An electromagnetic multipole expansion beyond the long-wavelength approximation

TL;DR

This paper derives exact multipole moment expressions for dynamic electromagnetic sources beyond the long-wavelength approximation, revealing significant differences from traditional approximate methods, especially for non-trivial source shapes.

Contribution

It introduces exact formulas for electromagnetic multipole moments that improve upon traditional approximations, applicable to complex source geometries.

Findings

Exact expressions differ significantly from approximate ones for non-trivial shapes.

Validation against analytical solutions confirms accuracy of the new formulas.

Discrepancies are notable even for subwavelength sources.

Abstract

The multipole expansion is a key tool in the study of light-matter interactions. All the information about the radiation of and coupling to electromagnetic fields of a given charge-density distribution is condensed into few numbers: The multipole moments of the source. These numbers are frequently computed with expressions obtained after the long-wavelength approximation. Here, we derive exact expressions for the multipole moments of dynamic sources that resemble in their simplicity their approximate counterparts. We validate our new expressions against analytical results for a spherical source, and then use them to calculate the induced moments for some selected sources with a non-trivial shape. The comparison of the results to those obtained with approximate expressions shows a considerable disagreement even for sources of subwavelength size. Our expressions are relevant for any scientific area dealing with the interaction between the electromagnetic field and material systems.