Interferometry of a Monotonic Source?

The methodology for interferometry looks fairly hairy for most people who
may not be acquainted with Fourier analysis. I think that's almost a
requirement. Would the explanation be simplified and useful if the source
consisted of a single frequency for say two observers? For example, would
it substantially and meaningfully simplify the reconstruction from the
readings of a circular image of say 1/10 of a degree across?

Wayne T. Watson (Watson Adventures, Prop., Nevada City, CA)
(121.015 Deg. W, 39.262 Deg. N) GMT-8 hr std. time)
Obz Site: 39° 15' 7" N, 121° 2' 32" W, 2700 feet
--
"Nature invented space so that everything didn't
have to happen at Princeton." -- Martin Rees,
Britain's Royal Astronomer, in a lecture at Princeton

Web Page: home.earthlink.net/~mtnviews

W. Watson wrote:

The methodology for interferometry looks fairly hairy for most people
who may not be acquainted with Fourier analysis. I think that's almost a
requirement.

You can get a rough idea of how interferometers work by playing with the
Young's slit experiment in the lab and then imagining it being used in
reverse. Detectors with correlators placed behind the slits would only
be sensitive to signals coming from the bright fringes.

Would the explanation be simplified and useful if the
source consisted of a single frequency for say two observers?

Almost every interferometry text uses monochromatic radiation for the
derivation of the technique. The restriction is then relaxed in
practical cases provided that fractional bandwidth dF/F 1.

I presume you mean monochromatic rather than monotonic.

For
example, would it substantially and meaningfully simplify the
reconstruction from the readings of a circular image of say 1/10 of a
degree across?

Not at all. There are a handful of simple patterns that occur in the sky
and are easy to diagnose from the measured visibility fringes. Roughly
equal brightness double sources like Cygnus A being one such.

There is a toy Java interferometer on the web where you can see the
visibility fringes from various common sources.

http://www.narrabri.atnf.csiro.au/astronomy/vri.html

Regards,
Martin Brown

Martin Brown wrote:

W. Watson wrote:

The methodology for interferometry looks fairly hairy for most people
who may not be acquainted with Fourier analysis. I think that's almost
a requirement.


You can get a rough idea of how interferometers work by playing with the
Young's slit experiment in the lab and then imagining it being used in
reverse. Detectors with correlators placed behind the slits would only
be sensitive to signals coming from the bright fringes.

Would the explanation be simplified and useful if the source consisted
of a single frequency for say two observers?


Almost every interferometry text uses monochromatic radiation for the
derivation of the technique. The restriction is then relaxed in
practical cases provided that fractional bandwidth dF/F 1.

I presume you mean monochromatic rather than monotonic.

For example, would it substantially and meaningfully simplify the
reconstruction from the readings of a circular image of say 1/10 of a
degree across?


Not at all. There are a handful of simple patterns that occur in the sky
and are easy to diagnose from the measured visibility fringes. Roughly
equal brightness double sources like Cygnus A being one such.

There is a toy Java interferometer on the web where you can see the
visibility fringes from various common sources.

http://www.narrabri.atnf.csiro.au/astronomy/vri.html

Regards,
Martin Brown
Yes, monocrhomatic. Thanks very much for the response, and tips. That makes
sense in terms of re-construction.

I guess another thing that disturbs me about many explanations is how two
observers can be as effective resolution-wise as an instrument whose
diameter is the same as the distance between them, albiet seeing the object
as a dimmer. Intuitively it doesn't seem to make sense; however, it seems
demonstratable by masking a simple telescope's mirror. In the simplest case,
this seems possible to demonstrate by putting one's hand in front of the
tube of a Newtonian, and looking through the eyepiece. I would think a
better demonstration would be to mask the mirror instead by covering some
inner ring of the mirror.


