Waveguides

How a hollow tube guides a wave

An electromagnetic wave in free space can travel in any direction with any frequency. Put two parallel conducting walls in front of it and the wave must obey new boundary conditions: the tangential component of the electric field must vanish at the conductor surface. This single requirement turns out to be remarkably restrictive — only certain wave patterns can simultaneously satisfy Maxwell's equations and the boundary conditions, and only above a certain frequency.

Picture a rectangular metal tube of internal width a and height b. A wave entering one end will reflect off the four walls, bouncing in a zigzag pattern as it propagates down the tube. For the reflection geometry to "lock in" — to produce a coherent transverse pattern instead of a phase-incoherent mess — the wavelength must fit between the walls in a precise sense. The smallest pattern that fits in a rectangular guide is one transverse half-wavelength across the wide dimension: this is the TE₁₀ mode, named for its one half-cycle across a and zero variation along b.

The condition for the TE₁₀ mode to propagate is that the free-space wavelength be shorter than 2a. Equivalently, the operating frequency must exceed the cutoff frequency f_c = c/(2a). Below f_c the wave cannot fit a half-wave across the guide; the field exists, but as an evanescent solution that decays exponentially with distance along the tube. Above f_c the wave propagates with a longitudinal wavenumber k_z = (1/c)·√(ω² − ω_c²), corresponding to a group velocity vg = c·√(1 − (f_c/f)²) that approaches c at very high frequency and zero right at cutoff.

Modes and parameters of rectangular waveguides

Mode	Cutoff wavelength	Cutoff f (WR-90, a = 22.86 mm)	Field pattern	Usage
TE₁₀ (dominant)	λ_c = 2a	6.557 GHz	1 half-wave across width, uniform across height	Standard X-band, 8.2–12.4 GHz
TE₂₀	λ_c = a	13.11 GHz	1 full wave across width	Higher-order — avoided in single-mode operation
TE₀₁	λ_c = 2b	14.76 GHz (b = 10.16 mm)	1 half-wave across height	Suppressed in standard a:b = 2:1 aspect
TE₁₁	λ_c = 2/√((1/a)² + (1/b)²)	16.16 GHz	Half-wave in both	Filter design, mode converters
TM₁₁	same as TE₁₁ (degenerate)	16.16 GHz	Half-wave both, E_z component	Mode-pair degeneracy
TE₂₁	λ_c = 2/√((2/a)² + (1/b)²)	19.75 GHz	Two-by-one pattern	Higher-band reuse

The 2:1 aspect ratio of standard rectangular waveguide is chosen specifically so that the single-mode band (between the TE₁₀ cutoff and the next-mode TE₂₀ cutoff) covers exactly an octave of frequency. In WR-90 the single-mode range is 6.6–13.1 GHz, of which the standard operating band 8.2–12.4 GHz leaves comfortable margin from both cutoff edges.

The TE₁₀ mode in detail

Take the wide wall of the guide to lie in the x-direction with width a, the short wall in the y-direction with height b, and the propagation along z. The TE₁₀ mode fields (sinusoidal time dependence e^{jωt} suppressed) are

E_y = E₀ sin(πx/a) e^(−jk_z z)
H_x = −(k_z / ωμ₀) E₀ sin(πx/a) e^(−jk_z z)
H_z = (jπ/(ωμ₀ a)) E₀ cos(πx/a) e^(−jk_z z)
E_x = E_z = H_y = 0

The electric field is purely transverse and points across the short dimension, peaking at the center of the wide wall (x = a/2) and vanishing at the side walls (x = 0, x = a). The magnetic field circulates: a longitudinal H_z and a transverse H_x form closed loops in the x-z plane. Power flows down the guide with a Poynting vector S_z = ½ Re(E_y H_x*) that peaks at the center.

