11institutetext: Union College, Schenectady, NY, USA 22institutetext: MIT
Haystack Observatory, Westford, MA, USA 33institutetext: Bridgewater State
University, Bridgewater, MA, USA

# Demonstrating the Principles of Aperture Synthesis with Table-Top Laboratory
Exercises

J. M. Marr 11<EMAIL_ADDRESS>A. E. E. Rogers 22 V. L. Fish 22 F. P. Wilkin 11
M. B. Arndt 33 G. Holodak 11 K. Durkota 11

###### Abstract

Many undergraduate radio astronomy courses are unable to give a detailed
treatment of aperture synthesis due to time constraints and limited math
backgrounds of students. We have taken a laboratory-based approach to teaching
radio interferometry using a set of college-level, table-top exercises. These
are performed with the Very Small Radio Telescope (VSRT), an interferometer
developed at the Haystack Observatory using satellite TV electronics as
detectors and compact fluorescent light bulbs as microwave signal sources. The
hands-on experience provided by the VSRT in these labs allows students to gain
a conceptual understanding of radio interferometry and aperture synthesis
without the rigorous mathematical background traditionally required.

The data are quickly and easily processed using a user-friendly data analysis
Java package, VSRTI_Plotter.jar. This software can also be used in the absence
of the equipment as an interactive computer activity to demonstrate an
interferometer’s responses to assorted surface brightness distributions. The
students also gain some familiarity with Fourier transforms and an
appreciation for the Fourier relations in interferometry using another Java
package, the Tool for Interactive Fourier Transforms (TIFT). We have
successfully used these tools in multiple offerings of our radio astronomy
course at Union College

##  1 Introduction

The radio astronomical technique of aperture synthesis, in which high-
resolution images are produced from interferometer arrays, has been a
productive tool for astronomers for decades. From its initial development in
the 1950s and early 1960s, aperture synthesis was quickly recognized as a
significant advancement for science. Sir Martin Ryle shared the Nobel Prize
for Physics in 1974 “for his observations and inventions, in particular of the
aperture synthesis technique” Nobel . The best resolution attained with
single-dish radio telescopes has historically been of order 20 arcseconds,
while that achieved with aperture synthesis can be four orders of magnitude
better. With the subsequent construction of the (now renamed) Jansky Very
Large Array111The Very Large Array is operated by the National Radio Astronomy
Observatory, which is a facility of the National Science Foundation operated
under cooperative agreement by Associated Universities, Inc. and continent-
scale arrays for very long baseline interferometry, radio astronomers have
been able to produce images with much higher angular resolution than has been
possible at other wavelengths. Yet, because of the complexity of the math
involved, aperture synthesis is often excluded from undergraduate curricula in
physics and astronomy.

We introduce here a set of labs at the undergraduate level which provide
hands-on experience with the basics of aperture synthesis observations and how
the data reveal information about the size and structure of the observed
sources. These labs are designed to give students a conceptual understanding
of aperture synthesis without the need for the dense mathematical formalism
that usually accompanies the topic in graduate-level classes.

##  2 The basic concept of aperture synthesis

Successful completion of the labs does not require that the students, or
instructors, know the mathematical formalism. As such, in this section, we
give a brief discussion of what an aperture synthesis observation entails,
touching on only those aspects necessary to comprehend the principles
demonstrated in these labs. Details about interferometry can be found in
textbooks MSK ; WRH ; BGS ; TMS ; Kraus .

In aperture synthesis a number of antennas, arranged in a particular pattern,
or “array,” receive the radiation from a celestial source simultaneously and
the signals are combined pairwise. The method of signal combination in most
modern interferometers is a cross-correlation, but the signals can also be
added. The response of each pair of antennas contains an amplitude and a phase
which, customarily, are represented as a complex number. (The “amplitude” in a
radio interferometer’s output is called the “modulus” of a complex number in
standard mathematical usage.) For a single point of emission, the amplitude is
proportional to the flux density of the source while the phase is related to
the difference in path lengths to the two antennas, as depicted in Figure 1.
For a general source of arbitrary structure, the detected amplitude and phase
are the complex superposition of the responses due to each unresolved point of
emission within the field of view.

