# Gyrotropy-induced symmetry breaking in RF-specific plasma wall interaction: Mechanisms and mitigation strategies

W. Tierens, A. Kumar, J. Lore, K. Fujii, G. Urbanczyk, R. Diab

► To cite this version:

W. Tierens, A. Kumar, J. Lore, K. Fujii, G. Urbanczyk, et al. Gyrotropy-induced symmetry breaking in RF-specific plasma wall interaction: Mechanisms and mitigation strategies. Physics of Plasmas, 2026, 33 (1), pp.010701. 10.1063/5.0305554. cea-05463981

HAL Id: cea-05463981

https://cea.hal.science/cea-05463981v1

Submitted on 18 Jan 2026

**HAL** is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire **HAL**, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Gyrotropy-Induced Symmetry Breaking in RF-specific Plasma Wall Interaction: Mechanisms and Mitigation Strategies.

W. Tierens,<fnref n="1" /> A. Kumar,<fnref n="1" /> J. Lore,<fnref n="1" /> K. Fujii,<fnref n="1" /> G. Urbanczyk,<fnref n="2" /> R. Diab,<fnref n="3" /> and The WEST Team<fnref n="4" />

Ion Cyclotron Range of Frequencies (ICRF) heating is essential for creating plasma in next-generation fusion devices. ICRF antennas often produce localized wall erosion, reducing reliability, material survivability, and overall plasma performance. We show that the wave physics' gyrotropy is among the causes of toroidal and poloidal asymmetries in these erosion spots, and that this source of asymmetry can be compensated for either by controlling the poloidal phasing or by modifying the limiter shapes.ᵃ

Ion Cyclotron Range of Frequencies (ICRF) is a leading technique for plasma heating and current drive in magnetic confinement fusion experiments, able to bring the plasma to reactor-relevant conditions<fnref n="1" />,<fnref n="2" />. Other applications include plasma-based space propulsion systems<fnref n="3" />,<fnref n="4" /> and the study of plasma-material interactions⁵. In 2022, ICRF alone contributed 20% to the record 59MJ fusion energy produced during the last deuterium-tritium campaign at JET⁶. In present-day fusion devices, ICRF performance depends critically on minimizing impurity generation and damage to plasma-facing components through optimized antenna design and operation. Careful control of the antenna phasing and power ratio⁷,⁸ has proven effective in reducing material sputtering and impurity sources, making ICRF compatible even with the high-Z environment⁹ of most planned fusion devices¹⁰. However, emerging experimental¹¹ and modeling results¹²,¹³ indicate that these mitigation strategies may not fully capture certain asymmetries in the wave physics, which can cause uneven heat loads on the limiters, threatening material survivability and overall plasma performance.

Spectroscopic measurements of tungsten (W) and deuterium (D) fluxes on the WEST tokamak, shown in Figure 1, show highly asymmetric erosion patterns. These results are based on W and D line emission signals from chords viewing the antenna limiters during the ICRF-heated discharges #61683 - #61689 (though it should be noted that such optical diagnostics may be affected by light that originates elsewhere being reflected off the plasma-facing component¹⁴, and this too introduces asymmetries in the measurement). This is consistent with other published measurements on WEST¹⁵. Cases

<figure><img src="image_2.png" /><figcaption>FIG. 1. Sputtering yields during a power scan from 0MW through 1.7MW on the WEST antenna limiters, shots #61683 - #61689 (blue through cyan: left limiter, orange through red: right limiter), computed from the Tungsten ($\Gamma_W$) and Deuterium ($\Gamma_D$) fluxes observed using visible spectroscopy. The sputtering is both horizontally and vertically asymmetric. The (total) port voltages for shot #61683 are 22kV on the upper ports and 27 kV on the lower ports, and for shot #61689 18kV on the upper ports and 19 kV on the lower ports. This work does not give a complete account of all sources of asymmetry in the experiment, instead we show that the wave physics itself still breaks both horizontal and vertical symmetry even if the antenna and the plasma are perfectly symmetric.</figcaption></figure>

where symmetric heat loads were observed exist¹⁶, but they are the exception rather than the rule. Similar asymmetric erosion has also been observed in other fusion devices (e.g. AUG¹⁷ and EAST¹⁸,¹⁹), demonstrating the universal nature of this behavior in ICRF-heated magnetic fusion experiments. These observations are also consistent with recent numerical modeling¹²,¹³,²⁰,²¹, which likewise predicts asymmetric erosion hotspots at the antenna limiters. In this letter, we show that an unavoidable asymmetry is inherent in the gyrotropy of the RF wave physics, persisting even when all other causes of asymmetry are removed (including the asymmetry in the an-

