Weak Lacunae of Electromagnetic Waves in Dilute Plasma

The propagation of waves is said to be diffusionless, and the corresponding governing PDE (or system) is said to satisfy Huygens’ principle if the waves due to compactly supported sources have sharp aft fronts. The areas of no disturbance behind the aft fronts are called lacunae. Diffusionless propagation of waves is rare, whereas its opposite—diffusive propagation accompanied by aftereffects—is common. Nonetheless, lacunae can still be observed in a number of important applications, including the Maxwell equations in vacuum or in dielectrics with static response. In the framework of these applications, lacunae can be efficiently exploited for the numerical simulation of unsteady waves, and considerable progress has been made toward the development of lacunae-based methods for computational electromagnetism. Maxwell equations in vacuum are Huygens’ because they reduce to a set of d’Alembert equations. Besides d’Alembert equations, there are no other scalar Huygens’ equations in the standard 3 + 1-dimensional Minkowski space-time. In terms of physics, this means that the mechanisms of dissipation and dispersion destroy the lacunae. In fact, all conventional low-frequency electromagnetic models, such as metals with Ohm conductivity, semiconductors, and magnetohydrodynamic media, are diffusive. An important case of the propagation of high-frequency electromagnetic waves in plasma is governed by the Klein–Gordon equation. It does not reduce to the d’Alembert equation either, and therefore the corresponding propagation is diffusive as well. However, one can still identify “weak lacunae” in the solutions of the Klein–Gordon equation, with the aft fronts that can be clearly observed, although they may not be as sharp as in the pure Huygens’ case. Moreover, one can show that the “depth” of a weak lacuna is controlled by the dimensionless ratio of the Langmuir frequency to the primary carrying frequency of the waves.