# Feedthrough coupling FDTD convergence report

## What was tested

1. **Runtime convergence**: same mesh (12 steps/λ), `run_time` = 48 / 96 / 128 ps
2. **Time-window convergence**: multiple late windows within each run (last 10–50%, mid, early)
3. **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

1. **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.

2. **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.”

3. **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.

4. **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.