Non-uniqueness 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 discretized Maxwell's operator. In this paper, we consider a dielectric permittivity that is a N-pole Lorentz function of the pulsation $\omega$. We propose a common formalism and obtain different formulas for the modal expansion. The non-uniqueness of the excitation coeffcient is due to a choice of the linearization of Maxwell's equation with respect to $\omega$ and of the form of the source term. We make the link between the numerical discrete modal expansion and analytical formulas that can be found in the literature. We detail the formulation of dispersive Perfectly Matched Layers (PML) in order to keep a linear eigenvalue problem. We also give an algorithm to regain an orthogonal basis for degenerate modes. Numerical results validate the different formulas and compare their accuracy.