Thinned array curse

# Thinned array curse

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Encyclopedia
The thinned array curse is a theorem in electromagnetic
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

theory of transmitter
Transmitter
In electronics and telecommunications a transmitter or radio transmitter is an electronic device which, with the aid of an antenna, produces radio waves. The transmitter itself generates a radio frequency alternating current, which is applied to the antenna. When excited by this alternating...

s. It states that a transmitting aperture which is synthesized
Aperture synthesis
Aperture synthesis or synthesis imaging is a type of interferometry that mixes signals from a collection of telescopes to produce images having the same angular resolution as an instrument the size of the entire collection...

by a coherent phased array
Phased array
In wave theory, a phased array is an array of antennas in which the relative phases of the respective signals feeding the antennas are varied in such a way that the effective radiation pattern of the array is reinforced in a desired direction and suppressed in undesired directions.An antenna array...

of smaller apertures that are spaced apart will have a smaller minimum beam spot size (typically, the main lobe
Main lobe
The main lobe, or main beam, of an antenna radiation pattern is the lobe containing the maximum power. This is the lobe that exhibits the greatest field strength....

has a solid angle
Solid angle
The solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...

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 Robert L. Forward cites use of the term in unpublished Hughes Research Laboratories
Hughes Research Laboratories
HRL Laboratories , was the research arm of the Hughes Aircraft Company. Its dedicated research center was established in 1960 in Malibu...

reports dating from 1976.

## 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 microwave
Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...

s 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 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 proportionally to the diameter of the synthesized array squared), but the power density at the ground is unchanged.

Thus:
1. 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).
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
Side lobe
In antenna engineering, side lobes or sidelobes are the lobes of the far field radiation pattern that are not the main 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 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.

## 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 a 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 comprises multiple individual transmitters, some of which fail, the power lost from the main lobe will exceed the power of the lost transmitter, because power will be also be diverted into the side lobes.

The thinned array curse has consequences for microwave power transmission and wireless energy transfer
Wireless energy transfer
Wireless energy transfer or wireless power is the transmission of electrical energy from a power source to an electrical load without artificial interconnecting conductors. Wireless transmission is useful in cases where interconnecting wires are inconvenient, hazardous, or impossible...

concepts such as solar power satellites; it suggests that it is not possible to make a smaller beam and hence reduce the size of a receiver (called a rectenna
Rectenna
A rectenna is a rectifying antenna, a special type of antenna that is used to directly convert microwave energy into DC electricity. Its elements are usually arranged in a multi element phased array with a mesh pattern reflector element to make it directional...

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
Interstellar probe
An interstellar probe is a space probe that has left—or is expected to leave—the Solar System and enter interstellar space, typically defined as the region beyond the heliopause...

(an application of beam-powered propulsion
Beam-powered propulsion
Beam-powered propulsion is a class of aircraft or spacecraft propulsion mechanisms that use energy beamed to the spacecraft from a remote power plant to provide energy...

), can be found in Robert Forward's paper "Roundtrip Interstellar Travel Using Laser Pushed Lightsails."