A Michelson-type radio interferometer for university education

The future of radio astronomy relies strongly on interferometers (e.g., ALMA, VLA, VLTI, aperture masking techniques).^1–3 From our experience at interferometer summer schools at the Nobeyama Radio Observatory⁴ and at the CARMA Observatory,⁵ we are convinced that hands-on experiments are critical to a full understanding of the concepts of interferometry. It is difficult, if not impossible, to obtain access to professional interferometers for university courses. Therefore, we built a low-cost radio interferometer for the purpose of education and developed corresponding syllabi for undergraduate and graduate astronomy lab courses.

This experiment teaches the basic concept of interferometry using the technique developed by Michelson and Peace in the early 20th century.⁶ They measured the diameter of Betelgeuse, one of the brightest stars in the sky, with a simple optical interferometer. Optical interferometry, however, needs high precision telescope optics. The same experiment becomes much easier when measuring the diameter of the Sun at radio wavelength, since the acceptable errors in the optics scale with the wavelength.

Figure 1 shows a conceptual sketch of our Michelson radio interferometer. This type of interferometer, adding signals instead of multiplying them, is appropriately called an “adding interferometer.” We discuss the mathematical background of such an interferometer in Sec. II, the design and development of the telescope and receiver system in Sec. III, the telescope setup and measurements in Sec. IV, and the results from a student lab report in Sec. V. A set of laboratory instructions are also available as supplementary material.⁷ What we present here is only one realization of the concept. Creative readers can modify any part of this project to meet the educational needs and constraints at their own institutions. For example, the astronomical measurement and the construction and tests of the telescope, receiver system, and other components could all be separate lab projects.

The best known Michelson interferometer is the one used for the Michelson–Morley experiment,⁸ one of the most important classical experiments taught in both lecture and laboratory courses.^9–12 Many studies and applications of the Michelson Interferometer have appeared in this journal,^13–23 and it is even currently being applied to the detection of gravitational waves.^24–26 The Michelson stellar interferometer is an application of the same physical concept of interference, in this case to a light source in the sky.

The theoretical basis of the Michelson stellar interferometer was already established in Michelson and Peace's original work,⁶ and has been used in radio interferometry, particularly in its early history.^27–30 This adding interferometer is the type used in modern astronomy at optical and near-infrared wavelengths,^31,32 though modern radio interferometers are of a different type, multiplying signals instead of adding them.^33,34 For educational purposes, some studies in this journal showed that the concept of the stellar interferometer could be demonstrated in an indoor laboratory setup using an artificial light source.^35,36 In professional optical astronomy, the technique is now being applied for advanced research.^31,32,37

The mathematical basis of the stellar interferometer is presented in Michelson and Peace's original work⁶ and can be found in textbooks.^27–30 Here, we describe the basic equations at a mathematical level that college students can follow.

We start from the geometric delay calculation in Sec. II A, and explain the total power, the parameter that we measure, in Sec. II B. We then show an example of point source (a commercial broadcast satellite) in Sec. II C and discuss the case of an extended source. Following this, we discuss how an interferometer measures Fourier components and define visibility in Sec. II D. Finally, we explain how visibility is measured with our interferometer, and how the Sun's diameter is derived in Sec. II E.

Interferometers mix signals received at two different positions, e.g., positions 1 and 2 in Fig. 2. In our radio interferometer, the signals that arrive at the two side mirrors (Fig. 1) are guided to the antenna and mixed. The separation between the two mirrors, called the baseline length B, causes a time delay τ in the arrival of the signal at position 2 because of the geometry (Fig. 2). Using the angle θ of telescope pointing and the angle θ₀ to an object in the sky, a simple geometric calculation provides the delay

where c is the speed of light. The small angle approximation $sin(θ−θ0)≈θ−θ0$ is used here because most astronomical objects have a small angular size.

Radio signals are electromagnetic radiation and can be described in terms of an electric field and a magnetic field. For simplicity, we consider only the electric field E in the following calculations, though this simplification does not limit the generality of the discussion. If we define the radio signal at frequency ν that is detected at position 1 (or reflected, if a mirror is there) at time t as

then the signal that is detected at position 2 at the same time is

where τ is the geometric time delay.

