Feedthrough coupling FDTD convergence report¶

What was tested¶

Runtime convergence: same mesh (12 steps/λ), run_time = 48 / 96 / 128 ps
Time-window convergence: multiple late windows within each run (last 10–50%, mid, early)
ModeMonitor at ResonanceFinder pole: multi-frequency ModeMonitor + spectral flux grid (±6 nm, 7 samples); analysis picks the frequency nearest the RF pole from the same ringdown

Cost: prior 48 ps pair ~0.155 FC; convergence suite ~0.518 FC; total ~0.673 FC.

Primary observable: closed-surface end-face fraction¶

\[\eta_x = (P_{+x}+P_{-x}) / \sum_{\mathrm{faces}} P\]

Runtime table (`η_x` from last 25% of each run)¶

Design	T=48 ps	T=96 ps	T=128 ps	Spread
initial	82.21%	82.21%	—	0.0010 pp
final	59.57%	59.57%	59.57%	0.0045 pp

Window table (max |Δη| across windows within each run)¶

initial T48: η_x ∈ [82.205%, 82.205%], span 0.0001 pp over 7 windows
initial T96: η_x ∈ [82.205%, 82.207%], span 0.0023 pp over 8 windows
final T48: η_x ∈ [59.572%, 59.575%], span 0.0033 pp over 7 windows
final T128: η_x ∈ [59.568%, 59.573%], span 0.0047 pp over 8 windows
final T96: η_x ∈ [59.572%, 59.575%], span 0.0035 pp over 8 windows

Spectral @ pole vs time-domain¶

| Design | T | η_x time last25% | η_x spectral@pole | |Δ| | |---|---:|---:|---:|---:| | initial | T48 | 82.21% | 82.21% | 0.002 pp | | initial | T96 | 82.21% | 82.18% | 0.025 pp | | final | T48 | 59.57% | 59.56% | 0.010 pp | | final | T128 | 59.57% | 59.65% | 0.078 pp | | final | T96 | 59.57% | 59.65% | 0.079 pp |

Q / wavelength stability¶

Design	T	Q	λ (nm)
initial	T48	1.3126e+04	1503.675
initial	T96	1.3125e+04	1503.675
final	T48	4.2681e+05	1492.523
final	T128	4.2654e+05	1492.523
final	T96	4.2638e+05	1492.523

ModeMonitor (TE0) at the RF pole¶

All runs find a clean fundamental TE0 branch:

Design	n_eff	TE frac	P_in/P_out	P_net/P_x	valid?
initial_T48	2.5885	0.995	0.264	1.223	False
final_T48	2.5964	0.995	0.329	0.866	False
final_T128	2.5970	0.995	0.330	0.866	False
final_T96	2.5970	0.995	0.330	0.866	False
initial_T96	2.5873	0.995	0.264	1.224	False

Validation failed in every case (Design23-style gates): - Seed: P_net(TE0)/P_x ≈ 1.22 > 1 and P_in/P_out ≈ 0.26 - Peak: P_in/P_out ≈ 0.33 (threshold 0.25), though P_net/P_x ≈ 0.87 is plausible - Tuning the ModeMonitor frequency to the RF pole (vs campaign λ) does not fix the incoming-wave contamination - Therefore no trustworthy TE0-only η is claimed; only the closed-surface η_x is.

Verdict¶

Trustworthy¶

η_x (closed-surface free-decay power fraction) is extremely well converged:
Initial: 82.21% (stable to <0.01 pp across 48→96 ps and all late windows)
Optimized: 59.57% (stable to <0.01 pp across 48→96→128 ps and all late windows)
Time-domain and spectral@pole agree to ~0.1 pp
Q and λ are stable across runtimes (coarse mesh; absolute Q still ~15–20% below campaign for peak)

Not trustworthy (yet)¶

Mode-resolved TE0 feed fraction from monochromatic ModeMonitor on a free-decay ringdown:
Large reverse-wave content on the TE0 branch (in/out ~25–33%)
Seed TE0 net power exceeds net face flux (unphysical as a pure subset)
Longer time and pole tuning do not remove this → issue is method, not short runtime

Intuition¶

Why η_x is so flat: After the source dies, both end-face and side radiation are dominated by the same complex pole. Their ratio is a property of the mode’s far-field partition and does not require waiting many lifetimes of a high-Q mode — only that continuum from the drive has left. Here that happens by ~24 ps for both designs.
Why peak η_x is lower than seed (~60% vs ~82%): Optimization raised total Q a lot, but the fraction of power through the waveguide ends dropped. Physically that means Q_feed rose more than Q_rad (side/top), so a larger share of the remaining loss is non-end-face radiation. This is consistent with “better mirrors / less end leakage” as a Q-raising mechanism, not with “more feedthrough coupling.”
Why analytical ~99% ≠ FDTD η_x ~60–82%: Different quantities. Analytical guided_external_decay_fraction is the guided-channel residue of the linewidth. FDTD η_x is geometric power through ±x monitor planes (guided + continuum through those planes). Side radiation in FDTD is a real open-system channel; the residue model can attribute almost all linewidth to the guided pole even when geometric side flux is nonzero.
Why ModeMonitor TE0 stays dirty: Free-decay + finite domain + PML produces standing-wave-like content and continuum on the port plane. The monochromatic mode basis then reports substantial a_- even for a “pure” outgoing decay. Longer ringdowns do not eliminate that when the high-Q mode has barely decayed (peak: power almost flat over 24–128 ps). A cleaner TE0 η would need a different protocol (e.g. longer pure single-mode window with exponential isolation, or driven CW port decomposition, or complex-frequency mode projection).

Recommendation for the deck¶

Safe to quote later (if desired): FDTD η_x ≈ 82.2% (seed) → 59.6% (optimized), with the closed-surface definition and the convergence above.
Do not quote TE0-only feed fraction from these runs.
Keep analytical ~99% labeled as the analytical guided-residue metric, not FDTD.