Wayne T. Watson (Watson Adventures, Prop., Nevada City, CA)
(121.015 Deg. W, 39.262 Deg. N) GMT-8 hr std. time)
Obz Site: 39° 15' 7" N, 121° 2' 32" W, 2700 feet
--
"Nature invented space so that everything didn't
have to happen at Princeton." -- Martin Rees,
Britain's Royal Astronomer, in a lecture at Princeton

Web Page: home.earthlink.net/~mtnviews

W. Watson wrote:

Martin Brown wrote:

W. Watson wrote:

The methodology for interferometry looks fairly hairy for most people
who may not be acquainted with Fourier analysis.

There is a toy Java interferometer on the web where you can see the
visibility fringes from various common sources.

http://www.narrabri.atnf.csiro.au/astronomy/vri.html

I guess another thing that disturbs me about many explanations is how
two observers can be as effective resolution-wise as an instrument whose
diameter is the same as the distance between them, albiet seeing the
object as a dimmer. Intuitively it doesn't seem to make sense;

That is because it doesn't. You need a suitable set of observations in
visibility space to compute an aperture synthesis image. Measuring a
single spatial frequency on its own is not enough to form an image.

To compute an image you need a combination of obervations of the
visibility fringes made across a whole set of baselines to fill in the
equivalent aperture. The earliest version of Earth rotation aperture
synthesis used a perfect E-W baseline (Cambridge One Mile) to use the
Earths rotation to collect a pair of baselines with one movable scope.
The 5km scope relaxed the E-W requirement a bit, and the VLA using
superior computer power broke free and is able to handle a Y shaped set
of baselines allowing snapshots as well as deeo synthesis.

however,
it seems demonstratable by masking a simple telescope's mirror. In the
simplest case, this seems possible to demonstrate by putting one's hand
in front of the tube of a Newtonian, and looking through the eyepiece. I
would think a better demonstration would be to mask the mirror instead
by covering some inner ring of the mirror.

To see what aperture synthesis fringe measurements look like in the
optical you need a pair of smallish holes in a cardboard mask and the
ability to rotate it. Point the whole lot at a near equal brightness
double star and with any luck you will see the interference fringes vary
in intensity as you rotate the mask relative to the sky.

Let me ask some questions about the Java program. It's been quite a while
since I thought about Fourier transforms, but I have a reasonable idea of
what they are about. I've looked at the documentation and guide. After I
formulated the questions below, I happened to glance at the end of the
guide, and see there's more documentation. Maybe you can take a shot at
these questions, and I'll rummage through the additional documentation.
Whoops, a quick survey of those items shows I've got about all there is
doc-wise.

The display is divided into four area. Upper left is the image, Upper right
is some sort of positional display of antennas. The lower right is
uv-coverage. The lower left is the uv components such as the real and
imaginary part, phase and magnitude.

It looks like this program is set up to operate for several specific
observatories, ATCA, Merlin, etc. Suppose I want to define my own two
station observatory. How can that be done?

What is uv coverage? If I play around the controls of that area, I see what
look like red dots. They are distributed differently for the obs choice.

What do the buttons in the lower right corner do? Apply, Add, Accumulate
and Clear. Blue, red, hide? How about at the upper right? Default and
Station Lock.

How does this actually operate? I guess I give it an object like a disk or
point, and somehow start an observing session. If it simulates a period of
operation (if specifiable), what do I actually end up with?





Wayne T. Watson (Watson Adventures, Prop., Nevada City, CA)
(121.015 Deg. W, 39.262 Deg. N) GMT-8 hr std. time)
Obz Site: 39° 15' 7" N, 121° 2' 32" W, 2700 feet
--
"Nature invented space so that everything didn't
have to happen at Princeton." -- Martin Rees,
Britain's Royal Astronomer, in a lecture at Princeton

Web Page: home.earthlink.net/~mtnviews

W. Watson wrote:

Let me ask some questions about the Java program. It's been quite a
while since I thought about Fourier transforms, but I have a reasonable
idea of what they are about. I've looked at the documentation and
guide. After I formulated the questions below, I happened to glance at
the end of the guide, and see there's more documentation.

