Waveguide Cutoff Frequency Calculator

Introduction to waveguide cutoff and modal propagation

Waveguide cutoff is the frequency boundary at which a particular electromagnetic field pattern changes from decaying along a metal guide to propagating through it. Unlike coaxial cable, a hollow conducting waveguide does not support a TEM mode. Its walls force the fields into discrete transverse patterns called modes. Each mode has a family, TE or TM, and indices that describe its transverse variation. Below that mode’s cutoff, the axial field is evanescent; above it, the mode can carry power.

This calculator evaluates ideal rectangular and circular guides. It finds the selected TE or TM cutoff, identifies low-order modes, and, when an operating frequency is supplied, estimates guide wavelength, phase velocity, group velocity, or below-cutoff decay. It is useful for checking an aperture, understanding a standard WR size, and seeing whether a proposed frequency is near cutoff or in an overmoded region.

Cutoff is a boundary-condition result rather than a complete component rating. Launches, flanges, bends, windows, surface finish, material loss, and tolerance all affect a finished assembly. The result is therefore a first-pass modal calculation: it tells you what an ideal uniform section permits, not the insertion loss or return loss of an entire system.

How to use the waveguide cutoff inputs

Choose the cross-section before entering dimensions. For rectangular guide, enter the finished internal broad-wall dimension a and narrow-wall dimension b; a standard preset can fill these values. For circular guide, enter one internal size and explicitly choose whether it is a radius or diameter. A radius-versus-diameter mistake changes the calculated cutoff by a factor of two.

Select TE or TM and enter whole-number indices. In a rectangular guide, TE modes may have one zero index but TE₀₀ does not exist. Rectangular TM modes require both indices to be at least one. In a circular guide, m is azimuthal order and n selects a Bessel root, so the root number begins at one. Use ε_r = 1 and µ_r = 1 for air or vacuum; positive values describe a uniform, lossless filling.

The operating frequency is optional. Leave it blank to obtain only cutoff and the mode spectrum. Enter it to see whether the selected mode propagates. The generated spectrum is important: a frequency can be above the desired mode’s cutoff while also being above a competing mode’s cutoff, making the guide overmoded.

Waveguide cutoff formulas for rectangular and circular guides

For a rectangular guide, the transverse eigenvalue is set by standing-wave variation across the two walls:

When the axial propagation constant reaches zero, the rectangular cutoff frequency becomes:

Here c is 299 792 458 m/s. For air-filled TE₁₀, the narrow-wall term is zero, so the familiar result depends only on the broad wall:

Circular walls require cylindrical solutions. TE modes use roots of the derivative of the Bessel function, while TM modes use roots of the Bessel function itself:

The dominant ideal circular mode is TE₁₁, with first derivative root about 1.84118. The lowest circular TM mode is TM₀₁, with root about 2.40483. A uniform filling lowers every ideal cutoff by √(ε_rµ_r), but does not widen the relative spacing between modes.

For a frequency above cutoff, the calculator uses the plane-wave velocity v in the filling to calculate dispersion:

Phase velocity can exceed the plane-wave velocity without sending energy or information faster than light. Near cutoff, guide wavelength and phase velocity become large while group velocity becomes small. Below cutoff, the corresponding ideal attenuation constant is:

Worked example: WR-90 TE₁₀ at 10 GHz

WR-90 has an internal aperture of 22.86 mm × 10.16 mm. With air filling, TE₁₀ uses m = 1 and n = 0. The broad wall therefore gives a cutoff of approximately 6.557 GHz. At 10 GHz, the ratio f/f_c is about 1.525, so the selected mode propagates with reasonable separation from cutoff. Its guide wavelength is roughly 39.7 mm, longer than the roughly 30.0 mm free-space wavelength.

The next distinct threshold is TE₂₀ near 13.114 GHz for this aperture. Therefore 10 GHz is inside the ideal single-mode interval. The usual WR-90 band of 8.2–12.4 GHz adds practical margin below the next mode and above dominant cutoff. If the broad wall becomes one percent wider, TE₁₀ cutoff becomes approximately one percent lower; higher modes can depend on both walls.

Interpreting cutoff margin and the mode spectrum

A calculated propagating state means a mode is physically allowed, not that it is strongly excited. Launch symmetry determines coupling, while bends, joints, steps, and misalignment can convert power into other permitted modes. When operating frequency exceeds the next distinct cutoff, several patterns can propagate with different phase constants. Their interference may produce ripple, polarization changes, and unstable measurements.

Just above cutoff, dispersion is strong and group velocity is low. For ordinary transmission hardware, cutoff itself is not a useful band edge. Designers often seek a comfortable margin above the dominant mode and remain below the next-mode threshold. The calculator’s suggested range is only a broad orientation aid; the spectrum for the entered dimensions and the manufacturer’s component ratings are the more useful design checks.

A below-cutoff section can still have a purpose. Filters, attenuators, coupling structures, and electromagnetic feedthroughs use evanescent fields through finite lengths of guide. The displayed decay is the ideal rate for a uniform section. It does not include reflections or reactive energy at transitions, so it is not a complete insertion-loss or shielding-effectiveness prediction.

Practical aperture, material, and measurement checks

Rectangular TE₁₀ is controlled by the broad wall for cutoff, but the narrow wall remains important for higher-mode spacing, impedance, loss, power handling, and mechanical compatibility. A conventional two-to-one aspect ratio tends to place TE₂₀ and TE₀₁ near twice the TE₁₀ cutoff. Other aspect ratios can bring a different higher mode closer, which is why the generated mode table is preferable to assuming a standard ratio.

For circular guide, all cutoffs scale inversely with radius. Circular TE₁₁ is the dominant mode, and many circular modes have two ideal angular orientations. Ovality, seams, or a non-symmetric launcher can split or mix those polarizations. The calculator evaluates a bounded low-order set of circular roots, sufficient for dominant-mode checks but not for a highly overmoded mode-converter design.

Uniform material filling is an essential assumption. A dielectric slab, support bead, air gap, partial liquid fill, ridge, corrugation, or substrate changes the eigenvalue problem; averaging permittivity is generally unreliable because the field energy is not uniformly distributed. Real dielectric permittivity and magnetic permeability can also vary with frequency and temperature. Use measured material data and an eigenmode solver for a narrowband or critical design.

When measurements disagree with a calculation, first check units, final internal dimensions, radius versus diameter, and mode notation. Then examine plating thickness, corner radii, seams, material properties, and launch symmetry. A vector network analyser measures scattering parameters rather than cutoff directly, and a short guide section can transmit measurable evanescent energy. Longer uniform sections and carefully chosen calibration planes usually make the modal transition easier to interpret.

Limitations of this ideal waveguide model

This waveguide cutoff calculator assumes a straight, uniform guide with perfectly conducting walls and a homogeneous, lossless fill. It does not calculate conductor loss, surface roughness, bends, flanges, tapers, ridges, corrugations, partial loading, power breakdown, or a transition’s excitation efficiency. Those effects can be decisive in millimetre-wave, high-power, vacuum, cryogenic, or precision phase systems.

Use the result as a modal threshold and a dimensional sanity check. For procurement and fabrication, consult the applicable controlled mechanical standard. For a critical assembly, compare the ideal result with a full-wave model and calibrated network-analyser data. The most conservative usable range is usually the intersection of the guide, transitions, bends, windows, loads, and instrument specifications.

Waveguide cutoff questions and concise answers