Figure 1 shows the radial Fourier plane distributions which are obtained with a ring array and a filled array. The central peak in the case of the ring array is inevitable for a large number of antennas. The radial Fourier plane distribution of the ring array is basically uniform, as the central peak does not cover very much of the 2-D (u,v) area. Hence, we justify approximating the ring array as a uniform Fourier plane coverage. The filled array gives a centrally condensed Fourier plane coverage which we approximate as a function which decreases linearly with (u,v) distance.
I have simulated (u,v) data with abstract Fourier plane distributions, ie, Fourier coverages which are not associated with any physical distribution of antennas, and investigated some of the properties of these Fourier plane distributions. The radial profiles of uniform and centrally condensed (linearly decreasing) Fourier plane distributions are shown in Figure 2. To make the comparison more fair to the uniform Fourier distribution case, we stretched the centrally condensed distribution by 30% so the naturally weighted beam would be the same size as for the uniform distribution. (If we make the maximum baseline of the two distributions the same, rather than the resolution, the relative increase in SNR of the centrally condensed Fourier plane distribution upon tapering is exaggerated.)
Other moderately centrally condensed Fourier plane distributions have been studied and give similar noise curves after normalizing the width of the distribution to give the same natural resolution. I made simulations of 28,000 random (u,v) points which were consistent with these distributions. On a 128 by 64 Fourier plane grid, essentially every cell within the maximum (u,v) distance had one or more (u,v) samples for each of these distributions. These simulated (u,v) data sets were gridded with natural weighting, tapered to the desired resolution, and Fourier transformed to yield the synthesized beam. Untapered robust and uniform weightings were also investigated. I explored tapers as large as 4 times the full resolution beam. (Larger tapers won't be used often, as they provide the same resolution as the next smaller array configuration, but much less efficiently.)
The two features of the synthesized beam which we study here are the amount of sensitivity lost in the tapering, and the level to which the main lobe of the synthesized beam conforms to a Gaussian. The extent to which the beam mimics a Gaussian is important to imaging and analysis algorithms but is not fundamental. Since almost every radio astronomical researcher convolves their clean components or maximum entropy images with a Gaussian beam, we will assume that a Gaussian beam is desirable. If the synthesized beam falls off faster than a Gaussian, the sensitivity on the longer baselines is wasted as we convolve with a Gaussian with wider wings. On the other hand, a highly centrally condensed Fourier plane distribution, such as is obtained from the VLA, causes a synthesized beam with very broad wings which results in imaging problems since most of the array sensitivity is on very short spacings.
Figure 3 illustrates the point source sensitivity loss as a function of taper for the uniform and centrally condensed distributions. Consistent with intuition, the centrally condensed distribution does not lose sensitivity with taper as quickly as the uniform distribution. At 4 arcsec taper (almost a factor of 2 lower resolution than naturally weighted), the uniform distribution has 16% higher noise than the centrally condensed case; at 6 arcsec taper, the uniform distribution has 21% higher noise than the centrally condensed case. The upturned spurs at the high resolution end of Figure 3 are the robust and uniformly weighted results. While the centrally condensed Fourier plane distribution suffers much in sensitivity when robust or uniform weighting are used, it is able to achieve much higher resolution than its naturally weighted case. With uniform Fourier plane coverage almost no increase in resolution and little loss of sensitivity occur.
Figure 4 shows the fractional difference between the integrals of the synthesized beam and its best fit Gaussian, down to the 0.10 level of the Gaussian. Both uniform and centrally condensed distributions show significant departures from Gaussian beams at the highest resolution, but the uniform distribution produces much higher deviations. When the Fourier distribution is highly tapered, both beams are very similar to a Gaussian, the uniform distribution being a few tenths of a percent away from Gaussian and the centrally condensed distribution being about 1% away from Gaussian. The nearly Gaussian beams which result upon tapering indicate that the relative noise improvements of the centrally condensed Fourier plane coverage over the uniform Fourier plane coverage are not at the expense of imaging quality.
What is the optimal Fourier plane distribution? Assuming a Gaussian beam in the final image is desired to aid in the interpretation and analysis of the image, it is clear that a uniform Fourier distribution is NOT optimal. Figure 5 shows cuts through the uniform distribution's beam (very nearly a J1 Bessel function) and its ``best fit'' Gaussian. The Gaussian has much broader wings than the uniform distribution's beam, or the uniform distribution has too many long baselines to yield a nice beam. If the uniform coverage is tapered to produce a beam which is more Gaussian, about 30% of the sensitivity is lost. While I can't say at this point which Fourier plane distribution is optimal, I can say that the uniform distribution is sub-optimal with respect to resulting beam shape and noise behavior upon tapering.
All of our reasoning has been based on the assumption that we have so many antennas and (u,v) samples that every cell which needs to be sampled can be sampled, and we are then dealing with the issue of where the extra (u,v) samples should go. The conclusions we draw for the 40 element MMA, for which this will often be true, will be quite different from the conclusions that might be drawn from a 6 element instrument, which is really stretching to fill as many cells as possible. A typical MMA field observed with the D array will be filled with complicated structure and requires complete (u,v) coverage. The centrally condensed Fourier plane distribution which results from the requirement of maximum surface brightness sensitivity in the D array also meets the complete (u,v) coverage requirement. In C array, either uniform or the linear distributions will usually fill most cells, permitting good imaging of complex objects. If typical fields observed by the MMA's larger configurations are not filled with complicated structure, but are splattered with regions of complicated structure, we may relax the goal of putting a (u,v) sample in absolutely every cell, and we may instead ask where we should put the (u,v) samples to do the most good.