Rectangular waveguide cutoff frequency (TE/TM modes)
This page is a practical waveguide cutoff frequency calculator for an air-filled rectangular waveguide. Rectangular guides support discrete electromagnetic field patterns called modes. Each mode is identified by two integers, m and n, and each mode has a cutoff frequency fc. If your operating frequency is below fc, that mode becomes evanescent: the fields decay exponentially along the guide and the mode cannot transport power.
Engineers use cutoff calculations to pick a waveguide size that supports a desired band with margin. For example, you might want the dominant mode TE10 to propagate comfortably, while keeping the next higher modes below cutoff to avoid multimode behavior. This calculator helps you do that quickly: enter the internal dimensions and the mode indices, then read the cutoff in GHz.
Inputs (what each field means)
- Width a (cm): the internal broad-wall dimension of the waveguide cross-section. Enter in centimeters.
- Height b (cm): the internal narrow-wall dimension of the waveguide cross-section. Enter in centimeters.
- Mode index m: number of half-wave field variations across a (integer ≥ 0).
- Mode index n: number of half-wave field variations across b (integer ≥ 0).
The common dominant mode for most rectangular waveguides is TE10 (m = 1, n = 0). The combination (m = 0, n = 0) is not a propagating mode and is rejected by the calculator.
Tip for real hardware: use internal dimensions, not external dimensions. If you are working from a datasheet, confirm whether the listed size is internal, external, or a nominal standard. Small dimensional differences matter more as you approach cutoff.
Formula used
The calculator uses the standard rectangular waveguide cutoff relationship for an air-filled guide with ideal conducting walls. Internally it converts centimeters to meters. The cutoff frequency for mode (m, n) is:
fc = (c / 2) · √[(m/a)2 + (n/b)2]
where c is the speed of light in vacuum (299,792,458 m/s), and a and b are the waveguide dimensions in meters. The output is displayed in GHz. The formula is the same for TE and TM families in a rectangular guide; what changes between TE and TM is which (m, n) combinations are physically allowed and how the fields look, not the cutoff expression itself.
If you are using a dielectric-filled guide, the phase velocity changes and the cutoff shifts. A common first-order approximation is: fc,dielectric ≈ fc,air / √εr, where εr is the relative permittivity. This page does not apply that correction automatically, but the relationship is useful when you are estimating trends.
Worked example (TE10 in a 1 in × 0.5 in guide)
A common lab example is a waveguide with internal dimensions a = 2.54 cm and b = 1.27 cm (approximately 1 inch by 0.5 inch). For the dominant mode TE10 (m = 1, n = 0), the cutoff is:
- a = 2.54 cm → 0.0254 m
- b = 1.27 cm → 0.0127 m
- m = 1, n = 0
Plugging into the formula gives a cutoff near 5.9 GHz. In practice, engineers often operate with margin (for example, 1.25× cutoff) to reduce dispersion and attenuation that increase as you approach cutoff.
Second worked example: comparing TE10 and TE20
Cutoff calculations are especially useful when you want to avoid higher-order modes. Using the same dimensions as above (a = 2.54 cm, b = 1.27 cm), compare TE10 (m = 1, n = 0) to TE20 (m = 2, n = 0). Because the TE20 mode has twice the half-wave variation across a, its cutoff is roughly doubled relative to TE10 when n = 0.
That means if TE10 cuts on around 5.9 GHz, TE20 will cut on around 11.8 GHz for the same guide. If your system operates at 9 GHz, TE10 propagates but TE20 is still below cutoff, which supports single-mode operation (assuming other modes like TE01 or TE11 are also below cutoff for your geometry).
Sanity check: for modes with n = 0, cutoff scales approximately as m/a. Increasing a lowers cutoff; increasing m raises cutoff. For modes with m = 0, cutoff scales approximately as n/b.
Common rectangular waveguide modes (quick reference)
The calculator accepts any nonnegative integers (m, n) except (0, 0). In practice, a few modes show up repeatedly in design discussions. The table below summarizes what the indices mean physically and how they tend to affect cutoff. Use it as a guide for choosing inputs and for interpreting results.
| Mode label | (m, n) | Field variation across a and b | Typical design meaning |
|---|---|---|---|
| TE10 | (1, 0) | One half-wave across a, none across b | Dominant mode in most standard rectangular waveguides; lowest cutoff for typical a > b. |
| TE01 | (0, 1) | None across a, one half-wave across b | Often the next competing mode when b is not much smaller than a; can limit single-mode bandwidth. |
| TE11 / TM11 | (1, 1) | One half-wave across both a and b | Higher-order mode; cutoff depends on both dimensions; can appear in bends, transitions, and overmoded sections. |
| TE20 | (2, 0) | Two half-waves across a, none across b | Common higher-order competitor above TE10; cutoff roughly doubles when n = 0. |
| TE21 | (2, 1) | Two across a, one across b | Even higher cutoff; relevant in overmoded waveguides and wideband components. |
Remember: the cutoff formula gives the threshold for propagation, but it does not tell you how strongly a mode is excited. In real components, discontinuities (steps, irises, bends, imperfect launches) can couple energy into higher-order modes even if you intended to excite only TE10. That is one reason designers keep a comfortable margin below the next-mode cutoff.
