(
) is a baseline, 
being its maximum;
the block M is multiplier, I is integrator (time averager), and the
block is phase shifter.
Next: Image Quality Assessment Using the Modulation Transfer Function
Previous: Correcting Some DRAO Synthesis Telescope Wide-Field Imaging Problems
Table of Contents --- Search ---
PS reprint
Nailong Wu
Dept. of Physics and Astronomy, York University,
4700 Keele St., North York, Ontario, M3J 1P3, Canada
In an aperture synthesis radio telescope, a correlator (Figure 1) is used for each baseline to generate a Fourier component of the distribution of power density (brightness) of the field of view in the sky under observation. The map (distribution) is reconstructed by the Fourier transform technique.
The resolvability of a telescope depends mainly on the maximum baseline, although it may be improved to some extent utilizing sophisticated image processing techniques.
In this presentation a new type of interferometer, called the High Order Harmonic Interferometer (HOHI), is proposed to increase the resolvability of a telescope by a factor of n without extending its maximum baseline.
In the following one-dimensional notation is used for simplicity and clarity.
In Figure 1 for the conventional (base or first harmonic) interferometer (n=1)
and Figure 2 for the second harmonic interferometer as an example of HOHI
(n=2), () is a baseline, being its maximum;
the block M is multiplier, I is integrator (time averager), and the
block is phase shifter.
Suppose in the field of view there are K point sources
having amplitudes 
and random phases 
, 
.
The radiations of angular frequency 
from a single point source
arrive at the angle 
with respect to the baseline.
Then the phase difference is 
between the two signals in the left and right branches for the baseline
, where 
is the wavelength.
Denote 
by x and use it as the coordinate. Let 
,
. Use 
(
) for the cosinusoidal signals to facilitate calculations.
Then the two signals are respectively, ignoring any additional phase shift and
amplitude change,
and
where t is the time.
The feasibility of HOHI is shown in the following four subsections. The basic idea, as its name implies, is to generate high order harmonics for each baseline.
The following complex mathematical operations are performed in the n-th order (complex) correlator:
where the angular brackets denote time average and ellipses in
each of exponents 
represent the expression in
the parentheses of its counterpart 
.
Equation (1) can be reduced on the assumption that the point sources
are spatially incoherent, i.e., 
, 
are
independent random phases. For the conventional
correlator, n=1, the result is easily found to be
The reduction of Equation (1) for 
is quite complicated. For the
second order (complex) correlator C
as shown in Figure 2,
the result is,
Note 
in the exponent of the first term in the braces.
We see in Equation (3) that the output from C
consists of the wanted second
harmonic and the unwanted cross terms. The latter can be eliminated
by operations as shown in the following:
The (complex) visibility 
in Equation (2) represents the base or
first harmonic (the spatial frequency being 
)
while 
in Equation (4) represents the second harmonic
(the frequency being 
). In general HOHI of order n
at the baseline 
generates the visibility 
representing the n-th harmonic (the frequency being 
).
In this way the maximum baseline is virtually increases by a factor of n:
.
HOHI of order n includes the correlators of order up to n and the operations to eliminate the cross terms, namely,
The correlator of order n can be constructed following
calculation rules of complex numbers in the reduction of Equation (1).
For example, C
in Figure 2 is constructed according to the formula
.
In the case of continuous distribution 
over x in the
field of view, Equation (5) becomes
where the integration is over the field of view, and
is the density such that
With the conventional interferometers, from 
, 
we reconstruct a map of 
or 
, which is power or its density
in an ordinary sense. On the other hand, with the second harmonic
interferometers, for example, what we reconstruct from 
,
is a map of 
or 
, which is ``power-square''
or its density. Their physical meanings are not clear. Then what is the use
of the latter map? First, we could extract morphological information from it,
taking advantage of higher resolution. Second, we could (approximately) derive
a map of 
or 
from the map of 
or 
.
The interpretation of maps would be increasingly more difficult
with increasing n.
We do not get something for nothing. Instead of increasing the maximum baseline, we do pay other prices for higher resolvability. The most important one is the increase of observation time.
While the maximum baseline is ``stretched'' by a factor of n, the increments of baselines are also ``stretched'' by a factor of n. In order to avoid the possible problem of undersampling, the increments after being stretched must be kept as originally designed. This can be achieved, with an antenna array having movable elements, by carrying out more observations at shorter (physical) baselines. That is to say, the higher resolvability is achieved by increasing the observation time proportionally.
The theoretical analysis has proven the feasibility of HOHI with the second order as an example. (Analysis for the third order has been completed but is not presented here due to the limited space.) However, much more work remains to be done before putting HOHI in practical use.
Theoretical analysis: Mathematical analysis of HOHI
for 
; technical details, performance analysis, etc.
Computer simulation: A necessary intermediate stage between the theoretical analysis and practical test.
Practical test: Existing antenna arrays with movable elements are suitable for this purpose, e.g., facilities at DRAO, NRAO and C.S.I.R.O.
Rapid progress in the research can be made provided that sufficient time and necessary financial support are available.
