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{{Short description|Theorem in electromagnetic theory of antennas}}

The '''thinned array curse''' (sometimes, ''sparse array curse'') is a theorem in electromagnetic theory of transmitters. It states that a transmitting aperture which is synthesized by a coherent array of smaller apertures will have a smaller minimum beam spot size (typically, a main lobe of a solid angle that is smaller by an amount proportional to the ratio of the area of the synthesized array divided by the total area of the individual apertures), but the amount of power that is beamed into this main lobe is reduced by an exactly proportional amount, so that the total power density in the beam is constant.

{{Infobox theorem|name=Thinned-array Curse|type=electromagnetic theory of antennas|field=electromagnetic|first stated date=1976|first stated by=[[Robert L. Forward]]|statement=A transmitting antenna which is synthesized from a coherent phased array of smaller antenna apertures that are spaced apart will have a smaller minimum beam spot size.}}

The '''thinned-array curse''' (sometimes, '''sparse-array curse''') is a theorem in [[electromagnetic radiation|electromagnetic]] theory of [[antenna (radio)|antenna]]s. It states that a transmitting antenna which is [[Aperture synthesis|synthesized]] from a coherent [[phased array]] of smaller antenna apertures that are spaced apart will have a smaller minimum beam spot size, but the amount of power that is beamed into this [[main lobe]] is reduced by an exactly proportional amount, so that the total [[power density]] in the beam is constant.<ref>{{Cite journal|last1=Glennon|first1=Eamonn P|last2=Dempster|first2=Andrew G|last3=Aboutanios|first3=Elias|date=2018-07-07|title=Distributed Beamforming Architectures: Taxonomy, Requirements & Synergies|url=https://www.ignss2018.unsw.edu.au/sites/ignss2018/files/u80/Papers/IGNSS2018_paper_29.pdf|journal=International Global Navigation Satellite Systems Association|volume=IGNSS Conference 2018|pages=11}}</ref>

Consider the case that of a number of small apertures that are mutually adjacent to one another, so that they form a filled aperture array. Now, supose you separate these (but keep the mutually phased) so as to synthesize a larger aperture (that is, like a [[radiotelescope]] array). The spot size on the ground is reduced in size proportionally the diameter of the synthesized array (and hence the area reduced proportionate to the diameter of the synthesized array squared), *but the power density at the ground is unchanged.*

The origin of the term is not clear. [[Robert L. Forward]] cites use of the term in unpublished [[Hughes Research Laboratories]] reports dating from 1976.<ref>T. R. O'Meara, ''The Thinned Array Curse Theorems,'' Hughes Research Laboratories, unpublished internal report, Malibu CA Dec. 1976</ref><ref>W. B. Bridges, ''Looking at the Thinned Array Curse from a Slightly Different View,'' Hughes Research Laboratories, unpublished internal report, Malibu CA April 1976</ref>

==Example==

Consider the case that of a number of small sub-apertures that are mutually adjacent to one another, so that they form a filled aperture array. Suppose that they are in orbit, beaming [[microwave]]s at a spot on the ground. Now, suposesuppose you separatehold theseconstant the number of sub-apertures and the power emitted by each, (but keepseparate the sub-apertures (while keeping them mutually phased) so as to [[aperture synthesis|synthesize a larger aperture (that is, like a [[radiotelescope]] array). The spot size on the ground is reduced in size proportionally to the diameter of the synthesized array (and hence the area is reduced proportionateproportionally to the diameter of the synthesized array squared), *but the power density at the ground is unchanged.*

Thus:

1.# ItThe array is radiating the same amount of power (since theeach individual sub-aperturesaperture making the array don'tradiates carea constant amount of power whether they'reor adjacentnot tooit is adjacent the next aperture or not).

2.# It has the same power per unit area at the center of the receiving spot on the ground.

3. # The receiving spot on the ground is smaller.

From these three facts, it is trivial to now derive the factclear that if the synthesized aperture has an area ''A'', and the total area of thisit that is filled by active transmitters is ''a'', then onlyat most a fraction ''a''/''A'' of the radiated power reaches the target, and the fraction (1 - ''a''/''A)'' is lost. This loss shows up in the form of power in [[side lobeslobe]]s.

