Critical frequency and sensitivity

Dense aperture arrays consist of many small receiver elements. The size and distance between these receiver elements determines the critical frequency of the system, which is the frequency at which the array performs best. For a dense array, that implies the elements are connected, and their size is half the critical frequency. In math notation: frequency = c/wavelength = c/(2*width), with c the speed of light.

For example, take a system with elements that are 10cm wide. All the elements are connected, so the spacing between two elements is the same as their width. The critical frequency of the system is then 20cm, equivalent to 1500MHz.

The sensitivity goes as the collecting area squared, in other words: given a fixed element size, the number of elements determines the sensitivity. More elements makes the telescope larger, and thus more sensitive. For a reasonable telescope, you need about 100.000 elements. Assuming 100x1000 elements at 10cm spacings this is a collecting area of 1000 square meters.

But what happens if you start playing with these numbers? This is where things become interesting. Let's assume that you keep the number of elements fixed to 100.000, but you increase the element width to 20cm. Now the critical frequency is the one corresponding to 40cm, or 750MHz.

The impact on the sensitivity is enormous. The size of the array doubles in both directions, 100x1000 elements at 20cm results in 4000 square meters, 4 times larger. The sensitivity of the telescope goes as the collecting area squared, so this increases by a factor of 16! By merely making the elements twice as large, an astronomer can observe the same point-source 4 times faster.

The big caveat is that this only holds for the frequencies below the critical frequency. At higher frequencies the receivers behave as a sparse array. This means that the only way to increase the sensitivity at higher frequencies is to add more receiver elements, which will increase the costs of an array.

Figure 1. The point-source sensitivity (expressed in RMS noise level) of a dense aperture array with two different critical frequencies (f), but the same number of elements. N denotes the number of stations used for the layout of the instrument. A station is equivalent to a dish for a classical interferometer telescope such as VLA and WSRT. The figure illustrates the enormous impact of changing the critical frequency, while the number of stations has limited impact. Copyright: Ilse van Bemmel, ASTRON.

From this mental exercise it also becomes obvious why the elements need to be packed as dense as possible. Spacing the 10cm wide elements at 20cm apart would lower the critical frequency to 750MHz, but the collecting area is still only 1000 square meters. Now the sensitivity at the lower frequencies has marginally improved, but the higher frequencies are now 16 times less sensitive. In the figure above, this is equivalent to shifting the blue and green curves horizontally to the left.
 
Lesson learned: the critical frequency of a dense aperture array is an important driver for the sensitivity as a function of frequency, but needs to be carefully balanced against the number of elements. The sensitivity of the instrument needs to be assessed over the full frequency range to optimize a future telescope for multiple scientific purposes.