An adding interferometer adds the two signals, giving a total electric field of

and measures the total power. The radio frequency ν is typically large compared to a data sampling rate. Hence, the total power P(θ) detected by a receiver is a time average (or integration). Using the notation $⟨⋯⟩$ for the time average, we obtain

which leads to

In going from Eq. (6) to Eq. (7), we used standard trig identities along with the fact that terms such as $⟨ cos(4πνt)⟩$ and $⟨ sin(4πνt)⟩$ vanish when time averaged. Substituting from Eq. (1) with the small angle approximation, this result becomes

where B_λ ≡ B/λ is a normalized baseline length, with λ = c/ν the wavelength.

Equation (8) can be generalized for an extended object to

where $E(θ0)$ is an intensity/energy density distribution of the object. Our adding interferometer measures P(θ). In use, we slew the telescope across the object in the azimuthal direction and obtain fringes, i.e., variations in the power as a function of θ.

Commercial broadcast satellites are very small in angle and approximate point sources. The energy density of a point source is a δ-function at the position of the object, or when θ₀ = θ_c. By adopting a coordinate system so that θ_c = 0, $E$ becomes

When combined with Eq. (9), we obtain

As we sweep the telescope from one side of this object to the other, we should see a sinusoidal power response as a function of θ.

The top panel of Fig. 3 shows the theoretical fringe pattern for a point source. Our satellite dish (and any other radio telescope) has a strong directional sensitivity, so its response pattern tapers off away from the center. The pattern that we actually obtain is attenuated by the dish response pattern—the beam pattern—as shown in the bottom panel of Fig. 3.

Fringe measurements are useful in determining the baseline length B_λ. The total power is zero when the normalized baseline is B_λ(θ − θ₀) = n + 1/2, where n is an integer. The separation between adjacent null positions is thus δθ = 1/B_λ = λ/B.

An astronomical object often has an extended size. In general, an interferometer measures the Fourier transform of the energy density distribution $E(θ0)$⁠. Here, we prove this statement.

We define the visibility V₀(B_λ) by writing Eq. (9) as

where

and

Here, the visibility V₀(B_λ) and the phase shift Δθ are given by

and

which leads to

The first term $ei2πBλΔθ$ is a phase shift of a complex visibility; the visibility amplitude is therefore

This expression is a Fourier component of the object $E(θ0)$ at a baseline length of B_λ. The inverse length 1/B_λ is the angular size of the Fourier component in radians. Observations at long baseline lengths detect structures of small angular size (i.e., Fourier components corresponding to small angular structures), while those at short baseline lengths capture structures of large angular size.

Using the sun as the source, we measure P(θ) and calculate the visibility amplitude $|V0(Bλ)|$⁠. From Eqs. (13) and (17), we have

The top panel of Fig. 4 shows the theoretical fringe pattern for the “top-hat” function, which represents the Sun's disk as we scan across it. As before, this fringe pattern will be attenuated by the beam pattern, so the bottom panel of Fig. 4 is what we would expect to see by slewing the telescope across the sun. Although the fringe pattern will be attenuated, we assume that the antenna response is approximately constant near the peak. Thus, the maximum and minimum power are given by

and

From these, we calculate

which is the visibility amplitude at a baseline length of B_λ.

The two side mirrors slide on the ladder in Fig. 1 and change the baseline length. We repeat measurements of $|V0(Bλ)|$ at different baseline lengths and make a plot of $|V0(Bλ)|$ as a function of B_λ. Because $|V0(Bλ)|$ is a Fourier component of $E(θ0)$⁠, we should see the Fourier transformation of the emission distribution in the plot.

Assuming the Sun's diameter is α, the function $E(θ0)$ can be approximated as a top-hat function (rectangle function in one dimension)

with Fourier transform

This visibility amplitude is shown in Fig. 5 assuming a Sun diameter of α = 0.5°. By fitting this function to experimental data, the diameter of the sun can be deduced. (As mentioned, this presentation is for a one-dimensional approximation of the Sun's shape. A more ambitious exercise would be to treat the Sun's shape two dimensionally.)

Here, we describe the construction of the telescope and receiver system. Since budget is often the main limitation in the development of student lab experiments, we utilized low-cost parts and materials and used a commercial broadcast satellite dish and feedhorn operating at radio X-band. The system was constructed in our machine and electronics shops. Fabrication of the components could be offered as a student lab project.

