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BORAY-3D: A ray tracing code for three-dimensional magnetized plasma configurations

BORAY-3D is a versatile ray tracing code that unifies broad radio-frequency coverage, arbitrary 2D/3D magnetic configuration handling, and fully relativistic electron-cyclotron absorption/emission modeling, having been systematically benchmarked against established codes and experimental data for various plasma applications.

Original authors: Yuxuan Wang, Huasheng Xie

Published 2026-08-07

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CC BY 4.0

Original authors: Yuxuan Wang, Huasheng Xie

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to navigate a city where the streets are made of invisible, twisting magnetic ribbons, and the cars are waves of energy zipping through the air. This is the world of plasma physics, the study of super-hot, electrically charged gas that powers the stars and, if we can master it, could provide limitless clean energy for Earth. To keep this fiery gas from melting its container, scientists use powerful magnets to create a "cage" that holds the plasma in a specific shape. But to heat this gas up to the temperatures needed for fusion, they shoot radio waves into it, much like a microwave heats food. However, these waves don't just travel in straight lines; they bend, twist, and get absorbed by the plasma in complex ways depending on the shape of the magnetic cage and the speed of the particles inside. To predict exactly where the energy goes, scientists use "ray tracing," a computer method that acts like a GPS for these invisible waves, calculating their path step-by-step to ensure they hit the right spot to heat the fuel efficiently.

Enter BORAY-3D, a new piece of software developed by researchers Yuxuan Wang and Huasheng Xie that acts as a super-powered GPS for these plasma waves. Before this tool, scientists had to use different maps for different types of waves or different shapes of magnetic cages. Some programs could handle the complex, 3D twists of a "stellarator" (a donut-shaped machine with a twisted core) but only for high-frequency waves. Others could handle a wide range of frequencies but only for simpler, perfectly round "tokamak" shapes. BORAY-3D is the first to combine everything into one package. It can track waves from very low frequencies (like the hum of a radio) all the way up to extremely high frequencies (like the light from a laser), and it can do this in both simple, symmetrical shapes and wildly complex, 3D magnetic landscapes.

The researchers tested this new code by comparing it against other trusted programs and real-world data from existing fusion experiments. They simulated how waves travel through a tokamak with a "ripple" in its magnetic field (caused by the gaps between the magnets) and found their results matched perfectly with established codes. They also tested it on the Wendelstein 7-X (W7-X) stellarator, a massive machine in Germany with a very twisted magnetic shape. In these simulations, BORAY-3D showed that ignoring the "relativistic" effects—where particles move so fast they act according to Einstein's theory of relativity—can lead to big mistakes in predicting how much energy is absorbed. By including these effects, the code accurately predicted how the waves would heat the plasma and how the plasma would glow back at the detectors. Whether it's simulating 13.56 MHz waves in a small experimental device or 220 GHz waves in a giant stellarator, BORAY-3D proves that a single, flexible tool can handle the entire spectrum of fusion heating challenges, helping scientists design better, more efficient fusion reactors for the future.

Technical Summary of BORAY-3D: A Ray Tracing Code for Three-Dimensional Magnetized Plasma Configurations

Problem Statement
Radio frequency (RF) waves are essential for plasma heating and current drive in magnetic confinement devices, spanning frequencies from megahertz (ion-cyclotron, helicon, lower-hybrid) to gigahertz (electron-cyclotron). While ray tracing codes are standard tools for modeling wave propagation under the geometric-optics approximation, existing tools often lack a unified capability to handle the full spectrum of RF waves across diverse plasma geometries. Specifically, many codes either offer broad frequency coverage but are restricted to axisymmetric equilibria, or provide three-dimensional (3D) capabilities specialized only for electron-cyclotron (EC) applications. Furthermore, original implementations often omit fully relativistic absorption effects and rely on flux-function-based coordinate systems, which limits their applicability to open field-line regions or complex 3D configurations like stellarators and tokamaks with toroidal-field ripple.

