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 δ being the asymmetry factor of the Fano component and the other parameter η 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.

Figure 1

Examples of photonic systems showing first-order resonances. From top to bottom: Fabry-Perot cavity (F-P), ring resonator (Ring), and waveguide grating (WG).

Figure 2

First-order resonant line shapes. Horizontal axis in all graphs shows normalized frequency ε. Vertical axis shows scattering intensity $T$(ε) calculated according to (6) without normalization ($A$ = 1). The graphs are arranged as follows: from left to right, η = 0 (Lorentz), 1/4, 1/2, 3/4, 1 (Fano); from top to bottom, δ = 0, 1/2, 1, 2. Line shapes for negative values of δ can be obtained by flipping the axis of normalized frequency: $T$(ε,η,−δ) = $T$(−ε,η,δ).

Figure 3

Fiber taper transmission spectra of a 1D silicon photonic crystal nanobeam coupled to a fiber taper displaying a typical asymmetric line shape spectra (black) and a Fano-Lorentz fit (red). Inset: SEM of the photonic crystal cavity nanobeam. Length of the nanobeam is 10 μm.