# Hollow rectangular waveguide (WR-90, TE10)

Channel: release; package: 1.8.0; source: 7d1e97331656dd74ad0f5471653ac0c5effdff70

Page: https://remilab.ai/rfx/versions/v1.8.0/gallery/waveguide_wr90/
Source: https://github.com/bk-squared/rfx/blob/7d1e97331656dd74ad0f5471653ac0c5effdff70/docs/public/gallery/waveguide_wr90.mdx

A rectangular waveguide is a hollow metal pipe that carries one electromagnetic
mode at a time above a cutoff frequency. This case drives the dominant TE10 mode
down a standard WR-90 guide (X-band) and reads the two-port S-parameters. An
ideal empty guide should pass the mode with almost no reflection — and because
the matched empty-guide answer is known in closed form, it is a clean absolute
check on the waveguide port.

## The structure

![The hollow rectangular waveguide](https://remilab.ai/rfx/versions/v1.8.0/gallery/assets/waveguide_wr90/geometry.png)

A WR-90 guide: broad wall a = 22.86 mm, narrow wall b = 10.16 mm, metal (PEC)
walls, hollow air inside. The mode is launched at the left port and collected at
the right port. Both end faces are absorbing (CPML); the four side walls are PEC.

## What this is

The dominant mode of a rectangular guide is TE10. It only propagates above the
cutoff frequency set by the broad-wall width: `f_c = c / (2·a)`. For WR-90,
`a = 22.86 mm` gives `f_c ≈ 6.557 GHz`, so the X-band (8.2–12.4 GHz) sits
comfortably above cutoff and the TE10 mode travels cleanly. Below `f_c` the mode
is evanescent and the guide rejects it. Filling the guide with a dielectric `εr`
lowers the cutoff to `f_c,d = f_c / √εr` — the lever the autodiff section below
differentiates through.

## What you'll learn

- Launching the **TE10 dominant mode** with a mode-matched waveguide port (a
  one-sided mode injection, not a soft point source).
- The **cutoff relation** `f_c = c/(2·a)` and why X-band propagates in WR-90
  while lower frequencies would not.
- Reading two-port **|S11| (reflection) and |S21| (transmission)** against the
  exact matched empty-guide answer (`|S11| = 0`, `|S21| = 1`).
- Taking a **gradient through the FDTD solver** with `jax.grad`: how `|S21|`
  responds to the guide's dielectric fill, validated against finite differences.

## Geometry & materials

```python
import numpy as np
C0 = 299_792_458.0

a_wg = 22.86e-3        # WR-90 broad wall (sets the TE10 cutoff)
b_wg = 10.16e-3        # WR-90 narrow wall
f_c  = C0 / (2 * a_wg) # TE10 cutoff: c / (2a) -> 6.557 GHz
dx   = 1.0e-3          # 1 mm cells -> ~30 cells/wavelength at 10 GHz
freqs = np.linspace(8.2e9, 12.4e9, 21)   # X-band sweep
```

The guide is hollow air inside PEC walls. The only material knob is the optional
dielectric fill used in the gradient demonstration below.

## Source, ports, boundaries, mesh

- **Two TE10 waveguide ports.** The left port (`direction="+x"`) launches the
  mode; the right port (`direction="-x"`) acts as a matched load. Each port
  injects the discrete Yee-grid mode profile, so the launcher back-reflection is
  minimal.
- **CPML on the propagation axis only (20 cells).** Twenty layers drive the
  residual guided-mode reflection floor to essentially zero (`|S11| ≈ 0`);
  thinning to the default 10–12 layers raises it toward ~0.09 (a known trap).
  Only the two x faces absorb.
- **PEC side walls.** The four y/z walls are PEC — that is what defines a
  rectangular guide and its cutoff.
- **Uniform 1 mm mesh.** Ten cells span the narrow wall, about 23 the broad
  wall, and roughly 30 cells per wavelength at the top of the band — enough to
  resolve the TE10 half-sine across `a` and keep numerical dispersion small.

With the thick 20-layer CPML the empty-guide |S11| sits at essentially zero. Drop
the absorber to the default 10–12 layers and the floor rises toward about 0.09 —
pure absorber back-reflection, not a solver error. Above cutoff the guide is
nearly lossless, so any residual reflection is set by the CPML thickness.