<footnote n="1">¹) *Oak Ridge National Laboratory, 1 Bethel Valley Road, Oak Ridge, TN 37830, USA*</footnote>

<footnote n="2">²) *Institut Jean Lamour UMR 7198 CNRS-Université de Lorraine, 2 allée André Guinier, F-54011 Nancy, France*</footnote>

<footnote n="3">³) *MIT Plasma Science and Fusion Center, Cambridge, Massachusetts 02139, USA*</footnote>

<footnote n="4">⁴) See http://west.cea.fr/WESTteam</footnote>

<footnote>(*Electronic mail: tierenswv@ornl.gov)</footnote>

<footnote>(Dated: 13 January 2026)</footnote>

<footnote>ᵃ) Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (https://www.energy.gov/doi-public-access-plan).</footnote>
<figure><img src="image_3.png" /><figcaption>FIG. 2. The symmetric plasma density in front of the symmetric antenna. Left: toroidal cross section, right: poloidal cross section. This is the same 1D density profile used in¹³, i.e. one derived from WEST shot 57877, but extended to 3D in a fully symmetric manner.</figcaption></figure>

antenna geometry, plasma shape, antenna-plasma misalignment, asymmetric excitation of the antenna ports, and the prefer- ential poloidal direction of motion of convective cells²²,²³ (E × B drift, typically upward during RF operation in WEST)). This fundamentally limits the effectiveness of symmetric or flux-surface-aligned limiters. We therefore propose that non- aligned limiters, compensating for the gyrotropy of the wave physics, may outperform the current state of the art.

Numerically, it is possible to disentangle the various sources of asymmetry, or to single out one of them as we do here. We have constructed a fully symmetric WEST-like finite element antenna model based on the one used in refer- ence¹³, removing all asymmetric elements, and ensuring the bottom half is the mirror image of the top half. We then con- sider a perfectly symmetric plasma using the density from figure 2, and a purely toroidal magnetic field (no tilt angle). We include sheath rectification physics using the dielectric layer sheath model²⁴, which gives us both an RF potential V<sub>RF</sub> and a rectified DC potential V<sub>DC</sub> on the limiters. The relation between the two is given by the nonlinear but lo- cal sheath model of reference²⁵, which in practice²⁶ usually reduces to V<sub>DC</sub> ≈ 3k<sub>B</sub>T<sub>e</sub> + 0.6V<sub>RF</sub>. In our ideally symmet- ric scenario, all asymmetries of V<sub>RF</sub> are also asymmetries of V<sub>DC</sub> (though in a generic scenario, the relation between V<sub>RF</sub> and V<sub>DC</sub> will itself introduce new asymmetries, for ex- ample through the dependence on the angle between the sur- face normal and the confining magnetic field). From V<sub>DC</sub>, we compute the sputtering yield using the sputtering curve Y<sub>eff</sub>(V<sub>DC</sub>) for a WEST-like plasma given in reference¹³: sput- tering yield= Y<sub>eff</sub>(V<sub>DC</sub>)n<sub>i</sub>√3T<sub>i</sub>/m<sub>i</sub>. Because Y<sub>eff</sub>(V<sub>DC</sub>) varies rapidly with V<sub>DC</sub>, asymmetries in the sputtering pattern may appear exaggerated compared to those of V<sub>RF</sub>.

