Conical Horn
How the Conical Flare Shapes the Beam
A conical horn begins where a circular waveguide carrying the dominant TE11 mode opens into a cone. As the cross-section grows, the guide supports the same field distribution but at progressively larger transverse dimensions, so the energy that was confined to the throat spreads smoothly toward the aperture. The transition is gradual enough that very little energy reflects back into the feed, giving the horn a low return loss across a broad band, often better than 20 dB over a full waveguide octave. The flare also slows the rate at which the impedance changes, which is the practical reason horns are far more broadband than abruptly terminated open waveguides.
The defining trade in conical horn design is between aperture size and phase error. Because the wave radiating from the apex of the cone arrives at the aperture plane on a spherical wavefront, the field at the rim lags the field at the center. This quadratic phase error broadens the main beam and lowers the realized efficiency. Designers cap the path-length difference at the aperture edge, conventionally s = δ/λ ≈ 0.4 for an optimum-gain horn, which fixes the relationship between aperture radius and slant length. Push the diameter larger without lengthening the cone and gain actually falls, because the added phase error outweighs the extra collecting area.
For systems that demand equal beamwidths in both planes and very low cross-polarization, the smooth-wall conical horn is extended into a corrugated horn, whose inner grooves force a hybrid HE11 mode. The smooth-wall version remains the most economical choice for general reflector feeds, gain standards, and point-to-point millimeter-wave links where its rotational symmetry is the key advantage.
Governing Equations
G (dBi) ≈ 10 log10[ εap × (πd / λ)2 ] with εap ≈ 0.51
Engineering shortcut (optimum flare):
G (dBi) ≈ 20 log10(d / λ) + 7.0
Aperture phase-error parameter:
s = δ / λ = a2 / (2 λ L) (optimum ≈ 0.375)
Half-power beamwidth (optimum design):
HPBW ≈ 1.15 × (λ / d) × 57.3°
Where d = aperture diameter, a = aperture radius (d/2), λ = free-space wavelength, L = slant length from apex to aperture, δ = center-to-edge path difference, εap = aperture efficiency. Example: d = 60 mm at 30 GHz (λ = 10 mm) → d/λ = 6 → G ≈ 22.6 dBi.
Conical Horn vs. Other Horn Types
| Horn Type | Aperture Geometry | Dominant Mode | Aperture Efficiency | Cross-Pol | Best Application |
|---|---|---|---|---|---|
| Conical (smooth wall) | Circular | TE11 | ≈ 51% | Moderate (-20 to -25 dB) | Reflector feeds, gain standards |
| Corrugated conical | Circular, grooved | HE11 | ≈ 75 to 84% | Very low (< -30 dB) | Satellite & radio-astronomy feeds |
| Pyramidal | Rectangular | TE10 | ≈ 51% | Low in principal planes | Gain standards, linear polarization |
| Sectoral (E/H plane) | Rectangular, 1-axis flare | TE10 | ≈ 0.5 to 0.65 | Plane dependent | Fan beams, slotted-array feeds |
| Diagonal | Square (45°) | TE10+TE01 | ≈ 0.5 | Higher | Compact dual-pol feeds |
Frequently Asked Questions
How do you calculate the gain of an optimum conical horn?
Use G (dBi) ≈ 10 log10[εap × (πd/λ)2] with εap ≈ 0.51 for the optimum flare, or the algebraically equivalent shortcut G ≈ 20 log10(d/λ) + 7.0 dB. A 60 mm aperture at 30 GHz (λ = 10 mm) gives d/λ = 6 and about 22.6 dBi. Enlarging the diameter without lengthening the cone does not raise gain, because aperture phase error then dominates.
What is the optimum flare angle and slant length for a conical horn?
The optimum design fixes the edge phase error at s = δ/λ ≈ 0.375, so the aperture radius satisfies a2 ≈ 0.75 × λ × L, with L the slant length from the apex to the aperture. The half-flare angle typically lands between 15° and 30°: tighter angles need a longer horn for the same aperture, while wider angles raise phase error and drop efficiency below the 51% optimum.
Why is a conical horn preferred over a pyramidal horn for reflector feeds?
The TE11 conical pattern is nearly rotationally symmetric, so its E- and H-plane beamwidths match and it illuminates a circular dish more uniformly than a rectangular pyramidal horn. That symmetry improves reflector aperture efficiency, cuts spillover, and supports circular or dual polarization through circular waveguide. For the lowest cross-polarization and exactly equal beamwidths, the conical geometry is corrugated to carry the HE11 mode.