Topological electromagnetics in anisotropic media: Theory and application to magnetized plasmas

This Tutorial reviews the mathematical foundations of topological electromagnetics in anisotropic continuum media, using magnetized plasmas as a representative case study. Particular emphasis is placed on the mathematical tools and physical concepts commonly employed in the analysis of topological phenomena in complex electromagnetic materials, and on providing a unified introduction to concepts that are often presented from different perspectives in the literature. We begin by reviewing core concepts in differential geometry and introduce topological invariants in real-space geometry. We then discuss their relevance to topological quantities in electromagnetics, such as Berry curvature and Chern numbers, while emphasizing that these quantities differ in their underlying definitions. The framework is applied to the computation of bulk band topology and surface-wave dispersion relations by constructing decaying field solutions and enforcing boundary conditions. Examples are presented for surface-wave propagation at interfaces between a magnetized plasma and air, dielectric, or metallic media, in both two-dimensional and three-dimensional configurations. This Tutorial offers a foundation for future investigations, including experiments, and supports the development of analytical and numerical methods for other anisotropic, dispersive, and topologically nontrivial media.