Radio Interferometry and Aperture Synthesis

Transcription

1 Radio Interferometry and Aperture Synthesis

2 Phil gave a detailed picture of homodyne interferometry Have to combine the light beams physically for interference Imposes many stringent conditions on the instrument Heterodyne receivers retain phase information about the photons in the IF signal Interference can be achieved using just the electronic signals It can even be achieved by recording the signals and combining them later in some different place We ended the discussion of radio telescopes by pointing out that it was impossible to achieve high resolution images by making bigger telescopes The ease and flexibility of interferometry solve this problem

3 Because heterodyne signals can be combined electronically with any phase delay we want, many of the stringent conditions for homodyne interferometry disappear. We no longer need optical delay lines, nor do we have to follow the conditions for coherent combination of the light from the entrance apertures:

4 Here is a radio interferometer: There are two telescopes separated a distance B. Their outputs are combined in the correlator. If the signal voltages are V S1 and V S2 and the noise voltages are V R1 and V R2, then the correlator output is The simplification is possible because the noise terms average to zero. If we expand the extreme right hand expression and simplify, we find = R C is defined as the cosine response. As θ varies, the correlated signal has a sinusoidal behavior on the sky, i.e., the response is in the form of fringes. However, the response is also subject to the diffraction-limited beam of the individual telescopes, which defines the primary beam of the interferometer. If we define a beam as half the fringe period, its diameter is

5 The situation we have just described is shown in (a). If we add more telescopes, their outputs can be combined to yield N(N-1)/2 baselines. Thus, the field patterns (b) and (d) yield rapidly improving images (c) and (e). Remember that these sharper images are just in the direction along the interferometer; perpendicular to this direction, the image is just the primary beam of the telescopes. That is, the image is long and thin.

6 We derived the cosine response, R C, but there is another term with equal power in the sine response, R S. This signal is captured by dividing the outputs of the telescopes and adding a phase shift. A correlator that does this is called a complex correlator and the complex visibility is V= R C j R S = A e i with amplitude A = (R 2 C + R S2 ) and phase = tan -1 (R S /R C ) and (the van-cittert Zernike theorem) 2 i b s c V( b) R ir I ( s) d / C S e where b is the baseline. These illustrations (from R. Perley) may help illustrate. To the left is R C and to the right R S ; above is a long baseline and below a short one. The envelope is the primary beam characteristic.

7 An illustration of how adding visibilities reproduces the source image: where is the source direction relative to the line perpendicular to the plane on which the interferometer operates. This follows from the vc-z Theorem. Above is the effect of adding the visibilities for equally spaced baselines on a point source. To the right is the same effect for visibilities of a rectangular profile source (from R. Perley).

8 To make a better image we need to rotate the interferometer. We think this in terms of the uv plane, defined as a coordinate system perpendicular to the direction toward the source. As the interferometer is rotated (a), it yields a corresponding a set of baselines extending in all directions around the origin of the UV plane (b). By combining the results from all these baselines, we can synthesize a highresolution beam.

9 But how are we going to rotate a row of heavy, large radio telescopes? We let the earth do the work. Of course, this means that the rotation is in only one specific pattern, so typically the uv plane coverage will not be in the circles we would like, but will be in an ellipse, and the ellipse will be interrupted where the earth blocks the view of the source.

10 The uv plane coverage for a linear interferometer is far from ideal. Well, at the celestial pole, it is the circle we would like, but at the equator it collapses to a line and close to the equator it is not much better. As a result, the synthesized beam for a source near the equator is just the width of the primary beam in one direction.

11 To fix this issue, the JVLA antennas are placed in a Y configuration (left). One observation gives the baselines indicated in the middle, and a full track on a source at the equator gives the ensemble of baselines to the right. The imaging of the interferometer can be modified by changing the spacing of its elements; note the different spacings along the arms of the Y. The Y is a popular pattern, but others have their uses, and ALMA can move the antennas to give a variety of patterns depending on the science goals.

12 The Y shape of the VLA gives good coverage at all declinations (except far south).

13 Recent upgrades make interferometers even more powerful 1.) conveying the IF signals by optical fibers (rather than, say, waveguides) greatly increases the bandwidth 2.) Receivers, particularly at high frequencies, have been improved, including features such as sideband separation 3.) and of course, ALMA

14 ALMA has 66 antennas at 5000 m elevation (very low water vapor), each with extremely high performance mm- and submm-wave receivers (up to ten sets, most likely initially from 90 to 850 GHz). At 1mm (300 GHz) the resolution is up to 0.02.

15 The future (?): interferometric arrays of many small telescopes Current-generation interferometers have relatively few, large antennae. For example, the VLA has 27 antennae, each 25 meters in diameter. The result is that the primary beam, over which the VLA images, is relatively small (l/d for the antenna size, about 0.5 degree at 21 cm). With cheaper correlators the electronics that link the antenna signals it is possible to have more, smaller antennae. The Allen Telescope Array (top) is to have meter telescopes, so 4 times the field of the VLA with about the same collecting area. The Square Kilometer Array (not funded) is planned to have thousands of antennae. The issue is that the number of correlators goes as N(N-1), where N is the number of antennae, so the SKA would be prohibitively expensive with today s technology.

16 And the ultimate resolution through combination of signals from telescopes around the world in Very Large Baseline Interferometry (VLBI). Of course, even greater resolution would be possible with an antenna in orbit or on the moon.

