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Focus in chaos

In open space, waves can be focused, for example by a parabola. The 2D numerical simulation below shows an omnidirectional emitter on the right and a wire-frame parabola on the left. The parabola focuses the neat wave emitted by the emitter. This produces the bright magenta spot in front of it. (The four colors green, magenta, yellow and cyan code the wave phase, the brightness codes the wave amplitude.)

parabolic antenna focusing a wave

The purpose is to put a receiving antenna at that focus point (the bright magenta spot). That way, being located in a zone where the wave is strong, the receiving antenna will have a strong signal. In the picture below, the antenna in the magenta spot gets a much stronger signal than the other one, despite the fact that they are located at the same distance from the emitter:

parabola focusing on an antena

Below is a calculation of the wave echoing in a virtual room with metallic pillars. In-between he pillars, there is no more neat wavefront that can be focused by a parabolic reflector:

waves resonating in a room with pillars

This is a problem, for example for devices like WiFi receivers or cell phones, that are typically used inside buildings or amongst dense city structures. They cannot focus the radio waves using a parabola...

But there is a solution, using "phased elements" like in high-tech radar systems.

In order to increase the number of persons that will be able to follow, here is an analogy, to make the phase thing more palpable:

It is being told that in 212 BC, Archimedes helped Syracuse defeat the Roman navy. He made the soldiers that defended Syracuse use polished bronze shields to focus the Sun towards the Roman ships. The shields acted like a solar furnace and put the ships on fire one by one. Many historians now rather the believe that the bronze mirrors were only used to blind the sailors and that the ships were put on fire by conventional means, like incendiary bombs thrown by catapults.

But let's stick to the idea that the ships were indeed put on fire by the focused sunlight and that each soldier along the shore turned his mirror accurately to direct the sun rays towards the same spot on a ship (a magenta spot...)

Now suppose that the Romans had arrived on a cloudy day and there was no visible sun... Archimedes still could have disturbed them using soldiers and their shields placed along the shore. He could have asked each soldier to push his shield violently into the water to emit a shockwave on the surface of the sea. One shockwave, sent by one soldier, would have been inoffensive when it reached a ship a hundred meters away. But what if a hundred soldiers pushed their shields in the water... All the small shockwaves arriving altogether on one ship would sum up and form a little tsunami. That would have made a real strong splash on the side of the ship and if at that same time the general leading the invasion was dining aboard the ship and a few droplets fell on his table, that would have ruined his fighting spirit, lead him to nervous breakdown and make the conquest of Syracuse abort.

But... in order for this psychological warfare to operate properly, the shockwaves need to arrive on the target at the same time; in order to unite. There is a problem here, because the soldiers along the shore are not all at the same distance from the ship. If they all plunge their shields in the water at the same time, say when a drum is hit, the shockwaves will start at the same time but they will arrive on the ship at different times, hence there will be no effect.

Each soldier must be told to wait a given time after the drum hit, before plunging his shield in the water. The closer a soldier is to the ship, the longer he has to wait. Let's call this delay a "time shift".

To target a different enemy ship, situated at a different position, the time shifts will be different.

Now let's suppose that Archimedes asked the soldiers to do something different. Rather than plunge their shields in the water one sudden time, they have to oscillate the shields back and forth in the water at a steady frequency. Push the shield, then pull it back, then push it again, then pull it back again, push, pull, push-pull... This creates a continuous wave on the water surface.

Each push and pull (a "period") has to span one second. A drummer is asked to hit his drum once a second, so that the soldiers can be synchronized.

That won't do... because the waves generated by the soldiers do no arrive on the target ship "in phase". When the wave generated by one soldier pushes water towards the flank of the ship, the wave generated by another soldier pulls the water away from the ship. The two waves cancel each other out... On the average most waves will cancel each other out and the enemy ship will be only slightly disturbed.

For this to work, each soldier needs to adapt his "phase", in such a way that when all the waves arrive on the target, they are "in phase" (all pulling or all pushing). To achieve this, Archimedes can give one of these two instructions to each soldier:
So, each soldier will have the appropriate "phase shift" to have the waves arrive "in phase" on the target.

This is what radars that use phased elements do, in order to have their radar beam directed towards a given target.

