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[Min Yun]

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Date: Fri, 17 Mar 2000 14:03:33 -0500 (EST)
From: Eric Keto <keto at dogstar.harvard.edu>
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Min,

To answer your question I will try to explain why uniform 
coverage and high resolution result in the same configuration. 
The concept is not difficult, but I find my e-mail explanations 
are usually not as clear as I would like.

The high resolution of course comes from the longest baselines.
Suppose one wanted a multi-element interferometer with only
long baselines and no short baselines to achieve the
highest sensitivity at the highest resolution. How would one
design it? It would seem that no matter the location of the 
antennas, there are always more shorter baselines than 
long ones. Try this with 4 antennas.

However, Tim Cornwell, myself, and others before and after have
shown that one can have a nearly flat distribution of baselines
lengths. By this I mean nearly uniform UV coverage, or
roughly equal numbers of baselines of all lengths between
some minimum and some maximum. For a ring-like configuration,
either a circle or Reuleaux triangle, the maximum baseline
is the diameter, and the minimum baseline is the circumference
divided by the number of antennas. It would appear that this
flat distribution is about the limit that one can achieve in
terms of the least number of shorter baselines and the
largest number of longer ones. There is an explanation in terms
of the auto-correlation function in the first column of 
page 849 in my paper (1997 ApJ 475 843). This explanation 
discusses the possibility of making a centrally evacuated 
(the opposite of condensed) distribution for earth-rotation 
synthesis. To repeat that explanation in 3 sentences: 
The UV distribution is the auto-correlation function of 
the antenna pattern. The auto-correlation function can 
be visualized as the overlap of the antenna pattern with 
itself at different positions.  Starting with the two 
copies of the pattern one on top of the other, it seems 
unlikely that there is any shape whose overlap with itself 
increases as the separation increases.

Dick Thompson pointed out to me that if one has a ring of some 
finite thickness, the auto-correlation function or UV distribution
does have some rise close to the boundary or longest baselines
due to the thickness of the ring. So this is one shape whose
auto-correlation function does increase at least momentarily with 
separation. Of course the thicker one makes the ring, the greater 
is the increase of the auto-correlation function at the shortest 
lags or separations. 

In conclusion, if one poses the design criterion as either 
the most long baselines for the least number of short ones, which is
to say the highest sensitivity at the maximum angular resolution, or 
one asks for the most uniform distribution of baselines, then in 
either case we arrive at the ring configurations of which the
Reuleaux triangle has some advantages.

The hard edge of the UV distribution of any of the ring configurations
does cause some annoying ringing in the image plane. Even though one 
would suppose that the ringing artifacts could be removed by an 
image processing algorithm designed to do so, it is reasonable to 
ask whether some shaping of the edge of the UV distribution should 
be done. It is a particularly interesting question in that the 
angular resolution is more a function of the flatness of the UV 
distribution (given the discussion above) while the ringing is more 
a function of the edge. For example, we know that the famous prolate 
spheroidal wave function, like the Gaussian, has no sidelobes in 
either Fourier or image space, but the PSWF also has the minimum 
extent or width in both spaces simultaneously. This means that a 
UV distribution with this form could be a nice compromise between 
angular resolution and power in the main beam. What does this mean 
for the configuration? If one plots the eigenvalues of the 1D prolate 
spheroidal wave function one finds a flat distribution for about 2/3 
of the extent with a smooth roll-off approximately in the shape of a 
tangent function in the final third. If sidelobe level is a criterion, 
and low sidelobe level is certainly one desirable feature, then perhaps 
a configuration similar to the PSWF would be appropriate. One probably 
does not need to worry too much about the exact shape of the PSWF since 
the finite number of antennas will make the UV distribution approximate 
in any case. But a goal of a UV distribution which is 2/3 flat and 1/3 
roll-off might be achievable. 

In our study for the SMA, and in my ApJ paper, we did not fully 
investigate the options for shaping the edge of the distribution, 
though it was very briefly discussed in the paper. The SMA has enough 
antennas to approximate a Reuleaux triangle as opposed to a straight 
sided triangle, but not have enough antennas to do any significant 
shaping of the UV edge. It might be the case that ALMA does have 
enough antennas to allow one to do so. 

-Eric Keto


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