Lens to interferometer

1
Lens to interferometer
Suppose the small boxes are very small, then the
phase shift Introduced by the lens is constant
across the box and the same on both holes, so
irrelevant. Thus the cos/sinc pattern we saw in
the last slide of lecture 2 is the same if there
is no lens, i.e. we just have a standard Young
type interferometer. We dont measure the
electric field in the image plane, but its
average square, the received power, which in the
case of two small holes looks like Sinc2cos2
For more complicated apertures (remember the
Besel Functions) we remember from Fourier
transformations that the FT of an absolute square
is the autocorrelation function of the FT
itself. Thus the power psf is the FT of the
autocorrelation function of the aperture pattern.
2
Power versus correlation
Up until now we have assumed that our detection
equipment measures the total light power received
in the image plane. In other words I(p)
ltE(p)E(p)gt
Where f is the phase angle between E1 and E2. The
high resolution part of this signal is the last
term which oscillates like cos (kpDX/f) while
the first two terms represent the total
power coming through the 1st and 2nd slits.
These may contain very large terms due to sky
radiation that have nothing to do with the
target, so it would be nice to get rid of them.
We can do this if we have a Correlator rather
than a detector. A correlator measures
the average produce of two signals
Ill describe later how some correlators work.
3
Correlation
So now we can abstract our optical system even
further , throw away the focal system behind the
aperture and replace it with a correlator. Then,
if I have two small slits looking at a point
source, then the correlated flux is
Where D is the Optical Path Delay(OPD), the
difference in path length from the source to slit
2 versus slit 1. If the two slits are
separated by a baseline vector B, and the source
is in the direction n, then
or

• is the angle between the baseline and the
  source.
• Note that C does not depend on the position of
  the
• receivers on the ground but only on their
  separation
• vector.

4
Correlation
Now lets consider what happens if we have more
than one source of radiation on the sky. Then
antenna 1 receives not only electric field E1
from one source but also, say, F1 from the other
source, with similar E2 and F2 at the other
receiver. Then the formulas for I and C should
contain complicated terms like
reflecting the difference in position between the
two sources. This would be very messy, but
fortunately astronomical sources are incoherent,
that is, the phase difference between two
unrelated sources is never completely constant,
but drifts quickly or slowly with time (to be
described later). So a term like
actually shows up as
where J is a random number
and so the cos averages to zero and can be
ignored. This would not be true if there were
coherent sources (like lasers) distributed
over the sky.
5
Correlation
So, surprisingly or not, if I have a complicated
source on the sky, the response of the
interferometer is determined by the sum of
the intensities from the individual components
rather than the electric fields. Symbolically Wh
ere is the emitted intensity as a
function of the spatial position. This looks
sort of like a Fourier Transform but not quite.
The term is non-linear in the
position n on the sky because of the complicated
spherical trigonometry. But in a small piece of
sky near some reference point Lo this can be
linearized
is the interferometric delay as a function of
position in
the field, L is the vector position relative to
Lo. (U,V) are the UV coordinates of the
baseline and equal the physical vector
baseline projected onto the sky at Lo. For
narrow band (e.g. radio) measurements the
wavenumber is included in
the definition of U.
6
If we are smart enough to design a fully complex
correlator, that also measures
then we can write more directly
which looks exactly like a fourier transform.
The easiest way to get the imaginary part of the
correlation is to insert a quarter wave (l/4)
extra delay in the path, although this can be
tricky for a wide band system. Ideally this
would be all there is to interferometry if you
measure a whole lot of baselines you get an
estimate of C over a complete piece of the
UV-plane, and by a simple numerical Fourier
Transform you can reconstruct I(L). This is
called Aperture Synthesis. The goal of measuring
many UV points is partly achieved by sitting down
and letting the rotation of the Earth change the
projection of the baseline on the sky, and partly
by having many telescopes at different positions
to form into pairs, or moving around
the telescopes you have.