Best to read the documentation too, but to play with it and get a feel
for what an aperture synthesis instrument does it is invaluable as a
simulator (mirrors exist at Jodrell Bank UK).

The display is divided into four area. Upper left is the image, Upper
right is some sort of positional display of antennas. The lower right is
uv-coverage. The lower left is the uv components such as the real and
imaginary part, phase and magnitude.

Top Left is the real space image (x,y)
(choose an object - wide double is good)
Click FFT and it will show you the visibility

Bottom Left is now the FFT of the image (u,v)

Top Right allows you to position your observatory antennae
(place them at equal separations in a straight horizontal line)
Bottom Right are the measurements they will make (u,v)
(set observation range HA from -6h to +6h,
and dec to -90 for nice easy circular baseline tracks)

Press "Reset" to make the observations (set of red rings)

Now press apply to mask off the FFT of the true sky to leave only the
measurments that your interferometer would have made.

Press "FFT-1" and you have the raw image from aperture synthesis.

It looks like this program is set up to operate for several specific
observatories, ATCA, Merlin, etc. Suppose I want to define my own two
station observatory. How can that be done?

What is uv coverage? If I play around the controls of that area, I see
what look like red dots. They are distributed differently for the obs
choice.

u-v coverage is where the baselines have measured the spatial
frequencies of the sky brightness distribution.

What do the buttons in the lower right corner do? Apply, Add,
Accumulate and Clear. Blue, red, hide? How about at the upper right?
Default and Station Lock.

How does this actually operate? I guess I give it an object like a disk
or point, and somehow start an observing session. If it simulates a
period of operation (if specifiable), what do I actually end up with?

A good idea of what the image from aperture synthesis would look like.

Once you have set up the model you can load different target sources and
pretend to observe each one with VRI by clicking

Source
FFT
Apply
FFT-1

Hope this makes some sense. You can play with much more complex
locations of antennae but interpretaion becomes harder. VLA style Y is fun.

Regards,
Martin Brown

Martin Brown wrote:

W. Watson wrote:

Let me ask some questions about the Java program. It's been quite a
while since I thought about Fourier transforms, but I have a
reasonable idea of what they are about. I've looked at the
documentation and guide. After I formulated the questions below, I
happened to glance at the end of the guide, and see there's more
documentation.


Best to read the documentation too, but to play with it and get a feel
for what an aperture synthesis instrument does it is invaluable as a
simulator (mirrors exist at Jodrell Bank UK).
As it turns out, that's all the documentation there is. The above reference
was just back to stuff they had referenced earlier.

The display is divided into four area. Upper left is the image, Upper
right is some sort of positional display of antennas. The lower right
is uv-coverage. The lower left is the uv components such as the real
and imaginary part, phase and magnitude.


Top Left is the real space image (x,y)
(choose an object - wide double is good)
Click FFT and it will show you the visibility
Visibility? real, imag, phase, mag in the lower left? Ah, I suspect you mean
what the selection looks like, e.g., two stars widely separated. Using wide
double and FFT for ampl, the lower right shows vertical (hazy, unsharp) lines.

Bottom Left is now the FFT of the image (u,v)

Top Right allows you to position your observatory antennae
(place them at equal separations in a straight horizontal line)
I'm using ATCA and spread the six antenna out on a line equadistantly. The
central area shows some small circles along the innermost lines.
Bottom Right are the measurements they will make (u,v)
(set observation range HA from -6h to +6h,
and dec to -90 for nice easy circular baseline tracks)

Press "Reset" to make the observations (set of red rings)
Which reset? I do see now some semi-circular rings in the uv-coverage. Ah,
it looks like it was cranking through and now has red circles? Initially the
uv coverage showed a line of red dots or dashes.