Our antenna has four coaxial ports, in each of which we can specify the amplitudes and phases of the incident wave. We excite the antenna in dipole phasing¹⁵ (wave phases 0, π, π, 0, red in figure 3), with the same power on the top and bot- tom ports. This matches the idealized operating regime on WEST, though in reality the matching system determines

<figure><img src="image_4.png" /><figcaption>FIG. 3. Left: Numerically predicted RF potential at 1MW coupled power, and (right) corresponding sputtering yield in a completely symmetric WEST-like antenna. Despite the symmetry, hot spots preferentially appear at the top. The total (integrated) sputtering yield is 5.2 · 10¹⁷ atoms per second. The phase of the incident wave in each of the four coaxial ports is indicated in red.</figcaption></figure>

these parameters²⁷, and they may deviate from this ideal dipole. Within these constraints, only a single amplitude de- gree of freedom remains, which we always choose (for all re- sults in this work) such that 1MW of power is coupled to the plasma. Even in this maximally symmetric scenario, numer- ical results still predict up-down asymmetries in the rectified potential and the sputtering yield at the antenna limiters, as shown in figure 3. This up-down asymmetry in the erosion pattern is robust to small changes in the vertical location of the plasma and small changes in the magnetic field tilt an- gle. Only by turning the confining magnetic field all the way around (reversing the direction of the currents in the toroidal field coils), which is from an electromagnetic point of view equivalent to mirroring the situation since B is a pseudovec- tor, do the hot spots end up on the bottom instead. Reference¹¹ points out that such B-reversal experiments were tried on Tore Supra, and that they indeed found the heat loads to turn up- side down. They attributed this to E × B drifts, but since we see this numerically in wave physics simulations which do not contain E × B drifts, the wave physics itself must also exhibit this asymmetry.

Recall the electromagnetic frequency-domain wave equa- tion in Cartesian coordinates

$$
\nabla \times \nabla \times E - k_0^2 \epsilon E = 0 \quad (1)
$$

where k₀ = ω/c is the vacuum wavenumber, ω is the wave (angular) frequency, c is the speed of light, and the cold plasma dielectric tensor (in the absence of a magnetic field angle, with the confining magnetic field along z) is

$$
\epsilon = I - \sum_s \begin{bmatrix} \frac{\omega_s^2}{\omega^2 - \Omega_s^2} & \frac{\Omega_s}{-i\omega} & \frac{\omega_s^2}{\omega^2 - \Omega_s^2} & 0 \\ \frac{\Omega_s}{i\omega} & \frac{\omega_s^2}{\omega^2 - \Omega_s^2} & \frac{\omega_s^2}{\omega^2 - \Omega_s^2} & 0 \\ 0 & 0 & \frac{\omega_s^2}{\omega^2} & \frac{\Omega_s^2}{\omega^2} \end{bmatrix} \quad (2)
$$

where I is the unit matrix, and ω<sub>s</sub>, Ω<sub>s</sub> are the plasma frequency and the cyclotron frequency of particle species s, respectively.
The presence of off-diagonal elements in (2) is referred to as gyrotropy, which vanishes in the highly magnetized limit$^{28}$.

We can see that if $\epsilon_{12} = -\epsilon_{21} = 0$ (the non-gyrotropic case), the wave equation (1) possesses a mirror symmetry: if $\epsilon$ is vertically symmetric ($\epsilon(x,y) = \epsilon(x,-y)$) and

$$E = [E_x(x,y), E_y(x,y), E_z(x,y)]^T \exp(in_z z \omega/c)$$

is a solution of (1), then so is its vertical mirror image

$$E = [E_x(x, -y), -E_y(x, -y), E_z(x, -y)]^T \exp(in_z z \omega/c)$$

This means that, if excited symmetrically in the absence of gyrotropy, the wave equation (1) has a vertically symmetric solution with corresponding symmetric $V_{RF}$. Thus, the absence of gyrotropy suffices to restore the vertical symmetry in our scenario.

A deeper insight can be obtained by considering the two wave modes supported in the cold magnetized plasma: the fast wave and the slow wave$^{29}$. For the fast wave, we have $E_z \approx 0$. Taking $E = [E_x(x), E_y(x), 0]^T \exp(i(n_x x + n_y y + n_z z)\omega/c)$, and $\epsilon = \epsilon(x)$, we may derive an approximate expression for $E_y(x)$, proceeding along the lines of$^{30}$:

$$\frac{A}{k_0^2} \frac{d^2 E_y}{dx^2} + \frac{B}{k_0} \frac{d E_y}{dx} + CE_y = 0 \quad (3)$$