## Geometry check (before running)

The render above confirms the guide before committing compute: broad wall a,
narrow wall b, the two port planes set back from the absorbers, and clearance to
the CPML on both ends. An empty WR-90 run at 1 mm (about 46k cells, a few hundred
steps) finishes in seconds on a CPU; peak memory is a few tens of megabytes.

## Run

```python
import numpy as np, jax.numpy as jnp
from rfx.api import Simulation
from rfx.boundaries.spec import Boundary, BoundarySpec

a_wg, b_wg, dx = 22.86e-3, 10.16e-3, 1.0e-3
freqs = np.linspace(8.2e9, 12.4e9, 21)
f0 = float(freqs.mean()); bandwidth = 0.5
domain_x = 0.200

sim = Simulation(
    freq_max=float(freqs[-1]) * 1.1,
    domain=(domain_x, a_wg, b_wg), dx=dx,
    boundary=BoundarySpec(x=Boundary(lo="cpml", hi="cpml"),
                          y=Boundary(lo="pec", hi="pec"),
                          z=Boundary(lo="pec", hi="pec")),
    cpml_layers=20,
)
port_freqs = jnp.asarray(freqs)
sim.add_waveguide_port(0.040, direction="+x", mode=(1, 0), mode_type="TE",
    freqs=port_freqs, f0=f0, bandwidth=bandwidth,
    waveform="modulated_gaussian", reference_plane=0.050, name="left")
sim.add_waveguide_port(domain_x - 0.040, direction="-x", mode=(1, 0), mode_type="TE",
    freqs=port_freqs, f0=f0, bandwidth=bandwidth,
    waveform="modulated_gaussian", reference_plane=domain_x - 0.050, name="right")

result = sim.compute_waveguide_s_matrix(num_periods=200, normalize=True)
```

Run it from the command line:

```bash
python scripts/precompute_gallery_artifacts.py --case waveguide_wr90
```

## Fields

E_z on the broad-wall mid-plane of a source-fed empty guide, captured once the
pulse has propagated to mid-guide. The field is a half-sine across the broad wall
— peaking at the centre and vanishing at the PEC side walls — travelling in the
+x direction, with the next half-cycle of opposite sign following behind. That is
the TE10 mode moving down the guide.

![TE10 mode E_z on the broad-wall mid-plane of the WR-90 guide](https://remilab.ai/rfx/versions/v1.8.0/gallery/assets/waveguide_wr90/field_te10.png)

The same mode in motion: the TE10 wave packet propagates in the +x direction down
the guide and into the matched absorber, holding its half-sine profile across the
broad wall as it travels.

![Animation: the TE10 mode travelling down the WR-90 guide](https://remilab.ai/rfx/versions/v1.8.0/gallery/assets/waveguide_wr90/field_anim.gif)

## Result

Across the X-band, |S21| sits at about 0 dB and |S11| is essentially zero
(`max|S11| = 0.0000` with the thick CPML) — so the empty matched guide passes the
TE10 mode essentially perfectly, which is exactly what an ideal guide above cutoff
should do. (Thinning the absorber raises the reflection floor toward ~0.09; see
the limit note under Validation.)

![Reflection and transmission vs frequency for the empty WR-90 guide](https://remilab.ai/rfx/versions/v1.8.0/gallery/assets/waveguide_wr90/sparams.png)

[S-parameter data (Touchstone .s2p)](https://remilab.ai/rfx/versions/v1.8.0/gallery/assets/waveguide_wr90/sparams.s2p)

## Validation & limits

The empty matched guide has an exact answer: `|S11| = 0` and `|S21| = 1` at every
frequency above cutoff. The figure overlays the rfx values on that reference.

| Observable | rfx | Reference | Tolerance | Pass |
|---|---|---|---|---|
| max\|S11\| | 0.0000 | 0 (analytic matched-load) | `< 0.02` | yes |
| min\|S21\| | 1.0000 | 1 (analytic matched-load) | `> 0.97` | yes |

![|S11| and |S21| vs the exact matched empty-guide answer](https://remilab.ai/rfx/versions/v1.8.0/gallery/assets/waveguide_wr90/validation.png)

Reference: analytic matched-load (`|S11| = 0`, `|S21| = 1` above cutoff),
cross-checked against MEEP/OpenEMS/Palace. With the thick
20-layer CPML this run produces, the published result is `max|S11| = 0.0000` and
`min|S21| = 1.0000` to the figure's precision — the empty matched guide passes
the TE10 mode essentially perfectly, as an ideal guide above cutoff should.

