Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/multilayer_nanobeam_phase2/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Multilayer nanobeam track — Phase 2

Phase 2 reconstructs the regular electric Weyl dyadic from the exact TE/TM stack reflections implemented in Phase 1. It analytically adds the local longitudinal contact term and projects the result on normalized Legendre functions across the 500 x 300 nm Si3N4 core.

This is the first operator in the new track that directly changes the uniform-cell eigenproblem. It retains both z parities; the asymmetric stack generates nonzero even/odd z coupling.

Run:

python -m unittest multilayer_nanobeam_phase2/test_phase2.py -v
python multilayer_nanobeam_phase2/run_phase2_cross_section.py \
  --output multilayer_nanobeam_phase2/RESULTS_PHASE2.json

The next phase solves the asymmetric uniform-waveguide beta pole and performs basis/cutoff continuation. Only after that pole is stable should the existing longitudinal hole determinant and QNM residue be reconnected.