Source
engines/design23_v1/package/design23_recentered_optimizer/fryett_q_series_phase6/STATUS_PHASE6.md · assembled 2026-07-29 15:57 UTC.
Phase 6 status¶
Implemented¶
- Exact real-Zernike parity under x and y reflection.
- Exact Legendre and vector-component parity under z reflection.
- Orthonormal x-paired projectors for all eight Maxwell mirror sectors.
- Completeness and reduced-operator equivalence tests.
- Exact sector-vector lift between increasing hole-basis degrees.
Numerical result¶
The Phase-5 seed at
has weight 0.9999999999999998 in the exact sector
(s_x,s_y,s_z)=(+1,+1,-1). The reduced 45-dimensional degree-0 solve
reproduces the full 270-dimensional pole and Q=351.9587242.
| hole Zernike degree | waveguide y degree | sector dimension | Re(omega/omega_ref) | Im(omega/omega_ref) | Q |
|---|---|---|---|---|---|
| 0 | 1 | 45 | 1.0214498812 | -0.0014510933 | 351.9587 |
| 1 | 1 | 90 | 1.0217010366 | -0.0051715492 | 98.7809 |
| 2 | 2 | 180 | 1.0280515955 | -0.0025834990 | 198.9650 |
| 3 | 3 | 270 | 1.0280553641 | -0.0025868167 | 198.7105 |
Every row meets the joint eigenvalue and smallest-singular-value acceptance
criterion. Reciprocity errors remain below 3.2e-16. Degree 2 to degree 3
changes Q by only -0.1279%, compared with order-one changes at the first two
increments. This is the first credible in-plane basis plateau for this branch.
Important branch correction¶
At zero thickness degree, sector (+,+,-) retains the even-z E_z component.
The field shown for the Fryett cavity is E_y-like. An even-z E_y mode has
s_z=+1, and its fundamental even scalar profile in y gives s_y=-1 after
the polar-vector mirror sign is included. Thus the reported cavity branch must
be sought in one of the two x-parity sectors (+,-,+) or (-,-,+).
The converged value near 199 is therefore not a failed approximation to
Q_paper ~ 100,000; it is a numerically resolved pole of the wrong polarization
sector, about a factor of 503 below the paper value.
Next numerical task¶
Search the two E_y-compatible sectors for all poles around the normalized
740-nm frequency, classify them by overlap, and only then restart the p=0..3
basis continuation. Thickness degree, transverse cutoff, contour deformation,
and the pinned geometry verification remain open after that branch is found.