with

$$A = \frac{n_z^2 - \epsilon_{11}(x)}{n_y^2 + n_z^2 - \epsilon_{11}(x)}$$

$$B = -n_y^2 \frac{\frac{d}{dx} \epsilon_{11}(x)}{k_0(n_y^2 + n_z^2 - \epsilon_{11}(x))^2}$$

$$C = \frac{-n_y/(ik_0)}{n_y^2 + n_z^2 - \epsilon_{11}(x)} \left( \frac{d}{dx} \epsilon_{21}(x) + \frac{\epsilon_{21}(x) \frac{d}{dx} \epsilon_{11}(x)}{n_y^2 + n_z^2 - \epsilon_{11}(x)} \right) - (n_z^2 - \epsilon_{11}(x)) - \frac{\epsilon_{21}(x)^2}{n_y^2 + n_z^2 - \epsilon_{11}(x)}$$

All coefficients in (3) are symmetric in $n_y$, with the exception of $C$, which is symmetric in $n_y$ only if the plasma is non-gyrotropic ($\epsilon_{21} = 0$), or if there are no gradients in the $x$ (radial) direction. Numerical experiments in figure 4 show that only the first option (no gyrotropy) completely restores the symmetry. In fact, the mere presence of a plasma-vacuum interface (the antenna is kept in vacuum in all calculations in this work, hence there is a discontinuous plasma-vacuum interface) suffices to break the vertical symmetry, even if the plasma is otherwise uniform, which can be seen from earlier studies of the scattering problem on this interface$^{31}$.

Any parasitically excited slow waves, which may play an important role in the sheath excitation physics$^{32,33}$, also have this asymmetry. The electric field of the slow wave can be approximated using a potential $\phi^{34}$. The electric field is $E =$

<figure><img src="image_5.png" /><figcaption>FIG. 4. Eq. (3) suggests two ways of restoring the vertical symmetry absent in figure 3. Left: by artificially removing the gyrotropy. Right: by removing the gradients (constant density $n_e = 10^{18}m^{-3}$ and magnetic field strength $B = 3.2$T, though the antenna is still in vacuum). Only removing the gyrotropy succeeds in completely restoring the symmetry.</figcaption></figure>

$-\nabla(\phi(x) \exp(i(n_y y + n_z z)\omega/c))$, and it obeys $\nabla \cdot (\epsilon E) = 0$:

$$\begin{aligned} k_0^2 \phi(x) (n_y^2 \epsilon_{11}(x) + n_z^2 \epsilon_{33}(x)) = & \left(\frac{d}{dx} \epsilon_{11}(x)\right) \left(\frac{d}{dx} \phi(x)\right) \\ & + \epsilon_{11}(x) \frac{d^2}{dx^2} \phi(x) \\ & + ik_0 n_y \phi(x) \frac{d}{dx} \epsilon_{12}(x) \end{aligned} \quad (4)$$

Once again, all coefficients are symmetric in $n_y$ provided that either $\epsilon_{12}(x) = 0$ (no gyrotropy) or $\frac{d}{dx} \epsilon_{12}(x) = 0$ (no gradients).

Returning to the horizontal symmetry, which is also absent in experiment (figure 1), we consider under what circumstances the wave equation (1) possesses a horizontal mirror symmetry. With the dielectric tensor (2) a horizontal mirror symmetry exists even in gyrotropic plasma, but this symmetry is broken by the introduction of a nonzero magnetic field tilt angle (typically $\sim 5^\circ$ in the WEST edge plasma). Thus, in the experimentally relevant scenario where the plasma is both gyrotropic and the confining magnetic field is not purely toroidal, we should expect neither horizontal nor vertical symmetry.

Finally, let us consider some possible mitigation strategies. Hot spots like those in figure 3 can be reduced by controlling the phasing and power balance, as is often done to great effect in the toroidal direction$^7$. Here, we have to do it in the poloidal direction. This is not possible on current devices, though benefits of having this capability were pointed out as early as the 1980s: the poloidal spectrum affects both how much power reaches the core$^{30,35,36}$, and how efficiently the power is absorbed in the core$^{37}$. The ITER-like antenna (ILA) in JET had this capability, as will the under construction WEST Traveling Wave Antenna$^{38}$. Figure 5 suggests that the peak RF potential can be reduced by $\sim 20\%$ following this approach.

