FDTD verification¶

This page defines when cavity \(Q\), mode volume, and mode-resolved decay fractions may be published as trustworthy FDTD observables. It also records the current Design23 40-hole implementation and its known limits.

Claim and interpretation¶

The reported FDTD \(\beta_{\mathrm{TE0}}\) is the fraction of the resonant cavity mode's total outward decay power carried by the fundamental TE0 waveguide mode through both x ports:

\[ \beta_{\mathrm{TE0}} =\frac{P_{+\mathrm{TE0}}+P_{-\mathrm{TE0}}}{P_\mathrm{all}}. \]

It is a cavity-decay branching ratio. It is not driven transmission, loading efficiency, or automatically the total spontaneous-emission beta of an emitter. Interpreting it as an emitter beta additionally assumes that emission is dominated by the fitted cavity pole and that direct/background emission is negligible or accounted for separately.

For the validated symmetric 40-hole checkpoint, the current numerical claim is

\[ \beta_{\mathrm{TE0}}=0.8828\pm0.018 \]

or approximately 2% relative numerical uncertainty. Equivalently, the calculation is numerically resolved to about 98% under the controls tested. This is an engineering error bound, not a 98% statistical confidence probability and not an experimental tolerance. Material-index uncertainty, fabrication disorder, emitter position/orientation, and background emission are outside this bound.

Preflight before submission¶

The simulation must be constructible and serializable without cloud access. Review:

exact material stack, geometry units, and symmetry assumptions;
PML clearance and all six loss-monitor planes;
source position, polarization, bandwidth, and shutoff;
cavity field sampling at every reported mode-volume reference point;
a true mode monitor at a uniform feedthrough cross-section;
run time and frequency resolution relative to the expected Q;
mesh and spatial downsampling near material interfaces;
maximum estimated cost and remaining conservative budget.

Save the simulation JSON before upload. The uploaded task ID must enter durable state before the job is started.

The routine low-Q Design23 diagnostic uses:

Tidy3D automatic grid with 14 minimum steps per wavelength;
no mesh-override regions;
16 ps maximum run time;
automatic field-decay shutoff at \(10^{-5}\);
a separate mesh-16 calculation for convergence rather than routine production.

Resonance extraction¶

Fit the late-time signal to a passive complex frequency and record the fit window, residual, and competing modes. For a pole (\tilde\omega=\omega_r-i\gamma), report

\[ Q=\frac{\omega_r}{2\gamma}. \]

Field plots and energy integrals must be coherently demodulated at the fitted frequency. An arbitrary final time slice mixes phase, numerical noise, and neighboring modes.

Use the same clean post-source time interval and the same window function for the field projection, every outgoing-power channel, and every mode-port projection. A result assembled from a ringdown fit in one interval and residual field or flux in a much later interval is not a target-pole measurement.

For mode volume, compute the numerator and reference-point denominator from the same projected field. A maximum taken from an unprojected late residual can collapse the reported volume around a numerical hotspot and must be treated as invalid.

The production Design23 diagnostic is deliberately two-stage:

A broadband source excites the cavity. Five point probes record the source-free ringdown and determine one passive pole.
A second simulation of the identical geometry places all E/H, permittivity, closed-flux, six-face flux, and mode-decomposition monitors at that fitted pole. Every observable monitor uses the same apodization interval.

The second stage prevents a mode volume, flux, or port amplitude from being combined with a pole fitted in a different time interval.

Loss closure¶

Compute outward power on every boundary face. For stored energy (U), each channel obeys

\[ Q_c=\frac{\omega U}{P_c}, \qquad \frac{1}{Q_\mathrm{loss}}=\sum_c\frac{1}{Q_c}. \]

Compare the summed directional loss Q with the fitted ringdown Q. A mismatch is a diagnostic, not a correction to hide. Check monitor orientation, incoming contamination, integration coverage, time/frequency normalization, and fields inside PML. The diagnostic dashboards require 0.8 <= Q_loss / Q_ringdown <= 1.2 before publishing absolute decay fractions or projected mode volume; failed measurements remain available only as raw audit evidence.

For nondispersive materials, the stored energy used by the coherent diagnostic is

\[ U=\frac14\int \left(\epsilon_0\epsilon|\mathbf E|^2+\mu_0|\mathbf H|^2\right)dV. \]

Electric and magnetic energies must agree within 20%. The implementation also compares two nested closed flux boxes and compares their flux with the signed sum of six independently oriented planes. Absolute values must not be used to hide an inward face.

Feedthrough mode decomposition¶

At a uniform x-normal port, select the continued fundamental propagating Ey-like branch and verify real effective index, propagation direction, polarization fraction, and power closure. The two-sided cavity decay fraction is

\[ \beta_{\mathrm{TE0}}=\frac{P_{+\mathrm{TE0}}+P_{-\mathrm{TE0}}}{P_\mathrm{all}} =\frac{Q_\mathrm{total}}{Q_\mathrm{TE0}}. \]

The direct implementation first computes

\[ \eta_{+\mathrm{TE0}}=\frac{P_{+\mathrm{TE0}}}{P_{x+}}. \]

For the present exactly x-even geometry,

\[ P_\mathrm{TE0} =\eta_{+\mathrm{TE0}}(P_{x+}+P_{x-}). \]

The x-face powers already belong to \(P_\mathrm{all}\); TE0 power is a subset and must not be added to the closed-box power again. An asymmetric geometry must use explicit mode monitors at both ports instead of this symmetry inference.

As an independent construction,

\[ Q_\mathrm{TE0}=\frac{\omega U}{P_\mathrm{TE0}},\qquad \beta_Q=\frac{Q_\mathrm{ring}}{Q_\mathrm{TE0}}. \]

The direct power ratio and the Q-ratio construction must agree within 10%.

