Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/multilayer_nanobeam_phase13/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Multilayer nanobeam — Phase 13

This phase differentiates the complete complex-frequency multilayer cavity operator with respect to all packed gaps and ellipse radii. The regular complex-beta continuum and off-diagonal guided-residue propagation use analytic Fourier/Zernike geometry derivatives. Exact finite-hole guided self blocks are differentiated locally, without repeating the contour or a QNM solve.

This is a fixed-frequency operator JVP. The next phase contracts it with the left/right QNM vectors and applies the implicit pole formula to obtain ds/dgeometry and dQ/dgeometry without differentiating through the secant iterations.