2. It has the same power per unit area at the center of the receiving spot on the ground.

This theorem can also be derived in more detail by considering a partially filled transmitter array as being the superposition of a fully filled array plus an array consisting of only the gaps, broadcasting exactly out of phase with the filled array. The [[Interference (wave propagation)|interference]] pattern between the two reduces the power in the main beam lobe by exactly the factor 1 - ''a''/''A''.

3. The receiving spot on the ground is smaller.

Note that the thinned array curse applies only to [[mutual coherence (physics)|mutually coherent]] sources. If the transmitting sources are not mutually coherent, the size of the ground spot does not depend on the relationship of the individual sources to one another, but is simply the sum of the individual spots from each source.

From these three facts, it is trivial to now derive the fact that if the synthesized aperture has an area A, and the total area of this that is filled is a, then only a fraction a/A of the radiated power reaches the target, and the fraction (1-a/A) is lost. This loss shows up in the form of power in side lobes.

==Consequences==

The thinned array curse means that while synthesized apertures are useful for narrow-beam receivers with high angular resolution, butthey are not useful for power transmitters. It also means that if ana filled array transmitter has gaps between individual elements, the main lobe of the beam will lose an amount of power proprotionalproportional to the area of the gaps. Likewise, if a transmitter comprisingcomprises multiple individual individual transmitters, hassome subaperturesof which have fail, the power lost willfrom notthe merelymain lobe will equalexceed the power of the lost transmitter, butbecause power will also havebe andiverted equalinto amountthe ofside power lost from the beamlobes.

The thinned array curse has consequences for [[microwave power transmission]] and [[wireless energy transfer]] concepts such as [[solar power satellite]]s; it suggests that it is not possible to make a smaller beam and hence reduce the size of a receiver (called a ''[[rectenna]]'' for microwave power beaming) by phasing together beams from many small satellites.

A short derivation of the thinned array curse, focusing on the implications for use of [[lasers]] to provide impulse for an [[interstellar probe]] (an application of [[beam-powered propulsion]]), can be found in Robert Forward's paper "Roundtrip Interstellar Travel Using Laser Pushed Lightsails."<ref name="Forward 1984 pp. 187–195">{{cite journal | last=Forward | first=Robert L. | title=Roundtrip interstellar travel using laser-pushed lightsails | journal=Journal of Spacecraft and Rockets | publisher=American Institute of Aeronautics and Astronautics (AIAA) | volume=21 | issue=2 | year=1984 | issn=0022-4650 | doi=10.2514/3.8632 | pages=187–195 | bibcode=1984JSpRo..21..187F | citeseerx=10.1.1.1079.9524 }}</ref>

==See also==

*[[Radiation pattern]]

==Notes==

A good reference to the thinned array curse, focussing on the implications for use of lasers to provide impulse for an interstellar probe, can be found in [[Robert Forward]]'s paper "Roundtrip Interstellar Travel Using Laser Pushed LIghtsails<ref>Robert L. Forward, "Roundtrip Interstellar Travel Using Laser Pushed LIghtsails," ''J. Spacecraft and Rockets, Vol. 21,'' No. 2, Mar-Apr 1984, pp. 190.</ref>."

{{Reflist}}

==References==

{{Refbegin}}

<references />

*The general theory of phased array antennas, from which the thinned- array curse can be derived, can be found in [http://www.ece.rutgers.edu/~orfanidi/ewa/ch18ch19.pdf Chapter 819] of Sophocles J. Orfanidis, ''Electromagnetic Waves and Antennas'' (electronic version accessed MayJuly 2220, 20072009).

*See also Constantine A. Balanis: “Antenna Theory, Analysis and Design”, John Wiley & Sons, Inc., 2nd ed. 1982 {{ISBN|0-471-59268-4}}

{{Refend}}

{{DEFAULTSORT:Thinned-Array Curse}}

*The general theory of phased array antennas, from which the thinned-array curse can be derived, can be found in [http://www.ece.rutgers.edu/~orfanidi/ewa/ch18.pdf Chapter 8] of Sophocles J. Orfanidis, ''Electromagnetic Waves and Antennas'' (electronic version accessed May 22, 2007)

[[Category:Interferometry]]

[[Category:Electromagnetic radiation]]