Optical analogy for stellarators: Ridges as caustics and coils as singularities

TL;DR

This paper develops an analytical theory linking ridges in stellarator flux surfaces to optical caustics, unifying plasma and coil structures through geometrical optics principles.

Contribution

It introduces a novel optical analogy framework for understanding ridge formation and coil placement in stellarators, grounded in geometrical optics and topological constraints.

Findings

Ridges correspond to magnetic field line caustics.

Coil and ridge locations are constrained by a zero-determinant manifold.

More compact devices naturally develop sharp ridges on the inboard side.

Abstract

A common feature of most numerically optimized stellarator geometries is the presence of sharp ridges on outer flux surfaces, irrespective of the rotational transform. Despite their importance, an analytical theory for their existence has been lacking. In this work, we demonstrate that ridges are not artifacts but mathematical necessities. We develop such a theory for devices with quasisymmetry (QS). We demonstrate that QS exhibits close connections with the theory of geometrical optics, following Parker's ``optical analogy" (E.N. Parker, Geophys. Astrophys. Fluid Dyn, 1989). By mapping vacuum QS to the eikonal equation of geometrical optics, we derive the conditions for ridge formation, identified as field line caustics where magnetic field lines focus. Furthermore, we prove a geometric theorem for stellarator coil design: both ridges and filamentary coils must lie on the zero-determinant manifold of the magnetic gradient tensor. This topological constraint unifies the description of plasma ridges and external coils, providing a precise criterion for identifying valid coil locations and explaining the efficacy of the magnetic gradient lengthscale (J. Kappel et al., Plasma Phys. Control. Fusion, 2024) as a coil optimization parameter. We demonstrate that as the device becomes more compact, sharp ridges naturally form on the inboard side in quasiaxisymmetry. We support our analytical theory with extensive numerical evidence.