17 However, we have to remember the fundamental limitations of interferometers: missing low spatial frequencies and image artifacts. To start with low spatial frequencies, for example the best JVLA images of M33 are missing 85% of the total flux from the galaxy (around 1 GHz). Here is the VLA image at 6 cm (~5 GHz):

18 Here, to the left, is a filled dish image with the Effelsberg 100m telescope (~ 2 GHz). The image to the right is at 160 microns. It does not match the VLA image at all but corresponds closely to the Effelsberg one. The far IR appears to arise from the same regions as the low surface brightness extended emission that is invisible to the VLA.

19 Image artifacts arise because the uv plane is not covered in full. The gaps in coverage produce structure in the image of a point source that departs significantly from the ideal Airy pattern: This image is a snapshot obtained in a single observation without the benefit of extended tracking and earth rotation. The sidelobes reach 20% of the peak response. Full track observations improve on this PSF, but there are still significant artifacts. These untreated images are called dirty

20 The standard way to remove the artifacts is with CLEAN 1.) assume that the image can be approximated by a field of point sources; 2.) locate the position of the brightest point in the dirty map; 3.) subtract a scaled version of the dirty beam from this position; the subtraction should account for only a modest fraction of the brightness at this point; 4.) Record the position and subtracted intensity in a CLEAN component file; 5.) Find the brightest position in the dirty map left from the subtraction; 6.) repeats steps 3.) 5.) until no subtraction is possible without making part of the dirty map negative.

21 The above discussion assumes that the phases of the signals are completely unmodified by the atmosphere. In general, this is not correct, particularly at high frequencies. The closure phase can be used to mitigate this issue. From the diagram, and if the error phases introduced by the atmosphere are ε 1, ε 2, and ε 3, and the phase differences from a celestial source are, e.g., Φ 12, then the observed phases, e.g. Φ 12 are: (9.24) And in the sum of these three observed phases, the error terms cancel! By modeling the closure phases, it is possible to deduce the source structure independent of the atmospheric effects.

22 Self-calibration is another useful approach. If there is a point source in the primary beam that is bright enough to be detected at reasonable signal to noise in a coherence time (the time when the atmospheric effects can be taken to be constant), then corrections can be bootstrapped from the imaging of this source. Another class of problem occurs when a bright source lies outside of the official primary beam, but it is SO bright that its artifacts contaminate the observations even if they are observed at low efficiency. These artifacts are not dealt with using CLEAN because there is no point source within the image to associate them with. To deal with this issue, observations often include a quick excursion to the bright source periodically to obtain an image of it under the conditions that prevail at that time, allowing its artifacts to be removed accurately. Nonetheless, because of the constantly changing atmospheric effects, reduction of deep radio interferometric images is very challenging.

23 CLEAN intrinsically should not work so well for extended sources. Nonetheless, if the sources are only a few beams in size, it can be used successfully. For more extended sources approaches based on the maximum entropy method (MEM) are used. This is a general example of deconvolution. For simplicity, we discuss the problem in one dimension, e.g., for a spectrum. All the principles are the same for images (2D), but the math is more complex. We describe the true spectrum as (l) (a function of wavelength, l), the observed one as O(l), and the instrumental line profile (analogous to the PSF) as P(l). Then O(l) is the convolution of (l) and P(l): Is usual it is easier to work with the convolution in Fourier space, in which case we get a restored image (corrected for smoothing represented by P(l)) by dividing the transform of the observed image by that of the PSF and inverse transforming:

24 In principle, the effect of the beam can be corrected by dividing the observed MTF by that of the PSF. In practice this does not work well: The deconvolved image can include physically unrealistic negative sources Noise is amplified unacceptably near the MTF cutoff The abrupt cutoff in the reconstructed frequency spectrum results in strong ringing

25 The Wiener Filter applies a weighting dependent on the signal to noise as a function of spatial frequency: This suppresses ringing and controls the amplification of noise. However it is not clear that the resulting reconstruction is unique, or even the best possible. The deconvolution method with the best claim to be unique or best is the Maximum Entropy Method (MEM). It gives the smoothest solution that is consistent with the data. There are many alternatives. Here is a comparison of CLEAN, a method known as nonnegative least squares, and MEM. Note the strange scalloped structures with CLEAN missing in the other two.

26 Another issue in radio interferometry is how to account for the missing flux that would be seen with a filled aperture telescope. Total power obtained from a single dish telescope can be: Added in uv plane (MIRIAD: invert). Single dish image must be Fourier transformed to create simulated uv coverage Feathered with an interferometer image after both images are made (AIPS++: image.feather, MIRIAD: immerge). IF there is sufficient uv overlap between interferometer and single dish data (VLA+GBT, OVRO/BIMA+IRAM, ATCA+Parkes). Used as a starting model in deconvolution (AIPS++: imager makemodelfromsd with subsequent clean). Model created from a single dish image is used as an initial model during deconvolution. The model is improved where uv coverage overlaps.

27 An example from Debra Shepherd (Summer Synthesis Imaging Workshop 2006): VLA mosaic of central region, 9 fields. Deconvolved with MEM in AIPS resolution. Feathered GBT+VLA mosaic using AIPS++. Image looks pretty but fidelity (quality) is low due to disparate 90 and 8.4 resolutions. GBT+VLA mosaic GBT image input as a model and then deconvolved with multi-scale CLEAN. Final image fidelity significantly better.