Om-Theory of Macroscopic Electromagnetism: Greener Vibes for Isotropy-Broken Media

Abstract

The applicability ranges of macroscopic and microscopic electromagnetisms are opposite. While microscopic electromagnetism deals with point sources, singular fields, and discrete atomistic materials, macroscopic electromagnetism concerns smooth average distributions of sources, fields, and homogenized effective metamaterials. Greens function method - GFM - involves finding fields of point sources and applying superposition principle to find fields of distributed sources. When utilized to solve microscopic problems GFM is perfectly within the applicability range. Extension of GFM to simple macroscopic problems is convenient, but not fully logically sound, since point sources and singular fields are technically not a subject of macroscopic electromagnetism. This explains the difficulty of both finding the Greens functions and applying superposition principle in complex isotropy-broken media, which are very different from microscopic environments. In this manuscript, we lay out a path to solution of macroscopic Maxwells equations for distributed sources bypassing GFM, by introducing inverse approach and a method based on Om-potential which we describe here. To the researchers of electromagnetism this provides access to powerful analytical tools and a broad new space of solutions for Maxwells equations.