Waveguide Engineering

Cutoff Wavelength

Q: How do you calculate the cutoff wavelength of a rectangular waveguide?

For a rectangular waveguide with broad-wall dimension a and narrow-wall dimension b, the cutoff wavelength of the TE_mn or TM_mn mode is lambda_c = 2 / sqrt((m/a)^2 + (n/b)^2). For the dominant TE10 mode this simplifies to lambda_c = 2a, exactly twice the broad-wall width. A WR-90 guide has a = 22.86 mm, so lambda_c = 45.72 mm, which corresponds to a cutoff frequency of 6.557 GHz. Because lambda_c depends only on geometry and not on the dielectric fill, the same guide always cuts off at the same wavelength regardless of the material inside it.

Q: Why does a signal below the cutoff frequency not propagate through a waveguide?

When the free-space wavelength exceeds the cutoff wavelength, the propagation constant becomes purely imaginary and the field reduces to an evanescent (non-propagating) mode. Instead of traveling down the guide, the field amplitude decays exponentially with distance as exp(-alpha z), where alpha = (2 pi / lambda_c) times sqrt(1 - (lambda_c / lambda)^2). For the TE10 mode well below cutoff, attenuation approaches roughly 27.3 dB per guide width a, so an inch or two of below-cutoff guide acts as an extremely high-rejection lowpass barrier. This is the operating principle of a waveguide-below-cutoff (WBCO) attenuator and of the honeycomb vents used in EMI-shielded enclosures.

Q: What is the relationship between cutoff wavelength and guide wavelength?

The guide wavelength lambda_g is always longer than the free-space wavelength and is related to the cutoff wavelength by lambda_g = lambda / sqrt(1 - (lambda / lambda_c)^2). As the operating frequency approaches cutoff (lambda approaching lambda_c), the term under the radical approaches zero and lambda_g grows without bound, which is why phase velocity becomes very large and group velocity collapses near cutoff. Far above cutoff, lambda_g approaches the free-space wavelength. This is why practical waveguide operation stays in the 1.25 to 1.9 times cutoff frequency window, roughly the 0.53 to 0.8 lambda_c range, to keep dispersion and loss manageable.

/KUHT-awf WAYV-length/ (symbol λ_c)

The longest free-space wavelength that a particular waveguide mode will carry, fixed entirely by the cross-section geometry rather than the dielectric fill. A wave whose free-space wavelength exceeds λ_c (equivalently, a frequency below the cutoff frequency) cannot propagate; its propagation constant turns purely imaginary and the field collapses into an evanescent mode that decays exponentially down the guide. For the dominant TE₁₀ mode of a rectangular guide the rule is simply λ_c = 2a, twice the broad-wall width, so a WR-90 guide (a = 22.86 mm) cuts off at 45.72 mm, or 6.557 GHz.

Category: Waveguide Engineering

Symbol: λ_c

TE₁₀ rule: λ_c = 2a

How Geometry Sets the Cutoff

A hollow metallic waveguide behaves as a high-pass structure: it transmits energy only above a sharply defined threshold and rejects everything below it. That threshold is most naturally expressed as a wavelength because, for a given mode, it depends only on the guide's physical cross-section. The cutoff wavelength is the boundary case where the transverse field pattern exactly fits the guide walls with zero longitudinal phase progress, meaning the wave bounces purely sideways and makes no forward headway. Any free-space wavelength longer than λ_c over-fills the cross-section and can no longer satisfy the boundary conditions as a traveling wave.

Each mode (TE₁₀, TE₂₀, TM₁₁, and so on) has its own cutoff wavelength. The mode with the longest λ_c, and therefore the lowest cutoff frequency, is the dominant mode; for standard rectangular guide that is TE₁₀. The usable single-mode bandwidth of a guide is the span between the dominant-mode cutoff and the next higher-order cutoff (TE₂₀ at λ_c = a, or half the dominant value). Engineers deliberately operate inside this window so that only one well-behaved field pattern carries the energy, avoiding the multimode dispersion and mode beating that would otherwise corrupt the signal.

Because λ_c is set by geometry and not by frequency or material, it is the most stable design anchor a waveguide engineer has. Filling the guide with a dielectric of relative permittivity ε_r leaves the cutoff wavelength unchanged but lowers the cutoff frequency by a factor of √ε_r, a trick used to shrink components without changing the metalwork.

