# Why is the linear phased array pattern a sinc (visual interpretation)?

It has for sure a very obvious answer, but I must admit I find it very difficult to visualize how a linear phased array can focus the beam with a sinc pattern. I'm not talking about the mathematical proof of the array factor shape, which can be easily found everywhere on the web, but just the intuitive analysis.

Let's consider the following picture:

It says the main beam of the sinc array factor points along the array normal direction. It appears to justify this by saying the single radiating elements spherical wavefronts are aligned in such a direction. However, I do not understand that. Why is the main beam direction orthogonal to the horizontal line connecting the single spherical wavefronts?

Will there be another constructive interference along the following direction in blue?

Why are there the side lobes? How can we visualize their cause?

• Why is the main beam direction orthogonal to the horizontal line connecting the single spherical wavefronts? Have a look at Huygens Fresnel wavelets, and then come back with a more researched question Jan 23, 2022 at 10:30
• Will there be another constructive interference along the following direction in blue? Yes, if the antenna spacing is around one wavelength. That's why in a normal phased array, antenna spacing is maintained well below one wavelength. You see strong reinforcement in many directions in optical diffraction gratings, where the pitch of the grating ruling is often many optical wavelengths. Jan 23, 2022 at 10:41
• @Neil_UK can you explain me how can we deduce which is the strongest direction of interference once we have sketched the wavefronts? Do we sketch all the tangent lines to each point of them and then? All the people tell me it is obvious, but none told me which is actually the procedure. Jan 23, 2022 at 11:03
• Form each antenna, sketch the circle of each phase repeat, so every 360 degrees, and join all those with tangents. This will give you all of the diretions from an optical diffraction grating. However, with a properly proportioned phased array, the antenna spacing will be such that the tangents to the same phase are the only ones you can draw practically. For why Sinc, brush up on your Forruier transforms of rectangular windows. Jan 23, 2022 at 15:20
• A square lens will focus a beam of light to a sinc() pattern by delaying the wavefront using a thickness of glass. If you tilt the lens you will deflect that sinc(). The phased array simply delays each emitter by the same amount at each point as the lens thickness would, so it does the same thing as the lens. Jan 23, 2022 at 19:33

Why is the main beam direction orthogonal to the horizontal line connecting the single spherical wavefronts?

If the antenna feeds are synchronous (behavior identical to a diffraction grating with "linear" slots), it is obvious that the wave fronts will strengthen in the direction perpendicular to the antenna grating (because the phase differences are obviously zero in this direction case and there is therefore reinforcement of the "fields", "constructive" interference).

One should try first with only 2 sources to understand ...

https://www.animations.physics.unsw.edu.au/light/interference/index.html#4.5

But not only ...

If another direction is such that the wavelets also add (waves in phase), there will also be a lobe in this direction.

So, a linear array of antennas can emit in any direction. Just adjust the phases of the different antennas...

From "Balanis_Antenna_Theory_ Analysis and Design",

A great application of the "linear antennas" network is ... the circular "linear antennas" network ... used as VOR in aviation, where the phase is proportional to the angle of radiation. And which makes it possible to generate a "rotating" electromagnetic field ... quite as is done for an electric "motor".

https://en.wikipedia.org/wiki/VHF_omnidirectional_range#/media/File:D-VOR_PEK.JPG

If you want intuition, it's nice to play with a model. Try this Ripple Tank Simulation from Falstad.com. This has been set up with just 4 elements at a λ/2 spacing.

You can see how:

• Orthogonal to the array, the signals all reinforce each other.
• At an angle off the center (orthogonal) axis, there is a null.
• Further off-angle, the signal shows up again but it is less strong.

Then try this:

• Right-click in the wave tank and select "add point source" to add yet another source into the array.
• Click and drag the point source into the array. Your spacing doesn't have to be perfect, just get it about the same spacing by eye.
• Repeat a few more times to take this 4-element array up to a 6- or 8-element array.
• Observe how the main beam gets narrower, and there are more nulls. (This is now a tighter sinc function.)

You can also drag the elements to a wider spacing -- if you stretch them out to 1λ spacing you will see that strong sideways signal you mentioned. (You can hover your mouse to see the wavelength and element position then drag the element to the proper position. I did not have the patience to do this for the 1λ spacing but it's how I set up the λ/2.)

If you are looking for intuition, I believe there is nothing better than running a simulation and adjusting it by hand to observe the effects. Play around with this for a while, it should be a big help to your intuitive understanding.