Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/quan_loncar_series_phase28/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Quan--Loncar series, Phase 28

Phase 28 converts the validated electric Green residue into a generalized complex mode volume. The Green tensor used by this project solves

[curl curl - k0^2 epsilon] G = I delta

and the nonlinear eigenfrequency variable is s=omega/omega0. For real unit polarization e, the convention-consistent inverse volume is

1/Vtilde = -2 epsilon(r0) stilde k0_ref^2 e^T Res_s(G) e.

The effective volume entering the high-Q Purcell formula is

1/Veff = Re(1/Vtilde).

The implementation validates the prefactor by comparing the mode-volume Purcell factor with both the single-pole Green tensor and a direct dressed Green inversion at the real resonance frequency.