Historical derivation — preserved in full

Source: engines/design23_v1/package/design23_recentered_optimizer/multilayer_nanobeam_phase7/DERIVATION_PHASE7.md
Snapshot: Design23 recentered optimizer package. The body below is unabridged.

Layered Si3N4 strip and feed-pole subtraction

Let G_b(beta) be the projected outgoing Green matrix of the complete planar background from Phases 5–6. Adding the finite-width Si3N4 strip is a compact Dyson problem,

D(beta) = I - k0^2 Delta_epsilon_core G_b(beta),
G_wg(beta) = D(beta)^(-1) G_b(beta).

The physical y=-1 block contains the Ey-dominant feed mode. If beta_f is a simple zero of D, with normalized right/left null vectors r,l, then

G_wg(beta) = R_f/(beta-beta_f) + regular,

R_f = r [l^H G_b(beta_f)] / [l^H D'(beta_f) r].

At the real closed-background pole the operator is Hermitian, so l=r. The derivative is taken with respect to physical beta, giving a residue denominator with units of length. Reciprocity gives

R_- = -R_+^T

at the negative feed pole. Both Laurent terms are removed before numerical longitudinal integration and restored in real space as exact right/left-going feedthrough propagation. A symmetric two-sided Laurent estimate cancels the linear regular term and therefore converges quadratically to R_f.