  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.
