Historical derivation — preserved in full

Source: engines/quan_loncar_v1/lineage/phases_01_28/quan_loncar_complete_lineage_phases_01_28/fryett_q_series_phase8/DERIVATION_PHASE8.md
Snapshot: Quan–Lončar complete lineage, phases 1–28. The body below is unabridged.

Phase 8 derivation: through-thickness convergence

1. Exact Legendre overlap

The hole perturbation fills the same thickness interval as the rectangular waveguide basis. With normalized Legendre functions, the z projection is

\[ \int_{-T/2}^{T/2}\phi_\ell(z)\phi_j(z)\,dz=\delta_{\ell j}. \]

Consequently, retaining hole degree ell=1 requires waveguide degree z at least one. No numerical z quadrature is introduced.

2. Controlled sequence

At fixed in-plane degree three and sector (-1,-1,+1), evaluate

1. hole z degree 0, waveguide z degree 0;
2. hole z degree 0, waveguide z degree 1;
3. hole z degree 1, waveguide z degree 1.

The first difference is the background-waveguide truncation effect. The second is the contribution of new physical hole-field coefficients.

For z-even sector sign s_z=+1, the old degree-zero modes contain the allowed E_x and E_y components. The new odd scalar Legendre modes can contribute only through E_z, whose polar-vector mirror sign supplies the second minus sign. Exact mirror reduction changes the sector dimension from 450 to 630, instead of doubling it blindly to 900.

3. Numerical result

At the common in-plane degree-three baseline,

\[ Q_{(\ell_h,\ell_{wg})=(0,0)}=508.7904, \]

\[ Q_{(0,1)}=510.6369, \qquad Q_{(1,1)}=511.0266. \]

The separate relative increments are

\[ \frac{Q_{(0,1)}-Q_{(0,0)}}{Q_{(0,0)}}=3.6291\times10^{-3}, \]

\[ \frac{Q_{(1,1)}-Q_{(0,1)}}{Q_{(0,1)}}=7.6325\times10^{-4}. \]

Thus degree-one through-thickness hole structure is not the source of the missing high Q. The result is not yet a thickness convergence proof because Legendre degree two has different mirror/component content: it introduces even-z E_x and E_y functions rather than another copy of the odd-z E_z subspace.