Thinned-array curse

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.

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.*

Thus:

1. It is radiating the same amount of power (since the individual sub-apertures making the array don't care whether they're adjacent too 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 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 synthesized apertures are useful for narrow-beam receivers, but are not useful for power transmitters. It also means that if an filled array has gaps between individual elements, the main lobe of the beam will lose an amount of power proprotional to the area of the gaps. Likewise, if a transmitter comprising multiple individual individual transmitters has subapertures which have fail, the power lost will not merely equal the power of the lost transmitter, but will also have an equal amount of power lost from the beam.

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."

References

• The general theory of phased array antennas, from which the thinned-array curse can be derived, can be found in Chapter 8 of Sophocles J. Orfanidis, Electromagnetic Waves and Antennas (electronic version accessed May 22, 2007)