The longitudinal wavenumber satisfies k_z² = (ω/c)² − (π/a)² = (ω/c)² − ω_c²/c². When ω > ω_c, k_z is real and the field oscillates along z; when ω < ω_c, k_z is imaginary and the field decays as e^(−|k_z|z). The decay constant for evanescent fields at, say, 60 % of cutoff is roughly 1.4/(λ_c/2π) — meaning the field falls to 1/e in about λ_c/9, a fraction of the guide width.

Worked example: WR-90 X-band guide

WR-90 (a = 22.86 mm, b = 10.16 mm) is the standard rectangular guide for X-band radar and satellite uplinks. Compute the TE₁₀ parameters at the operating frequency f = 10 GHz:

Cutoff frequency:
  f_c = c / (2a) = 3 × 10⁸ / (0.04572) = 6.557 GHz

Free-space wavelength at 10 GHz:
  λ₀ = c/f = 30 mm

Guide wavelength (longer than free-space because k_z < k₀):
  λ_g = λ₀ / √(1 − (f_c/f)²) = 30 / √(1 − 0.4299) = 39.7 mm

Group velocity:
  vg = c · √(1 − (f_c/f)²) = c · 0.755 = 2.27 × 10⁸ m/s

Phase velocity (> c, but carries no energy):
  vφ = c / √(1 − (f_c/f)²) = c / 0.755 = 3.97 × 10⁸ m/s

Characteristic wave impedance:
  Z_TE₁₀ = η₀ / √(1 − (f_c/f)²) = 376.7 / 0.755 = 499 Ω

At 6.557 GHz exactly, k_z = 0, the group velocity is zero, and the wave impedance is infinite — energy cannot propagate. At 7 GHz, just above cutoff, vg ≈ 0.36 c — the wave crawls. By 10 GHz it has accelerated to 0.76 c, and approaches c asymptotically at very high frequency where the cutoff becomes a small correction. The frequency-dependent group velocity makes waveguides intrinsically dispersive: short pulses spread out, which is why pulsed radar typically uses chirped signals to compensate.

Below cutoff: evanescent fields and quantum tunneling

Drive a waveguide at f < f_c and the wave does not vanish — it becomes evanescent. The fields exist for a few skin-depth-like lengths into the guide, decaying as e^(−αz) where α = √(ω_c² − ω²)/c. They carry no time-averaged power; the Poynting vector oscillates but its time average is zero. Energy injected into one end of a below-cutoff section bounces back like a closed reactive load — the section behaves as a high-impedance attenuator.

This is the microwave analog of quantum-mechanical tunneling. A wave packet incident on a below-cutoff section sees an exponentially decaying field; if the section is short compared to 1/α, some amplitude leaks through and continues on the far side. Microwave "tunneling" experiments have measured group delays of these evanescent transitions and reported superluminal apparent velocities — though no actual information travels faster than c. The wavefunction analogy is exact: the wave equation in the below-cutoff guide is mathematically identical to the Schrödinger equation in a potential barrier.

The exponential decay below cutoff is also the principle of below-cutoff filters: short sections of guide narrowed so far that the operating frequency is below TE₁₀ cutoff produce sharp reactive responses ideal for high-Q microwave filter design.

