Resonances and quality factor

Nonlinear outgoing problem

The source-free frequency-domain Maxwell equation is

\[ \nabla\times\mu^{-1}(\mathbf r,\omega)\nabla\times\mathbf E -\omega^2\varepsilon(\mathbf r,\omega)\mathbf E=0 \]

with outgoing radiation conditions. After device-specific projection, the quasinormal mode is a nonlinear null problem

\[ A(\tilde\omega,\theta)u=0, \]

where (\theta) is the physical geometry vector. Green-function formulations move the open boundary condition into an outgoing background resolvent; transfer/scattering formulations impose it through the exterior channel basis. cavitygrad shares the pole protocol, not the form of (A).

Q identities

For (\tilde\omega=\omega_r-i\gamma), modal energy decays as (e^{-2\gamma t}), so

\[ Q=\frac{\omega_r}{2\gamma} =-\frac{\Re\tilde\omega}{2\Im\tilde\omega}. \]

Independently, a channel with time-averaged outward power (P_c) and stored energy (U) has (Q_c=\omega_r U/P_c). Additive independent losses satisfy

\[ Q_\mathrm{total}^{-1}=\sum_c Q_c^{-1}. \]

The pole Q, ringdown Q, and flux-closure Q should agree within demonstrated numerical uncertainty. Their disagreement is one of the most valuable FDTD diagnostics.

Pole selection

The smallest singular value alone does not identify the desired mode. Refine a candidate complex pole, enforce passivity, evaluate the nonlinear residual, and continue the left/right eigenspace from the previous accepted geometry. Near degeneracy, track the invariant subspace instead of an arbitrarily phased single vector.

The complete device-specific outgoing-contour and determinant derivations are preserved in the derivation archive.