Linear systems approach to describing and classifying Fano resonances

We show that a generalized asymmetric resonant line shape derived elsewhere from rigorous electromagnetic calculations [Gallinet and Martin, Phys. Rev. B 83, 235427 (2011)] and from the two-oscillators model [Joe et al., Phys. Scr. 74, 259 (2006)] can also be obtained using a very general assumption that the spectral dependence of the scattering amplitudes is given by the transfer function of a linear system. We reformulate the line shape equation and show that in the case of a first-order transfer function all possible line shapes can be presented by a weighted sum of the original Fano and Lorentzian line shapes. We propose a new two-parameter classification scheme for asymmetric resonances with one parameter \ensuremath{\delta} being the asymmetry factor of the Fano component and the other parameter \ensuremath{\eta} quantifying the relative weight of the Fano and Lorentzian components of the line shape. The proposed formula is used to fit experimental spectra of a silicon photonic crystal cavity nanobeam interrogated using a fiber taper probe.