Extracting complete resonance characteristics from the phase of physical signals

Resonances are ubiquitous in wave physics and shape the spectral response of every linear physical system. Yet their description remains fragmented, through partial and sometimes competing definitions. Here, we show that the resonance properties are fully encoded in the phase of the response functions of physical systems and can be decoded by deriving the spectral function of the phase. The frequency derivative of the phase is related to the Wigner-Smith time-delay operator, but is applied to individual elements of the scattering matrix. By deriving the meromorphic expansion of this scalar element-wise time delay in terms of poles and zeros in the complex frequency plane obtained from the knowledge of both phase and amplitude of the response function, we demonstrate that it yields a quality function whose peaks reveal both the strength and the position of each resonance. This expression proves that poles only are not sufficient to fully characterize resonances and that zeros bring the other part of the information. The numerical calculation of the quality function together with the analytical and balanced treatment of poles and zeros provide a rigorous framework for identifying and quantifying resonances without any prior knowledge of the physical system.