Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/fryett_q_series_phase6/DERIVATION_PHASE6.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Phase 6 derivation: exact symmetry reduction of the full hole basis¶

1. Mirror action on vector hole modes¶

The legacy Fryett reconstruction is invariant under reflections in x, y, and z. A polar vector changes sign in the component normal to the reflecting plane. For a local basis mode

\[ Z_{nm}^{\sigma}(\rho,\theta)P_\ell(2z/T)\mathbf e_c, \]

the scalar disk parities are

angular mode	x reflection	y reflection
`cos(m theta)`	`(-1)^m`	`+1`
`sin(m theta)`	`(-1)^(m+1)`	`-1`

The thickness parity is (-1)^ell. Multiplying by -1 when component c is normal to the mirror gives the complete local vector parity.

2. Pairing positive and negative holes¶

Let p_x(mu) be the local x-mirror sign of mode mu. For global x parity s_x, the normalized paired basis vector is

\[ |j,\mu;s_x\rangle=\frac{1}{\sqrt2} \left(|+j,\mu\rangle+s_xp_x(\mu)|-j,\mu\rangle\right). \]

The y and z mirrors act within each hole, so a mode is retained only when its local y and z signs equal the chosen global sector signs. The resulting sparse matrix S obeys

\[ S^TS=I. \]

Summing S S^T over all eight sectors gives the identity in the full hole basis. Hence this is an exact block diagonalization, not a physical approximation.

3. Reduced pole operator¶

At every beta node the full projection C(beta) is replaced by

\[ C_s(\beta)=C(\beta)S_s. \]

The sector pole operator is

\[ A_s(\omega)=I-k_0^2\Delta\epsilon \int_{\mathcal C_+}\frac{d\beta}{2\pi} C_s(-\beta)^T G_{\rm wg}(\beta,\omega)C_s(\beta). \]

Equivalently, A_s=S_s^T A S_s. This equality is tested directly against the unreduced operator.

4. Basis continuation¶

For a sector vector at degree p, the algorithm:

expands it to the full degree-p coefficients with S_p;
embeds those coefficients into the degree-p+1 local mode ordering;
projects with S_(p+1)^T.

This lift preserves norm and every old coefficient exactly. The complex Newton solver then follows the same null vector at the higher degree.

5. Numerical convergence of the classified branch¶

For all 90 holes, the Phase-5 seed is purely in (s_x,s_y,s_z)=(+1,+1,-1) to roundoff. The symmetry-reduced degree-0 pole matches the unreduced pole, and degree 1 matches the earlier 810-dimensional calculation. Raising the in-plane hole and waveguide degrees gives

\[ Q_0=351.9587,\qquad Q_1=98.7809,\qquad Q_2=198.9650,\qquad Q_3=198.7105. \]

The relative degree-2 to degree-3 change is

\[ \frac{Q_3-Q_2}{Q_2}=-1.2789\times10^{-3}. \]

This does not prove convergence of the complete series: the thickness degree, transverse momentum cutoff, and Sommerfeld contour remain fixed. It does show that the in-plane local basis has reached a plausible plateau for this pole.

6. Why this is not yet the Fryett mode¶

For thickness degree zero, an even-z E_z basis function has physical mirror eigenvalue s_z=-1, while an even-z E_y basis function has s_z=+1. The classified (+,+,-) pole is consequently E_z-like. The Fryett paper's mode is E_y-like. With an even scalar profile across y, the vector sign of the normal E_y component makes s_y=-1; its x parity is not fixed a priori.

The target search therefore belongs in the two sectors

\[ (s_x,s_y,s_z)=(+1,-1,+1)\quad\text{and}\quad(-1,-1,+1). \]

Exact symmetry has changed the interpretation of the earlier "best-Q" seed: it is a valid outgoing pole, but it is not the target polarization. Searching only the physically compatible blocks is the next step and avoids confusing a well-converged wrong branch with a poor approximation to the reported Q.