Computational and Analytical Optimization of Helicon Antennas with a Fast Full Wave Solver Exploiting Azimuthal Fourier Decomposition

TL;DR

This work tackles the computational challenge of modeling helicon‑driven plasmas for the MAP/AWAKE context by developing a 3D full‑wave helicon solver that uses azimuthal Fourier decomposition to reduce the problem to a handful of 2D axisymmetric simulations. By reconstructing 3D fields from these mode solutions, the approach enables rapid exploration of hundreds of density/antenna‑length scenarios while accurately predicting wavefields and power deposition. The study identifies that low‑order azimuthal modes, especially $m=\pm1$, dominate the fields and that the simulated spectra align with the helicon dispersion relation, which explains the existence of density‑dependent optimum antenna lengths. An analytic expression for the optimal antenna length, $L_{ideal}$, is derived from the overlap between the antenna power spectrum and the dispersion relation, providing a practical design rule for minimizing RF power while achieving target densities. The framework substantially reduces computational cost and is applicable to any linear helicon device with a sufficiently homogeneous magnetic field, enabling optimized plasma profiles and more efficient helicon operation.

Abstract

Plasma wakefield accelerators (PWA), such as AWAKE, require homogenous high-density plasmas. The Madison AWAKE Prototype (MAP) has been built to create a uniform argon plasma in the $10^{20}\,\mathrm{m^{-3}}$ density range using helicon waves. Computational optimization of MAP plasmas requires calculating the helicon wavefields and power deposition. This task is computationally expensive due to the geometry of high-performance half-helical antennas and the small wavelengths involved. We show here for the first time how the 3D wavefields can be accurately calculated from a small number of 2D-axisymmetric simulations. Our approach exploits an azimuthal Fourier decomposition of the non-axisymmetric antenna currents to massively reduce computational cost and is implemented in the Comsol finite-element framework. This new tool allows us to calculate the power deposition profiles for 800 combinations of plasma density, antenna length, and radial density profile shape. The results show the existence of an optimally coupling antenna length in dependence on the plasma density. This finding is independent of the exact radial profile shape. We are able to explain this relationship physically through a comparison of the antenna power spectrum with the helicon dispersion relation. The result is a simple analytical expression that enables power coupling and density optimization in any linear helicon device by means of antenna length shaping.

Figures (14)

• Figure 1: CAD model of the core components of the Madison AWAKE Prototype (MAP) . Shown are the magnetic field coils, vacuum vessel with argon plasma, and helicon antenna (center). The antennas used in our simulations are either left-helical (LH) or right-helical (RH). Both versions are shown at the bottom along with the spatial rotation and propagation directions of the different azimuthal modes denoted by wave number $m$. Modes that rotate in a right-handed (red) or left-handed (blue) sense around $\bm{\hat{z}}$ are launched in opposite directions. These launch directions are reversed for opposite antenna helicities. The direction of current flow during one half of the RF cycle is indicated in green.
• Figure 2: Helicon-TG dispersion relation with corresponding axial and radial wave lengths in a uniform plasma with $B = 50 \mathrm{\; mT}$, $n_e = 5\cdot 10^{19}\mathrm{\; m^{-3}}$ at $f = 13.56 \mathrm{\; MHz}$. The dispersion relation splits into the helicon and Trivelpiece-Gould (TG) branches at the minimum axial wave number. Axial wavelengths are an order of magnitude larger than radial wavelengths over most of the dispersion relation and up to two orders of magnitudes larger in the high $\beta$ parts of the dispersion relation.
• Figure 3: Flattened geometry of a right-handed half-helical antenna of the type shown on the bottom right in \ref{['fig:MAPCAD']}. RF current directions and magnitudes are shown in green, coordinate system directions and origin in blue, and general antenna shape and dimensions in black.
• Figure 4: Real and imaginary parts of the $K_{\phi}$ and $K_{z}$ current densities for the right-handed antenna on the lower right in \ref{['fig:MAPCAD']} for azimuthal mode number m=1, calculated with \ref{['eq:antCurZ', 'eq:antCurPhi']}.
• Figure 5: Reconstruction of azimuthal antenna currents $K_{\phi}$ in $(\phi, z)$ space for the ten lowest order azimuthal modes and comparison to the real antenna currents. The imaginary parts of the currents in every $\pm m$ pair cancel each other. The currents have been calculated using \ref{['eq:antCurZ', 'eq:antCurPhi']} and summed from $m=-5$ to $m=5$ using the Fourier series in \ref{['eq:mphiFtDef2']}.