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Historical derivation — preserved in full

Source: engines/quan_loncar_v1/lineage/phases_01_28/quan_loncar_complete_lineage_phases_01_28/fryett_q_series_phase11/DERIVATION_PHASE11.md
Snapshot: Quan–Lončar complete lineage, phases 1–28. The body below is unabridged.

Phase 11 derivation: even thickness degree four and polarization audit

Controlled continuation

At fixed in-plane Zernike degree 3, waveguide y degree 3, transverse rule, Sommerfeld contour, and disk quadrature, compute

  1. hole z degree 3, waveguide z degree 3;
  2. hole z degree 3, waveguide z degree 4;
  3. hole z degree 4, waveguide z degree 4.

This separates the degree-four correction to the bare-waveguide Green tensor from the correction due to new physical hole-field basis functions. For the selected sector (s_x,s_y,s_z)=(-1,-1,+1), even thickness degree four admits E_x and E_y, but not E_z. The symmetry-block dimension grows from 1260 to 1710.

Polarization diagnostic

Let S be the orthonormal symmetry embedding and v the reduced right null vector. With u=S v, define

\[ w_j=\frac{\sum_{a\equiv j\pmod 3}|u_a|^2}{\|u\|_2^2}, \qquad j\in\{x,y,z\}. \]

The scalar hole basis and S are orthonormal, so the three weights sum to one and are invariant under reduced-coordinate ordering. They quantify the vector component content of the represented field inside the holes. They are not an electromagnetic energy partition for a non-Hermitian quasinormal mode.

The target mirror signs s_y=-1,s_z=+1 are the signs of an E_y field with a scalar envelope even in width and thickness. The chosen s_x=-1 selects the longitudinally odd member. Phase 7 chose it because it was the resolved E_y-compatible pole nearest omega/omega_ref=1; the paper/geometry audit needed to fix longitudinal parity remains deliberately pinned.

Numerical result

\[ Q_{(3,3)}=577.6291,\qquad Q_{(3,4)}=578.2189,\qquad Q_{(4,4)}=578.3588. \]

The separated relative changes are +0.102115% and +0.024190%, with a net degree-three to degree-four change of +0.126330%. Seed-to-mode overlaps are 0.999999 and 0.999727. The final coefficient fractions are

\[ (w_x,w_y,w_z)=(0.0033181,0.9955990,0.0010829). \]

Thus both even- and odd-thickness parity subsequences are provisionally settled at the present in-plane, cutoff, contour, and quadrature settings. This is convergence of one truncation axis, not a claim that Q=578.36 is the Fryett cavity's final physical Q.