This is not driven transmission and not input-to-cavity loading efficiency. An S-parameter experiment with a port source is a different recipe.

Publication gates¶

The dashboard publishes absolute beta and mode volume only when every applicable gate passes:

Gate	Required condition
signed plane powers	all six are outward
plane/box closure	six-plane sum agrees with closed box within 10%
monitor-position closure	inner and outer flux boxes agree within 10%
energy balance	\(0.8\le U_E/U_H\le1.2\)
ringdown/flux closure	\(0.8\le Q_\mathrm{flux}/Q_\mathrm{ring}\le1.2\)
pole alignment	observable monitor is within 0.2 fitted linewidth
TE0 subset	\(0\le P_\mathrm{TE0}/P_x\le1.10\)
modal closure	all propagating modes recover x-face flux within 10%
branch identity	selected branch is propagating and Ey-like
incoming contamination	\(P_\mathrm{in}/P_\mathrm{out}\le0.10\)
beta range	\(0\le\beta_{\mathrm{TE0}}\le1\)
independent beta	direct-power and Q-ratio beta agree within 10%
time-window sensitivity	named observables change by at most 5%
mesh convergence	wavelength, Q, volume, and beta meet the thresholds below

If any gate fails, the dashboard sets fdtd_beta_valid and/or fdtd_mode_volume_valid false. Raw values remain audit evidence but are not quotable results.

Required convergence checks¶

Before production status, vary one control at a time around a frozen geometry:

spatial grid density;
run time and fit window;
field-monitor downsampling;
simulation-domain and PML clearance;
mode-monitor position and mode count.

Report relative changes for wavelength, Q, every named volume, total loss closure, and TE0 decay. Convergence of Q alone is insufficient.

The current Design23 auto-14 to auto-16 gates are:

Observable	Allowed relative change
wavelength	0.2%
ringdown Q	5%
peak mode volume	5%
fixed-point core mode volume	5%
absolute two-port TE0 beta	10%

Current Design23 evidence¶

The mesh-converged auto-14 recommendation is:

Quantity	Value
wavelength	770.621274 nm
ringdown Q	1127.4177
peak \(V/(\lambda/n_\mathrm{SiN})^3\)	2.601441
core-center Ey \(V/(\lambda/n_\mathrm{SiN})^3\)	2.874723
anthracene-center Ey \(V/(\lambda/n_\mathrm{Anth})^3\)	6.211477
absolute two-port TE0 beta	0.882810

The checks supporting the beta claim are:

Check	Observed
auto-14 to auto-16 beta change	0.291%
direct-power vs Q-ratio beta difference	0.963%
all-mode/x-face closure error	1.336%
inner/outer flux-box difference	0.869%
\(Q_\mathrm{flux}/Q_\mathrm{ring}\)	0.990371
\(U_E/U_H\)	1.03476
incoming/outgoing TE0 power	\(1.00\times10^{-8}\)
early/main-window beta change	0.000434%
monitor detuning	0.000147 linewidth

The largest observed internal discrepancy is 1.34%. Rounding the combined mesh, modal-closure, flux-position, and independent-construction evidence to a 2% engineering envelope gives the quoted \(\pm0.018\) beta bound.

The already-completed auto-14 validation task did not store direct mode fields. Its TE0 branch was cross-identified against a mode calculation on the exact same geometry and grid: TE fraction 0.99391 and effective-index change 0.0213%. The auto-16 validation and all future production tasks store direct mode fields; the auto-16 TE fraction is 0.99399.

Historical failure and why it is retained¶

The old 24 ps diagnostics sampled volume fields and face fluxes near 23.98 ps, more than 50 energy lifetimes after the source. Ringdown Q was still recoverable from the earlier probes, but the late volume and flux were residual-dominated. At step 3, \(Q_\mathrm{flux}/Q_\mathrm{ring}\approx18.4\); the old mode volume and beta therefore fail closure and are invalid.

Those records remain in history as failure evidence. The validated coherent analysis is a new linked record; it does not overwrite the raw cloud result.

Remaining limits¶

The present 2% numerical bound does not cover:

a separate simulation-domain/PML-clearance sweep;
explicit independent mode decomposition at the x-minus port;
additional mesh levels beyond auto-14 and auto-16;
material dispersion or uncertainty;
fabrication disorder;
emitter placement, dipole orientation, or direct non-cavity emission;
generalized complex-QNM normalization for strongly leaky or dispersive systems.

Until domain/PML and explicit two-port sweeps are complete, quote the result as FDTD cavity-decay beta_TE0 = 0.883 ± 0.018 (numerical, symmetric model). Do not quote it as an experimental emitter beta.

Reproducibility and tests¶

The production implementation is in:

engines/design23_v1/campaigns/pure_bfgs_maxQ_diagFDTD/diagnostic_fdtd.py;
engines/design23_v1/campaigns/composite_qv_beta_40holes/fdtd_observables_validation/run_validation.py.

The evidence bundle is under:

engines/design23_v1/runs/20260728T013605Z_design23_40hole_composite_bfgs/diagnostics/fdtd_observables_validation/.

Offline tests verify:

fixed 16 ps runtime, auto-14 grid, \(10^{-5}\) shutoff, and no overrides;
identical pole frequency and apodization across coherent monitors;
complete monitor names for E/H, nested boxes, six faces, and mode port;
mapping only fully valid coherent data into dashboard beta/volume fields;
rejection of invalid automatic-shutoff settings;
dashboard invalidation when any physical gate fails;
zero charge for HDF5 reuse;
diagnostic cadence across interrupted/resumed campaigns.

Run:

python -m pytest tests/test_fdtd_runtime_policy.py \
  tests/test_dashboard_publication.py