---
title: Quasinormal Modal Expansion Method (QMEM)
url: https://www.emergentmind.com/topics/quasinormal-modal-expansion-method-qmem
type: topic
---

# Quasinormal Modal Expansion Method (QMEM)

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 [1311.3244], scattering matrices [1609.03902, 2105.01749], far-field observables [2003.11305], dispersive resonators including absorbing dielectrics [2101.00968, 2312.11048], quantum-surface-response models [2105.06328], and regularized quadratic observables [2212.11117]. More recent work extends the method to physically agnostic nonlinear eigenvalue formulations [2410.03631], finite topological waveguides [2508.07767], Floquet slabs [2507.02784], and user-oriented numerical workflows [2602.18067].

## 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 $\tilde{\omega}_m$ [2101.00968, 1609.03902]. 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) [2101.00968].

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 $\tilde{\omega}_m$ [2101.00968]. 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 [2101.00968].

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

Source: https://www.emergentmind.com/topics/quasinormal-modal-expansion-method-qmem