Shifted cooperativity optimization architecture

Purpose and scientific status

This architecture accelerates proposal generation after a source-coupled resonance already exists. It adapts the moving-resonance idea of Shaker et al., Eigenvalue-accelerated LDOS optimization of high-Q optical resonances, to this repository's nonlinear exact periodic-lead boundary.

It does not evaluate the surface Green recursion at a lower-half-plane frequency. The tracked pole and its topology derivative come entirely from a frozen-subspace Loewner continuation of retarded real-axis response samples. The wide/local/confirmatory shared-pole validator and the transverse-PML gate remain authoritative.

Continuous objective

Let the moving evaluation frequency be

\[ \Omega(p)=\operatorname{Re}\widetilde f_p(p), \]

where \(\widetilde f_p\) is the local continuation of the selected source-coupled pole. The provisional shifted cooperativity objective is

\[ J_s(p)=\log\left[ \frac{\operatorname{Im}d_a^T A(\Omega(p),p)^{-1}d_a} {\operatorname{Im}d_a^T A_{\rm hom}(\Omega(p))^{-1}d_a} \right]. \]

The physical atom-frequency quantity remains

\[ F_a(p)=\frac{\operatorname{Im}d_a^TA(f_a,p)^{-1}d_a} {\operatorname{Im}d_a^TA_{\rm hom}(f_a)^{-1}d_a}. \]

An accepted step must improve both quantities and shared-pole Q. Thus the moving objective cannot trade away coupling to the actual atomic transition.

Discrete estimator and normalization

Sample the centered-dipole complex response on the existing pole-centered real-frequency grid using the retarded surface recursion.
Build a scalar fixed-subspace Loewner pencil and select the pole nearest the independently validated shared pole.
Set \(\Omega=\operatorname{Re}\widetilde f_p\).
Solve the device and same-grid homogeneous reference at real \(\Omega\).
Estimate the two log-LDOS frequency derivatives by centered real-axis solves at \(\Omega\pm\Delta f\), with the default \(\Delta f=10^{-4}\).

The homogeneous calculation preserves grid, source, PML, and longitudinal boundary machinery and changes only n_core == n_clad. Its frequency derivative is included; treating the denominator as fixed would give the wrong moving-frequency gradient.

Gradient

For the frozen Loewner compression subspaces,

\[ \frac{d\widetilde f_p}{dp_j} =\sum_i \frac{d\widetilde f_p}{dh_i}\frac{dh_i}{dp_j}. \]

Every \(dh_i/dp_j\) uses the reciprocal transpose bilinear form and the exact transpose material/filter maps. The total shifted gradient is

\[ \frac{dJ_s}{dp}= \left.\frac{\partial\log\operatorname{LDOS}_{\rm dev}}{\partial p} \right|_{\Omega} +\left( \frac{\partial\log\operatorname{LDOS}_{\rm dev}}{\partial\Omega} -\frac{\partial\log\operatorname{LDOS}_{\rm hom}}{\partial\Omega} \right) \operatorname{Re}\frac{d\widetilde f_p}{dp}. \]

For signed atom detuning in pole half-widths,

\[ \delta_a=\frac{\operatorname{Re}\widetilde f_p-f_a} {-\operatorname{Im}\widetilde f_p}, \]

the implemented derivative is

\[ \frac{d\delta_a}{dp}= \frac{\operatorname{Re}(d\widetilde f_p/dp)}{\gamma} +\frac{(\operatorname{Re}\widetilde f_p-f_a) \operatorname{Im}(d\widetilde f_p/dp)}{\gamma^2}, \qquad \gamma=-\operatorname{Im}\widetilde f_p. \]

Directional differences test the shifted chain rule on the smallest 3D Yee grid. Separate regressions already test the Maxwell response gradient and the frozen-subspace Loewner pole derivative.

Constrained optimizer and trust region

One provisional inner iteration solves a conservative separable moving-asymptote subproblem. It maximizes predicted \(J_s\), subject to local nondecrease constraints for \(\log F_a\) and \(\log Q\), two inequalities for the signed atom-detuning band, physical [0,1] density bounds, and an outer L-infinity trust region. The dual dimension is four regardless of topology size.

Three provisional inner iterations are the production default. They refresh the real-axis pole and all gradients, but they do not constitute accepted topology steps. The cumulative candidate stays inside one outer trust box. The authoritative validator is rebuilt only for the cumulative candidate.

After an accepted outer step, agreement between predicted and observed shifted-objective gain grows, retains, or shrinks the next trust radius. Any backtrack, rejection, or poor agreement shrinks it. The current allowed range is 0.00025 to 0.008.

Independent acceptance

The provisional MMA result is accepted only if a completely rebuilt trial:

passes sheet-local-v2 wide/local/confirmatory shared-pole validation;
retains the tracked mode within the topology pole-motion limit;
strictly improves atom-frequency normalized LDOS;
strictly improves shifted normalized LDOS evaluated at the authoritative shared pole's real part;
strictly improves independently fitted shared-pole Q;
stays within two pole half-widths of the atomic frequency, or monotonically restores detuning if the starting checkpoint is outside that band;
has maximum_density_in_air_strip == 0; and
subsequently passes its checkpoint-bound transverse-PML campaign before another outer topology step begins.

This remains a controlled-grid cooperativity optimization. It is not a physical Q, bare g, beta-factor, grid-converged, or device claim.

Production use

scripts/run_multistep_cooperativity.py enables the shifted constrained-MMA path, carries the recommended trust radius through restartable checkpoints, and leaves PML worker settings unchanged. scripts/run_long_cooperativity.sh remains the unattended entry point.