Example
Consider 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 microwaves at a spot on the ground. Now, suppose you separate these (but keep them mutually phased) so as to synthesize a larger aperture (such as a radio telescope 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 proportionally to the diameter of the synthesized array squared), but the power density at the ground is unchanged.
Thus:
- The array is radiating the same amount of power (since each individual sub-aperture making the array radiates a constant amount of power whether or not it is adjacent the next aperture).
- It has the same power per unit area at the center of the receiving spot on the ground.
- 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 lobes.
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 factor 1 - a/A.
Note that the thinned array curse applies only to 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.
Read more about this topic: Thinned-array Curse
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