Fresnel zone: Difference between revisions

A '''Fresnel zone''' ({{IPAc-en|lang|f|r|eɪ|ˈ|n|ɛ|l}} {{respell|fray|NEL|'}}), named after physicist [[Augustin-Jean Fresnel]], is one of a series of confocal [[prolate spheroid|prolate]] [[ellipsoid]]al regions of space between and around a transmitter and a receiver. TransmittedThe primary wave will travel in a relative straight line from the transmitter to the receiver. Aberrant transmitted radio, sound, or light waves which are transmitted at the same time [[Multipath propagation|can follow slightly different paths before reaching a receiver]], especially if there are obstructions or reflectingdeflecting objects between the two. The two waves can arrive at the receiver at slightly different times and willthe beaberrant slightlywave may arrive out of phase with the primary wave due to the different path lengths. Depending on the magnitude of the phase shiftdifference between the two waves, [[Wave interference| the waves can interfere constructively or destructively]]. The size of the calculated Fresnel zone at any particular distance from the transmitter and receiver can help to predict whether obstructions or discontinuities along the path will cause significant interference.

In any wave-propagated transmission between a transmitter and receiver, some amount of the radiated wave propagates off-axis (not on the line-of-sight path between transmitter and receiver). This can then [[Reflection (physics)|deflect]] off of objects and then radiate to the receiver. However, the direct-path wave and the deflected-path wave may arrive out of [[Phase (waves)|phase]], leading to [[destructive interference]] when the phase difference is a [[half- an odd integer]] (<math>{(2z+1)/2, z \in \mathbb Z}</math>) multiple of the [[frequency|period]]. The n-th Fresnel zone is defined as the locus of points in 3D space such that a 2-segment path from the transmitter to the receiver that deflects off a point on that surface will be between n-1 and n half-wavelengths out of phase with the straight-line path. The boundaries of these zones will be ellipsoids with foci at the transmitter and receiver. In order to ensure limited interference, such transmission paths are designed with a certain clearance distance determined by a Fresnel-zone analysis.

The dependence on the interference on clearance is the cause of the [[picket-fencing]] effect when either the radio transmitter or receiver is moving, and the high and low signal strength zones are above and below the receiver's [[Cut-off (electronics)|cut-off threshold]]. The extreme variations of signal strength at the receiver can cause interruptions in the communications link, or even prevent a signal from being received at all.

Fresnel zones are seen in [[optics]], [[radio]] [[telecommunication|communications]], [[electrodynamics]], [[Banana Doughnut theory|seismology]], [[acoustics]], [[gravitational radiation]], and other situations involving the radiation of waves and [[multipath propagation]]. Fresnel zone computations are used to anticipate obstacle clearances required when designing highly directive systems such as [[Microwave transmission|microwave]] [[parabolic antenna]] systems. Although intuitively, clear line-of-sight between transmitter and receiver seemsmay seem to be all that is required for a strong antenna system, but because of the complex nature of radio waves, obstructions within the first Fresnel zone can cause significant weakness, even if those obstructions are not blocking the apparent line-of-sight signal path. For this reason, it is valuable to do a calculation of the size of the 1st, or primary, Fresnel zone for a given antenna system. Doing this will enable the antenna installer to decide if an obstacle, such as a tree, is going to make a significant impact on signal strength. The rule of thumb is that the primary Fresnel zone would ideally be 80% clear of obstacles, but must be at least 60% clear.

Fresnel zones are confocal [[prolate spheroid|prolate]] [[ellipsoid]]al shaped regions in space (e.g. 1, 2, 3), centered around the line of the direct transmission path (path AB on the diagram). The first region includes the ellipsoidal space which the direct line-of-sight signal passes through. If a stray component of the transmitted signal bounces off an object within this region and then arrives at the receiving antenna, the [[phase shift]] will be something less than a quarter-length wave, or less than a 90º shift (path ACB on the diagram). The effect regarding phase-shift alone will be minimal. Therefore, this bounced signal can potentially result in having a positive impact on the receiver, as it is receiving a stronger signal than it would have without the deflection, and the additional signal will potentially be mostly in-phase. However, the positive attributes of this deflection also depends on the polarization of the signal relative to the object (see the section on polarization below).

