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engines/majumdar_lab/package/multilayer_autonomous_optimizer/quan_loncar_series_phase25/STATUS_PHASE23_25.md · assembled 2026-07-29 15:57 UTC.

Quan--Loncar analytic-Q status after Phase 25

Outcome

The unstable Cartesian transverse-momentum quadrature has been replaced by a shell-subtracted polar representation of the compact Maxwell Green term. At fixed hole and waveguide bases, the N=15 dielectric-centered Ey pole now passes separated radial-order, angular-order, and cutoff convergence gates.

The final checked pole is

omega / omega_0 = 0.9884159171540067 - 1.7346214406701478e-6 i
Q               = 284908.2497135932

The K=12 to K=14 cutoff increment changes the energy decay rate by 1.741%, and the A=48 to A=56 angular increment changes it by 0.355%. Every row is in the lower half-plane. The successive mode overlaps are at least 0.99999984, and 98.1506% of the retained hole-coefficient weight is Ey.

This is a converged result for the present finite Galerkin truncation, not yet a claim of agreement with the paper's multi-billion radiation-limited Q.

Differentiation checkpoint

For A(omega,p)x=0, Phase 25 implements the simple-pole derivative

d omega / d p = -(y^H A_p x) / (y^H A_omega x).

It was validated for the taper center filling fraction against independently re-solved p+h and p-h cavities at K=10,A=40. The relative complex-pole gradient error was 0.0324%, the Q-gradient error was 2.853%, and both perturbed mode overlaps exceeded 0.99996.

At the final K=14,A=56 pole,

d(omega/omega_0)/d f_center
    = 0.7668516674509804 - 6.603352808135260e-4 i

dQ/d f_center = -1.0823776234107599e8

The local derivative should not be extrapolated over a 0.01 filling-fraction change because the predicted Q change is larger than the base Q.

The pole solve now has a custom differentiation boundary, but A_p is still formed by a centered matrix difference. Replacing that partial with analytic or automatic JVPs is the next AD implementation task.

Remaining physical convergence

Before mode volume or port coupling is treated as quantitatively trustworthy, the polar engine must be checked under:

  1. hole basis P1 to P2 (and then only if needed P3);
  2. waveguide and hole thickness degree z=0 to z=1;
  3. mirror/terminal-pair continuation for the paper's high-Q geometry.

Mode volume is especially sensitive to field reconstruction, so passing only the present eigenfrequency quadrature gate is insufficient for that observable.

Course-abandonment criterion

Do not abandon the volume-Galerkin route now. Abandon it in favor of a different QNM backend only if, after P2/z1 basis refinement, either passivity or the Ey branch cannot be preserved, or if a channel-resolved residue fails both rank-one Green-function and linewidth-sum tests under systematic quadrature refinement.