Figure 1: A pair of antennas receive radiation from a celestial source in a
direction $\theta$ relative to the mid-plane position. The path from the
source to antenna 2 contains the extra distance $s$, which causes a phase
difference when the signals are combined. The projected baseline (component
perpendicular to the source direction) is shown as the length $b$.

In an aperture synthesis observation, the fully calibrated response of each
pair of antennas, termed a “visibility,” is a function of the separation of
the antennas, generally referred to as the “baseline,” $\vec{b}$. If the
source is not located along the mid-plane of the baseline, $\vec{b}$ is the
component of the baseline perpendicular to the direction of the source.

The visibility for any particular antenna pair is most sensitive to source
structure on an angular scale proportional to $\lambda/b$, where $\lambda$ is
the wavelength of the observation. This is a familiar concept from optics, in
which the resolution of a telescope is proportional to $\lambda/D$, where $D$
is the diameter of the objective lens or mirror. With an array of more than
two antennas, the visibility for each pair is obtained and hence the
visibility function can be probed over a wide range of baseline lengths and
orientations. In this way, information about the source structure on multiple
angular scales enables one to produce an image of the source. When written as
complex values, the visibility function is related to the image of the source
via a two-dimensional Fourier transform.

In the labs described here, the presentation of the important principles is
simplified in two ways. First, since the Very Small Radio Telescope (VSRT)
measures only the amplitude of the visibilities and not the phase, complex
numbers can be avoided. Second, the labs are performed with all sources and
antennas in the horizontal plane and so the analysis is reduced to one-
dimension. Students are introduced to the visibility function by measuring the
interferometer response as a function of antenna separation and discover the
relationship between visibilities and simple source structure. By
extrapolation, the students gain an appreciation of how information about
source structure can be recovered from data obtained with many pairs of
antennas.

The visibility as a function of baseline is analogous to the diffraction and
interference patterns produced in the single and double slit experiments.
Radiation from each point in the observed source is detected by each antenna
and the different path lengths result in constructive or destructive
interference depending on the point’s position in the sky and the baseline
geometry when the signals are combined. For a source of arbitrary brightness
pattern, the resulting visibility function is the sum of the interferometer
responses for all points of emission in the source. When written as complex
values, the visibility function, with $\vec{b}/\lambda$ as the independent
variable, is related to the image of the source, i.e. intensity as a function
of angle, $\vec{\theta}$, on the sky, via a Fourier transform. In practice,
the radio astronomer obtains an image of a source with software which performs
an inverse Fourier transform on the visibility data.

##  3 Equipment

The laboratory exercises described herein use the VSRT (Very Small Radio
Telescope), a laboratory interferometer developed by MIT Haystack Observatory.
A description of the operation of the VSRT as an additive interferometer is
provided online AR . Composed of commercially-available electronics and
satellite TV equipment, the VSRT is a low-cost instrument designed to
demonstrate the principles of radio waves and interferometry in high school
and college-level labs DFN . The VSRT is stable, reliable, and easy to
assemble, operate and manipulate in the lab room. (Full details of the system
can be found at the VSRT project web page VSRTurl .) The VSRT uses Ku-band
satellite TV feeds to receive radiation near 12 GHz. The feeds can be
installed in satellite dishes to increase the collecting area and
directionality of the instrument, as is done for an experiment measuring the
solar diameter Fish . Alternatively, the feeds can be used without the dishes
to observe strong radio sources in the lab. The exercises discussed here use
the instrument in this manner, with compact fluorescent light bulbs (CFLs)
serving as radio sources (see Figure 2).

In addition to visible light, CFLs produce broadband radio emission from 100
MHz up to 100 GHz when the free electrons in the plasma collide with the glass
walls of the bulb MS . The CFLs can be hidden by an optically-opaque material
that is transparent to radio waves (e.g., a cardboard box) to demonstrate that
the feeds are sensitive to the bulbs’ radio emission, not their visible light.

Figure 2:  Union College students working with the VSRT. Two “triple” DirecTV
feeds, taken from TV satellite dishes, used here as radio antennas, receive
and detect the radio emission from compact fluorescent light bulbs (CFLs).
Only one feed of each triple is active. The CFLs can be moved to assorted
separation distances. The output spectrum from the interferometer is displayed
on the laptop screen, and the data files are recorded.

