User:Summermolasses/Fano resonannce

Plot of scattering cross-section versus normalized energy for various values of the parameter q illustrating the asymmetric Fano line-shape. The parameter q is the ratio of resonant scattering to direct scattering.

A Fano resonance is a type of resonant scattering phenomena that gives rise to an asymmetric line-shape. Interference between a background process and a resonant scattering process produces the asymmetric profile. Italian physicist Ugo Fano is credited with giving the first theoretical explanation for it. Because it is a general wave phenomena examples can be found in across many areas of physics and engineering.

History

The explanation of the Fano line-shape first appeared in the context of inelastic electron scattering by helium and autoionization. The incident electron doubly excites the atom to the $2s2p$ state. The doubly excited atom spontaneously decays by ejecting one of the excited electrons. Fano showed that interference between the amplitude to simply scatter the incident electron and the amplitude to scatter via autoionization creates an asymmetric scattering line-shape around the autoionization energy with a line-width very close to the inverse of the autoionization lifetime.

Explanation

The Fano resonance line-shape is due to interference between two scattering amplitudes, one due to scattering within a continuum of states (the background process) and the second due to a excitation of a discrete state (the resonant process). It is important to note that the resonant state energy lie in the continuum of the background states. Near the resonant energy, the background scattering amplitude typical varies slowly with energy while the resonant scattering amplitude changes both in magnitude and phase quickly. It is this variation that creates the asymmetric profile.

For energies far from the resonant energy the background scattering process dominates. Within $2\Gamma _{res}$ of the resonant energy, the phase of the resonant scattering amplitude changes by $\pi$. It is this rapid variation in phase that creates the asymmetric line-shape.

Fano showed that the total scattering cross-section assumes the following form,

${\frac {\left(q\Gamma _{res}/2+E-E_{res}\right)^{2}}{\left(\Gamma _{res}/2\right)^{2}+\left(E-E_{res}\right)^{2}}}$

where q, the Fano parameter, measures the ratio of resonant scattering to the direct (background) scattering amplitude. (This is consistent with the interpretation within the Feshbach–Fano partitioning theory.) In the case the direct scattering amplitude vanishes, the q parameter becomes infinite and the Fano formula boils down to the usual Breit–Wigner (Lorentzian) formula:

${\frac {\left(\Gamma _{res}/2\right)^{2}}{\left(\Gamma _{res}/2\right)^{2}+\left(E-E_{res}\right)^{2}}}$

Examples

Examples of Fano resonances can be found in atomic physics, nuclear physics, condensed matter physics, circuits, microwave engineering, and nanophotonics.