---
title: Composite waveguide air-gap sweep
---

# Composite SiN/SiO₂ waveguide air-gap sweep

**Status:** `complete` · **completed:** 205/205 FDTD points ·
**mean wall time:** 14.8 s/point.

This is the seed study for the next optimization campaign. Two identical,
coaxial square waveguides face one another across air at 780 nm. The composite
slices use a square Si₃N₄ core (n=2.0) embedded in a finite square SiO₂ guide
(n=1.44); the SiO₂-only controls remove that core. Air surrounds every guide.
The incident field is the y-polarized fundamental mode of the complete guide;
the output observable is the same forward mode in the opposite guide.

![Geometry and port cross-sections](../_static/generated/composite_waveguide_gap_sweep_schematic.png)

## Requested 1 µm SiO₂ slice

![One-micron oxide sweep](../_static/generated/composite_waveguide_gap_sweep_sio2_1p0um.png)

![Best 150 nm / 1 µm signed Ey field](../_static/generated/composite_waveguide_gap_sweep_field_sin150_sio2_1um.png)

## Requested 2 µm SiO₂ slice

![Two-micron oxide sweep](../_static/generated/composite_waveguide_gap_sweep_sio2_2p0um.png)

![Best 150 nm / 2 µm signed Ey field](../_static/generated/composite_waveguide_gap_sweep_field_sin150_sio2_2um.png)

## Direct comparison

![Insertion-loss comparison](../_static/generated/composite_waveguide_gap_sweep_comparison.png)

## Added 100 nm SiN / 3 µm SiO₂ slice

![Three-micron oxide sweep](../_static/generated/composite_waveguide_gap_sweep_sin100_sio2_3p0um.png)

![Best 100 nm / 3 µm signed Ey field](../_static/generated/composite_waveguide_gap_sweep_field_sin100_sio2_3um.png)

This follow-up uses the convergence-qualified 0.20 ps tape and compact 4.75 µm
non-PML x domain. Its 3 µm square oxide still has 0.5 µm of air before the
transverse PML on every side.

## SiO₂-only controls

These controls remove the Si₃N₄ core while retaining the same square-guide,
fixed-port measurement.

### 1 µm square SiO₂

![One-micron SiO2-only sweep](../_static/generated/composite_waveguide_gap_sweep_sio2_only_1um.png)

![Best 1 µm SiO2-only signed Ey field](../_static/generated/composite_waveguide_gap_sweep_field_sio2_only_1um.png)

### 2 µm square SiO₂

![Two-micron SiO2-only sweep](../_static/generated/composite_waveguide_gap_sweep_sio2_only_2um.png)

![Best 2 µm SiO2-only signed Ey field](../_static/generated/composite_waveguide_gap_sweep_field_sio2_only_2um.png)

## Lowest sampled loss

| SiN side (µm) | SiO₂ side (µm) | air gap (µm) | T (%) | loss (dB) | same-mode R (%) | radiation + other modes (%) |
|---:|---:|---:|---:|---:|---:|---:|
| 0.150 | 1.000 | 1.000 | 46.894 | 3.2888 | 7.316 | 45.790 |
| 0.150 | 2.000 | 1.175 | 81.162 | 0.9065 | 1.049 | 17.789 |
| 0.100 | 3.000 | 1.175 | 96.809 | 0.1408 | 0.126 | 3.064 |
| none | 1.000 | 1.000 | 59.490 | 2.2556 | 7.035 | 33.475 |
| none | 2.000 | 1.200 | 91.364 | 0.3922 | 0.060 | 8.576 |

The optimum in this table is the best **sampled** 25 nm grid point, not a
continuous fit. Transmission is normalized by a directional input-mode
monitor. The residual is reported as raw `1-T-R`; a negative value would flag
a numerical normalization problem rather than being silently clipped.

## Field check

![Best sampled center-frequency fields](../_static/generated/composite_waveguide_gap_sweep_fields.png)

## Numerical controls

- Continuous-guide calibration: **100.033% T** and
  **0.00012% R**. The compact 100 nm / 3 µm guide gives **100.000% T** and **0.00294% R**. The SiO₂-only guides give **100.125% T** (1 µm) and **100.041% T** (2 µm).
- Full-domain versus y/z-symmetric transmission difference at the 1 µm gap:
  **1.192%**.
- 0.20 ps versus 0.40 ps transmission difference at the 2 µm SiO₂ / 1.2 µm
  gap reference: **0.020%**.
- Compact- versus original-x-domain transmission difference at that reference:
  **0.038%**.

The square guide has a degenerate fundamental polarization pair. The y-PEC and
z-PMC walls select its y-polarized member. For the unrestricted full-domain
audit, TE/TM sorting is disabled and the highest-effective-index fundamental
pair is selected directly; this avoids accidentally comparing against a lower
mode branch.

## Numerical method

- Fixed 25 nm Yee grid; the 150 nm and 100 nm nitride cores span six and four
  cells, respectively, while the controls contain no nitride.
- Component-aware Yee material discretization with subpixel-averaged facets.
- Twelve-cell PML and y/z mirror symmetry; the normalized S-parameter ratio is
  retained by the modal overlaps. A one-way left-port source forbids direct x
  symmetry; a correct even/odd x decomposition would require two half-domain
  simulations.
- A broadband Gaussian mode pulse is propagated for at most 0.40 ps in the
  original slices and 0.20 ps in the convergence-qualified compact extension,
  with relative-energy early termination in both cases.
- Source and detector planes do not move with the air gap. This keeps port
  normalization identical across the sweep.
- The artifact is three-dimensional in `(SiN side, SiO₂ side, air gap)`, so
  the remaining two parameter directions can be expanded without changing
  the measurement definition.

The complete setup and motivation are recorded in
[`METHOD.md`](https://github.com/hoodlab/fdtdx-hoodlab/blob/main/benchmarks/cases/composite_waveguide_gap_sweep/METHOD.md).