Extracting Complete Resonance Characteristics From the Phase of Physical Signals

Resonances are common in wave physics and their full and rigorous characterization is crucial to correctly tailor the response of a system in both time and frequency domains. However, they have been conventionally described by the quality factor, a real-valued number quantifying the sharpness of a single peak in the amplitude spectrum, and associated with a singularity in the complex frequency plane. But the amplitude of a physical signal does not hold all the information on the resonance and it has not been established that even the knowledge of the full distribution of singularities carries this information. Here we derive a dimensionless quality function that fully characterizes resonances from the knowledge of the phase spectrum of the signal. This function is driven by the spectral derivative of the phase. It is equivalent to the imaginary part of the Wigner-Smith time delay but it has the advantage of being valid for arbitrary response functions, including the components of the S-matrix. The spectral derivative of the phase can be calculated numerically from simulations or experimental acquisitions of the phase spectrum. Alternatively, it can be retrieved from the distribution of poles and zeros in the complex frequency plane through an analytic expression, which demonstrates that singularities do not suffice to fully characterize the resonances and that both singularities and zeros must be taken into account to retrieve the quality function. This approach permits to extract all the characteristics of resonances from arbitrary spectral response functions without a priori knowledge on the physical system.