Quasinormal Mode Excitation Coefficients

Updated 10 December 2025

Quasinormal mode excitation coefficients are residue-based factors that quantify how external sources trigger discrete resonant modes in dissipative systems.
They are computed using techniques such as continued-fraction solvers in black hole perturbation theory and biorthogonal projection in optical resonators.
Their applications span gravitational-wave analysis and photonic device modeling, effectively linking theoretical formulations with observable phenomena.

Quasinormal mode excitation coefficients quantify how external sources or initial data excite the discrete, complex-frequency resonant modes (“quasinormal modes”, or QNMs) of dissipative wave systems, such as black holes, optical resonators, and dispersive electromagnetic cavities. These coefficients are central to modal decomposition techniques for linearized systems where the response is expressed as a sum over QNMs, each weighted by an excitation factor that encapsulates both intrinsic system properties and the coupling to the source. The formalism extends across gravitational wave astrophysics, photonic materials, and more generally, any system where open or absorbing boundaries lead to non-Hermitian eigenvalue problems.

1. Mathematical Definition and Residue Formulation

Excitation coefficients are rigorously defined as residues of the frequency-domain Green’s function at the QNM poles. In canonical notation, for a system governed by a linear differential operator $\mathcal{L}[\psi] = S(x,\omega)$ , the Green’s function $G(x,x';\omega)$ has isolated simple poles at discrete complex QNM frequencies $\omega_n$ , such that

$G(x, x'; \omega) \sim \frac{\psi_n(x)\psi_n(x')}{\omega - \omega_n} \;\;\text{as}\;\;\omega\to\omega_n\, .$

The excitation coefficient (or excitation factor, QNEF) for mode $n$ is extracted as

$B_n = \left. \frac{A_\mathrm{out}(\omega)}{2\omega \, \frac{d A_\mathrm{in}}{d \omega}} \right|_{\omega=\omega_n}$

where $A_\mathrm{in/out}$ are the asymptotic amplitudes of homogeneous solutions at infinity or the horizon (or, more generally, outgoing/ingoing boundaries) (Lo et al., 31 Mar 2025, Kubota et al., 8 Sep 2025, Dolan et al., 2011, Chen et al., 2024, Silva et al., 2024). These coefficients are independent of the source specifics—fully intrinsic to the system—but the full observable amplitude also involves an overlap integral coupling the source or initial data to the mode: $C_n = B_n\,I_n \quad , \quad I_n = \int \psi_n(x')\,S(x',\omega_n)\,dx'$ where $C_n$ is the complete mode amplitude.

2. Computational Frameworks

Black Hole Perturbation Theory

In Kerr and Schwarzschild black holes, QNMs are solutions of the Teukolsky, Regge–Wheeler, or Sasaki–Nakamura equations with outgoing/ingoing boundary conditions. The excitation factors are computed via continued-fraction solvers (Leaver–Nollert), series methods (Mano–Suzuki–Takasugi, MST), or direct integration with complex scaling to regulate exponential behavior at boundaries (Lo et al., 31 Mar 2025, Zhang et al., 2013, Kubota et al., 8 Sep 2025, Green et al., 2022).
The amplitude of the QNM in waveforms is

$C_{lmn} = B_{lmn}\,I_{lmn}$

where $B_{lmn}$ is extracted via the residue of the Green’s function at $\omega_{lmn}$ and $I_{lmn}$ is a source-dependent overlap (Rocca et al., 8 Dec 2025, Zhang et al., 2013).

Optical Resonators and Dispersive Media

Electromagnetic resonators employ a discretized Maxwell operator linearized with auxiliary fields for dispersive materials. The excitation coefficients $\alpha_m$ in such systems are similarly computed through biorthogonal projection:

$\alpha_m(\omega) = \frac{\langle F_h, x_m^\perp \rangle}{i(\tilde\omega_m - \omega)}$

with $F_h$ detailing the source, $x_m^\perp$ the left eigenvector, and $\tilde\omega_m$ the mode frequency. Multiple algebraic forms exist depending on the linearization and source representation (Gras et al., 2019).

Nonlinear and EFT Models

In effective-field-theory extensions of GR or in nonlinear QNM couplings, excitation coefficients are computed by modified residue formulas after analytical continuation, often requiring complex contour integration and careful regulator prescriptions to handle non-Hermitian features and avoid divergences (Silva et al., 2024, Ma et al., 2024).

Quasinormal mode (QNM) excitation coefficients govern how strongly each mode is present in the system’s response following a perturbation. In gravitational wave astronomy, the observed ringdown signal is modeled as a sum over damped sinusoids,

$h(t) = \sum_{l,m,n} C_{lmn} e^{-i\omega_{lmn} t}$

where $C_{lmn}$ is determined by both the universal excitation factor $B_{lmn}$ (residue at the QNM pole) and a process-dependent overlap with the source. The mode structure determines sensitivity to parameters such as mass, spin, and source configuration (Gao et al., 21 Feb 2025, Rocca et al., 8 Dec 2025, Amicis et al., 26 Jun 2025).

