# Zhang *et al.* semi-2D PhC cavity

This page reproduces the unloaded cavity in [Zhang *et al.*, *Scientific
Reports* 6, 26038 (2016)](https://www.nature.com/articles/srep26038) and asks a
new cavity-QED question: what LDOS does an atom see at the top edge of the
now-removed coupling waveguide?

```{admonition} Answer
:class: important

For a **y-polarized atom** exactly **500 nm below the cavity edge**, with the
bus waveguide absent, the resonant LDOS is **82.04×
vacuum**. This consists of a broadband continuum of
**1.02×** plus a cavity-mode contribution of
**81.02×**.
```

```{admonition} Polarization convention
:class: note

The paper designs and plots the TE **Ey** cavity mode, so the cavity-enhanced
number below is the Ey-projected LDOS. At the x=0, z=0 symmetry point, the Ex
and Ez components of this fundamental mode vanish by parity; an out-of-plane
Ez transition therefore does not receive this resonant Purcell term.
```

## Literal geometry

```{figure} ../_static/generated/paper_semi2d_geometry.png
:alt: Reproduced semi-2D photonic-crystal cavity and atom location

Gold is the 220 nm suspended silicon membrane, pale blue is air, and every hole
is part of the actual rasterized FDTD mask. The dashed bus is only a positional
reference and is absent from the solve.
```

| Published input | Value used |
|---|---:|
| Silicon slab | 220 nm, n=3.46 |
| Triangular-lattice period / radius | 420 / 134 nm |
| Rows / rasterized holes | 10 / 335 |
| Center → outer edge width | 550 → 400 nm, quadratic |
| Taper / total length | 31a / 33a |
| Paper fundamental | 1547 nm, Q=6.7e+07 |
| Bus-waveguide gap used only for atom position | 500 nm |
| FDTDX grid / symmetry | 20 nm conformal fractional-cell / x-PMC and z-PMC |

## Results


| Quantity | FDTDX result |
|---|---:|
| Resonance wavelength | 1.540689 µm |
| Ringdown Q | 6.3655e+05 |
| Energy/flux Q | 7.6424e+05 |
| Conservative Q used | 6.3655e+05 |
| Emitter-oriented mode volume | 2183.4 µm³ |
| Continuum Ey LDOS | 1.0201× vacuum |
| Resonant modal Purcell term | 81.022× vacuum |
| **Total resonant Ey LDOS** | **82.042× vacuum** |
| Paper-Q projection (not an FDTDX measurement) | 8528.9× vacuum |
| Raw finite-8 ps value at the pole | 1.7255× vacuum |



```{figure} ../_static/generated/paper_semi2d_mode_ldos.png
:alt: Reproduced cavity Ey mode and atom LDOS spectrum

The geometry contour and late-time Ey mode (left); direct finite-tape LDOS and
pole-extrapolated resonant value (right).
```

```{figure} ../_static/generated/paper_semi2d_ringdown.png
:alt: Short FDTDX cavity ringdown

The short ringdown is fitted in the time domain; the reported ultra-narrow pole
is not claimed from the spacing of frequency samples.
```


## Why the raw 8 ps DFT is not the answer

The paper's ideal linewidth is only about 23 femtometres. An 8 ps Fourier tape
cannot resolve that linewidth, so its peak height is a transient lower bound.
The reported resonant LDOS instead uses the fitted pole and the late-time
emitter-oriented mode volume,

\[
F_{\mathrm{cav}} = \frac{3}{4\pi^2}\frac{\lambda^3 Q}{V_{\mathrm{eff},y}},
\]

then adds the independently measured off-resonant continuum. The Q audit uses
both harmonic inversion and the modal energy-to-radiated-power ratio. The
paper-Q projection answers the separate counterfactual “what if this exact
remote field profile retained the article's 6.7e7 Q?”; it is labeled and never
substituted for the FDTDX-measured result. The
complete arrays and metrics are stored in
`benchmarks/artifacts/paper_semi2d_phc_cavity_ldos_v1.npz`.

## Reproduce

```bash
.venv/bin/python -m benchmarks.cases.paper_semi2d_phc_cavity.run
.venv/bin/python scripts/generate_paper_semi2d_report.py
```