Multipoles, directional scattering and the Kerker conditions

The Mie series is a sum over multipoles, and every term has a physical identity: $a_\ell$ is the electric multipole of order $\ell$ (dipole, quadrupole, …), $b_\ell$ the magnetic one. Taking a spectrum apart term by term turns a curve with many bumps into a short list of resonances, each with its own field pattern and its own radiation diagram, and the interference of those diagrams is what makes a particle scatter forward, backward, or not at all. This is the vocabulary of all-dielectric nanophotonics, and it is a few lines of algebra on the Mie coefficients.

Each order's share

Because the vector spherical harmonics are orthogonal over the sphere, the scattered power is a plain sum over orders and types, with no cross terms:

Each term is bounded: a single channel cannot scatter more than $\frac{2}{x^2}(2\ell+1)$, the limit reached when $|a_\ell| = 1$, which for a lossless sphere is where the coefficient's phase passes through $\pi/2$ at a resonance. In cross-section terms the bound is $(2\ell+1)\lambda^2/(2\pi)$ per channel: a dipole, however resonant, cannot scatter more than $3\lambda^2/2\pi$, which is why a 10 nm silver particle at its plasmon has a cross section a hundred times its geometric one and no more. Extinction splits the same way, through $\mathrm{Re}\,a_\ell$ and $\mathrm{Re}\,b_\ell$; the calculator's multipole panel draws these shares, one colour per order, electric dashed and magnetic dot-dashed.

The small sphere: what a_1 and b_1 are made of

Expanding the coefficients in the size parameter shows what each resonance is physically. To leading orders,

The electric dipole is the quasi-static polarizability of a sphere, with the Clausius–Mossotti denominator and its plasmon pole at $m^2 = -2$; the magnetic dipole has no pole in the quasi-static limit and starts two powers of $x$ later, at the same order as the electric quadrupole. For a metal the first term is everything and the particle is an electric dipole; for a dielectric the $x^2$ correction inside the bracket and the $x^5$ terms grow quickly with $m$, and at $mx \sim \pi$ the expansion fails altogether, because the magnetic dipole resonates.

A magnetic dipole without magnetism

Silicon has $\mu = 1$, yet a silicon sphere has a strong magnetic-dipole resonance. When the wavelength inside the particle, $\lambda/n$, is close to its diameter, the electric field inside circulates around an axis perpendicular to the incident field. That loop of displacement current is a magnetic dipole, and in a high-index sphere it is the first resonance on the long-wavelength side: the condition is $\lambda_\mathrm{MD} \approx n_s\, d$, the first zero of the magnetic denominator at $mx \approx \pi$ for large $m$ (Evlyukhin et al. 2010), so a 150 nm silicon sphere resonates near 600 nm. The electric dipole sits at shorter wavelength ($mx \approx 4.5$), the magnetic quadrupole between them. Kuznetsov et al. (2012) saw the magnetic dipole move across the visible as the particles grew, giving them their colours, and Geffrin et al. (2012) measured the same resonances in the microwave range with centimetre-sized ceramic spheres, where the far-field pattern can be mapped directly.

import numpy as np
import pystratify as ps
from pystratify.docs import material  # n + ik of a refractiveindex.info page

lam = (430 + 2 * np.arange(186)) * 1e-3  # wavelength, µm
N = np.column_stack([material('crosscheck/classics/data/Si_Green-2008.yml', lam), np.full(lam.shape, (1+0j))])  # n + ik: core, then the host
cs = ps.cross_sections(ps.solve([0.091], N, lam))  # outer radii, µm
q_ext, q_sca = cs.q_ext, cs.q_sca
b1 = cs.sca_by_order[:, ps.TE, 0] / cs.geometric  # magnetic dipole part of Q_sca

Move the radius and watch the order: magnetic dipole first, then electric dipole and magnetic quadrupole close together, then the electric quadrupole. The electric dipole is broad because it also radiates from the particle's polarization outside the resonance; the magnetic modes, confined inside the sphere, are narrower.

Forward and backward

The far field in the forward and backward directions follows from the same coefficients, because at $\theta = 0$ and $\pi$ the angular functions $\pi_\ell$ and $\tau_\ell$ both equal $\pm\ell(\ell+1)/2$:

Electric and magnetic multipoles of the same order add in the forward direction and subtract backwards: an electric dipole and a magnetic dipole, crossed and in phase, radiate like a Huygens source, forward only. If only the dipoles matter and $a_1 = b_1$, the backscattering vanishes: the first Kerker condition, found by Kerker, Wang and Giles (1983) for spheres with $\varepsilon = \mu$, where every $a_\ell$ equals its $b_\ell$ and the sphere does not backscatter at any size. A dielectric sphere has no $\mu$, but it reaches $a_1 = b_1$ at one wavelength where its electric- and magnetic-dipole curves cross on the long-wavelength side of the magnetic resonance, and there it scatters forward almost only (Fu et al. 2013; Geffrin et al. 2012 saw the same in the microwave). The second Kerker condition, $a_1 = -b_1$, minimises forward scattering, but cannot null it: the optical theorem ties forward scattering to the extinction, which is never zero. With quadrupoles the conditions generalise to the full sums, $\sum_\ell (2\ell+1)(-1)^\ell(a_\ell - b_\ell) = 0$ for zero backscattering, which can be met at more than one wavelength and with a sharper directivity; the forward-to-backward ratio of the classic card is the number to watch.

From one sphere to a Huygens surface

A sphere at its first Kerker condition transmits all the light it intercepts, with a phase delay set by the two resonances. An array of such spheres, one per sub-wavelength cell, is a Huygens metasurface (Staude et al. 2013): a flat sheet with near-unit transmittance whose phase can be chosen from $0$ to $2\pi$ by the particle size, the building block of flat lenses and holograms. Its design reduces to placing the electric and magnetic dipole resonances of the single particle, which is the calculation above, with the lattice adding a correction that the single-sphere calculation leaves out.

Beyond the dipoles

When an electric dipole and a toroidal dipole of the same strength radiate in antiphase, their far fields cancel and the electric-dipole scattering nearly vanishes while the field inside stays strong: the anapole (Miroshnichenko et al., 2015). In the Mie picture it is simply a zero of $a_1$, visible in the decomposition as a dip of the electric dipole between resonances: the full coefficient contains both the Cartesian electric dipole and the toroidal moment, and the multipole expansion in vector spherical harmonics, being complete, needs no separate toroidal term. The distinction matters when one wants to know what is going on inside the particle (a toroidal field pattern) rather than what it radiates, and the near-field panel of the calculator shows exactly that circulating pattern at the anapole wavelength.