Short note: Pyramidal horn antenna

A pyramidal horn is a rectangular waveguide flared in both the E-plane and the H-plane, so that its mouth is a larger rectangle (a_H × a_E). It is the most widely used horn antenna and the standard reference for gain measurement at microwave frequencies.

 waveguide a x b        aperture a_H x a_E
   ____         ___________
  |____|=====> |           |  ===> beam
  TE₁₀ mode    |___________|
           flare in both planes, length L

Working: the TE₁₀ wave in the guide spreads gradually through the flare, which matches the waveguide impedance to free space (377 Ω). The aperture field (uniform in E-plane, cosine in H-plane) radiates a directive beam along the axis.

Design

• Path-difference (phase error) limits for an optimum horn: δ ≈ 0.25λ (E-plane), δ ≈ 0.4λ (H-plane), with δ ≈ a²/(8L).
• Both flares must give the same throat point so the horn can be built (physical realizability).

Performance (optimum horn)

Gain       G ≈ 7.5 a_E a_H / λ²   (η ≈ 0.6)
HPBW (E) ≈ 56 λ/a_E deg,  HPBW (H) ≈ 67 λ/a_H deg

Advantages: high gain (15–25 dBi), wide bandwidth, low VSWR, simple, accurately predictable gain. Applications: gain standard, feed for reflectors, radar, microwave test benches.