It works in the opposite direction too. Suppose that there is a strong mist over the sea. The enemy fleet cannot be seen but Archimeds wants to know where an enemy ship is situated anyway. Suppose one persons aboard the ship falls into the water, which triggers a shockwave that will ultimately reach the shore. When a soldier along the shore sees the shockwave, he rises his hand. Archimedes sees when each soldier rises his hand and immediately feels where the shockwave was generated, hence where the enemy ship is. And if no Roman falls into the water, Archimedes still can generate a strong shockwave from the shore. A part of it will bounce back on the ship and the soldiers can raise their hands when the reflection arrives back. The time shift for each hand is the clue to the position of the ship.

Radars rather use a continuous wave and rely on the phase shift...

And by the way, the wave that bounces back and arrives on the shore, is ways to weak for a soldier to notice it, because the sea is naturally undulating with waves. But modern electronics manage to detect the weak signal anyway, "by summing up the observations of all soldiers and over some timespan".

Now back to our pillars. A set of antennas can be placed somewhere between them:

antenas between pillars

They are wired towards a central device that makes the sum of their signals, in order to get one single and stronger signal:

The problem is that each antenna receives the wave with a different phase. For example, the  lowest antenna on the image and the one left of it, are situated in a zone with the same wave phase (cyan color...) Summing up their signals will yield a stronger signal. But other antennas are in zones with different phases, hence simply summing up all their signals will yield a poor result. A phase shift element has to be inserted for each antenna, in order to have the signals arrive with the same phase in the summing device:

Each phase shift element has to be tuned. This can be done by software. One antenna can be chosen as reference, the other ones are inhibited, then each one in turn is activated and its shift element is tuned to get an optimal signal.

The antennas that yield a low signal can then be kept inhibited to decrease the noise.

Practically, this would resemble those WiFi routers that have two antennas at the back. But there would be much more than two antennas. Maybe ten or even hundred. This will make the router have some volume... It would not be practical on a cell phone... But it will still be practical in many cases and the volume will decrease over a few decades, when shorter wavelengths are being used.

Phased elements used in radars are top technology and can be very expensive. But for the application discussed here, rudimentary phase shifters are sufficient and their cost is negligible. It's like the pistons of a trumpet, that change the pitch of the instrument by changing the length of the resonating tube. Two "pistons" would be enough to appropriately change the phase of a signal. Each "piston" would be a few SMD components...

One advantage of parabolic antennas is that the same frequency can be used by different pairs of communicating antennas in the same array. Each antenna only "sees" the one it has to and will not receive the signal from another emitter. Or to the least, the signal from the intended correspondent will be much stronger. This can also be achieved between the pillars. The animation below shows two wave patterns. One is created by an emitter in a given place and the other one by an emitter slightly aside of it. If the set of phase shifters of a receiver is tuned for one antenna, it will not receive the other antenna very well...

different wave patterns between pillars

Reciprocally, any change in the surrounding, even a person moving, can modify the wave pattern and make the phased elements need to readjust. This implies that the choice for the positions of the emitters and the receivers should take into account the probability of being disturbed by movements.

The trick works both ways. An emitter too, can use a set of phased elements, in order to create different wave patterns and privilege one receiver or another between the pillars. Even if the receiver has a single antenna, the emitter can tune its phased elements so that the receiver is in a zone where the wave is strong.

If the elements of the emitter are close enough to each other, then it can achieve some directivity, which in many cases allows to increase the power on the receiver, even if the receiver is not directly seen by the emitter.

If the array with the pillars was completely surrounded by a dense mesh of elements, then it would be possible to create a single zone where the wave is strong; where the receiver is situated. This would be optimal... but if one is ready to place that much elements, then it would probably be more useful to simply place one independent element on each pillar... And if elements are situated everywhere and are always in sight, then better use harmless infrared or visible light modulated at high frequency, rather than radio waves... (A shell of elements can have other applications anyway, like in tomography, 3D manufacturing, memories...)

Overall, this should allow to create less electromagnetic pollution for a same volume of data transfers.

Norwegian translation of this page provided by Globe Views

Eric Brasseur  -  May 8 2012