Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/fryett_q_series_phase3/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Fryett Q series: Phase 3

This checkpoint connects the rectangular-waveguide Green tensor to the actual elliptical-hole geometry and produces the first finite cavity determinant.

The new object is

\[ D_N(\omega;g)=\det\!\left[ I-k_0(\omega)^2\Delta\epsilon\, P_NG_{\rm wg}^+(\omega)P_N \right]. \]

The longitudinal inverse transform is evaluated on a dimensionless outgoing Sommerfeld contour. Because the contour is scaled by the complex cladding wavenumber, it supplies a concrete homotopy from causal frequency to a quasinormal pole in the lower half-plane. It includes guided, radiation, and evanescent contributions in one integral; it does not insert an empirical radiation loss.

At this checkpoint, P_N contains only the normalized constant x-, y-, and z-vector mode in every ellipse. Higher Zernike--Legendre modes, higher waveguide polynomial degree, transverse quadrature order, and contour order are the four explicit convergence axes still to be raised.

Run:

PYTHONPATH=../fryett_q_series_phase1:../fryett_q_series_phase2 \
  python run_phase3_checkpoint.py
pytest -q

The pole-refinement routine is present, but a returned pole must not be called the Fryett result until the four-axis convergence table and geometry ambiguity are resolved.