The 2nd region surrounds the 1st region but excludes it. If a reflective object is located in the 2nd region, the stray sine-wave which has bounced from this object and has been captured by the receiver will be shifted more than 90º but less than 270º because of the increased path length, and will potentially be received out-of-phase. Generally this is unfavorable. But again, this depends on polarization. Use of same [[Circular_polarization#Reflection|circular polarization]] (explainede.g. belowright). in both ends, will eliminate odd number of reflections (including one).

Although radio waves generally travel in a relative straight line, fog and even humidity can cause some of the signal in certain frequencies to scatter or bend before reaching the receiver. This means that objects thatwhich are clear of the line of sight path will still potentially block parts of the signal. To maximize signal strength, one needs to minimize the effect of obstruction loss by removing obstacles from both the direct radio frequency [[Line-of-sight propagation|line of sight]] (RF LoS) line and also the area around it within the primary Fresnel zone. The strongest signals are on the direct line between transmitter and receiver and always lie in the first Fresnel zone.

In the early 19th century, French scientist [[Augustin-Jean Fresnel]] discoveredcreated a method to calculate where the zones are — that is, whether a given obstacle will cause mostly in-phase or mostly out-of-phase deflections between the transmitter and the receiver.

==Clearance calculation==

The cross sectional radius of each Fresnel zone is the longest at the midpoint of the RF LoS, shrinking to a point at each vertex (which coincide with the antenna on each end, withinbehind the approximation made)antennas.

To obtain the radius <math>r_n</math> of zone <math>n</math>, note that the volume of the zone is delimited by all points for which the difference in distances, between the directreflected wave (<math>D=d_1\overline{AP} +d_2 \overline{PB}</math>) and the reflecteddirect wave (<math>\overline{AP} D=d_1+ \overline{PB}d_2</math>) is the constant <math>n\frac{\lambda}{2}</math> (multiples of half a [[wavelength]]). This effectively defines an ellipsoid with the major axis along <math>\overline{AB}</math> and foci at the antennas (points A and B). So:

:<math>\sqrt{d_1^2+r_n^2}+\sqrt{d_2^2+r_n^2}-D(d_1+d_2)=n\frac{\lambda}{2}</math>

Assuming the distances between the antennas and the point <math>P</math> are much larger than the radius, expandingand applying the roots[[binomial inapproximation]] seriesfor andthe retainingsquare theroot, <math>\sqrt{1+x} \approx first1+x/2</math> two(for terms''x''≪1), the expression simplifies to:

:<math>\frac{r_n^2}{2}\left(\frac{1}{d_1}+\frac{1}{d_2}\right)=\approx n\frac{\lambda}{2}</math>

:<math>r_n=\approx\sqrt{n\frac{d_1\ d_2}{D}\lambda},\quad d_1, d_2 \gg n\lambda,</math>

For a satellite-to-Earth link, <math>d_2\approxit D</math>further andsimplifies to:<ref name="Braasch 2017 pp. 443–468">{{cite book | last=Braasch | first=Michael S. | title=Springer Handbook of Global Navigation Satellite Systems | chapter=Multipath | publisher=Springer International Publishing | location=Cham | year=2017 | isbn=978-3-319-42926-7 | doi=10.1007/978-3-319-42928-1_15 | pages=443–468}}</ref>

:<math>r_n\approx \sqrt{n d_1 \lambda},\quad d_1 \gg n\lambda,\quad d_2\approx D</math>

Note that when <math>d_1=0</math> or <math>d_2=0\implies r_n=0</math>, which implies that the foci seem to coincide with the [[vertex (curve)|vertices]] of the ellipsoid. This is not correct and it's a consequence of the approximation made.

Setting the point <math>P</math> to one of the vertices (behind an antenna), it's possible to obtain the error <math>\epsilon</math> of this approximation:

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*[http://www.tapr.org/ve3jf.dcc97.html VHF/UHF/Microwave Radio Propagation: A Primer for Digital Experimenters] {{Webarchive|url=https://web.archive.org/web/20180216110440/http://www.tapr.org/ve3jf.dcc97.html |date=2018-02-16 }}