Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/multilayer_nanobeam_phase8/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Multilayer nanobeam — Phase 8

This phase instantiates the exact 50 elliptical holes used in the supplied Majumdar FDTD benchmark. The independent geometry vector contains 25 positive gaps, 25 x radii, and 25 y radii. Reflection constructs the negative side.

Only the physical x=+1, y=-1 symmetry is applied. Both local z parities are retained because the substrate/anthracene/PVA stack breaks z reflection. The exact right/left-going Si3N4 feed residues from Phase 7 are projected onto the finite holes, including sign-ordered same-hole propagation rather than a point-hole approximation.