Nonuniqueness of the quasinormal mode expansion of electromagnetic Lorentz dispersive materials

Any optical structure possesses resonance modes, and its response to an excitation can be decomposed onto the quasinormal and numerical modes of a discretized Maxwell operator. In this paper, we consider a dielectric permittivity that is an N-pole Lorentz function of the frequency. Even for discretized operators, the literature proposes different formulas for the coefficients of the quasinormal-mode expansion, and this comes as a surprise. We propose a general formalism, based on auxiliary fields, which explains why and evidences that there is, in fact, an infinity of mathematically sound possible expansion coefficients. The nonuniqueness is due to a choice of the linearization of Maxwell's equations with respect to frequency and of the choice of the form of the source term. Numerical results validate the different formulas and compare their accuracy.