Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/multilayer_nanobeam_phase11/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Multilayer nanobeam — Phase 11

The complete 50-hole operator now accepts complex s = omega / omega_reference. Every evaluation:

1. Newton-continues the positive Si3N4 feed beta pole;
2. constructs the negative pole by exact reciprocity;
3. integrates the feed-subtracted layered strip on the positive half of the scaled complex-beta contour;
4. restores exact finite-hole right/left feed propagation; and
5. forms the air-hole Maxwell pole operator with (s k0_ref)^2.

At s=1, this route must reproduce the independently assembled real-frequency Phase 9 determinant. It is the operator used for the subsequent nonlinear QNM solve.