Fano resonances

An asymmetric resonance that appears when a narrow (discrete) mode interferes with a broad (continuum) one. On one side of the narrow mode the two add, on the other they cancel, so the line shape is a peak next to a dip. Ugo Fano explained it in 1961 for atomic spectra; in nanophotonics the broad mode is often a dipole and the narrow one a quadrupole, a whispering-gallery mode or a dark mode; at high intensities nonlinear Fano resonances shift and switch with the light itself.

σ(ε) = (q + ε)² / (1 + ε²), with the reduced energy ε = 2(E − E₀)/Γ (resonance position E₀, width Γ) and the asymmetry parameter q. In practice a background and an amplitude are added, and the formula is fitted to the measured or computed spectrum.

q compares the coupling of the light to the narrow mode with its coupling to the broad continuum. Large |q| gives an ordinary, symmetric Lorentzian peak; q = 0 a symmetric dip (an anti-resonance); |q| ≈ 1 the strongly asymmetric peak–dip shape. Its sign tells on which side the dip lies.

In particle clusters, dimers and nanoshells where a bright dipolar plasmon overlaps a dark or quadrupolar mode, and in arrays where particle plasmons meet lattice modes. The sharp dip makes them sensitive to the surrounding refractive index, so they are used in sensors; the calculator reports a sensing figure of merit.

A non-radiating state of a dielectric particle (its anapole moment): its electric dipole and toroidal dipole radiate equal and opposite fields, so they cancel in the far field and scattering drops sharply, while a strong field remains inside the particle. It shows up as a deep minimum in the electric-dipole scattering of high-index spheres and disks.

Fit the Fano formula plus a smooth background around the feature, with E₀, Γ, q and an amplitude as free parameters; start Γ from the visible width and q from the asymmetry. The calculator fits each Fano feature of an exact spectrum and reports q, width and position.