Another option is to modify the limiter shape. In light of the asymmetries inherent in the wave physics, we consider if an asymmetric limiter might outperform a symmetric one. We
<figure><img src="image_6.png" /><figcaption>FIG. 5. One way to compensate for the hot spots in figure 3 is to change the *poloidal* phasing and power balance. The idealized operating regime on WEST is $(\pi, 0)$ (circle) on this plot: same amplitude but opposite phase on the top and bottom ports. The optimum (star) has slightly less power on the top port and a phase difference slightly below $\pi$. The RF potentials in this plot are normalized to the value at $(\pi, 0)$.</figcaption></figure>

have parameterized the limiter shape in our model using three circular arcs, so we can retract it further away from the plasma at the location of the hot spots. Figure 6 (right) shows one possible (not necessarily optimal) modified limiter shape. This figure also shows the predicted sputtering for both proposed modifications (poloidal phasing, modified limiter shape). Optimal poloidal phasing gives a ~60% reduction in peak sputtering and a ~20% reduction in total sputtering w.r.t. the reference of figure 3. The modified limiter shape gives a factor ~3 reduction in peak sputtering and a factor ~2 reduction in total sputtering. Both cases show a more evenly distributed erosion pattern, and hence better material survivability. It is likely that the plasma shape can be modified while keeping the limiter shape constant to achieve a similar effect, which may be easier to achieve in practice.

The predicted effectiveness of this apparently minor limiter shape change can be understood in terms of two factors: first, the edge density gradient is very steep, so even a small displacement gives rise to a large difference in the incident ion flux. Second, we displace precisely the location of the hot spot, which accounted for most of the sputtering to begin with. This allows us to reduce sputtering while keeping the overall coupled power constant, which is difficult to achieve when one retracts the entire antenna instead (this is possible on WEST). In machines where the ICRF actuators are carefully aligned with the first wall (e.g. ITER), such limiter deformation is not possible. On the other hand, in machines like WEST, C-Mod, or SPARC, where objects with distinct limiters poke into the plasma, this deformation can be successfully used to mitigate sputtering.

It is now standard in ICRF operation to control the toroidal phasing and power balance, which minimizes the parallel electric field near the limiters and thereby the sheath

rectification$^7$, and which can also be used to mitigate convective cell-induced asymmetries in the particle flux to the ICRF antenna$^8$. Our results strongly indicate that a further decrease of the RF-specific impurity sputtering, beyond that already achieved by these measures, is achievable by poloidal modifications of the antenna, ranging from relatively simple poloidal phasing and power balance control, to modification of the plasma-limiter distance by changing the shape of the limiter or the plasma. This reduction is possible even in scenarios where the plasma is otherwise well-aligned with the antenna. This provides a pathway to mitigate asymmetric RF-induced erosion hotspots (critical for material survivability) while also reducing the net erosion flux by nearly 50%, thereby ensuring robust ICRH antenna operation, extending material lifetimes, and contributing to sustained reactor-grade plasma performance.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (https://www.energy.gov/doi-public-access-plan).

<footnote n="1">¹J. Garcia, Y. Kazakov, R. Coelho, M. Dreval, E. de la Luna, E. R. Solano, Ž. Štancar, J. Varela, M. Baruzzo, E. Belli, et al., “Stable deuterium-tritium plasmas with improved confinement in the presence of energetic-ion instabilities,” Nature Communications 15, 7846 (2024).</footnote>

<footnote n="2">²A. J. Creely, M. J. Greenwald, S. B. Ballinger, D. Brunner, J. Canik, J. Doody, T. Fülöp, D. T. Garnier, R. Granetz, T. K. Gray, and et al., “Overview of the SPARC tokamak,” Journal of Plasma Physics 86, 865860502 (2020).</footnote>

<footnote n="3">³K. Takahashi, “Thirty percent conversion efficiency from radiofrequency power to thrust energy in a magnetic nozzle plasma thruster,” Scientific reports 12, 18618 (2022).</footnote>

