Transcript Example of Wave Optics

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 don’t 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 Sinc^2*cos^2… 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.
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) = <E(p)E*(p)>
E E 1E 2
I  E 12  E 22  2 Re(E 1E *2)
I  E 12  E 22  2E 1E 2 cos( )
Where  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:
C  E 1E 2 
I’ll describe later how some correlators work.
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:
C  S cos( 2D /  )
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: D  B  n
or: D  B cos(  )
 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.
Correlation
Now let’s 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:  E 1F *2 
 
 
 
 
2
2
c  E cos(B  nE )  F cos(B  nF )  EF (cos(B  nE )  cos(B  nF ))
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 EF cos(B  nE )
 
actually shows up as EF cos(B  nE   ) where  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.
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: C  I (n) cos(kn  B )d 2n


Where I (n )
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:

  2
D
D
C (U ,V )   I (L ) cos(kU  L )d L;U 
,V 
L
M


is the interferometric delay as a function of position in
D  n B
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 k  2 / 
is included in the
definition of U.
If we are smart enough
 to design
  a2 fully complex correlator,
that also measures  I (L ) sin(kU  L )d Lthen we can write more

 
directly: Cˆ (U ,V )  I (L ) exp(ikU  L )d 2L

which looks exactly like a fourier transform. The easiest way to
get the imaginary part of the correlation is to insert a quarter
wave (/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.