Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/multilayer_nanobeam_phase10/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Multilayer nanobeam — Phase 10

The feedthrough pole is continued from the validated real-frequency mode to complex s = omega / omega_reference. At each Newton step, the non-Hermitian Dyson eigenpair is selected by overlap with the preceding right vector. The left/right eigenvectors give the exact complex-beta Laurent residue. The negative pole and residue follow from reciprocity rather than a second solve.

This continuation is required before a complex-frequency cavity pole can be trusted: simply scaling the real feed beta would corrupt both the port phase and the external linewidth.