Beyond Green’s Functions: Inverse Helmholtz and “Om” ॐ-Potential Methods for Macroscopic Electromagnetism in Isotropy-Broken Media

• Published in: Photonics Vol. 12; no. 7; p. 660
• Main Author: Durach, Maxim
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.07.2025
• Subjects: Electromagnetism; Green's functions; Helmholtz equations; Isotropy; Maxwell's equations; Metamaterials; Methods; Photonics; Quantum field theory; Superposition (mathematics)
• ISSN: 2304-6732; 2304-6732
• DOI: 10.3390/photonics12070660

The applicability ranges of macroscopic and microscopic electromagnetism are contrasting. 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. Green’s function method (GFM) involves finding fields of point sources and applying the superposition principle to find fields of distributed sources. When utilized to solve microscopic problems, GFM is well 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 Green’s functions and applying the superposition principle in complex isotropy-broken media, which are very different from microscopic environments. In this manuscript, we lay out a path to the solution of macroscopic Maxwell’s equations for distributed sources, bypassing GFM by introducing an inverse approach and a method based on “Om” ॐ-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 Maxwell’s equations.