Mode volume¶

Closed-system reference¶

For a weakly dispersive, low-loss cavity and a dipole at \(\mathbf r_0\) with unit orientation \(\hat{\mathbf d}\), a familiar effective volume is

\[ V_\mathrm{eff}(\mathbf r_0,\hat{\mathbf d})= \frac{\int \varepsilon(\mathbf r)|\mathbf E|^2\,d^3r} {\varepsilon(\mathbf r_0)|\hat{\mathbf d}\cdot\mathbf E(\mathbf r_0)|^2}. \]

This expression is not directly convergent for an outgoing QNM and is insufficient for dispersive materials.

QNM residue definition¶

For the nonlinear projected operator (A(\omega)), let right and left null vectors satisfy (Au=0) and (v^\dagger A=0). The simple-pole residue contains

\[ \frac{u v^\dagger}{v^\dagger(\partial_\omega A)u}. \]

The device's point-evaluation maps convert this coefficient-space residue into the oriented electric Green-tensor residue at the emitter. The resulting complex QNM volume must state whether its real part, magnitude, or a derived Purcell observable is optimized. Current legacy runners report a normalized real-valued core-point quantity; their detailed map construction is archived with the phase-17/phase-28 material.

FDTD energy definition¶

At the fitted resonance, use coherently demodulated fields and the material- appropriate energy density. For dispersive media the electric term involves

\[ \frac{1}{4}\mathbf E^*\cdot \frac{\partial(\omega\varepsilon)}{\partial\omega}\mathbf E, \]

with the corresponding magnetic term. Report the exact finite integration region, interface interpolation, downsampling, and emitter denominator.

Required labels¶

Never publish a bare number called “mode volume.” Record:

reference point and dipole orientation;
local material and index used for normalization;
normalization unit, commonly \((\lambda/n)^3\);
analytical QNM or FDTD energy convention;
peak, core-center, and molecular-layer-center values as distinct keys;
convergence and any complex-to-real reduction.