Now press apply to mask off the FFT of the true sky to leave only the
measurments that your interferometer would have made.
The bottom left now contains concentric circles that are slighly broken up.
What did I mask off?

Press "FFT-1" and you have the raw image from aperture synthesis.
Pressing FFT-1 doesn't produce anything -- yet. Five minutes later I don't
see any change anywhere in the four areas.


It looks like this program is set up to operate for several specific
observatories, ATCA, Merlin, etc. Suppose I want to define my own two
station observatory. How can that be done?

What is uv coverage? If I play around the controls of that area, I see
what look like red dots. They are distributed differently for the obs
choice.


u-v coverage is where the baselines have measured the spatial
frequencies of the sky brightness distribution.
So this is sort of like getting some idea of the noise for the entire sky?
Are those red dots in the initial horizontal line associated with the array
pattern?

What do the buttons in the lower right corner do? Apply, Add,
Accumulate and Clear. Blue, red, hide? How about at the upper right?
Default and Station Lock.
????

How does this actually operate? I guess I give it an object like a
disk or point, and somehow start an observing session. If it simulates
a period of operation (if specifiable), what do I actually end up with?


A good idea of what the image from aperture synthesis would look like.
I'm thinking in terms of what steps really occur to make that happen. I
suspect I need a book like 'Apert Synth for Dummies". Anything on the web? I
have this vague notion that I start with with an image, mess it with noise,
convert it to the u-v domain, take observations some how that are in the u-v
comain, fiddle with collecting them over a 12 hour period, and the convert a
mess of stuff back to the spatial domain.

What should I have expected from above? The wide doubles as seen by the
particular array given some sky noise? Is there a resulting image in one of
the panels?


Once you have set up the model you can load different target sources and
pretend to observe each one with VRI by clicking

Source
FFT
Apply
FFT-1

Hope this makes some sense. You can play with much more complex
locations of antennae but interpretaion becomes harder. VLA style Y is fun.
Which choice is VLA?

Regards,
Martin Brown



Wayne T. Watson (Watson Adventures, Prop., Nevada City, CA)
(121.015 Deg. W, 39.262 Deg. N) GMT-8 hr std. time)
Obz Site: 39° 15' 7" N, 121° 2' 32" W, 2700 feet
--
"Nature invented space so that everything didn't
have to happen at Princeton." -- Martin Rees,
Britain's Royal Astronomer, in a lecture at Princeton

Web Page: home.earthlink.net/~mtnviews

W. Watson wrote:

Martin Brown wrote:

Now press apply to mask off the FFT of the true sky to leave only the
measurments that your interferometer would have made.

The bottom left now contains concentric circles that are slighly broken
up. What did I mask off?

Convolution in real space becomes multiplication in Fourier space. The
baselines determine which spatial frequencies were measured everywhere
else gets set to zero.

Press "FFT-1" and you have the raw image from aperture synthesis.

Pressing FFT-1 doesn't produce anything -- yet. Five minutes later I
don't see any change anywhere in the four areas.

Once you have masked the left bottom side with the u-v coverage FFT-1
should take about as long as FFT to perform the inverse. If you haven't
applied the u-v mask to the FFT of the original image FFT-1 will return
the original image (nothing obvious happens).

u-v coverage is where the baselines have measured the spatial
frequencies of the sky brightness distribution.

So this is sort of like getting some idea of the noise for the entire
sky? Are those red dots in the initial horizontal line associated with
the array pattern?

Try the example in Jodrells guide again:
http://www.jb.man.ac.uk/vri/vri/guide.html

What should I have expected from above? The wide doubles as seen by the
particular array given some sky noise? Is there a resulting image in
one of the panels?

The original ideal image should be replaced by the result of your
observations.

Once you have set up the model you can load different target sources
and pretend to observe each one with VRI by clicking

Source
FFT
Apply
FFT-1

Hope this makes some sense. You can play with much more complex
locations of antennae but interpretaion becomes harder. VLA style Y is
fun.

