Mie Resonances & the Kerker Effect

Nanosphere

Radius 120 nm

Wavelength λ 550 nm

Refractive index n 3.50

Dipole Response

Size parameter x

1.37

Qsca (dipole)

0.00

|a1| electric dipole

0.00

|b1| magnetic dipole

0.00

Forward / backward scattering

0.0 dB

How it works

A dielectric nanosphere much smaller than the wavelength scatters light through Mie theory. Keeping only the lowest two multipoles — the electric dipole a₁ and magnetic dipole b₁ — already explains most of the physics of a sub-wavelength resonator:

a1 = [m ψ1(mx)ψ1'(x) − ψ1(x)ψ1'(mx)] / [m ψ1(mx)ξ1'(x) − ξ1(x)ψ1'(mx)]
b1 = [ψ1(mx)ψ1'(x) − m ψ1(x)ψ1'(mx)] / [ψ1(mx)ξ1'(x) − m ξ1(x)ψ1'(mx)]

x  = 2π r / λ        (size parameter)
m  = n_particle / n_medium
ψn, ξn = Riccati-Bessel functions

Unlike a plasmonic metal nanoparticle, a high-index dielectric sphere (Si, TiO₂, GaAs) supports both an electric and a magnetic dipole resonance at nanoscale sizes. Their interference sets the scattering direction:

S1(θ) = 1.5·[a1 + b1·cosθ]
S2(θ) = 1.5·[a1·cosθ + b1]
I(θ)  = ½(|S1(θ)|² + |S2(θ)|²)

Forward (θ=0):  I ∝ |a1 + b1|²
Backward(θ=π):  I ∝ |a1 − b1|²

First Kerker condition (a1 ≈ b1, in phase) — backscattering nearly vanishes; almost all light is thrown forward through the particle.
Second / anti-Kerker condition (a1 ≈ −b1) — forward scattering nearly vanishes instead, throwing light backward.
The 3D lobe on the right is the scattered-intensity surface I(θ), rotated about the incident-beam axis (vertical); its front/back asymmetry is exactly this a1/b1 interference.

This directional control from an all-dielectric, non-absorbing nanoparticle — no metal, no ohmic loss — is the working principle behind Huygens' metasurfaces, high-efficiency dielectric antireflection coatings and directional nanoantennas in modern nanophotonics.

Scattered intensity I(θ) Incident plane wave

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