Source
engines/design23_v1/package/design23_recentered_optimizer/multilayer_nanobeam_phase11/DERIVATION_PHASE11.md · assembled 2026-07-29 15:57 UTC.
Complete complex-frequency QNM operator¶
At complex s=omega/omega_ref, every dimensional wave number scales by s.
The beta contour is scaled by k0_ref s, while its dimensionless topology and
outgoing deformation remain fixed. The positive feed pole is Newton-continued
at the same s; the negative pole is supplied by reciprocity.
The complete nonlinear pole problem is
where A(s) contains the feed-subtracted complex-beta strip remainder and the
exact finite-hole feed residues. For a tracked simple operator eigenvalue
lambda(s), a complex secant iteration solves lambda(s)=0. Branch identity
is maintained by overlap of successive right eigenvectors, and each cavity
evaluation likewise uses the preceding feed pole as its continuation seed.
The quality factor and wavelength are recovered from
for the exp(-i omega t) convention, where a decaying QNM has Im(s)<0.