Quasinormal Modal Expansion Method (QMEM)

• QMEM is a modal-analysis framework that expands wave responses in open systems using complex-frequency eigenstates with outgoing-wave conditions.
• It replaces frequency-by-frequency simulations with a reduced-order representation based on complex poles, modal residues, and background contributions.
• Applications of QMEM span electromagnetics, nanophotonics, elastic plates, and topological waveguides, improving efficiency in analyzing radiative systems.

Searching arXiv for recent and foundational papers on QMEM and related quasinormal-mode expansions. arXiv search query: "quasinormal modal expansion method quasinormal mode expansion" Quasinormal Modal Expansion Method (QMEM) is a family of modal-analysis frameworks for open, radiative, and often dispersive wave systems in which the driven response is expanded over quasinormal modes (QNMs), that is, complex-frequency eigenstates satisfying outgoing-wave conditions. Across electromagnetics, nanophotonics, elastic plates, and finite topological waveguides, QMEM replaces direct frequency-by-frequency simulation by a reduced-order representation in terms of complex poles, modal residues, and, where required, nonresonant or background contributions. In the electromagnetic setting, QMEM has been developed for open two-dimensional structures (Vial et al., 2013), scattering matrices (Alpeggiani et al., 2016, Benzaouia et al., 2021), far-field observables (Binkowski et al., 2020), dispersive resonators including absorbing dielectrics (Sauvan, 2021, Ming, 2023), quantum-surface-response models (Zhou et al., 2021), and regularized quadratic observables (Betz et al., 2022). More recent work extends the method to physically agnostic nonlinear eigenvalue formulations (Nicolet et al., 2024), finite topological waveguides (Martí-Sabaté et al., 11 Aug 2025), Floquet slabs (Vial et al., 3 Jul 2025), and user-oriented numerical workflows (Wu et al., 20 Feb 2026).

1. Conceptual definition and mathematical setting

QMEM starts from the observation that resonant scattering, radiation, absorption, and near-field enhancement in open systems are governed by discrete complex resonances rather than by the real-frequency normal modes of closed cavities. In the electromagnetic case, QNMs are source-free solutions of Maxwell’s equations with outgoing radiation boundary conditions and complex eigenfrequencies $\tilde{\omega}_m$ (Sauvan, 2021, Alpeggiani et al., 2016). A standard source-free formulation for nonmagnetic resonators is

$$\nabla \times \tilde{\mathbf{E}}_m = i\tilde{\omega}_m \mu_0 \tilde{\mathbf{H}}_m,\qquad \nabla \times \tilde{\mathbf{H}}_m = -i\tilde{\omega}_m \varepsilon(\mathbf{r},\tilde{\omega}_m)\tilde{\mathbf{E}}_m,$$

or equivalently

$$\nabla\times\nabla\times \tilde{\mathbf{E}}_m(\mathbf{r}) - \frac{\tilde{\omega}_m^2}{c^2}\,\varepsilon(\mathbf{r},\tilde{\omega}_m)\,\tilde{\mathbf{E}}_m(\mathbf{r}) = 0$$

with outgoing-wave conditions implemented numerically by perfectly matched layers (PMLs) (Sauvan, 2021).

The fundamental expansion ansatz expresses the scattered field as a modal superposition. For a background/scattered decomposition $\mathbf{E}=\mathbf{E}_b+\mathbf{E}_s$, one common form is

$$\mathbf{E}_s(\mathbf{r},\omega)=\sum_m \alpha_m(\omega)\,\tilde{\mathbf{E}}_m(\mathbf{r}),$$

with modal coefficients that are meromorphic in frequency and inherit the poles at $\tilde{\omega}_m$ (Sauvan, 2021). In dispersive media described by rational models, one explicit expression is

$$\alpha_m(\omega)=\frac{\omega}{\omega-\tilde{\omega}_m}\,a_m(\omega),\qquad a_m(\omega)=-\iiint_V \Delta\varepsilon(\mathbf{r},\omega)\,\mathbf{E}_b(\mathbf{r},\omega)\cdot \tilde{\mathbf{E}}_m(\mathbf{r})\,d^3\mathbf{r},$$

which makes the pole structure explicit (Sauvan, 2021).

A recurring theme in the literature is that QMEM is not a single formula but a class of exact or approximate expansions whose precise form depends on the operator representation, linearization strategy, and treatment of background terms. This is explicit in dispersive formulations, where a continuous family of exact