Introduction to Interferometry
Lo and Cornwell
Copyright, 1996 © Dale Carnegie & Associates, Inc.
Wave-front and Source
• A point source emits a spherical wave which turns into a plane
wave at large distances
E(r, t) = Eo e
i(kr-t)
This describes a plane wave-front that has a constant E. If
we performs a two point correlation with any two points on
the wave-front, the result is a constant, |Eo|2.
<E(1)E(2)> = V(1-2) = V(u,v) = |Eo|2
where (u,v) are the components of the vector from point 1 to
point 2 on the wave-front. The Fourier Transform of V(u.v) in
this case is a delta function (k), which represents the
brightness distribution of the point source in the direction
k=0.
Wave-front and Source
•
A double point source emits two spherical waves which turns into
two plane waves in slightly different directions k1 and k2, = k±k, at
large distances
E(r, t) = E0 e
i(k1r-t)
= E0 [e
i(kr)
+ Eo e
+e
i(k2r-t)
i(-kr)
]e
i(kr-t)
E0 [e i(kr) + e i(-kr) ] describes a wave-front that varies with position
along the wavefront. If we performs a two point correlation with any
two points on the wave-front, the result in this case is.
<E(1)E(2)> = V(1-2) = V(u,v) = |Eo|2 [e
i(kb)
+e
i(-kb)
]
where (u,v) are the components of the vector b from point 1 to point
2 on the wave-front. The Fourier Transform of V(u.v) in this case
consists of two delta functions, which represent the brightness
distribution of the double point source.
Zernike-van Cittert Theorem
•
Basic optics theorem:
<E1E2> = F. T. of Intensity Distribution of source
•
Interferometer measures <E1E2> = visibilities
•
Interferometer array + earth rotation synthesis measures
<E1E2> = V(u,v) across the wavefront (uv plane)
•
The array design is aimed at uniform coverage of the (u,v) plane by
the pairs of the antennas in the array, and with the help of the earth
rotation if needed
•
Fourier Transform of V(u,v) yields the I(x,y), the intensity distribution
of the source, i.e. the image or picture
•
Because of the sampling theorem, it is not necessary to get complete
coverage of the (u.v) plane if we are interested in only a range of
angular scales (or angular frequency)
Single interferometer:
to measure <E1E2>
<E1E2>
Aperture Synthesis
• The basic relationship in aperture synthesis is
V(u, v) =  I(x, y) Pn(x,y) ei2(ux + vy)/ dxdy
where V is the visibility that the interferometer array
measures, I is the intensity distribution (image) we want to
determine and Pn is the antenna pattern of the individual
antenna of the interferometer array.
• The image or map we want can then be obtained by the
inverse FT
M(x, y) = I(x, y) Pn(x,y) =  V(u, v)ei2(ux + vy)/ dudv
• Note that the map M we determine is the true I multiplied by
Pn . Thus, the beam-width of the individual telescopes in the
interferometer array defines the Field of View (FOV) of the
interferometer array.
In practice…….
1. Use many antennas (VLA has 27):
351 (u,v)’s at a given time
2. Amplify signals
3. Digitize
4. Send to correlator in a central location
5. Perform cross-correlation to yield
V(u.v) = <E1E2>
6. Take advantage of earth rotation to fill
in “aperture”
7. Inverse Fourier Transform of V (u,v) to
get an image: P(x,y) I(x,y)
8. Correct for limited number of
antennas: CLEAN
9. Correct for imperfections in “telescope”
e.g. calibration errors: Self Calibration
10. Make a beautiful image….
Incomplete Sampling of uv-plane
• The incomplete sampling of V(u,v) can be represented by a
sampling function S(u,v) which is 1 at (u,v) with a
measurement and 0 elsewhere. So the Md we obtain is
Md(x, y) =  V(u, v) S(u,v) ei2(ux + vy)/ dudv
• FT of two products is the product of the two FT:
Md(x, y) = M(x, y)  Bd(x, y)
where Bd(x, y) =  S(u,v) ei2(ux + vy)/ dudv is the Dirty Beam,
which is basically the transfer function of the interferometer
array, defined by the (u,v) coverage of the measurements,
S(u,v). Sometimes, it is called Point Spread Function. The map
obtained this way is called a Dirty Map
• CLEAN is a process to deconvolve the dirty image by replacing
Bd by a clean beam Bc to produce a clean map:
Mc(x, y) = M(x, y)  Bc(x, y)
CLEAN
• CLEAN is an iterative process in which the image of the source
is represented by a sum of delta-functions by picking out the
peaks in the images successively
• Then a clean beam (typically Gaussian) is fitted to the main
beam of the dirty beam
• Finally, the clean beam is convolved with the collection of
delta functions to produce the CLEAN map
• Not a unique process, as in all deconvolution, and affected by
S/N. Need always to compare Clean map with Dirty map to
make sure no artificial features have been introduced in the
process
• BUT, it works very well in practice.
• Other restorative processes include Maximum Entropy…
The power of Image Processing
Example of imaging a complex source
• VLBA simulated observations of M87-like jet source
• Will show
•
•
•
•
•
UV coverage
Correlation function
Point Spread Function
“Dirty” image
Best Clean image
Original and smoothed model
Fourier plane sampling
UV Sampling  Point Spread Function
S(u,v)
Dirty Beam
Point Spread Function
Dirty Beam
Cross cut of Dirty Beam
Original model and Dirty image
Dirty Map
Original model and best image
Clean Map
Correcting for limited number of antennas
• Sky is not arbitrarily complex: can exploit this to improve the
imaging
– CLEAN:
• sky is composed of point sources on a dark sky
• sky is composed of resolved sources of known extent on a dark sky
– Multi-scale CLEAN:
• sky is composed of smooth, limited extent blobs on a dark sky
– Maximum Entropy Method:
• sky is smooth and positive
– Non-negative least squares:
• sky is non-negative and compact
– Hybrid algorithms:
• Some combination of the above...
