Exact electromagnetic multipole expansion using elementary current multipoles

Abstract

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.

Figures (8)

• Figure 1: Examples of current configurations in the excitations of current multipoles of different orders $l$ in the point-multipole approximation: a current dipole ($l$ = 1), quadrupole ($l$ = 2), octupole ($l$ = 3), hexadecapole ($l$ = 4), and triacontadipole ($l$ = 5), corresponding to moments $p_x$, $Q_{yx}$, $O_{zxy}$, $H_{xxyz}$, and $T_{xxxyz}$, respectively. Each arrow represents a single linear current element (see Ref. grahn2012). Current multipoles of high orders, such as 4 and 5, are relatively common, constituting classical multipoles of lower orders. In particular, current hexadecapoles are constituents of classical electric quadrupoles and magnetic octupoles, whereas current triacontadipoles appear in classical electric octupoles and magnetic hexadecapoles (see Eqs. (\ref{['eq:aE20']})-(\ref{['eq:aM44']})). In numerical calculations presented in this work, we consider classical multipoles up to the electric and magnetic hexacontatetrapoles ($l$ = 6) which include current multipoles of orders $l$ up to 8. Note that, in the used convention, the outermost current element at positive coordinates is positive if the multipole moment is positive.
• Figure 2: Numerical validation of the exact expression for the current multipole moments [Eq. (\ref{['eq:M_exact']})] using the optical response of a silicon sphere of diameter 600 nm embedded in PMMA, for which we evaluate the multipole contributions to the normalized scattering cross section $Q_{\text{scat}}$ (a, b) and absorption cross section $Q_{\text{abs}} = Q_{\text{ext}}-Q_{\text{scat}}$ (c, d) defined in Eqs. (\ref{['eq:scat']})-(\ref{['eq:abs']}). The dots correspond to the contributions of moments $a_{\text{E}}(l,m)$ and $a_{\text{M}}(l,m)$ obtained by numerically calculating the current multipole moments $M^{(l)}_{\text{exact}}$ in Comsol Multiphysics using Eqs. (\ref{['eq:J']}) and (\ref{['eq:M_exact']}) and converting them to the classical multipole moments using Eqs. (\ref{['eq:Mapping-for-aE0']})-(\ref{['eq:Mapping-for-aM1']}), (\ref{['eq:aE20']})-(\ref{['eq:aM44']}), and Ref. kolkowski2025. The solid lines are the cross sections obtained from the Mie theory. The insets show the peaks of electric and magnetic triacontadipoles ($l=5$) and hexacontatetrapoles ($l=6$). In the main plots, the curves for subsequent multipoles are vertically offset for clarity, and some of them are magnified by a constant factor, as indicated. The results of numerical multipole expansion and Mie theory overlap perfectly (up to the numerical precision of Comsol), which validates the expressions presented in this work.
• Figure 3: Numerical validation of the current multipole expansion using the optical response of a silver sphere of diameter 400 nm embedded in PMMA. Similarly to Fig. \ref{['fig:2']}, the solid lines represent the cross sections calculated using the Mie theory, while the dots correspond to the cross sections obtained numerically in Comsol Multiphysics using the current multipole expansion (Eqs. (\ref{['eq:J']}), (\ref{['eq:M_exact']}), (\ref{['eq:Mapping-for-aE0']})-(\ref{['eq:Mapping-for-aM1']}), (\ref{['eq:aE20']})-(\ref{['eq:aM44']}), and Ref. kolkowski2025). In the given example, the scattering cross section ($Q_{\text{scat}}$) has significant contributions of (a) the electric multipoles of order $l\leq6$ (up to hexacontatetrapoles) and (b) the magnetic multipoles of order $l\leq3$ (up to octupoles), while the absorption cross section ($Q_{\text{abs}}$) is dominated by the electric multipoles of $l\geq$4, as shown in (c). Similarly to Fig. \ref{['fig:2']}, we find an excellent agreement between the numerical multipole expansion and Mie theory.
• Figure 4: Anapole excitations in a silicon nanodisk: conventional analysis based on decomposition of the total electric dipole moment into the electric dipole and electric toroidal dipole moments in the point-scatterer approximation. In this example, we consider a disk of diameter 600 nm and thickness 60 nm, surrounded by vacuum, illuminated by a linearly polarized plane wave. The anapole excitations are revealed by peaks in the spectrum of volume-averaged electric energy density enhancement $\langle\eta\rangle$ (violet curve), coinciding with the dips of the total scattering cross section $Q_{\text{scat}}^{\text{tot}}$ (black) and the electric-dipole scattering cross section $Q_{\text{scat}}^{a_\text{E}(1,0)}$ (red). The pink curves in the upper plot show the contributions of $p_z^{\text{point}}$, $T_z^{\text{point}}$ and $p_z^{\text{point}}+ikT_z^{\text{point}}$ to the scatteting cross section (dotted, dashed, and solid curve, respectively; see Eqs. (\ref{['eq:pzsph']})-(\ref{['eq:Tz']})), whereas the phases of $p_z^{\text{point}}$ and $-ikT_z^{\text{point}}$ are shown in the lower plot (dotted and dashed, respectively). The insets show the spatial distributions of $\eta$ across the $xz$-plane inside the disk for each of the anapole excitations (at $f$ = 307, 434, and 530 THz).
• Figure 5: Selected contributions of classical multipole moments $a_{\text{E/M}}(l,\pm m)$ to the scattering cross section of a silicon nanodisk (the same as in Fig. \ref{['fig:4']}). Other contributions are significantly smaller or vanish due to symmetry.