<footnote n="4">⁴A. V. Arefiev and B. N. Breizman, “Theoretical components of the VASIMR plasma propulsion concept,” Physics of Plasmas 11, 2942–2949 (2004), https://pubs.aip.org/aip/pop/article-pdf/11/5/2942/19293021/2942_1_online.pdf.</footnote>

<footnote n="5">⁵J. Rapp, A. Lumsdaine, C. J. Beers, T. M. Biewer, T. S. Bigelow, J. F. Caneses, J. B. Caughman, R. H. Goulding, N. Kafle, C. H. Lau, et al., “Latest results from Proto-MPEX and the future plans for MPEX,” Fusion Science and Technology 75, 654–663 (2019).</footnote>

<footnote n="6">⁶M. Maslov, E. Lerche, F. Auriemma, E. Belli, C. Bourdelle, C. Chalis, A. Chomiczewska, A. D. Molin, J. Eriksson, J. Garcia, J. Hobirk, I. Ivanova-Stanik, P. Jacquet, A. Kappatou, Y. Kazakov, D. Keeling, D. King, V. Kiptily, K. Kirov, D. Kos, R. Lorenzini, E. D. L. Luna, C. Maggi, J. Mailloux, P. Mantica, M. Marin, G. Matthews, I. Monakhov, M. Nocente, G. Pucella, D. Rigamonti, F. Rimini, S. Saarelma, M. Salewski, E. Solano, Z. Štancar, G. Stankunas, H. Sun, M. Tardocchi, D. V. Eester, and J. Contributors, “JET D-T scenario with optimized non-thermal fusion,” Nuclear Fusion 63, 112002 (2023).</footnote>

<footnote n="7">⁷V. Bobkov, D. Aguiam, R. Bilato, S. Brezinsek, L. Colas, A. Czarnecka, P. Dumortier, R. Dux, H. Faugel, H. Fünfgelder, et al., “Impact of ICRF on the scrape-off layer and on plasma wall interactions: From present experiments to fusion reactor,” Nuclear Materials and Energy 18, 131–140 (2019).</footnote>
<figure><img src="image_7.png" /><figcaption>FIG. 6. Predicted sputtering on (left) the standard limiter with the optimal phasing from figure 5 (incident wave phase indicated in red for all four ports) and (middle) on a modified asymmetric limiter with standard phasing, whose geometry is shown on the right. The total (integrated) sputtering yield is 4.2·10¹⁷ (left) resp. 2.8·10¹⁷ (right) atoms per second.</figcaption></figure>

<footnote n="8">⁸R. Diab, S. G. Baek, A. Q. Kuang, Y. Lin, E. S. Marmar, J. L. Terry, and S. J. Wukitch, "Mitigation of ICRF-Edge plasma interaction in Alcator C-Mod," Nuclear Fusion (2025).</footnote>

<footnote n="9">⁹V. Bobkov, D. Aguiam, R. Bilato, S. Brezinsek, L. Colas, H. Faugel, H. Fünfgelder, A. Herrmann, J. Jacquot, A. Kallenbach, et al., "Making ICRF power compatible with a high-Z wall in ASDEX upgrade," Plasma Physics and Controlled Fusion 59, 014022 (2016).</footnote>

<footnote n="10">¹⁰P. Barabaschi, A. Fossen, A. Loarte, A. Becoulet, and L. Coblentz, "ITER progresses into new baseline," Fusion Engineering and Design 215, 114990 (2025).</footnote>

<footnote n="11">¹¹L. Colas, G. Urbanczyk, M. Goniche, J. Hillairet, J.-M. Bernard, C. Bourdelle, N. Fedorczak, C. Guillemaut, W. Helou, V. Bobkov, R. Ochoukov, P. Jacquet, E. Lerche, X. Zhang, C. Qin, C. Klepper, C. Lau, B. Van Compernolle, S. Wukitch, Y. Lin, M. Ono, J. Contributors, the ASDEX Upgrade Team, the EAST Team, the WEST Team, and I. IOS, "The geometry of the ICRF-induced wave-SOL interaction. a multi-machine experimental review in view of the ITER operation," Nuclear Fusion 62, 016014 (2021).</footnote>

