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Aperture synthesis (interferometry)

Although this stuff is my everyday bread and butter, this is a difficult page to write, which is why it took me so long...

Interferometry is an extremely clever way of overcoming the limited angular resolution of telescopes, which is caused by the so-called "diffraction" of light.

This section is sub-structured into:

Motivation

Radio astronomers were the first to try and find a way around diffraction-limited angular resolution of a telescope, because they were worst-hit by this effect due to the enormous wavelengths they deal with.

It is technically and financially impossible to build single radio telescopes much larger than 100 m diameter. But with a 100-m telescope, at the wavelength of the important neutral hydrogen (HI) emission line of about 20 cm, one has an angular resolution on the sky of only 8.3 arcminutes, which is comparable to the human eye's vision. Not very satisfactory, considering that the typical angular resolution of an optical image is around 1 arcsec (i.e., 500 times better)! Which means that, in order to achieve the same angular resolution at 20 cm wavelength, one would have to build a 50-km radio telescope...

However, the long wavelengths of radio waves can also help: exactly because of this fact interferometry is easiest (although still technically demanding) to achieve with radio waves.

How does interferometry work? Let me try to illustrate this based on first some basic principles and then a practical example, say a real observation with the Australia Telescope Compact Array (ATCA). The ATCA is not quite that as big as the 54 km mentioned above, but 6 km is not bad either. But wait, the information on the ATCA page says there are 6 22-m telescopes; so how can they form a 6-km antenna?

Basics of interferometry

The basics of interferometry can be presented based on experiments with optical light, because the rules of behaviour are the same for light all across the electromagnetic spectrum. An explanation of the diffraction patterns of a single and a double hole blend is given on a separate page.

Interferometry with astronomical telescopes

Scaling the whole setup of the double- (or multi-) slit experiment by orders of magnitude and by taking into account the equivalence of a hole blend and a telescope mirror as a light-collecting aperture, one can transfer the above scenario to the size scale of astronomical telescopes (for the sake of simplicity, since I have these examples available, make it radio telescopes, although the same rules apply all across the electromagnetic spectrum).

Interferometers as multiple double-slit experiments

Each combination of light registered by one telescope with that of another (called a "baseline") is the equivalent of a double-slit interference pattern, or rather double-hole blend, considering that telescopes are normally round. An interferometer array can be treated as a large number of baselines, i.e. either a large number of double-slit experiments, or by combining them, a multi-slit experiment. An Earth-bound interferometer, as seen from the sky, works like a rotating multi-hole blend.

Delay compensation

Trying to make many small telescopes simulate (or "synthesise") one enormously big one, they must form part of the hypothetical surface of that synthesis telescope. Obviously, they can do this only if they are all in one plane (the so-called "uv-plane", which is described further below).

However, if one imagines an array of several telescopes, all looking at the same region on the sky, signals (waves) will arrive at the individual telescopes at different times, just because the geometrical pathlengths from a source on the sky are different (see sketch below).

Geometrical delay, τg, in a two-element interferometer.

By adding a delay to the signal of the telescopes closest to the source so that they appear to arrive at a point (in time) when they can be compared with those received by other antennas at the same time, the telescopes are virtually "brought into one plane", the uv-plane of the synthesis telescope. By adding delays to the incoming signals from the telescopes closest to the source, they are "moved backwards" as far as necessary to line them up with the farthest antenna in the back.

Note that the delay compensation needs to be determined at short time intervals, because the relative distances of the various telescopes to the sources change as they track them across the sky. All telescopes then continue to trace the same wavefront. And keep in mind that being wrong by only 1 nanosecond (10-9 s) leads to an error of 30 cm! Compared to wavelengths between 1 cm and 1 m, typically, one would obviously lose phase coherence for all but the longest wavelengths already. So correct, high-precision delay compensation in an interferometer is very important.

In the optical one uses mirrors on carriages, as shown here. At radio frequencies this is done electronically, in delay units such as those displayed in the photo below.

Delay racks of the ATCA, with the "Delay Unit Control Computer", DUCC, which is an "incredibly powerful" machine (on a large-scale version of this photo one can read the CPU speed from its display of 33 MHz - it's been there since 1987...). There is one rack per antenna (the ATCA has six antennas).

Not really outwardly very imposing, these are the delay racks of the ATCA. In a modern radio telescope basically everything is done under computer control. And even if there is nothing to actively control, computers are used to at least monitor the systems.

The "uv-plane" of an interferometer

Imagine watching the two little holes of a double-slit experiment as shown above, as the plate with the hole blends is turned around slowly by 180 degrees. What will this look like in the end? One will see two half tracks, leading to the creation of a circle (or ellipse, if looked at a bit from the side).

The same happens in a synthesis telescope standing on the surface of the Earth. Just imagine what two telescopes will look like when seen from the sky (after delay compensation, as described above). Because of the Earth's rotation (assuming that the motion of the Earth through space has been taken into account properly), they will be rotating around each other slowly (over the course of 12 hours, or spanning an angle of 180 degrees). Again, a circle or ellipse will be created. Its radius is the projected distance between the two telescopes (called "baseline" length), its width is the diameter of the individual telescopes.

Typical interferometers have many individual telescopes, the outputs of each one of which are compared with every other. This means that lots of double-slit interferograms are recorded at the same time, and thus many little circles/ellipses ("uv-tracks") are created at the same time, thus starting to fill the surface (uv-plane) of the synthesised telescope.

