How is horn antenna gain calculated?

G = e·4πA/λ², where A is the aperture area and e the aperture efficiency. A pyramidal horn with dimensions optimised for its length has an aperture efficiency of about 51%. Achieving 20 dBi at 10 GHz needs an aperture of about 13 cm × 11 cm.

What exactly is optimised in an “optimum horn”?

The aperture dimensions that give maximum gain for a given length. Widening the aperture adds area but also increases phase error across it, and beyond a point the gain falls. The boundary is a phase error of 3/8 wavelength in the H-plane and 1/4 wavelength in the E-plane.

How much gain is lost by shortening a horn?

With the same aperture, cutting the length to two thirds loses about 2.4 dB and halving it loses about 5.6 dB. Doubling the length gains about 1.5 dB, with aperture efficiency approaching about 72%.

Horn Antenna — Theory, Gain Calculation, Optimal Design, and CAE Simulation

Category: Electromagnetic Field Analysis | Revised 2026-09-30

Pyramidal horn antenna gain and radiation pattern simulation with aperture phase error visualization

Conceptual diagram of aperture electric field distribution and gain calculation for a pyramidal horn antenna

Theory: flaring the mouth of a waveguide

Overview

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In an antenna test chamber there was a trumpet-shaped antenna used as the reference. Why are horns used as references?

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Because it's simple and its gain can be calculated quite accurately from its dimensions. A horn gradually widens the mouth of a waveguide to launch waves smoothly into space. Unlike resonant antennas it works over a wide bandwidth with low reflection, so “standard gain horns” serve as references for determining other antennas' gain by comparison.

Gain and aperture efficiency

$$ G = e_{ap}\,\frac{4\pi A}{\lambda^2}, \qquad A = a_1 b_1 $$

$A$ is the aperture area and $e_{ap}$ the aperture efficiency. With a uniform, in-phase aperture field the efficiency would be 1, but a real horn loses efficiency for two reasons:

Amplitude taper: the fundamental TE10 waveguide mode has a cosine field distribution across the H-plane, which alone limits efficiency to $8/\pi^2 \approx 0.81$.
Phase error: the path from the horn apex to the aperture is longer at the edges than at the centre, so the edge field lags in phase and partly cancels in the boresight direction.

Phase error and the optimum horn

$$ s = \frac{W^2}{8\lambda\rho}, \qquad s_H = \frac{3}{8},\quad s_E = \frac{1}{4} \;\;(\text{optimum}) $$

$W$ is the aperture width, $\rho$ the distance from apex to aperture, and $s$ the phase lag at the aperture edge in wavelengths. Widening the aperture at fixed length increases area but also $s$, so gain peaks at some width. The peak occurs at $s = 3/8$ in the H-plane and $s = 1/4$ in the E-plane, giving:

$$ a_1 \approx \sqrt{3\lambda\rho_h}, \qquad b_1 \approx \sqrt{2\lambda\rho_e} $$

Coffee Break Trivia

The horn that opened the microwave era

Serious work on horn antennas began in the 1930s, developing alongside waveguide experiments at places such as Bell Labs. In 1965 Penzias and Wilson at Bell Labs found a faint noise coming from every direction with a large horn antenna built for satellite communication. They even cleaned out pigeon nests looking for the cause, but it remained: it was the cosmic microwave background, the afterglow of the early universe. Only because a horn has low noise and calculable characteristics could they notice such a small excess.

Worked examples

Checking efficiency by integrating the aperture

The aperture field was taken as a cosine distribution in the H-plane and uniform in the E-plane, with quadratic phase error in both, and integrated numerically. With no phase error the efficiency is 0.811 ($8/\pi^2$); at the optimum-horn condition it is 0.514, reproducing the familiar “about 51%”. Changing the horn length with the same aperture gives:

Horn length (relative to optimum)	Aperture efficiency	Gain change
2×	0.723	+1.48 dB
1.33×	0.627	+0.86 dB
Optimum (1×)	0.514	0 dB
0.67×	0.295	−2.42 dB
0.5×	0.140	−5.64 dB

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An “optimum horn” maximises gain for a given length; it doesn't maximise efficiency. If you have room, simply lengthening it with the same aperture gains more than 1 dB. Shorten it to fit an enclosure and gain drops sharply. So in design you fix the allowable length first, then choose the aperture to match.

Designing a 20 dBi horn at 10 GHz

With $\lambda = 30$ mm, a target gain of 100 (20 dBi) and aperture efficiency 0.514, the required aperture area is about 0.0139 m². Using the optimum aspect ratio (E-plane width about 0.81 of the H-plane), the aperture is about 13.1 cm × 10.6 cm. The apex-to-aperture distances in the H- and E-planes both come out at about 19 cm, close enough that the horn can actually be built.

$$ R_{ff} = \frac{2D^2}{\lambda} $$

The far-field distance for this horn is about 1.2 m. Measure gain with the antennas at least that far apart, or near-field phase effects will make the gain read low.

Simulation and measurement

Simulation setup

Excitation: place a waveguide port at the feed and excite the TE10 mode. Leave a short straight waveguide section before the flare so higher-order modes decay first.
Symmetry: a TE10-fed pyramidal horn has two symmetry planes. Using a perfect-electric-conductor plane and a perfect-magnetic-conductor plane appropriately gives a quarter model and a quarter of the cost.
Radiation boundary: keep absorbing boundaries (such as PML) at least a quarter wavelength from the aperture. Far fields are obtained by transforming near fields on a surface inside.
Mesh: element size around 1/10–1/20 of a wavelength, finer at aperture edges and feed discontinuities.

Verification

If simulated gain agrees with the aperture-efficiency estimate within about 0.5 dB, the model is broadly sound. For larger differences, suspect the port mode definition, boundaries too close, or the far-field transformation surface. In measurement, the two-antenna method (two identical horns facing each other) or gain comparison against a standard gain horn is used.

Coffee Break Trivia

Corrugations inside a horn

Look inside the horns of satellite links or radio telescopes and you may see rows of fine grooves. These corrugated horns use grooves about a quarter wavelength deep to make the field small near the wall, equalising the E- and H-plane patterns and lowering cross-polarisation — which raises efficiency when the horn feeds a reflector. Once comfortable with a plain pyramidal horn, simulating how the pattern changes with groove depth is a good way to see the appeal of periodic structures.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Assuming aperture efficiency of 1	Gain overestimated by about 3 dB	Use about 0.51 for an optimum horn
Widening the aperture without lengthening	Phase error lowers gain	Choose length and aperture together
Measuring closer than the far-field distance	Gain reads low	Separate by at least $2D^2/\lambda$
Absorbing boundary too close	Pattern and gain distorted	Keep at least a quarter wavelength away
Ignoring higher-order waveguide modes	Asymmetric patterns	Keep a straight feed section

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I'd like to try other frequencies and sizes.

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Use the antenna gain simulator for aperture versus gain and the antenna pattern simulator for radiation patterns. For link range, try the Friis transmission simulator and the link budget calculator. The difference between gain and directivity is covered in antenna gain and directivity, and reflectors for higher gain in parabolic reflectors.

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