Exact electromagnetic multipole expansion using elementary current multipoles

Multipole expansion plays an important role in the description of electromagnetic scatterers, allowing them to be accurately characterized by a small set of expansion coefficients. However, to describe electromagnetic excitations inside a scatterer, the current density in it should be decomposed into current multipoles, which include nonradiating current configurations (anapoles) that are absent in the classical field-based expansion. Unfortunately, the use of current multipoles has so far been limited by the absence of an exact and general expression for the current multipole moments beyond their point-multipole approximation. Here, we derive such an expression and present the exact mapping relations between the classical and current multipole moments. We use our theory to calculate the scattering and extinction cross sections for large, wavelength-scale, optical scatterers supporting multipole excitations up to the sixth order, showing perfect agreement with the Mie theory. We also demonstrate the ability of current multipole expansion to describe anapole excitations beyond the small-scatterer approximation, which allows us to derive the exact anapole condition and reveal the actual current configurations and their contributions to scattering. Our theoretical framework is valid for electromagnetic scatterers of arbitrary sizes and shapes without restrictions on the multipole orders, complementing the existing theory of electromagnetic multipole expansion. The minimalistic and universal character of current multipoles makes them a convenient tool for characterizing and designing diverse electromagnetic scattering systems of arbitrary complexity.

💡 Research Summary

The paper introduces a rigorous framework for expanding the electromagnetic response of scattering objects directly in terms of their internal current density J(r), rather than the traditionally used field‑based electric and magnetic multipoles. While the classical multipole expansion accurately describes far‑field radiation, it fails to capture non‑radiating current configurations such as anapoles, which are invisible in the far field. To overcome this limitation, the authors define elementary current multipoles and derive an exact, size‑independent expression for their moments.

The central result is Equation (3):

M(l)exact = i ω (2l‑1)!! (l‑1)! ∫ J(r) r^{l‑1} j{l‑1}(k r) / (k r)^{l‑1} d³r,

where j_{l‑1} are spherical Bessel functions of the first kind and k is the wavenumber of the surrounding medium. This formula reduces to the familiar point‑multipole approximation (Equation 2) only in the long‑wavelength limit (kr ≪ 1) but remains valid for arbitrary particle sizes and shapes. Each current multipole of order l corresponds to a unique Cartesian monomial involving a component of the current density and, for l ≥ 2, products of normalized coordinates (x/r, y/r, z/r). Table I lists explicit expressions up to l = 5, illustrating the simplicity of the elementary configurations (e.g., dipole p_x, quadrupole Q_{yx}, octupole O_{zxy}, etc.).

A crucial contribution of the work is the exact mapping between these current multipoles and the conventional electric (a_E) and magnetic (a_M) multipole coefficients used in Mie theory and other scattering formalisms. The mapping is not one‑to‑one: an electric multipole of order l contains contributions from current multipoles of order l and l + 2, while a magnetic multipole of order l contains contributions from current multipoles of order l + 1. Equations (4)–(7) provide explicit examples for dipole moments, showing how the electric dipole coefficient a_E(1,0) is a linear combination of the current dipole p_z and the current octupole components O_{ijk}. Higher‑order mappings are derived in Appendix C, and a Python script is supplied to generate them automatically for any l.

The theoretical developments are validated numerically using COMSOL Multiphysics. Two benchmark systems are considered: a silicon sphere (diameter 600 nm) and a silver sphere (diameter 400 nm), both embedded in PMMA. The current density obtained from the simulated internal fields is inserted into Equation (3) to compute the exact current multipole tensors. These tensors are then transformed into a_E and a_M coefficients using the derived mapping, and the resulting partial scattering, extinction, and absorption cross‑sections (Equations 8–11) are compared with those obtained from exact Mie theory. The agreement is essentially perfect, confirming that the current‑multipole formalism works for wavelength‑scale particles where the point‑multipole approximation would fail.

The paper further demonstrates the power of the current‑multipole approach for analyzing anapole excitations. A silicon nanodisk (diameter 600 nm, thickness 60 nm) in vacuum is illuminated by a plane wave. Conventional analysis treats the anapole condition as the destructive interference between the electric dipole p and the toroidal dipole T, expressed as p + i k T ≈ 0 in the point‑dipole limit. Using the exact current multipoles, the authors show that the same condition holds without approximation, and they can explicitly identify the higher‑order current components (e.g., octupole contributions) that participate in the cancellation. The spectral positions of the anapole resonances are corroborated by peaks in the volume‑averaged electric energy density and simultaneous dips in the total and electric‑dipole scattering cross‑sections.

In summary, the authors provide (i) an exact, analytically tractable expression for current‑multipole moments valid for arbitrary scatterer size and geometry, (ii) a complete, unambiguous mapping to the traditional electric and magnetic multipole coefficients, (iii) numerical validation against Mie theory for realistic dielectric and metallic particles, and (iv) a generalized, exact description of anapole conditions that reveals the underlying current configurations. This framework bridges the gap between source‑based and field‑based descriptions of scattering, offering a versatile tool for the design of nanophotonic devices, metasurfaces, and metamaterials where control over both radiating and non‑radiating modes is essential.