3D air-strip nanocavity

Milestone 6: rational pole validation and Loewner pole-Q optimization

Outcome

This milestone closes the 19% discrepancy between the old first-order pole continuation and the real-axis linewidth. The corrected Q~162 topology has the AAA-continued pole

f = 0.99772656962 - 0.00255901270 i,

corresponding to Q = 194.9437. A local Fano linewidth fit gives Q = 195.2327, a relative difference of 0.148%. Four interleaved holdout fits predict withheld complex response samples with worst NRMSE 9.26e-5 and move the tracked pole by at most 9.67e-4 linewidth. The full one-wavelength air strip remains exactly air.

The milestone also demonstrates the first differentiable rational-pole topology steps. Two independently accepted steps give

AAA Q: 194.98 -> 195.35 -> 196.35.

At the final checkpoint, the rebuilt estimates are:

estimatorQ
AAA with holdout validation196.3483
rank-10 Loewner pencil196.3718
local Fano linewidth196.0630

The final holdout error is 8.94e-5, pole displacement is 5.27e-4 linewidth, and no backtracking was required.

Why two rational models are used

The infinitely periodic mirrors are evaluated only at real frequencies, where their surface Green functions unambiguously remain on the retarded sheet. The sampled complex dipole response is

\[ h(f)=d^T A(f)^{-1}d. \]

AAA is a robust nonlinear rational continuation and an excellent way to find the physical lower-half-plane poles. However, its greedy support-point choices can change discontinuously under infinitesimal data perturbations. Numerical differentiation through that choice was tested and rejected as unstable.

The optimizer therefore uses a Loewner pencil constructed from alternating left and right frequency samples. A rank-10 SVD compression is frozen during one local step. In that fixed subspace the reduced descriptor matrices depend linearly on the response data, so the selected generalized eigenvalue has the analytic sensitivity

\[ df_p=\frac{y^H(dA_r-f_p dE_r)x}{y^H E_r x}. \]

The compression is rebuilt after the topology changes. Most importantly, the Loewner objective never accepts its own step: a fresh AAA model, four holdout fits, and a local linewidth fit independently accept or reject the proposed geometry.

Maxwell-to-pole gradient

For a reciprocal colocated dipole,

\[ h=d^T E,\qquad A E=d, \]

and therefore

\[ dh=-E^T(dA)E. \]

The same forward field used for each complex-response sample supplies the complete complex topology derivative. These derivatives are combined with the Loewner pole sensitivities across all 25 frequencies and backpropagated through the density filter and projection. Reflection symmetry is imposed on both density and gradient, and the protected central strip is outside the design variables.

Verification

  • 15 regression tests pass.
  • Complex Maxwell response derivative agrees with finite differences.
  • Fixed-subspace Loewner pole sensitivity agrees with finite differences.
  • AAA pole is stable across four interleaved holdout fits.
  • AAA, Loewner, and local-linewidth Q agree at the accepted topology.
  • The prior corrected topology passed the transverse-PML gate; the latest two small Loewner updates still require their own rational-PML rerun.

Scientific status

This is the important algorithmic proof: a 3D defect attached to exact semi-infinite complex periodic mirrors can now be optimized using a pole of a retarded real-axis rational model, without evaluating the periodic boundary on the wrong complex-frequency sheet.

It is not yet evidence of a high-Q physical device. Q~196 is a coarse-grid development result, and the last topology must still pass rational PML, domain, and grid convergence. The next implementation step is a parallel rational-PML sweep followed by longer Loewner trust-region continuation. Only after those gates should the grid and defect region be enlarged.

References