Which choice is VLA?

Just place the antennae in a 120 degree Y configuration roughly equally
spaced. Doing this avoids having poor coverage for low declinations.

Regrards,
Martin Brown

The descriptions I've seen of the u-v coverage are pretty sketchy, to me at
least. Apparently, it is fomred by looking at a baseline pair as seen from
the source's view in space as the earth rotates. Somehow this view should
when plotted against time look like an ellipse. If I simply take two points
on a globe and connect them, and look at them as I rotate the globe, then I
don' see what might be taken as an ellipse. One description I saw talks
about projecting the earth points onto a plane perpendicular to the line of
sight. Maybe some warpage in the projection does cause part of an ellipse
to appear on that plane.

Isn't the u-v plane a frequency domain? How does this related to x-y (and
maybe z) on the earth?


Wayne T. Watson (Watson Adventures, Prop., Nevada City, CA)
(121.015 Deg. W, 39.262 Deg. N) GMT-8 hr std. time)
Obz Site: 39° 15' 7" N, 121° 2' 32" W, 2700 feet
--
"Nature invented space so that everything didn't
have to happen at Princeton." -- Martin Rees,
Britain's Royal Astronomer, in a lecture at Princeton

Web Page: home.earthlink.net/~mtnviews

Take the case where you have two antennas forming an E-W baseline.
An object rising due east at first will have about zero length projected
baseline. At the meridian you now have the full projected base line.
Thus the base line goes as co-sine of the hour angle of the object

The fringe visibility of an interferometer is a function of the source
structure and the baseline. Take a double source in the east-west
direction. If the source is bigger than the projected
baseline/wavelength fringe spacing then you can have several
fringes across the object and the fringe contrast goes to zero thus
fringe visibility is zero. In the example of an object in the east one
would start with zero fringe visibility until the number of fringes
across the source was less than one and in full viability for that
baseline and source.

If the object was now a double source in the north-south direction,
the E-W interferometer would not see a double source as in that
direction (N-S) The baseline projection is zero at the meridian.

So it takes a a base line in the N-S direction to see the source structure.

Each antenna pair form a spatial filter projected on that baseline.
As you increase the number of antennas, you increase the number of
spatial filters needed for reconstructing the objects image.

The VLA with its 120 degree "arms" sees the entire UV plane with out
going to four legs

In away its like trying to reconstruct any type of signal that has
been sampled. Take music for example. Start with 1 band pass filter
at a low frequency an listen to the result, can you name that tune ?
How many filters do you need up to say 1/3 the sampling frequency
before you can reconstruct the original detail ?

Of course music is not based on spatial frequency, and instead is an
example of processing in the frequency domain

U of A has built a "binocular telescope" that will be used as an
interferometer. The base line rotates with the azimuth so that in the
case of the object rising in the east, the interferometer will sample
the N-S visibilities best in the east and E-W at the meridian.

d.






W. Watson wrote:
The descriptions I've seen of the u-v coverage are pretty sketchy, to me
at least. Apparently, it is fomred by looking at a baseline pair as seen
from the source's view in space as the earth rotates. Somehow this view
should when plotted against time look like an ellipse. If I simply take
two points on a globe and connect them, and look at them as I rotate the
globe, then I don' see what might be taken as an ellipse. One
description I saw talks about projecting the earth points onto a plane
perpendicular to the line of sight. Maybe some warpage in the
projection does cause part of an ellipse to appear on that plane.

Isn't the u-v plane a frequency domain? How does this related to x-y
(and maybe z) on the earth?


Wayne T. Watson (Watson Adventures, Prop., Nevada City, CA)
(121.015 Deg. W, 39.262 Deg. N) GMT-8 hr std. time)
Obz Site: 39° 15' 7" N, 121° 2' 32" W, 2700 feet