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The '''thinned array curse''' (sometimes, ''sparse array curse'') is a theorem in [[electromagnetic radiation | electromagnetic]] theory of [[transmitter]]s. It states that a transmitting aperture which is [[Aperture synthesis|synthesized]] by a coherent [[phased array]] of smaller apertures that are spaced apart will have a smaller minimum beam spot size (typically, the [[main lobe]] ofhas 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.
The origin of the term "thinned array curse" 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 [[Robert L. Forward]] cites use of the term in unpublished [[Hughes Research Laboratories]] reports dating from 1976 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T. R. O'Meara, ''The Thinned Array Curse Theorems,'' Hughes Research Laboratories, unpublished internal report, Malibu CA Dec. 1976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W. B. Bridges, ''Looking at the Thinned Array Curse from a Slightly Different View,'' Hughes Research Laboratories, unpublished internal report, Malibu CA April 1976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 as an example the case of a number of small 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 separate these (but keep them mutually phased) so as to synthesize a larger aperture (thatsuch is, likeas 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 each individual sub-aperture making the array radiates thea sameconstant amount of power without regard to whether they're adjacent to or atnot ait diatanceis fromadjacent the next aperture).
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 clear that if the synthesized aperture has an area A, and the total area of it that is filled by active transmitters is a, then at 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 lobe]]s.
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]] pattern between the two reduces the power in the main beam lobe by exactly the ratiofactor (1 - a/A).
It is worth notingNote that the thinned array curse applies only to [[mutual coherence | mutually coherent]] sources. If the transmitting sources are not mutually coherent, the size of the ground spot does not depended on the relationship of the individual sources to one another, but is simply the sum of the individual spots from each source.
==Consequences==
The thinned array curse means that while synthesized apertures are useful for receivers with high angular resolution, they 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 proportional 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 equallobe will exceed the power of the lost transmitter, butbecause power will be also havebe andiverted equalinto amountthe ofside power lost from the beamlobes.
The thinned array curse has consequences to concepts for [[Microwavemicrowave power transmission]] and [[Wirelesswireless energy transfer]] concepts such as design of a [[Solarsolar power satellite]], in thats; it suggests that it is not possible to make a smaller beam and hence reduce the size of a receiver (orcalled a ''[[Rectennarectenna]]'', for the case of 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]] (that is, an application of [[Beambeam-powered propulsion]], 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>."
==References==
==See also==
*[[Aperture synthesis]]
*[[Radiation pattern]]
{{Uncategorized|date=June 2007}}
[[Category:Interferometry]]
[[Category:Electromagnetic radiation]]
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