Design notes: single-mode operation and margins
Cutoff frequency is used to choose a waveguide size so that your intended band is above the cutoff of the desired mode (often TE10) but below the cutoff of higher-order modes that can introduce multimode propagation. If you are designing for a single-mode band, compare your operating frequency not only to the desired mode cutoff, but also to the next higher mode cutoffs.
Near cutoff, wave impedance and group velocity change rapidly; losses and sensitivity to tolerances increase. This calculator provides the ideal cutoff point, which is the starting point for those engineering tradeoffs. A common rule of thumb is to operate at least 15–25% above cutoff for the intended mode, and to keep the operating band comfortably below the next-mode cutoff. The exact margin depends on insertion loss targets, component complexity, and how cleanly the mode is launched.
If you are troubleshooting a system, cutoff can also explain symptoms: unexpected attenuation at the low end of a band, strong frequency-dependent phase shift, or sensitivity to small mechanical changes can all be signs that you are operating too close to cutoff.
Understanding rectangular waveguides
Waveguides are hollow metallic structures that guide electromagnetic waves by reflecting fields from conducting walls. Rectangular waveguides are widely used in microwave engineering because they handle high power with low loss compared with many coaxial lines at the same frequency. Inside the guide, the fields form standing-wave patterns across the cross-section. Only patterns that satisfy boundary conditions can exist, which is why propagation happens in discrete modes.
Each mode has a cutoff frequency. Below cutoff, the fields decay exponentially along the guide and the mode cannot transport power. Above cutoff, the mode propagates with a phase constant that depends on frequency; as you approach cutoff from above, dispersion increases and attenuation typically rises. In other words, cutoff is not just a yes/no threshold; it is also a region where performance can change quickly.
The role of mode indices (m, n)
In a rectangular guide, the indices m and n count the number of half-wave variations across the width a and height b. The dominant mode in most standard waveguides is TE10, which has one half-wave variation across the broad wall and none across the narrow wall. Higher-order modes (for example TE20, TE11, TM11) have higher cutoff frequencies and more complex field patterns.
A practical way to build intuition is to imagine “fitting” half-wavelengths across the cross-section. If you make the guide wider (increase a), it becomes easier to fit the pattern across that dimension, so cutoff drops. If you increase the mode index (increase m or n), you are asking the fields to vary more rapidly across the same space, so cutoff rises.
How to interpret the result (and avoid common mistakes)
The result is the cutoff frequency for the specific (m, n) pair you entered. A few quick checks help prevent misinterpretation:
- Units check: a and b are entered in centimeters. If you accidentally enter millimeters as centimeters, your cutoff will be off by 10×.
- Geometry check: for most standard guides, a > b. If you swap them, you may compute a cutoff for a different physical guide.
- Mode check: TE10 is (1,0). TE01 is (0,1). TE11 is (1,1). These are easy to mix up.
- Margin check: if your operating frequency is only slightly above cutoff, expect higher loss and stronger dispersion than at mid-band.
If you are comparing scenarios, change one input at a time (for example, increase a by 5%) and observe how the cutoff shifts. That sensitivity analysis is often more valuable than a single number because it tells you which dimension or mode choice dominates your design.
Practical considerations (what this calculator does and does not do)
This page computes the ideal cutoff frequency for a chosen (m, n) pair. It does not compute attenuation, dispersion, power handling, or the cutoff of every possible mode automatically. Use it to quickly check whether a candidate waveguide size supports a desired band and to build intuition for how changing a, b, m, or n shifts cutoff.
For component design, you will typically pair cutoff calculations with standard waveguide band tables, electromagnetic simulation, and measurements. Still, the cutoff formula is the foundation: it explains why standard waveguides have published “recommended bands” and why those bands are not arbitrary.
Limitations and assumptions
- Air-filled guide: assumes relative permittivity εr ≈ 1. If filled with dielectric, cutoff decreases by approximately 1/√εr.
- Perfect conductors: ignores finite conductivity, surface roughness, and plating effects that increase loss (especially near cutoff).
- Internal dimensions: assumes the dimensions entered are the internal waveguide dimensions that set boundary conditions.
- Integer mode indices: m and n must be whole numbers ≥ 0; (0,0) is not a propagating mode.
- Near-cutoff behavior: the computed cutoff is accurate, but real systems often require margin above cutoff for acceptable loss and dispersion.
If you are designing hardware, confirm results against a waveguide standard table and consider manufacturing tolerances, temperature, and mechanical stress. For compliance or safety-critical work, treat this calculator as a fast estimator and validate with authoritative references.
Interactive intuition builder
Mode Runner · Stay above cutoff
Surf a carrier through a rectangular guide for ~80 seconds. Keep the operating frequency safely above the cutoff of the active TE/TM mode without drifting into multi-mode chaos. Every wobble in width a and height b changes the cutoff—feel the ratio instead of reading it.
Controls
A/D or ←/→ widen or narrow a. W/S or ↑/↓ stretch or shrink b. Touch buttons mirror the moves. Keep the teal band riding above the cyan cutoff rails.
Bandwidth feels best ~25–40% above cutoff; slip below and power walls out.