A real example from the VLA
• Sampled correlation function => “Dirty” image
A real example from the VLA
• Effective aperture is filled in and the diffraction patterns vanish
One last synthetic example
Model
“Dirty”
image
PSF
CLEAN
image
The VLBA
Ten 25m Antennas,
20 Station Correlator
327 MHz - 86 GHz
Correlator Roon
In Socorro
Variations: the Very Long Baseline Array
• Antennas very far apart
• resolution very high: milli-arcsecs
• Very Long Baseline Interferometry
• Record signals on tape
Connected elements versus tape recording
VLA + VLBA
• Zoom lens to reveal inner cores of radio galaxies
VLBA: Time-lapse imaging
Calibration
•
The measured visibility V’ is related to the source visibility V as
<E1E2> = V’(u,v)
= A’(u,v) ei[(u,v)] = g1g2 A(u,v) ei[(u,v)+(u,v)]
= g1g2 ei[(u,v)]  V(u,v)
where  is the measured phase,  is the true source phase and  is phase noise
due to the electronics, atmosphere and ionosphere
•
Calibration is to determine g1g2 ei[(u,v)] ,where the phase noise is typically antenna
based. i.e.
(12) = [e(1)  e(2)] + [a(1)  a(2)] + [i(1)  i(2)] …
•
Observe calibrations that are point sources of known flux S and known position ( =
0), and the measured
V’(u,v)/S = g1g2 ei[(u,v)] = G1G2*
where the complex G represents the amplitude and phase that needs to be removed
to yield the true source visibilities.
•
You measure (phase) calibrators regularly throughout the observations to provide
solutions (as a function of time) on N G’s from N(N-1)/2 (baseline) measurements.
The G(t) are then applied to the observations of the source.
Self Calibration
•
Standard calibration is not perfect typically because the time scale of
variation of G due to atmosphere is faster than the intervals in between
calibrator observations, and because suitable calibrators may not be
angularly close to the source under study. Self calibration solves most of
these problems.
•
As in Adaptive Optics in the OIR, if we know we are looking at a point
source, we know what to expect and all the variance in amplitude and
phase are due to (atmospheric) fluctuations. These fluctuations are
embodied in the G’s and in the visibility can be removed from the data.
This works well if there is a strong point source in the field of view.
•
Even without a point of source, we can get an estimate of the source
structure by the standard process. From the initial source structure, we can
predict what the visibility should be, and then remove the fluctuations from
the raw data to make an (presumably) improved image.
•
From the improved image, we can derive a better source structure from
which to predict what the visibilities should be and remove a new (and
presumably better) determinatioin of the fluctuations from the raw data.
And so on. This iterative process is what is called Self-Calibration, and has
been very successfully applied to both VLA and VLBA data, as well as other
interferometer data.
Self Calibration Imaging
•
Can image even if calibration is poor or nonexistent
•
Possible because there are N gains and N(N-1)/2 baselines
– Can determine both source structure and antenna gains
– Need at least 3 antennas for phase gains, 4 for amplitude gains
– Works better with many antennas
•
Iterative procedure:
– Use best available image to solve for gains (can start with point)
– Use gains to derive improved image
– Should converge quickly for simple sources
• Many iterations (~50-100) may be needed for complex sources
• May need to vary some imaging parameters between iterations
• Should reach near thermal noise in most cases
•
Does not preserve absolute position or flux density scale
– Gain normalization usually makes this problem minor
•
Historically called “Hybrid Mapping”. Based on “Closure Phase”.
•
Is required for highest dynamic ranges on all interferometers
VLBI Data Reduction
Example Self
Cal Imaging
Sequence
•
Start with phase only selfcal
•
Add amplitude cal when progress
slows
•
Vary parameters between
iterations
–
•
Taper, robustness, uvrange etc
Try to reach thermal noise
–
Should get close
PHASE REFERENCING
•
Use phase calibrator outside target source field
– Nodding calibrator (move antennas)
– In-beam calibrator (separate correlation pass)
– Multiple calibrators for most accurate results
•
Very similar to VLA calibration but:
– Geometric and atmospheric models worse
• Affected by totals between antennas, not just differentials
• Model errors usually dominate over fluctuations
• Scale with total error times source-target separation in radians
– Need to calibrate often (5 minute or faster cycle)
– Need calibrator close to target (< 5 deg)
•
•
Biggest problems:
–
Wet troposphere at high frequency
–
Ionosphere at low frequencies (20 cm is as bad as 1cm)
Use for weak sources and for position measurements
– Increases sensitivity by 1 to 2 orders of magnitude
– Used by about 30-50% of VLBA observations
EXAMPLE OF
REFERENCED PHASES
•
6 min cycle - 3 on each
source
•
Phases of one source
self-calibrated (near
zero)
•
Other source shifted by
same amount
Phase Referencing Example
1. With no phase calibration, source is not detected (no surprise)
2. With reference calibration, source is detected, but structure is distorted (targetcalibrator separation is probably not small)
3. Self-calibration of this strong source shows real structure
No Phase Calibration
Reference Calibration
Self-calibration