# Electromagnetic Waves in Annular Regions
## 1 Introduction  
We collect here some mathematical results concerned with the development of solitary electromagnetic waves confined in bounded regions of space having a toroid shape. We start by pointing out that exact analytic solutions are available in the case of an unbounded cylinder (which is, hypothetically, a toroid closed at infinity), whereas for bounded rings some numerical assistance is necessary. The structures here considered display a strong analogy with fluid dynamics vortex rings (many classical texts are available as possible references). Indeed, vortex tori are amazing substructures. The flow lies inside a well determined doughnut-shaped domain and follows a rotatory motion around the major circumference. Usually, sections are not circles and the internal hole may degenerate into a segment ([PERSON]'s spherical vortex). These rings exhibit exceptional stability properties (see [1, 2]), even in presence of viscosity. An incomplete list of papers dealing with fluid vortices and related phenomena is [3, 4, 5, 6].  
Ring formations of electromagnetic type may appear, intentionally or spontaneously, inside appropriate containers, like in the case of plasma as a consequence of the movement of charges. In other applications, pure electromagnetic waves may find constrained states of equilibrium, where light \"rotates\" similarly to a non-viscous fluid. These waves enjoy additional features with respect to material fluids, being subject to electrical and magnetic properties. The situation can be ideally replicated by wrapping a wire around a toroid surface and supply it with an alternating current (like in a toroidal transformer). In order to simulate periodic situations, the wavelength of the pulsating current must be a sub-multiple of the length of the perimeter's section. However, periodicity cannot be obtained for all the geometries. For the reasons we are going to detail, the determination of the ring section depends on the behavior of the spectrum of a suitable elliptic operator. In particular, the shape and the magnitude of this region must allow for the existence of at least one eigenvalue with multiplicity greater or equal to two. This paper reviews the main results, with the aim of offering a mathematical background in view of possible practical implementations.  
Rotating light is present in a multitude of phenomena. Whispering gallery resonators exist in nature or are built on purpose to force electromagnetic waves to remain trapped (so possibly\"memorized\") into cavities. These waves circulate inside suitable geometrical environments. They are guided through a continuous reflection to turn around, and they may return at the origin with no change of phase, thus forming periodic patterns in time and space. In particular, photonic crystals are complex molecular configurations that may host trapped electromagnetic signals in their reticuli. Optical confinement in microcavities is examined for instance in [7]. There is plenty of papers, either theoretical or practical, on the subject. Here, we only have space to provide a few titles. A direct connection with topological structures associated with rings can be found in [8; 9; 10; 11; 12]. Some optical waves trapped in media are studied in [13; 14; 15]. Other more conceptual results are given in [16; 17; 18; 19; 20].