Historical derivation — preserved in full

Source: engines/design23_v1/package/design23_recentered_optimizer/multilayer_nanobeam_phase9/DERIVATION_PHASE9.md
Snapshot: Design23 recentered optimizer package. The body below is unabridged.

Complete complex-beta cavity determinant

For complex beta on the outgoing Sommerfeld deformation, the finite-film poles and half-space branch endpoints move off the real-qy integration path. The layered background can therefore be evaluated by ordinary convergent qy quadrature. Dyson resummation gives the unperforated Si3N4 strip matrix G_wg(beta).

Before longitudinal integration, both feed poles are removed:

G_reg(beta) = G_wg(beta)
              - R_+/(beta-beta_f)
              - R_-/(beta+beta_f).

For the xy-reduced hole projection P(beta), the smooth contribution is

G_h,reg = integral_C d beta/(2 pi)
          P(-beta)^T G_reg(beta) P(beta).

The negative contour half need not be evaluated independently. Reciprocity gives its matrix as the transpose of the positive-half contribution, cutting the expensive layered evaluations in half and imposing exact complex symmetry. The exact finite-hole feed matrix from Phase 8 is then added.

The air-hole pole operator is

A = I - k0^2 (epsilon_air-epsilon_SiN)
        (G_h,reg + G_h,feed).

Longitudinal panel width is controlled by the full 11.3-micrometer hole span, not only by spectral thresholds. In the y2-z1/P0-z01 audit, reducing maximum dimensionless beta-panel width from 0.1 to 0.05 stabilizes the smallest singular value to roughly four significant figures. The resulting operator at 780.443613 nm is nonsingular; this is a fixed-frequency assembly validation, not yet a Q prediction.