**Limit — absorber floor.** The residual `|S11|` is a CPML back-reflection floor,
not a solver error. The 20-layer absorber holds it at the `≈0` published here;
*thinning* the CPML to the default 10–12 layers raises the reflection floor toward
about 0.09. So the floor is a sensitivity to the absorber thickness, not the
guide's reflection.

## Autodiff

Fill part of the guide with a dielectric and `|S21|` responds, and that response
is differentiable. The design variable is the fill permittivity `εr`, entered
through `eps_override`, and the scalar is `|S21|²`. A single `jax.value_and_grad`
call returns `∂|S21|²/∂εr` straight through `compute_waveguide_s_matrix(
normalize="flux", ...)` — the power-flux extraction is on the AD tape, so the
gradient flows through the full modal S-matrix assembly.

There is no closed form for the filled-guide `|S21|`, so the AD gradient is
compared against central finite differences (`h = 0.05`), asking for agreement
within 5% and a matching sign.

| Check | AD value | Compared against | rel. error | Sign agrees |
|---|---|---|---|---|
| `∂\|S21\|²/∂εr` | `−0.2201` | central FD `−0.2197` (h = 0.05) | 0.0017 (0.17%) | yes |

It agrees to 0.17% with matching sign — well inside the 5% asked of it. That
number validates the differentiation of this computation; it is not a
chain-closure verdict for the family. The rectangular-waveguide S-matrix row of
`docs/guides/support_matrix.md` (source checkout) records: "**Chain battery:
chain-closed (v1.8) for uniform single-mode S on the `normalize=False` and
`normalize="flux"` lanes**" — contract criteria 1 (forward identity) and 3(a)
(AD-vs-FD) read under `JAX_ENABLE_X64` on the flux lane, the forward default
still float32. The 0.17% figure above is a **float32** flux-lane gradient: the
declaration leaves a float32 gradient pipeline on that lane outside it (its
forward identity misses the contract bound at 1.08–1.76 of the allowance),
so read this page's number as an AD-vs-FD agreement on this computation, not
as a chain-closed gradient. Chain-closed is not "supported"; the row stays
**limited**.

![Transmission sensitivity to the dielectric fill — AD gradient vs central finite differences](https://remilab.ai/rfx/versions/v1.8.0/gallery/assets/waveguide_wr90/autodiff.png)

The AD gradient `∂|S21|²/∂εr ≈ −0.220` matches the central finite difference to
about `0.2 %`, with matching sign. The sign is physical: a dielectric fill lowers
the cutoff toward the band through `f_c,d = f_c / √εr` (so
`∂f_c,d/∂εr = −f_c / (2·εr^{3/2})`, about `−1.8 GHz`), which moves the band edge
and reduces transmission. The closed-form cutoff relation points the same way the
solver gradient does.

**Limits.** The reflection floor is the CPML, not the physics — the 20-layer
absorber drives the published `|S11|` to ~0, and thinning to the default 10–12
layers raises it toward ~0.09 (a known trap). The broad-wall width `a` is **not**
a `jax.grad` input: `a/dx` is a discrete cell count and the analytic mode profile
is rebuilt for each width, so its sensitivity is shown by re-meshing and finite
differences, alongside the analytic cutoff relation `f_c = c/(2·a)` — not by
automatic differentiation. The differentiable lanes are `normalize=False` and
`normalize="flux"`. `normalize=True` is the host-assembled two-run lane and is
outside the differentiable chain: it raises `NotImplementedError` for a traced
`eps_override`/`sigma_override` under `jax.grad`/`jax.jit`
. The Run block above passes a concrete
`normalize=True` and runs forward; the same call under `jax.grad` raises.

## Next

[Dielectric slab (normal incidence)](https://remilab.ai/rfx/versions/v1.8.0/gallery/multilayer_fresnel/)

Computed with rfx 1.6.5 · 1eb551b · generated 2026-06-30T03:22:31Z