Methodology
The authors developed BORAY-3D, a new ray tracing code that integrates broad-frequency modeling, arbitrary 2D/3D geometry handling, and relativistic physics into a single framework. The methodology involves:

Coordinate System and Ray Equations: The code extends the axisymmetric BORAY formulation by solving ray equations in cylindrical coordinates $(r, \phi, z)$ . Unlike flux-coordinate-based codes, BORAY-3D accepts magnetic field, density, and temperature data directly in $(r, \phi, z)$ without the constraint of flux functions. This allows for the unified treatment of both closed and open field-line regions. The toroidal mode number $n_\phi$ is allowed to vary, enabling the simulation of non-axisymmetric fields.
Dispersion and Absorption Models:
- Low-to-Medium Frequency: For ion-cyclotron (ICW), helicon, and lower-hybrid (LHW) waves, the code utilizes a non-relativistic hot-plasma model inherited from BORAY, based on the Maxwellian dispersion tensor.
- High Frequency (ECW/ECE): For electron-cyclotron waves, the code implements a fully relativistic Maxwellian absorption model. It calculates the absorption coefficient using a relativistic integral and employs the Krivenski–Orefice tensor for wave polarization.
- ECE Radiative Transfer: A reciprocal radiative transfer module is coupled to the relativistic absorption model to compute electron cyclotron emission (ECE) spectra, utilizing Kirchhoff's law in the Rayleigh–Jeans limit.
Numerical Implementation: The ray equations are integrated using a fourth-order Runge-Kutta scheme. The cold plasma dispersion relation is used for ray tracing, while the hot plasma dispersion tensor is solved iteratively to determine absorption along the ray path.

Key Contributions
The paper identifies three primary contributions that distinguish BORAY-3D from existing literature:

Broad-Frequency Validity: The code operates within a single theoretical framework covering a frequency range from 13.56 MHz (helicon) to 220 GHz (EC emission), handling ICW, helicon, LHW, and ECW with appropriate kinetic absorption treatments for each regime.
Unified Geometric Treatment: By utilizing cylindrical coordinates and avoiding flux-function constraints, the code can simulate arbitrary 2D axisymmetric and fully 3D magnetic configurations (including stellarators and tokamaks with ripple) and seamlessly incorporate both closed and open field-line regions.
Relativistic EC Physics: The implementation of fully relativistic Maxwellian absorption and reciprocal ECE radiative transfer removes the non-relativistic limitation of the original BORAY code, which is critical for accurate modeling of weakly absorbed higher harmonics and ECE spectra.

Results and Benchmarks
The code was systematically benchmarked against established codes and published experimental results across various scenarios:

Tokamak Toroidal Field Ripple: Benchmarked against GENRAY for a 3.7 GHz lower-hybrid wave in a tokamak with strong toroidal-field ripple. Ray trajectories agreed within 7.1 mm, and power absorption trends matched, with discrepancies attributed to different underlying hot-plasma models.
Stellarator ECW (W7-X): Compared against Raytrax and TRAVIS for 140 GHz O-mode and X-mode rays. Trajectories agreed within 5 mm. The study highlighted that non-relativistic models significantly underestimate absorption and deviate from relativistic curves, validating the necessity of the relativistic treatment.
Stellarator ECW (HSX): Validated against published results for 28 GHz X2 heating. The code reproduced the density dependence of first-pass absorption and central deposition, though absolute values differed slightly due to unavailable machine-readable field data in the reference.
LHD-like ICW: Simulated a 50 MHz fast wave in a scaled LHD configuration. The code reproduced physical trends reported in literature, including density-dependent parallel wavenumber upshifts and the transition from tritium to electron absorption dominance.
MUSE Helicon: Demonstrated application to a 13.56 MHz helicon wave in a permanent-magnet stellarator, showing distinct 3D trajectory effects based on launch symmetry planes.
W7-X ECE: Calculated X-mode emission spectra (115–220 GHz) for low-field and high-field sides. The relativistic model correctly predicted X3 optical depths, whereas non-relativistic models underestimated them by an order of magnitude.

Significance and Claims
The authors claim that BORAY-3D provides a versatile tool for studying RF wave propagation in magnetized plasmas by unifying capabilities previously scattered across different codes. Its significance lies in its ability to handle the full RF spectrum and complex 3D geometries (including open field lines and ripple effects) without requiring flux-coordinate transformations. The inclusion of fully relativistic EC absorption and ECE modeling addresses specific limitations of earlier versions, enabling more accurate predictions for high-frequency heating and diagnostics in modern stellarator and tokamak devices. The paper notes that while the geometric-optics approximation is robust, wave reflection at plasma edges and diffraction effects still require full-wave modeling for quantitative validation in those specific regimes. Future work is planned to address collisional damping, wall reflection, current drive, and non-Maxwellian distributions.

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