##  4 Laboratory Exercises

We now describe the labs, designed to help students develop an intuitive sense
of how an array of antennas can be used to infer aspects of the spatial
structure of radio sources. To facilitate data analysis, we have produced a
package of Java programs, named “VSRTI_Plotter,” available online plotter ;
VSRTurl . These programs also have links to lab instructions and can produce
overlays of theoretical models with adjustable parameters. For the sake of
simplicity the lab set-ups occur entirely in the horizontal plane so that the
maps of the sources, as seen from the position of the feeds, are simple plots
of intensity vs. position in the horizontal direction.

###  4.1 The primary beam

Astronomy students are usually familiar with the fact that the angular
resolution of a single telescope, ignoring the atmosphere, is limited by
diffraction and is approximately $\lambda/D$. With interferometers, the
central diffraction peak of each individual telescope is known as the “primary
beam.” When an astronomical source is not at the center of the primary beam,
the detected power is decreased. The primary beam size of the individual
antennas of an array, therefore, places an effective upper limit on the
maximum field of view in aperture synthesis.

In the first exercise, students measure the primary beam pattern of the VSRT
feeds by placing the active feeds one above the other, making a baseline with
zero horizontal length and placing a CFL two meters away at the mid-plane
position. Keeping the feeds fixed, they record data with the CFL placed at
various horizontal angles to the mid-plane. The data files can then be drag-
and-dropped into the VSRTI_Plotter “Plot Beam” program, producing a graph of
the data. The VSRTI_Plotter program can also overlay a theoretical beam plot,
yielding a fitted value of the effective diameter of a feed (see Figure 3).

Figure 3:  A plot obtained by undergraduate students of the primary beam of a
VSRT feed. The plot, produced with the VSRTI_Plotter, shows the detected power
as a function of the angular position of a CFL. The angular position is given
in units of degrees and the power in units of Kelvins, as is traditionally
done in radio astronomy. The model overlay yields a measure of the effective
diameter of the antenna feed of approximately 3.8 cm.

###  4.2 A single resolved source

The distance between the antennas (i.e., the baseline) determines the angular
sizes of structure in the radio source that the interferometer is most
sensitive to. In short, the baseline acts like a spatial filter. With longer
baselines, the interferometer response becomes more dependent on finer scale
structure in the source, while its sensitivity to extended structure
decreases. Flux distributed over larger angles is said to be “resolved out.”
If the baseline is too long, it will not detect the source at all. The
detected power of a resolved source, therefore, decreases with increasing
baseline length. One can use the rate of fall-off of detected power to
determine the angular size of the source.

In this lab, students use a single CFL located at the mid-plane position
between the two feeds and then vary the horizontal separation between the
feeds. They produce a plot of the measured power versus the baseline length,
finding that the power decreases as the baseline length increases. In this way
students become acquainted with the visibility function, in which the
independent variable is baseline length divided by wavelength.

Students are instructed to obscure the bottom half of the CFL using a metal
plate. They find that the power decreases by about half and verify that this
holds at a variety of baseline lengths. Next, they place two metal plates
vertically obscuring the sides of the CFL in order to make a source whose
apparent size is smaller. They find that the measured power at short baseline
lengths is smaller than for an unobscured CFL, but that the power at long
baseline lengths is actually greater. Figure 4 displays results of this lab.
The students discover that the rate of decrease of the measured power is
inversely related to the angular size of the source.

Figure 4:  The detected power from a single CFL as a function of baseline
length. The data show that with increasing baseline length more of the flux of
the source is resolved out. The graph on left results with a smaller source
and shows that less of the flux of the smaller source is resolved out than
with a larger source. The model curves (shown in red) indicate the angular
source sizes were 0.023 and 0.04 radians.

###  4.3 Double Sources

When the radio source contains two components separated by an angle resolved
by the interferometer, the detected power varies with baseline in an
oscillatory manner. In this exercise, students discover this by using two CFLs
with a fixed separation between them. The students move the VSRT feeds to
larger separations and discover that the measured power oscillates with
baseline length. The students then move the two CFLs farther apart and repeat
their measurements, discovering a reciprocal relationship between the angular
separation of the two sources and the distance between minima in the
visibility function (Figure 5).