<footnote n="12">¹²A. Kumar, W. Tierens, T. Younkin, C. Johnson, C. Klepper, A. Diaw, J. Lore, A. Grosjean, G. Urbanczyk, J. Hillairet, P. Tamain, L. Colas, C. Guillemaut, D. Curreli, S. Shiraiwa, N. Bertelli, and t. W. Team, "Integrated modeling of RF-Induced tungsten erosion at ICRH antenna structures in the WEST tokamak," Nuclear Fusion 65, 076039 (2025).</footnote>

<footnote n="13">¹³W. Tierens, A. Kumar, C. Klepper, J. Lore, J. R. Myra, J. Hillairet, G. Urbanczyk, W. Helou, L. Colas, A. Grosjean, et al., "Radiofrequency sheath rectification on WEST: application of the sheath-equivalent dielectric layer technique in tokamak geometry," Nuclear Fusion 64, 126039 (2024).</footnote>

<footnote n="14">¹⁴C. Johnson, A. Diaw, E. Unterberg, N. Fedorczak, and P. Tamain, "Analyzing the effects of reflections on optical diagnostics in the main chamber and divertor of WEST," Review of Scientific Instruments 95 (2024).</footnote>

<footnote n="15">¹⁵G. Urbanczyk, L. Colas, J. Hillairet, E. Lerche, N. Fedorczak, J. Morales, J. Gunn, V. Ostuni, S. Heuraux, D. Vezinet, et al., "RF wave coupling, plasma heating and characterization of induced plasma-material interactions in WEST L-mode discharges," Nuclear Fusion 61, 086027 (2021).</footnote>

<footnote n="16">¹⁶G. Urbanczyk, N. Fedorczak, J. Gunn, L. Colas, J. Li, K. Wang, W. Team, et al., "Perspective of analogy between heat loads and impurity production in L-mode discharges with ICRH in WEST," Nuclear Materials and Energy 26, 100925 (2021).</footnote>

<footnote n="17">¹⁷V. Bobkov, F. Braun, R. Dux, L. Giannone, A. Herrmann, A. Kallenbach, H. W. Müller, R. Neu, J.-M. Noterdaeme, T. Pütterich, et al., "Operation of ICRF antennas in a full tungsten environment in ASDEX upgrade," Journal of nuclear materials 390, 900-903 (2009).</footnote>

<footnote n="18">¹⁸X. Zhang, H. Yang, C. Qin, S. Yuan, Y. Zhao, Y. Wang, L. Liu, Y. Mao, Y. Cheng, X. Gong, et al., "First experimental results with new ICRF antenna in EAST," Nuclear Fusion 62, 086038 (2022).</footnote>

<footnote n="19">¹⁹H. Yang, W. Zhang, X. Zhang, L. H. Zheng, N. Yan, J. Qian, R. Ding, and M. Li, "Experimental investigation of ICRF-edge plasma interactions and</footnote>

<footnote>surface wave excitation during ICRF heating on EAST," Nuclear Fusion (2025).</footnote>

<footnote n="20">²⁰A. Kumar, W. Tierens, T. Younkin, C. Johnson, C. Klepper, A. Diaw, J. Lore, A. Grosjean, G. Urbanczyk, J. Hillairet, P. Tamain, L. Colas, C. Guillemaut, D. Current, S. Shiraiwa, N. Bertelli, and the WEST Team, "RFPPC Invited talk: Analysis of RF sheath-driven tungsten erosion at RF antenna in the WEST tokamak," (2025), arXiv:2507.06059 [physics.plasm-ph].</footnote>

<footnote n="21">²¹R. Ragona, V. Maquet, and J. Hillairet, "Near field analysis of the ICRF travelling wave array launcher for WEST," RFPPC conference (2025).</footnote>

<footnote n="22">²²I. Cziegler, J. Terry, S. Wukitch, M. Garrett, C. Lau, and Y. Lin, "Ion-cyclotron range of frequencies in the scrape-off-layer: fine structure radial electric fields," Plasma Physics and Controlled Fusion 54, 105019 (2012).</footnote>

<footnote n="23">²³M. Martin, W. Gekelman, B. Van Compernolle, P. Pribyl, and T. Carter, "Experimental observation of convective cell formation due to a fast wave antenna in the large plasma device," Physical Review Letters 119, 205002 (2017).</footnote>

