Description
Large-scale rectangular magnetrons are central to industrial high-throughput coating, yet their theoretical description remains challenging due to the complex topology of their non-axisymmetric magnetic fields. We construct field-aligned Clebsch-type flux coordinates for the confinement region above the cathode, starting from a physically admissible vacuum field generated by an optimized ensemble of sub-surface magnetic dipoles. The magnetic flux partitions the domain into core, return region, and halo, bounded by neutral curves that form the geometric backbone of the construction. In contrast to axisymmetric devices with a single apex null and magnetic dome, elongated rectangular configurations may, beyond a critical aspect ratio, undergo a symmetry-breaking bifurcation of the central null and develop additional off-axis null points. Below the corresponding bifurcation height we identify a regular domain in which field lines remain arc-like and connect core and return region. The technically relevant active region is a subset of this domain and comprises those arc-like field lines along which the magnetic field strength exceeds a prescribed threshold. Within the regular domain, a magnetic flux function and a generalized azimuth are constructed that label flux surfaces and individual field lines and reproduce the magnetic field in Clebsch form, while an arc-length coordinate completes the system.