Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/multilayer_nanobeam_phase9/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Multilayer nanobeam — Phase 9

This phase constructs the first complete finite determinant for the supplied multilayer cavity. The smooth layered-strip continuum is integrated on an outgoing complex-beta contour. The deformation keeps the real-qy integrals away from film poles and half-space branch endpoints. Both Si3N4 feed poles are subtracted from the contour integrand and restored through the exact finite-hole real-space matrix from Phase 8.

At the initial P0-disk/z01 truncation, the physical x=+1,y=-1 determinant is 50 by 50. Increasing the Zernike and z orders produces a nested convergent sequence without reinstating an invalid z mirror symmetry.