Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/multilayer_nanobeam_phase10/DERIVATION_PHASE10.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Complex-frequency feed-pole continuation

Write s=omega/omega_ref. At fixed complex s, the layered strip operator is

D(beta,s) = I - (s k0_ref)^2 Delta_epsilon_core G_b(beta,s).

Starting from the validated real pole, a right eigenvector overlap selects the same branch at every Newton step. With left/right eigenvectors l,r, the tracked eigenvalue derivative is

d lambda/d beta = (l^H D_beta r)/(l^H r),

and Newton updates beta by -lambda/lambda_beta. At the converged root, the strip Green residue is

R(s) = r [l^H G_b] / [l^H D_beta r].

Reciprocity remains exact for complex frequency:

beta_-(s) = -beta_+(s),
R_-(s) = -R_+(s)^T.

Thus only the positive feed pole is numerically continued. This avoids an independent negative-branch solve and guarantees that the later xy-reduced cavity determinant remains complex symmetric.