{ce}-2\varOmega}{\omega -l\varOmega } \left[ 1-\sqrt{1-4\frac{{\omega_{pe}}^{2}}{{\beta _{W}}^{2}c^{2}} \left( \frac{\omega -l\varOmega }{\omega _{ce}-2\varOmega}\right) ^{2}}\right] \approx \frac{{\omega_{pe}}^{2}}{\beta _{W} c^{2}}\frac{\omega -l\varOmega }{\omega _{ce}-2\varOmega }, \end{equation}

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 (2.3) for $\varOmega =0$, and also that $\beta _{W}( +l) \neq \beta _{W}(-l)$ for $\varOmega \neq 0$. Similarly to the analysis done above for TG modes, further examination shows that the $(\omega _{ce}-2\varOmega )$ terms are associated with the Coriolis effect in the plasma rest frame (Lehnert Reference Lehnert1962; Uberoi & Das Reference Uberoi and Das1970) while the $( \omega -l\varOmega )$ terms come from the Doppler effect in the plasma rest frame.

Now, by introducing

(5.14)\begin{equation} K_{l}=\frac{{\omega_{pe}}^{2}}{c^{2}}\frac{\omega -l\varOmega }{\omega _{ce}-2\varOmega } \end{equation}

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

(5.15)\begin{equation} K_{0}=\frac{{\omega_{pe}}^{2}}{c^{2}}\frac{\omega }{\omega _{ce}}, \end{equation}

one finds that the Maxwell–Ampère equation for the WH branch reduces to

(5.16)\begin{equation} \boldsymbol{\nabla }\times \boldsymbol{B} = \left( K_{l}/\beta \right) \boldsymbol{B}. \end{equation}

Note that $K_{\pm l}$ is here simply a convenient variable to write (5.16) and it should not be mistaken for a modified Bessel function. Finally, applying the operator $\boldsymbol {\nabla }\times$ to (5.16) leads to the wave equation

(5.17)\begin{equation} \Delta \boldsymbol{B}+\left( K_{l}^{2}/\beta ^{2}\right) \boldsymbol{B} =\boldsymbol{0} \end{equation}

whose $z$ component writes as

(5.18)\begin{equation} \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. \end{equation}

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 (5.18) of the form

(5.19)\begin{equation} B_{z}=B_{z0}[ J_{l}( \alpha r) \exp j( \omega t-l\theta -\beta _{l}z) +J_{{-}l}( \alpha r) \exp j( \omega t+l\theta -\beta _{{-}l}z) ], \end{equation}

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

(5.20)\begin{equation} \beta _{{\pm} l}^{{-}2}K_{{\pm} l}^{2}(\omega ,\omega _{ce},\omega _{pe},\varOmega)- \beta _{{\pm} l}^{2}=\alpha ^{2} \end{equation}

which guarantees that $B_z$ verifies (5.18). Separating the $\pm l$ contributions, (5.20) can be rewritten as

(5.21)\begin{equation} \beta _{l}^{{-}2}K_{l}^{2}-\beta _{l}^{2}=\beta _{{-}l}^{{-}2}K_{{-}l}^{2}-\beta_{{-}l}^{2}=\alpha ^{2} \end{equation}

whose solutions are

(5.22)\begin{equation} \beta _{{\pm} l}^{2}=\sqrt{\alpha ^{4}+4K_{{\pm} l}^{2}}-\alpha ^{2}. \end{equation}

5.3. Faraday–Fresnel rotation

Expanding $K_{\pm l}^{2}$ given in (5.14) for the small parameter $\varOmega /\omega$ and assuming WH ordering $\omega _{ce}\gg \omega$ yields

(5.23)\begin{equation} K_{{\pm} l}^{2}=K_{0}^{2}\left( 1\mp 2l\frac{\varOmega }{\omega}\right) + O\left[\left(\frac{\varOmega }{\omega }\right)^2\right]. \end{equation}

Meanwhile, from (5.22),

(5.24)\begin{equation} \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}}. \end{equation}

Plugging (5.23) into this last expression finally leads to the lowest order in $\varOmega /\omega$ estimate for the OAM phase shift

(5.25)\begin{equation} \left. \frac{\beta _{{+}l}-\beta _{{-}l}}{\beta _{{+}l}+\beta _{{-}l}}\right|_{WH}={-}l \frac{\varOmega }{\omega }F_{WH}\left( \omega _{pe},\omega _{ce},\omega\right), \end{equation}

where we have defined the form factor

(5.26)\begin{equation} F_{WH}=\frac{4K_{0}^{2}}{\alpha ^{4}+4K_{0}^{2}-\alpha ^{2}\sqrt{\alpha ^{4}+4K_{0}^{2}}}. \end{equation}

Similarly to (4.18) which was obtained for TG modes, (5.26) shows that plasma rotation introduces a phase shift between OAM-carrying WH eigenmodes with opposite azimuthal mode numbers, or in other words to FFR. In both instances, the OAM phase shift is proportional to the plasma angular frequency to wave angular frequency ratio $\varOmega /\omega$, and to the azimuthal wavenumber $l$. However, closer examination of (4.18) and (5.26) shows that the form factor $F$ characterising FFR 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 (5.19). This type of antenna has been extensively investigated, both theoretically and experimentally, in Stenzel & Urrutia (Reference Stenzel and Urrutia2014) and Urrutia & Stenzel (Reference Urrutia and Stenzel2015b), and we refer the readers to the review paper by Stenzel (Reference Stenzel2019) for an in-depth analysis of their properties.

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

(5.27)\begin{equation} \theta_{n}=n{\rm \pi} /l,\quad n\in [ 0,1,2,3\cdots2l-1], \end{equation}

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

(5.28)\begin{align} I_{n}( t) =B_{z0}J_{l}( \alpha a) [ \exp -jn{\rm \pi} +({-}1) ^{l}\exp +jn{\rm \pi} ] \exp j\omega t,\quad n\in \left[ 0,1,2,3\cdots2l-1\right], \end{align}

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

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.

Both to have a good impedance matching to the $\pm l$ WH modes components of (5.19) and to minimise 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 $| J_{l}( \alpha r)\cos (l\theta ) |$ illustrated in figure 2(a). Alternatively, an improved antenna array can be designed with a second ring of $2l$ current loops near the second extremum of $J_{| l| }( \alpha r)$, as illustrated in figure 2(c).