Obstructions imposed by the Poincaré–Brouwer (“hairy ball”) theorem on the propagation of electromagnetic waves

Abstract

The Poincaré–Brouwer (“hairy ball”) theorem is applied for the analysis of the propagation of electromagnetic waves, in the case when the wavefront forms a surface, topologically equivalent to a sphere (the surface possessing the Euler characteristic χ = 2). At least one point on the surface at which vectors of electric and magnetic fields equal zero will appear. The Poynting vector in this point is also zero. Restrictions imposed by the Poincaré–Brouwer theorem on the reflection and refraction of electromagnetic waves are discussed, in the case when the media are separated by the interface topologically equivalent to a sphere.