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.

The structure

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

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.

Note

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

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

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.

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.

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.)

S-parameter data (Touchstone .s2p) Reflection (S11) and transmission (S21) vs frequency, two-port Touchstone.

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

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.

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. Both normalize=True and normalize="flux" support the gradient; each costs two runs per port.

Next

Dielectric slab (normal incidence) Reflection and transmission of a dielectric slab vs the exact transfer-matrix solution, with an exact-derivative autodiff check.

Rectangular microstrip patch antenna (2.4 GHz, FR4) A printed resonator: matched return-loss dip, Harminv resonance, and a differentiable substrate sweep.

Computed with rfx · 2026-06-25