<footnote n="24">²⁴C. J. Beers, D. L. Green, C. Lau, J. Myra, J. Rapp, T. R. Younkin, and S. J. Zinkle, "RF sheath induced sputtering on Proto-MPEX. i. sheath equivalent dielectric layer for modeling the RF sheath," Physics of Plasmas 28 (2021).</footnote>

<footnote n="25">²⁵J. Myra, "Physics-based parametrization of the surface impedance for radio frequency sheaths," Physics of Plasmas 24 (2017).</footnote>

<footnote n="26">²⁶G. Urbanczyk, R. Ochoukov, V. Bobkov, S. Shiraiwa, R. Bilato, N. Bertelli, W. Tierens, L. Colas, R. Dux, M. Dreval, et al., "Characterization of W production during ICRF operations: experiments and modeling," Nuclear Fusion 65, 046018 (2025).</footnote>

<footnote n="27">²⁷G. Bosia, "High-power density ion cyclotron antennas for next step applications," Fusion science and technology 43, 153-160 (2003).</footnote>

<footnote n="28">²⁸E. Bécache, P. Joly, and M. Kachanovska, "Stable perfectly matched layers for a cold plasma in a strong background magnetic field," Journal of Computational Physics 341, 76-101 (2017).</footnote>

<footnote n="29">²⁹T. H. Stix, *Waves in plasmas* (Springer Science & Business Media, 1992).</footnote>

<footnote n="30">³⁰D. W. Swain, M. D. Carter, J. Wilson, P. M. Ryan, J. B. Wilgen, J. Hosea, and A. Rosenberg, "Loading and asymmetry measurements and modeling for the national spherical torus experiment ion cyclotron range of frequencies system," Fusion science and technology 43, 503-513 (2003).</footnote>

<footnote n="31">³¹W. Tierens, L. Colas, and E. M. Team, "Slab-geometry surface waves on steep gradients and the origin of related numerical issues in a variety of ICRF codes," Journal of Plasma Physics 87, 905870405 (2021).</footnote>

<footnote n="32">³²L. Colas, W. Helou, G. Urbanczyk, V. Bobkov, F. Calarco, N. Fedorczak, D. Milanesio, and J. Hillairet, "Numerical assessment of ICRF-specific plasma-wall interaction in the new ITER baseline using the SSWICH-SW code," Nuclear Materials and Energy 42, 101831 (2025).</footnote>

<footnote n="33">³³J. Myra, "A tutorial on radio frequency sheath physics for magnetically confined fusion devices," Journal of Plasma Physics 87, 905870504 (2021).</footnote>

<footnote n="34">³⁴P. M. Bellan, *Fundamentals of plasma physics* (Cambridge university press,</footnote>
2008).

<footnote n="35"><sup>35</sup>A. Ram and A. Bers, “Coupling theory for ICRF heating of large tokamaks,” Nuclear fusion **24**, 679 (1984).</footnote>

<footnote n="36"><sup>36</sup>I. Pavlov and J. Heikkinen, “Effect of antenna orientation and plasma anisotropy on the directivity of fast wave antenna radiation,” Physics of Plasmas **2**, 3573–3581 (1995).</footnote>

<footnote n="37"><sup>37</sup>P. Vallejos, T. Johnson, R. Ragona, T. Hellsten, and L. Frassinetti, “Effect</footnote>

<footnote>of poloidal phasing on ion cyclotron resonance heating power absorption,” Nuclear Fusion **59**, 076022 (2019).</footnote>

<footnote n="38"><sup>38</sup>R. Ragona, F. Durodié, A. Messiaen, J. Ongena, M. Van Schoor, S. Agzaf, T. Batal, J.-M. Bernard, X. Courtois, J.-M. Delaplanche, R. Dumont, F. Durand, F. Faisse, M. Firdaouss, L. Gargiulo, P. Garibaldi, J. Hillairet, T. Hoang, G. Lombard, P. Mollard, Z. Chen, Y. Song, H. Xu, Q. Yang, and C. Yu, “Status of the WEST travelling wave array antenna design and results from the high power mock-up,” Nuclear Fusion **62**, 026046 (2022).</footnote>