Figure 5:  The visibilities as a function of baseline length for a pair of
CFLs. For the left plot, the CFLs were separated by 30 cm and at a distance of
2 m, while for the right plot the CFLs were separated by 40 cm. Students
discover that the detected signal for a double source has an oscillating
dependence on baseline length and that the period of oscillation is inversely
dependent on the separation of the sources. The overall decrease in power with
baseline is due to the resolving of each CFL, as seen in Figure 4. (The
difference in the visibility values between the graphs results because
different lamps were used in the two experiments.)

###  4.4 Mystery Source

After completing the exercises discussed in Sections 4.2 and 4.3, the
students’ new abilities to infer the angular separations and angular sizes
from visibilities are tested by placing a cardboard box over a pair of CFLs.
After obtaining their data and inferring the source structure, the students
can check their answer by lifting the box.

A slight increase in complexity of source structure and a challenge to the
students’ analytical abilities can then be demonstrated with an exercise using
three CFLs hidden under a cardboard box. The instructor sets the CFLs with
equal spacings, so that CFLs 1 and 2 and CFLs 2 and 3 create two pairs
separated by angle $\theta$ and CFLs 1 and 3 make one pair with a separation
of $2\theta$. The visibility data then show the sum of two sine waves – one
with a larger amplitude and period $=1/\theta$ and the other with a smaller
amplitude and half the period (Figure 6). The students are not informed that
there are three CFLs in the box, and are asked to infer, working as team, the
source structure considering the principles they learned in the labs.

Figure 6:  The detected power as a function of baseline length for a “mystery
source” hidden inside a box 2 meters away. The box contained three CFLs with
equal separations of 20 cm between the middle and end CFLs, yielding two pairs
with angular separations of 0.1 radians and one pair with a separation of 0.2
radians.

###  4.5 Fourier Transform

In actual aperture synthesis observations one obtains an image of the source
by performing a Fourier transform on the complex visibilities. Since the VSRT
data contain only amplitudes, and no phases, we have provided another java
package which students can use to discover the relation between Fourier
function pairs. They find that this relation is identical to that between the
visibility function and source structure. Using the “Tool for Interactive
Fourier Transforms” (or TIFT), also available online plotter , students use a
simple click and drag operation to make functions representing the brightness
distributions of the CFLs in the previous labs. In a companion plot, they
discover that the amplitudes of the Fourier transforms are the same as the
visibility amplitudes they measured with the VSRT (Figure 7).

Figure 7:  TIFT screen shot of the simulation of observation of two resolved
sources. In the “f(t)-Magnitude” window, two signals of width 0.03 s and
separated by 0.2 are drawn. The “F($\nu$)-Magnitude” window shows that the
Fourier transform oscillates with a period equal to 5 and a decreasing
envelope and is similar to the visibility function found in the lab observing
double sources (see Figure 5).

###  4.6 Solar Diameter

As an exercise using the VSRT to make a measurement of an actual celestial
source, the VSRT can also be used to determine the diameter of the Sun. This
involves a slightly more complicated set-up but also provides a practical
culmination of the VSRT labs Fish . The results of this experiment are shown
in Figure 8.

Figure 8:  Plot of the visibilities in a VSRT observation of the Sun, with
overlay of curves for 3 different models.

##  5 Practical Experience in the Classroom

Starting in 2012, these labs have been incorporated into the radio astronomy
class at Union College. The class consists of seniors, juniors and sophomores
with majors in either physics or engineering. The coverage of aperture
synthesis in the course is accomplished using only the labs; no lectures are
provided. The effectiveness of the labs was tested in the first course they
were used. Both before and after performing the labs, the students were given
a quiz on the basics of aperture synthesis, containing the following
questions:

1\. Explain conceptually how receiving signals with a number of radio
telescopes, as with the VLA, contains information about the image of a radio
source.

2\. What is an “array” of telescopes and what are the important criteria in
designing the array?

3\. What is the “Visibility function?”

4\. Describe what a 1-dimensional visibility function looks like when
observing:
a) a single, unresolved source.
b) a single, resolved source. How does the shape of the visibility function
change as the angular size of the source increases?
c) a pair of unresolved sources. How does the shape of the the visibility
function change as the separation of the two sources increases?