Where waveguides matter today

• Radar transmit/receive chains. Every X-band airport surveillance radar (10 GHz) uses WR-90 between magnetron/klystron and antenna. The Patriot missile fire-control radar runs C/X-band with WR-90 and WR-137 sections. Modern phased arrays still use waveguide between the high-power transmit module and the radiating element when peak power exceeds ~10 kW — coax cannot handle the breakdown.
• Satellite uplinks and gyrotrons. Ku-band uplinks (14 GHz) use WR-75, Ka-band (30 GHz) WR-28. ITER's electron-cyclotron heating system runs 1 MW gyrotrons at 170 GHz through corrugated waveguides; the corrugations support a low-loss HE₁₁ mode optimized for high-power transmission with minimal wall heating.
• Particle accelerator RF transport. SLAC's 50 GeV linear accelerator used 2.856 GHz S-band rectangular waveguide (WR-284) to transport megawatts of RF to thousands of cavity sections. CERN's linacs use S-band; the LHC superconducting cavities are fed by waveguide-to-cavity coupling at 400 MHz.
• 5G and millimeter-wave backhaul. 28 GHz and 39 GHz 5G NR small-cell backhaul links use waveguide because losses in coax are prohibitive at those frequencies. Microsoft's Project Natick subsea data centers route mm-wave through corrosion-resistant guides; Verizon's mmWave 5G rollout used standardized WR-28 between RRU and antenna.
• EHT and ALMA radio astronomy. The Event Horizon Telescope's 230 GHz receivers use waveguide between the feed horn and SIS mixer. ALMA's Band 7 receiver covers 275–373 GHz through WR-2.8 (0.71 × 0.355 mm) — at this size the guide is a precision-machined photonic-bandgap-like structure rather than a tube you can see by eye.

Variants and extensions

• Circular waveguide. A round metal tube supporting TE_nm and TM_nm modes labelled by Bessel-function indices. The dominant TE₁₁ has lowest cutoff but rotates polarization with mechanical twist; TE₀₁ has low loss and rotational symmetry but mode degeneracy issues. Used in millimeter-wave radar and gyrotron systems.
• Coaxial cable. Inner conductor inside outer shield, supporting TEM mode with no cutoff. Bandwidth runs from DC up to the TE₁₁ cutoff (set by mean diameter / dielectric). RG-58 coax to roughly 1 GHz; LMR-400 to 10 GHz; semi-rigid to 50 GHz.
• Microstrip and stripline. Planar transmission lines on PCB: a signal trace above a ground plane (microstrip) or sandwiched between two grounds (stripline). Quasi-TEM at low frequency; dispersive at high. The dominant interconnect on every modern RF circuit board.
• Optical fiber. Dielectric waveguide using total internal reflection between high-index core and lower-index cladding. Single-mode fiber at 1550 nm wavelength carries a single fundamental mode (LP₀₁); multi-mode fiber carries hundreds. Used in 99 % of long-haul telecom.
• Photonic-crystal waveguide. Periodic dielectric or metallic patterning with bandgap; guides light not by index contrast but by destructive interference outside a defect line. Used in silicon photonics chips for compact, low-loss optical routing.
• Dielectric waveguide / image guide. Open-air dielectric strip carrying surface-bound mode. Common at THz frequencies where conductor losses dominate and metallic guides are impractical.

Common pitfalls

• Operating right at cutoff. At f_c the group velocity is zero and the impedance infinite — you simply cannot launch power. Always operate at least 25 % above cutoff (1.25·f_c) to keep group velocity reasonable and impedance bounded. Standard waveguide bands always start above 1.25·f_c.
• Letting higher-order modes propagate. Once you exceed the TE₂₀ cutoff (f_c·2 = 13.1 GHz for WR-90), the next mode begins to propagate and any waveguide discontinuity will excite it. Multi-mode propagation kills phase reproducibility and adds dispersion. Stay below the second-mode cutoff for single-mode operation.
• Treating phase velocity as a physical speed. The phase velocity in a waveguide exceeds c — and yet no information travels faster than light. Information rides the group velocity vg = c√(1 − (f_c/f)²) which is always less than c. Confusing the two is one of the classic textbook errors.
• Using rectangular formulas for circular guides. The Bessel-function eigenvalues set the circular-guide cutoffs and they are not simple multiples of c/2a — the dominant TE₁₁ in a circular guide of radius r cuts off at f_c = 0.293c/r, not c/(2r). Use the correct geometry-specific formulas.
• Ignoring wall loss and surface finish. The skin depth in copper at 10 GHz is 0.66 μm; surface roughness on the same order spikes loss by 20–40 %. Electropolished aluminum guides routinely outperform unpolished copper at the same band. Plating standards spec ≥5 skin depths of gold or silver above 30 GHz.