The quantum and classical Fano parameter q

The Fano resonance has been regarded as an important phenomenon in atomic and molecular physics for more than half a century. Typically, a combination of one quantum bound state and one or more continuum result in an asymmetric peak in the ionization spectrum. The peak-shape, called the Fano profile, can be expressed by the simple formula derived by Fano in 1935. However the interpretation of its main parameter $q$, which represents the asymmetry of the peak in the formula, is not intuitively transparent. The Fano resonance is not necessarily a quantum effect, but it is a manifestation of a certain physical mechanism in various systems, both quantum and classical. We present three intuitively transparent classical pictures and rigorously derive their Fano profiles to properly formulate the physics of the Fano parameter.