The average score on the quiz before performing the labs was 0.3 points out of
18 (or 1.5%), demonstrating that none of the students had any prior knowledge
about aperture synthesis. Afterwards, the average quiz score was 11.9 (or
66%), indicating a normalized gain (defined by the fraction of the material
not known a priori that was learned by the time of the post-test) was 0.66.

The mystery source lab, in which three CFLs were hidden from view, was found
to be challenging by the students. However, with all students collaborating as
a team to brainstorm, within a class period and with extensive discussion, the
class succeeded in inferring the actual source structure correctly.

##  6 Concluding remarks

These labs have been incorporated in undergraduate radio astronomy courses at
Union College and the data for all the plots shown in the figures were
obtained by students in these classes. While single-dish radio observing
methods are sometimes included in undergraduate astronomy classes, aperture
synthesis is often de-emphasized due to the need to include high-level
mathematics. Using these labs, the students in these classes were able to gain
an intuitive understanding that data associated with pairs of antennas at
different spacings leads to a function from which one can infer the size of a
single source or the separation between sources in the sky.

These labs can be used to stimulate discussions about how the structural
details of even a complicated source can be extracted from aperture synthesis
data using standard observing setups and imaging algorithms. For instance,
after learning that extended sources can be resolved out on long baselines,
students can be asked about the importance of matching the angular resolution
of an observing array to the scale of the expected structure in an
astronomical source and asked to consider why the telescopes in the Jansky
Very Large Array and the Atacama Large Millimeter Array are designed to be
reconfigurable into arrays of different sizes. After the primary beam lab,
students can be asked to speculate why small dishes are used in arrays
designed for wide-field imaging, for example.

We are grateful for the significant contributions by Preethi Pratap and
Madeleine Needles in the development of the VSRT, for the careful lab work of
the radio astronomy Union College students in the years 2010-2016, as well as
George Hassel for ensuring that the labs ran smoothly at Siena.

The development of the VSRTs was funded by the National Science Foundation
CCLI grant DUE-0817136 and the development of the VSRTI_Plotter Java package
by the NSF IIS CPATH award 0722203.

## References

  * (1) Nobel Lectures, Physics 1971-1980 (World Scientific Publishing Co., Singapore, 1992).
  * (2) Marr, J. M., Snell, R. L., & Kurtz, S. E., Fundamentals of Radio Astronomy: Observational Methods (Taylor & Francis, Boca Raton, FL, USA 2015).
  * (3) Wilson, T. L., Rohlfs, K., & Hüttemeister, S., Tools of Radio Astronomy, 5th ed. (Springer-Verlag, Berlin-Heidelberg, 2009), pp. 201–234.
  * (4) Burke, B. F. & Graham-Smith, F., An Introduction to Radio Astronomy, 3rd ed. (Cambridge University Press, Cambridge, UK, 2010).
  * (5) Thompson, A. R., Moran, J. M., & Swenson, Jr., G. W., Interferometry and Synthesis in Radio Astronomy, 3rd ed. (Springer Nature, Cham, Switzerland, 2017), available online at https://link.springer.com/book/10.1007%2F978-3-319-44431-4
  * (6) Kraus, J. D., Radio Astronomy, 2nd ed., (Cygnus-Quasar, Powell, Ohio, 1986).
  * (7) Arndt, M. B., & Rogers, A. E. E. VSRT Introduction (http://www.haystack.mit.edu/edu/pcr/vsrt-ret/VSRT_Introduction_Summer2009.pdf, 2009).
  * (8) Doherty, M., Fish, V. L., & Needles, M., The Physics Teacher 49, 546 (2011).
  * (9) http://www.haystack.mit.edu/edu/undergrad/VSRT/index.html
  * (10) Fish, V. L., Measuring the Solar Diameter with the Very Small Radio Telescope to be submitted to Astronomy Education Review.
  * (11) Mumford, W. W. & Schafersman, IRE Transactions on Microwave Theory and Techniques, 3, 12–17 (1955).
  * (12) http://www1.union.edu/marrj/radioastro/labfiles.html.