Source

Engine design23_v1 · package package/design23_recentered_optimizer · assembled 2026-07-29 15:57 UTC.

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Pole/branch-subtracted absolute-Q gate

Date: 2026-07-22

Answer first

The previously specified stabilization step is implemented and the Design23 iteration-0 absolute-Q gate now passes.

The accepted real-axis pole is

\[ \widetilde\omega/\omega_0 =0.9992524494-6.8717390\times10^{-5}i, \]

which gives

\[ Q=7270.74,\qquad \lambda=796.28805\ {\rm nm}. \]

The corresponding FDTD result is Q=6176.55 and lambda=796.66450 nm. The absolute Q error is 17.72% and the wavelength error is -0.0473%, both inside the previously stated gates of factor-of-two Q agreement and 1% frequency agreement.

This result supersedes the rejected Q ~= 80 result in ABSOLUTE_QNM_INTEGRATION_GATE.md.

What changed

The failed v1.1 method evaluated the entire 84-hole Fourier kernel on one deformed complex-beta contour. A contour deep enough to clear every pole produced exponentially large reciprocal factors across the approximately 20-um cavity; a shallow contour intersected unresolved poles and branch cuts.

subtracted_real_axis_qnm.py instead evaluates the outgoing operator on the real-frequency boundary and extracts every singular term before longitudinal quadrature:

1. all propagating half-space TE/TM channels are integrated on light-line panels;
2. the planar-film TE pole at neff ~= 1.55246 is extracted and restored by its residue;
3. every closed SiN-strip pole in the physical y-odd block is enumerated from sign changes of the Hermitian Dyson eigenvalues;
4. the only such Design23 pole, neff ~= 1.76502 at the converged resonance, is subtracted from the beta-domain strip Green matrix;
5. its positive- and negative-beta residues are restored with the existing ordered real-space propagator; and
6. only the smooth reciprocal remainder is integrated over beta.

No factor grows as exp(abs(Im(beta))*cavity_length). The complete 336-dimensional cavity matrix stays reciprocal to approximately 2e-16.

Non-Hermitian pole extraction

At a real frequency s, the outgoing cavity operator has a tracked complex eigenvalue lambda(s). The nearby pole is continued analytically as

\[ s_{\rm pole}=s-\frac{\lambda(s)}{d\lambda/ds}, \qquad \frac{d\lambda}{ds} =\frac{l^\dagger (dD/ds)r}{l^\dagger r}. \]

This uses the full non-Hermitian left/right eigenvectors. The invalid v1.1 fallback that diagonalized the Hermitian part of a strongly non-normal operator is not used.

After two real-frequency Newton updates, the final on-axis eigenvalue is

1.3628e-5 + 1.90506e-3 i

at s=0.9992524494. Its real component is effectively zero; the remaining imaginary component generates the passive linewidth.

Numerical convergence

The long cavity requires resolving fine beta-domain interference fringes. The accepted rule uses 812 positive-beta nodes, maximum dimensionless panel width 0.025, four endpoint-clustered nodes per panel, and cutoff beta_max/k0=5.

Check	Coarser value	Accepted value	Change
max panel width	0.050	0.025	2x refinement
beta nodes	412	812	1.97x
Im tracked eigenvalue	0.00189930	0.00188793	0.60%
Q from directional-gradient run	7223.48	7270.74	0.65%
cutoff beta/k0	4	5	—
Im eigenvalue at final resonance	0.00190714	0.00190506	0.11%

Reciprocity remains between 1.1e-16 and 3.8e-16 in all accepted runs.

Checkpoint ranking

The converged frequency derivative from iteration 0 was contracted with the iteration-0, 15, and 30 on-axis eigenvalues. Reusing it is safe for this specific comparison because the maximum coordinate displacement at iteration 30 is only 0.0158 nm and the three linewidth eigenvalues agree to better than two parts per million.

Iteration	FDTD Q	Legacy analytical Q	Repaired absolute Q
0	6176.55	4701.94	7270.34
15	6179.20	7852.69	7270.38
30	6181.27	16948.57	7270.41

Endpoint ratios are:

FDTD                         1.0007628
legacy analytical            3.6045899
repaired absolute Q          1.0000101

The repaired solver therefore reproduces the scientifically important result: the saved optimizer direction is flat, not a 3.6x Q improvement.

Directional gradient gate

The saved iteration-0-to-30 geometry direction was magnified until its largest coordinate changed by 1 nm. A central finite difference at the coarser converged gradient rule gave:

Direction coordinate	Q	Branch overlap
-1	7217.44	0.99999942
0	7223.48	1.00000000
+1	7226.87	0.99999943

The actual saved iteration-30 displacement is only 1/63.39 of that test step. The rescaled predicted endpoint ratio is approximately 1.0000103, consistent with the direct checkpoint calculation. It has the same sign as FDTD but is correctly classified as flat rather than as a meaningful improvement.

Scope and next action

This gate establishes a converged absolute Design23 Q and a stable directional finite difference. It does not yet make the legacy autonomous optimizer safe to resume. The remaining production work is to differentiate the new subtracted operator itself, replace the legacy determinant and gradient in the trust-region loop, and validate a small set of deliberately amplified perturbations with FDTD.

No new FDTD run was required for this gate.

Reproduction

The converged iteration-0 pole evaluation is:

python run_subtracted_real_axis_qnm.py run_state.json \
  --iterations 0 \
  --center-ratio 0.9993220594083526 \
  --frequency-step 0.0001 \
  --maximum-panel-width 0.025 \
  --beta-order 4 \
  --beta-cutoff 5 \
  --branch nearest

The saved-direction central difference is:

python run_subtracted_directional_gradient.py run_state.json \
  --maximum-geometry-step-nm 1 \
  --maximum-panel-width 0.05 \
  --beta-order 4 \
  --beta-cutoff 5

The offline regression suite is:

python -m unittest \
  tests.test_offline \
  tests.test_radiation_channels \
  tests.test_subtracted_real_axis_qnm -v

All 18 tests pass.