Below Cutoff: The Evanescent Region

When a signal sits below cutoff, the guide does not simply reflect it at the mouth; the field penetrates a short distance and decays exponentially. The attenuation rate grows rapidly the further below cutoff the signal is, reaching about 27.3 dB per broad-wall width for a deeply sub-cutoff TE₁₀ mode. This predictable, frequency-stable rolloff is exploited in waveguide-below-cutoff (WBCO) attenuators, in piston attenuators used as precision laboratory standards, and in the honeycomb vent panels that let air through an EMI-shielded enclosure while blocking microwave leakage. The same physics defines how close to cutoff a practical guide can run before guide wavelength dispersion and attenuation become unacceptable.

Governing Equations

Cutoff wavelength, rectangular TE_mn/TM_mn:
λ_c = 2 / √[ (m/a)² + (n/b)² ]

Dominant TE₁₀ mode:
λ_c = 2a → f_c = c / 2a

Circular waveguide, TE₁₁ (dominant):
λ_c ≈ 1.706 × D (D = bore diameter)

Guide wavelength vs. cutoff:
λ_g = λ / √[ 1 − (λ / λ_c)² ]

Below-cutoff attenuation:
α = (2π / λ_c) × √[ 1 − (λ_c / λ)² ] Np/m

Where a, b = broad- and narrow-wall widths; m, n = mode indices; c ≈ 2.998 × 10⁸ m/s; λ = free-space wavelength. Example: WR-90 TE₁₀, a = 22.86 mm → λ_c = 45.72 mm, f_c = 6.557 GHz.

Standard Waveguide Cutoff Values (TE₁₀)

Waveguide	Broad wall a (mm)	Cutoff λ_c (mm)	Cutoff f_c (GHz)	Recommended band (GHz)	Band name
WR-340	86.36	172.7	1.736	2.20 to 3.30	S
WR-159	40.39	80.78	3.711	4.90 to 7.05	C
WR-90	22.86	45.72	6.557	8.20 to 12.4	X
WR-62	15.80	31.60	9.488	12.4 to 18.0	Ku
WR-28	7.112	14.22	21.08	26.5 to 40.0	Ka
WR-15	3.759	7.518	39.88	50.0 to 75.0	V
WR-10	2.540	5.080	59.01	75.0 to 110	W

Common Questions

Frequently Asked Questions

How do you calculate the cutoff wavelength of a rectangular waveguide?

Use λ_c = 2 / √[(m/a)² + (n/b)²], where a and b are the broad- and narrow-wall dimensions and m, n are the mode indices. For the dominant TE₁₀ mode this collapses to λ_c = 2a. A WR-90 guide has a = 22.86 mm, so λ_c = 45.72 mm, equivalent to a cutoff frequency of 6.557 GHz. Because λ_c depends only on geometry, the same guide cuts off at the same wavelength no matter what dielectric fills it.

Why does a signal below the cutoff frequency not propagate through a waveguide?

When the free-space wavelength exceeds λ_c, the propagation constant becomes purely imaginary and the mode is evanescent: the field decays as exp(−αz) instead of traveling. For TE₁₀ well below cutoff the rolloff is about 27.3 dB per broad-wall width, so a short length of below-cutoff guide is an extremely high-rejection barrier. That principle drives WBCO attenuators and the honeycomb EMI vents used in shielded enclosures.

What is the relationship between cutoff wavelength and guide wavelength?

They are linked by λ_g = λ / √[1 − (λ/λ_c)²]. As the operating wavelength approaches λ_c, the radical approaches zero and λ_g grows without bound, which is why phase velocity rises and group velocity collapses near cutoff. Far above cutoff, λ_g approaches the free-space wavelength. Practical guides therefore run roughly 1.25 to 1.9 times the cutoff frequency to keep dispersion and loss in check.

Does filling a waveguide with dielectric change its cutoff wavelength?

No. The cutoff wavelength is purely a geometric property of the cross-section, so it stays the same regardless of the fill. What changes is the cutoff frequency: f_c drops by a factor of √ε_r because the wave travels slower in the dielectric. Designers use this to shrink a guide's operating band downward without altering the metal dimensions, at the cost of higher loss tangent and some power-handling penalty.

Related Terms

← Cutoff Frequency (TL) Cutout →