Fresnel Zone - a simple presentation
This material was put together to provide details about the Fresnel Zone and to answer the question “Why is the gain of the antenna not taken into consideration in the Fresnel Zone calculations?” The short answer is that the Fresnel Zone calculations will determine what path should be free of obstacles.The path shape changes based on distance and wavelength. The strength of the source is based on energy/watts and antenna gain, which are totally different animals.
A Laser was used to prove the physics part of the electromagnetic wave, since light is both a wave and a particle.
Most of the pictures are from the internet, and the sites that provided the theory are the following:
http://theory.uwinnipeg.ca/mod_tech/node120.html
https://en.wikipedia.org/wiki/Gaussian_beam
https://en.wikipedia.org/wiki/Rayleigh_length
http://zoneplate.lbl.gov/theory
The rest is just common sense backed by education and wisdom on loan from God.
Chapter 1
1.1 Meet the Wave
What is the RF wave?“A wave is a transfer of energy from one point to another without the transfer of material between the two points.”
Figure 1. The wave in its glory
The wavelength is Lambda=Speed of light / frequency
Different frequencies have different wavelengths. For 2.4 GHz the wavelength is 0.125 m, or 12.5 cm. For 5 GHz the wavelength is 0.06 m, or 6 cm.
1.2 The Totally Destructive and Constructive Waves
When two waves meet, their amplitudes are added and the max of amplitudes, either the peak or trough, will generate a totally destructive, constructive or in between wave .
Figure 2. Constructive and destructive waves
When a Laser is reflecting off a surface, whatever surface it’s landing on, the laser spot looks granular. That is because of totally constructive and destructive waves. The dark spots are the result of two waves at 180 degrees shifted to each other. That shift is called phase.
The reason a flash light will not manifest the same granular effect is because white light is a combination of ALL wave lengths in the visible spectrum. Let’s say, if there is a dark spot where the blue wavelengths are totally destructive, other colors will be in full strength, filling up those dark spots.
Figure 3. Granular Laser spot due to totally constructive and destructive waves
Understanding this from a laser perspective it’s easy to see how multiple transceivers are used now by Cisco and other vendors to create the beam-forming technology with totally constructive waves to increase the signal strength from the AP to the client.
Chapter 2
2.1 The Gaussian Beam
Either light or radio waves, all beams have the shape shown in the diagram. There is NEVER a straight beam with parallel rays to infinity, but a curvy beam based on the aperture of the source at the starting point, and the wavelength.
Figure 4. The Gaussian beam
2.2 Rayleigh length
This guy Rayleigh came up with a way to calculate the radius, or waist, of the beam at distance z.
so that in scientific writing/calculator the formula would look like,
w(z)=w0(1+(z/((pi*w0^2)/lambda))^2)^.5
So, let’s say there is a 2.4 GHz dish antenna with a radius of .5m for WIFI, and all the rays are parallel, at 1000 m the WIFI beam will have a radius of 79.62 m. It is a natural thing and it’s based on the radius of the source, the distance and the wavelength.
Maxima application (Windows and Linux OS) did the muscle work, but any scientific calculator should be able to solve this kind of math.
/* [wxMaxima: input start ] */
.5*(1+(1000/((3.14*.5^2)/0.125))^2)^.5;
……………..
79.61940437942519
/* [wxMaxima: input end ] */
Figure 5. 1000 m beam of 2.4 GHz
Chapter 3
3.1 The Fresnel Zone
The weird thing about a beam is not only does it expand, but that it creates rings around the focus point. Look at the picture below.
Figure 6. Natural rings around the center dot
Figure 7. The Theory diagram of Fresnel Zone
Where r can be found out from the formula below
Now it makes sense why the first chapter focused on dark and bright spots of light. The same phenomenon happens to the WIFI transmissions: the center dot will have most energy, then there is the dark ring due to totally destructive waves, and then there is the ring of energy due to totally constructive waves, and so forth.
When there is a design for two towers to communicate with each other, the goal is to have the middle spot cleared of obstacles. So, let’s do some math. Looking at the picture below, the towers are at 2 km apart from each other. At 1000m, or half way, according to the Gaussian Beam, the radius of the beam is 79.62m for 2.4GHz band, but the most energy is actually focused in the middle spot with a radius of 7.9m.
Figure 8. The final picture
Solve([(1*.125/2)=(1000^2+x^2)^.5+(1000^2+x^2)^.5-1000-1000], [x]);
…….
(%o24) [x=7.905755913415231*%e^(1.0*%i*%pi),x=7.905755913415231*%e^(2.0*%i*%pi)]
or the radius = 7.906m.
For n=2, or the first ring,
solve([(2*.125/2)=(1000^2+x^2)^.5+(1000^2+x^2)^.5-1000-1000], [x]);
……..
(%o25) [x=11.18051457894491*%e^(1.0*%i*%pi),x=11.18051457894491*%e^(2.0*%i*%pi)]
or the radius = 11.18m.
And for n=3, or the second ring,
solve([(3*.125/2)=(1000^2+x^2)^.5+(1000^2+x^2)^.5-1000-1000], [x]);
……
(%o26) [x=13.69338486505435*%e^(1.0*%i*%pi),x=13.69338486505435*%e^(2.0*%i*%pi)]
or the radius =13.69m.
So, the area that needs to be cleared of obstacles has a max diameter of 28m.
Conclusion
1. The beam from a perfect antenna with a 0 degree angle of propagation will spread out according to its natural properties. The rate of spreading out is related to the diameter of source, wavelength and the distance.2. Fresnel Zone assumes that the source is a point, not a beam with a radius, that will create a center of energy with rings around it, based on the constructive and destructive wave properties. The size of the center dot and the radius of the outer rings are based on the distance and the wavelength.
3. Fresnel Zone will define the perfect path of the beam of energy, and it doesn't have in mind the loss of the medium. In order to have a successful link between the towers the power of the radios need to overcome the medium loss and to be received above the noise floor with enough energy, so it can be demodulated successfully.
4. Therefore, after the towers are aligned and there are no obstacles in the path defined by the Fresnel Zone, what comes in play are the antenna types, directional basically, with the highest gain ever. Omni directional antennas will work too, but remember that a lot of energy will just boil the clouds, heat the oceans and sterilize the bees.
5. Looking at the example above, out of the area defined by the Gaussian Beam, the energy is concentrated in only 17% of the area. So, if the center dot and the first two rings are blocked by obstacles, then the power has to be increased in the magnitude of hundreds to thousands manifold so that the third, fourth etc. rings of energy will carry enough understandable data.
THE END
About the author
He doesn’t believe that magnets of the speakers installed in the ceilings or walls will absorb or bend the WIFI. Only gravity can do that. And black holes. Or gravity...He doesn’t believe in World War Z.
He believes in being ready when the world will end.
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