I say "starting to fill" the uv-plane here, because even with a lot of individual telescopes in an interferometer, there are obviously more gaps than filled aperture. Therefore, one typically uses movable antennas that are placed on different stations over time, creating different subconfigurations (i.e., different baseline length or slit separations) over time. It the properties of the object that one observes do not change over time the data can later be combined in a computer and thereby the uv-plane filled better and better as more subconfigurations are added to the existing observations. An example of this is described on this separate page.

Correlation = creation of interferograms

The process of "combining" the dely-corrected incoming signals of one antenna with another is called "signal correlation". If one correlates the signals of two different antennas, the technical term is called "cross-correlation", correlating the output of one antenna with itself is called "auto-correlation". Each cross-correlated pair of incoming signals form an interferogram as shown here. The correlation of signals from pairs of antennas/telescopes in interferometers is normally strongly design-dependent. Which is the reason why the computers performing this task, the so-called "correlators", are usually customised machines that are unique. The ATCA's correlator is visible in the photo below.

The ATCA correlator, which produces the interferograms of each pair of antennas. On each of these boards, interference patterns as sketched here are created and then stored to hard disk.

Basic properties of an interferometer

Angular resolution (resolving power)

The diameter of an interferometer telescope is determined by the longest projected baseline length. And its diameter in turn defines its angular resolving power. The bigger, the better, with the caveat that, in order to recover all the radiation received from the sky, or at least be able to correct properly for the fact that there are still gaps, one needs to fill the uv-plane (almost) completely. Just like a single telescope, an interferometer as a whole is again limited in its ability to see details on the sky (its "angular resolution") by the process of diffraction. An interferometer has a point-spread function similar to the one shown above. Only now it does not depend on the size of the individual antennas, but is the Fourier transform of the distribution of uv-tracks in the uv-plane of the synthesised telescope.

The diffraction limit on the resolving power of telescopes has something to do with the relation between the size of the telescope and the wavelength observed with it. This dependence leads directly to a relation between the resolution and the observing frequency/wavelength. The higher the frequency (shorter the wavelength) of radiation measured with a telescope of given size, the higher its angular resolution (the narrower its point-spread function, i.e. the telescope's response to a point-like source on the sky).

The reason why many find interferometry hard to understand is that there is no intuitive, direct way of imaging. Rather, one needs a mathematical formalism to transform the registered (fringe) patterns into observed light distributions, which is not exactly a straight-forward process.

In addition, what is shown on the page about the basics of interferometry is the so-called "dirty beam" (or uncorrected point-spread function) of the interferometer. Every point source in an image created with this PSF will have that particular shape. But knowing the shape precisely, one can correct for it ("deconvolve" the image) and only then obtain a much better ("cleaner") view of the field-of-view of the telescope. Yet another level of complexity...

Field-of-view of an interferometer

While thus the point-spread function changes with uv-coverage, there is one thing that does not vary in an interferometer, irrespective of the way the uv-plane is filled: its field-of-view. This has always the size of the point-spread function of each individual telescope (assuming that all are identical). This is an interesting feature, because one can always naively assume that by using increasingly large interferometer arrays - and thereby resolving more and more detail on the sky - one is using different levels of zoom like in a tele-lens. However, in an interferometer, while zooming in or out, the size of the field-of-view does not change.

Sensitivity of an interferometer

Another "strange" feature of interferometry is that the sensitivity of a synthesis telescope, i.e. its ability to detect faint sources, does not scale with the number of antennas in the array, but with the number of baselines. In an array like the ATCA, with 6 antennas, the addition of a 7th dish would not increase its sensitivity by a mere 1/6 (or 16.6%), but (if the interference patterns of each antenna with every other in the array are recorded) from 15 to 21 baselines, and hence by 40%! Funny old world, isn't it?

Data structure

The individual recordings of an interferometer, which have so far been called interferograms, are measurements made over a short timespan. Such an individual, short-term recording is normally called a "visibility". This kind of data is not necessarily always transformed into an image. One can also create spectra and photometric series from the raw data. What exactly is done or possible depends on the flexibility of the instrument and on the specific choice of experiment by the observer. A second example of the modern, flexible way of obtaining and regrouping data points is available

here.

East-West arrays

An East-West array, as its name already implies, is a one-dimensional array, in which all antennas are aligned East-West. Such a configuration instantaneously provides high resolution only in the North-South direction. So as to obtain two-dimensional high-resolution imaging, an East-West array depends on the Earth's rotation. Examples of such arrays are the ATCA and the WSRT.

Two-dimensional arrays

Some arrays have been designed to span a two-dimensional plane and thereby be capable of high-resolution imaging instantaneously. An example of a two-dimensional array configuration is the VLA's upside-down "Y" shape.

Note, however, that although arrays such as the VLA are capable of "snapshot" imaging, they normally also make use of the Earth's rotation so as to fill their uv-plane more completely (which improves the quality of the imaging, plus, because of the longer time spent on the target, also the sensitivity of the observations).

Radio interferometer observatories

Links to various radio wavelength interferometers are provided on this page.

Resolving minute details

Returning to the comparison of radio with optical observations (above), it is interesting to note that the VLA in its largest configuration reaches an angular resolution at 20 cm wavelength of 1.5 arcsec, which in fact is similar to that of optical images. The VLA's maximum baseline is about 32 km long.

There are currently two types of interferometry that achieve the highest angular resolution on the sky and thereby allow us to create extremely sharp images of parts of the sky (of strong emitting sources):

These two techniques reach angular resolutions down to tens of microarcseconds, which is a little bit better than 0.5-1 arcsec typical optical seeing or even arcminutes (the typical resolution of a radio single dish at centimetric wavelengths).