Antenna arrays

The waves

from the antennas (the dots), in their plane: crests red, troughs cyan

Drag an element to move it; double-click to add one. Slide θ₀ in the panel to swing the beam, or click an element to set its own strength and phase.

Far-field pattern

power in each direction; −10 dB is a tenth of the beam’s

See also: Radio interferometer, the same array run backwards, receiving, to make a picture of the sky.

Experiments

Start here

Eight antennas in a row send out waves that add up into one beam. The polar plot on the right shows how much power goes in each direction.

Slide θ₀ in the Phases panel: the beam swings round with nothing moving but the timing of the elements. Drag an element to move it, double-click in the picture to add one, or change N and the spacing.

Look for the beam narrowing as you add elements, and for the smaller lobes beside it (the sidelobes). A radio telescope array is this run backwards: see Radio interferometers after the experiments.

Antenna arrays

The elements lie in the plane of the picture; lengths are in wavelengths. Each element's colour is its phase (a larger phase means a later signal), and its size its strength. Angles $\theta$ are measured from straight up, positive to the right. The numbers under the polar plot are measured from the pattern as you change things.

1. A line of eight makes a beam, and the phases steer it

The picture the page opens with: eight elements half a wavelength apart, each lagging its left-hand neighbour by $kd\sin\theta_0$ with $\theta_0=20°$, so that their waves arrive in step in that direction. Slide $\theta_0$: the beam follows, with nothing moving, as in a radar's phased array.

Look at the numbers under the polar plot. The beam is 13.6° wide between its half-power points; the highest sidelobe (the smaller lobes beside the beam; the red dot marks it) is 12.8 dB down, a twentieth of the beam's power; and the directivity is $D=8$: towards the beam the waves add to $8^2$ times one element's power, while the total power sent out is just 8 times one element's. Slide $\theta_0$ to 0: the beam points straight up (broadside), 12.8° wide. The polar plot also has a beam pointing down: a line of elements can't tell one side from the other, so that mirror image is not counted as a sidelobe.

2. Two elements: the waves add one way and cancel another

Two elements half a wavelength apart, in step. Straight up and down, their waves arrive together and add; along the line they arrive half a period apart and cancel. The pattern is $|AF|^2=4{\cos}^2(\tfrac{\pi}{2}\sin\theta)$: broadside, 60° wide.

Now make the right-hand element lag by half a period, and the beams turn to point along the line

Then put the elements a quarter wavelength apart, with a lag of a quarter period To the right, the lag makes up for the shorter path and the waves add; to the left, path and lag add up to half a period and the waves cancel. Two isotropic elements, and the power goes one way: $4{\cos}^2(\tfrac{\pi}{4}(1-\sin\theta))$.

3. More elements in the line: a narrower beam

Eight elements $\lambda/2$ apart, in step ($\theta_0=0$). The beam is 12.8° wide and the first sidelobes are 12.8 dB down. Change $N$: the beam narrows as $1/N$, close to $0.886\,\lambda/Nd$ radians (6.36° for sixteen), and the first sidelobe stays near −13 dB (−13.26 dB for many elements). The directivity is $D=N$: in the beam the waves add to $N^2$ times one element's power, while at this spacing the total power is just $N$ times one element's.

4. Steering by phase: the beam widens as it swings

Sixteen elements, each lagging its left-hand neighbour by $\alpha=kd\sin\theta_0$, so that their waves arrive in step in the direction $\theta_0$. Slide $\theta_0$ and the beam follows, with nothing moving: a phased array, as in radar. Away from broadside the beam widens, roughly as $1/\cos\theta_0$, because the array looks shorter from the side: sixteen elements give 6.36° at 0°, 7.35° at 30° and 13.0° at 60°. To swing the beam back and forth, as a radar does:

5. Grating lobes: a second beam when the spacing is too wide

Twelve elements 0.8λ apart. At $\theta_0=0$ there is one beam (and its mirror image). Steer to 30°: a second beam, as strong as the first, appears at $\sin\theta=\sin\theta_0-\lambda/d$, here −48.6°. The red triangles mark these grating lobes, at $\sin\theta=\sin\theta_0+m\lambda/d$; one comes into view once $d>\lambda/(1+|\sin\theta_0|)$, which is 0.667λ at 30°. The array samples its aperture too coarsely, and the pattern aliases, as a signal sampled too slowly does. Arrays that must steer all the way to ±90° keep $d\le\lambda/2$. At a spacing of 2λ there are grating lobes at every $\sin\theta=m/2$