In electromagnetic systems, the modal expansion for the scattered field follows analogous constructions, with the amplitude of each $x_m$ mode quantified by the excitation coefficient, providing a rigorous link between spectral decomposition and field response (Gras et al., 2019).

4. Resonances, Non-Uniqueness, and Degeneracies

Resonant Enhancement and Exceptional Points

As eigenvalues approach one another (e.g., via variation of system parameters like black hole spin), excitation coefficients can exhibit sharp resonant amplification or suppression, associated with exceptional points and branch-point singularities in the non-Hermitian spectrum. Near avoided crossings, the factors execute figure-eight trajectories (lemniscates) in the complex plane, and overtone hierarchy can invert; the sum of excitation coefficients across the resonance remains nearly invariant (Kubota et al., 8 Sep 2025, Lo et al., 31 Mar 2025).

In dispersive and photonic systems, multiple expressions for the excitation coefficient are mathematically equivalent when the full mode set is used but differ algorithmically depending on linearization and source representation choices. Non-uniqueness is benign for complete bases but critical when truncations or practical implementations are involved (Gras et al., 2019).

Degenerate and Non-Orthogonal Modes

Degeneracies in the mode spectrum require orthogonalization of the eigenbasis (e.g., via Gram–Schmidt), as the physical expansion coefficients become ill-defined in non-diagonalizable settings (Gras et al., 2019). In Kerr, the Petrov D property induces nontrivial symmetry and conserved currents, allowing for orthonormal modes under generalized bilinear forms even in non-Hermitian cases (Green et al., 2022).

5. Dynamical and Nonlinear Excitation: Time Dependence and Second-Order Couplings

Dynamical excitation during nonstationary processes (plunge, merger, inspiral) introduces time-dependent excitation coefficients,

$C_{lmn}(t) = B_{lmn}[c_{lmn}(t) + i_{lmn}(t)]$

that encode both “activation” (hereditary, source-history dependent) and “impulsive” (localized, instantaneous) pieces, subject to causality conditions and horizon redshift effects (Amicis et al., 26 Jun 2025). Late-time amplitudes settle to “stationary” ringdown values plus an infinite tower of redshift modes.

Quadratic (second-order) mode amplitudes arise from nonlinear couplings and scale as $b^{(2)}[A^{(1)}]^2$ where $b^{(2)}$ is strongly dependent on black hole spin and angular overlap structures. These modes persist even in the extremal Kerr limit, with the formalism extending to arbitrary harmonic couplings (Ma et al., 2024).

6. Numerical Implementation and Robustness

Extraction from Simulations

Extraction of QNM coefficients from waveforms (e.g., numerical relativity) is done via linear or nonlinear fitting algorithms, often employing iterative greedy approaches to fix robust fundamentals before overtone estimation, thus maximizing reliability and avoiding spurious contributions from mode-mixing or fitting artifacts. Robustness criteria include constancy over time windows, consistency across models, and sensitivity to remnant black-hole parameter estimation (Gao et al., 21 Feb 2025).

Semi-Analytic and Eikonal Regimes

For high multipolar order (eikonal limit), semi-analytic expansions (e.g., Dolan–Ottewill inverse-multipole techniques with high-order WKB corrections) yield rapid, percent-level accurate estimates for excitation factors, bridging the gap to fully numerical MST approaches (Chen et al., 2024, Dolan et al., 2011).
The dependence of excitation coefficients on system extensions—such as new length scales in effective field theories—shows that overtones are generically more sensitive than fundamentals, and isospectrality can be broken, requiring refined parametrizations for consistent interpretation (Silva et al., 2024).

7. Applications and Impact

Excitation coefficients are instrumental in gravitational-wave data analysis (black-hole spectroscopy, parameter estimation, probing fundamental physics via mode amplitude-phase relations), in photonic device modeling (spectral decomposition, resonance characterization), and in the systematic design of template banks for multimodal systems (Gao et al., 21 Feb 2025, Rocca et al., 8 Dec 2025, Gras et al., 2019). Their theoretical framework enables robust benchmarks against numerical and experimental data, clarifies the impact of nonlinear effects, and grounds the modal expansion in mathematically consistent residue theory.

In summary, quasinormal mode excitation coefficients represent universal, residue-based quantities that mediate between modal structure and physical excitation in dissipative wave systems. Their precise computation, interpretation, and extraction from data constitute a central pillar of open-system spectral theory across relativity, photonics, and condensed matter domains.