Rectangular magnetron fields described in Clebsch-type flux coordinates

Large-scale rectangular magnetrons are the workhorses of modern high-throughput coating and other industrial plasma-processing applications. Despite their importance, the theoretical modeling and numerical simulation of these devices remain underdeveloped, largely due to the intricate topology of their magnetic fields, which poses significant conceptual and computational challenges. This work addresses one central challenge: the construction of field-aligned flux coordinates for non-axisymmetric geometries. Focusing on the confinement region above the cathode, the magnetic field is shown to admit a global Clebsch-type representation B=∇Ψ×∇Θ, where Ψ(r) is a magnetic flux function whose constant values define nested flux surfaces, and Θ(r) is a generalized azimuth labeling individual magnetic field lines on each surface. Together, Ψ and Θ uniquely specify the spatial location of a field line within that region, and their gradients reproduce the magnetic field through their cross product, automatically ensuring ∇·B=0. A third coordinate S(r) measures the arc length along field lines and provides a natural parameterization. The triplet (Ψ,Θ,S) thus defines a system of field-aligned flux coordinates suited for describing plasma behavior in large non-axisymmetric rectangular magnetrons, including electron confinement, transport, and heating. The topological principles of this construction are discussed in detail and illustrated with a realistic rectangular magnetron field. The resulting mathematical framework provides a rigorous foundation for self-consistent modeling and simulation of plasma dynamics in field-aligned flux coordinates, with direct relevance to industrial-scale plasma processing.