Zhang et al. semi-2D PhC cavity

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

Answer

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×.

Polarization convention

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

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

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

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

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