6. Tapering: quieter sidelobes, a wider beam

Sixteen elements in step, all at full strength: the first sidelobes are 13.1 dB down. Now weaken the elements towards the ends, a taper (the buttons under Amplitudes). Hann: sidelobes at −31.5 dB, and a beam 9.72° wide instead of 6.36°. Dolph–Chebyshev: every sidelobe exactly at the level you choose (−30 dB), with a 7.98° beam, the narrowest any taper allows for that level; Taylor's taper comes close (−30.0 dB, 8.05°). Binomial: no sidelobes at all, and a 15.7° beam. An aperture with sharp edges has a Fourier transform that rings; tapering smooths the edges, and pays for quieter sidelobes with a wider beam and less directivity (16 uniform, 13.8 Chebyshev).

7. Dipoles: the element's own pattern multiplies the array's

Four short dipoles, lying along the line. A dipole sends nothing along its own axis: one element's pattern (dotted) is ${\sin}^2$ of the angle to its axis. The whole pattern (solid) is that times the array factor (dashed), the pattern the same layout of isotropic elements would have: pattern multiplication. Turn the dipole axis to 90°, across the line: the dipoles' nulls now point straight up and down, just where the array factor puts its beam, and in this plane the beam is gone. What is left is two lobes at ±50°, and the directivity towards them is only 0.31. The beam hasn't vanished: it points out of the screen, which the plane doesn't show. For dipoles the picture shows the magnetic field, which is perpendicular to the plane; it is exact at every distance, near field included.

8. A random sparse array: no grating lobes, a floor of sidelobes

Twenty-four elements scattered over 14λ, no two closer than 0.6λ, phased to send their waves straight up. The beam is as narrow as the array is wide: 3.71°. A regular array this sparse would have grating lobes. Here the waves that would have made them arrive with random phases, and spread into a floor of sidelobes averaging about $1/N$ of the beam's power (−13.8 dB for 24; −12.9 dB for this layout), with peaks above it. Press Scatter them again for another layout. Low-frequency radio telescopes such as LOFAR are built like this.

9. Focusing in the near field: the waves meet at a point

The phases $\varphi_n=-k|\vec r_f-\vec r_n|$ delay each element by its distance from the cross, so that every wave arrives there in step: the waves converge on it and spread out again past it. This is time reversal: record what a source at the cross would send to each element, and play it backwards. Drag the cross. The spot is about $\lambda/(\sin\alpha_1+\sin\alpha_2)$ from its middle to its first dark points, where $\alpha_1$ and $\alpha_2$ are the angles out to the array's two ends: never much less than $\lambda/2$. Its brightest point is half a wavelength nearer the array than the cross, because the nearer waves are stronger (the focal shift). Move the focus beyond about $D^2/\lambda$ and it fades into an ordinary beam. Ultrasound scanners focus like this.

How it works

Waves from several sources add up. Far away, in the direction $\hat r$, the waves from elements at $\vec r_n$ with complex amplitudes $a_n$ add up to the array factor $$AF(\hat r)=\sum_n a_n\,e^{-ik\hat r\cdot\vec r_n},\qquad k=\frac{2\pi}{\lambda}:$$ the Fourier transform of the layout, on the wavevectors of length $k$.

The picture is the field at one instant, $\mathrm{Re}\,\psi e^{-i\omega t}$, where $\psi$ adds up the elements' complex waves (for isotropic ones, $a_n e^{ik|\vec r-\vec r_n|}/k|\vec r-\vec r_n|$): crests red, troughs cyan. The polar plot is the far-field power pattern, in decibels down to −40 dB (or in plain power). Below it are the beam's direction, its width between the half-power points, the highest sidelobe, and the directivity $D=4\pi U/P$: the power per solid angle towards the beam, compared with the average over the whole sphere (radiation out of the plane counts too), also given in dBi, decibels above an isotropic radiator, $10\log_{10}D$.

What is simplified. The elements are ideal point sources or short dipoles, each radiating as if the others weren't there (no mutual coupling), with no ground. The pattern is drawn in the plane of the elements only.

New in 2026 for BrainFlux. Source: github.com/gallicchio/brainflux, MIT licence.