Hollow rectangular waveguide (WR-90, TE10)

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.

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.

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.

• 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.

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.

• 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.

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.

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:

python scripts/precompute_gallery_artifacts.py --case waveguide_wr90

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.

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.

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.)

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.

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.

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.

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.

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.

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