Figure 1 shows the design of the Michelson stellar radio interferometer. Radio signals from the Sun hit two flat mirrors at the sides and are reflected to a satellite dish antenna by the central flat mirrors. The signals from the two sides are mixed as detected. Figure 6 shows photos of the telescope, which was built with mostly commercial products and materials. A broadcast satellite dish and feedhorn [Fig. 1 and Figs. 6(a) and 6(b)] operates at a frequency of ν ∼ 11 GHz (λ ∼ 2.7 cm). The required accuracy of optics at this wavelength is about ∼3–5 mm, which is relatively easy to achieve with flat mirrors (without curvature).

The flat mirrors are made of fiberboard with wooden framing structures, as shown in Fig. 6(e). The mirror surfaces are all angled 45° from the optical path. We originally covered their surfaces with kitchen aluminum foil, which has an appropriate thickness with respect to the skin depth (∼0.8 μm) at the operating wavelength (reflectivity ∼ 96% from our lab measurements). Later, we replaced the aluminum foil with thin aluminum plates for “student-proofing” [Fig. 6(d)]. The two side mirrors slide on a ladder to change the baseline length.

The azimuth-elevation mount structure is made with plywood; the axes are driven by motors [Fig. 6(c)] that are controlled by a paddle [handset in Fig. 6(b)]. A protractor [Fig. 6(f)] is placed at the center of the bottom mount plate for measurement of the azimuthal angle of the telescope. Figure 6(a) shows a complete view of the entire structure. A metal pole is mounted perpendicular to the top mount plate [Fig. 6(b)] and aluminum frame [Fig. 6(c)] to support the dish. Note that the pole should be perpendicular to the mount plate, which makes the pointing adjustment easier (discussed later).

The azimuthal rotation is facilitated by greased handcrafted ball bearings in circular grooves around the azimuth shaft on the base [below the bottom mounting plate in Figs. 6(a) and 6(b)] and on the bottom mount plate.

Sweeping across the Sun in azimuth permits fringe measurements. The telescope can be converted to a single-dish telescope by rotating the satellite dish by 180° around the metal pole (see Fig. 6(b)). Both single-dish and interferometer measurements can be easily made and compared, which is essential for appreciation of the high angular resolution possible with the interferometer.

Table I lists the commercial product parts that we purchased. The other parts, mostly the support structure, are made in the machine shop.

The signal detection system in radio astronomy is a series of electronic components. Figure 7 shows the design and photos of the receiver. Again, these are mostly commercial products.

Signals from the sky are at too high a frequency (∼11 GHz) to be handled electronically. Hence the Low Noise Block Feedhorn (LNBF) down-converts the frequency to a lower frequency, called the intermediate frequency (IF, 950–1950 MHz) by mixing the sky signal with a reference signal at a slightly offset frequency and producing a signal at the beat frequency of the sky and reference signals. This is called heterodyne receiving. The LNBF works as a heterodyne mixer.

Figure 7 shows the flow of signal. In sequence, an amplifier, two attenuators, and bandpass filter adjust the signal amplitude to the input range of a square-law detector. We combined two commercially available attenuators to achieve the desired attenuation of ∼16 db. A 100-MHz-width filter narrows the frequency range, since the bandwidth of the IF (1 GHz at the operating frequency of ∼11 GHz) is too broad for detection of null fringes in interferometry. Output from the detector is then amplified to the whole dynamic range of the analog-to-digital (A/D) converter. We assembled all these components inside a metal box for protection. A power supply is also in the box to power to the LNBF and amplifiers.

The output from the receiver box goes to a commercial LabPro A/D converter. The LabPro is connected via USB to, and controlled by, a laptop computer with LabPro software installed. This computer and software control the time integration and the sampling rate for voltage measurements.

Table II lists the electronics components that we purchased. The square-law detector (Schottky diode detector) was purchased through eBay, and similar devices are readily available there. We then found and purchased the amplifier and attenuators to adjust the signal voltage amplitude to match the input range of the detector and the output range of the LNBF when the telescope is pointing toward the Sun and satellites.

The mount structure, ladder, and mirrors of the telescope (Fig. 6) are detached when it is stored in our physics building. We move them with a cart to the front of the building and assemble them there on the morning of experiment. Using a triangle, we make sure that the flat mirrors are angled at 45° with respect to the optical path and 90° vertically. We then attach the ladder and mirrors to the mount structure using clamps mounted on the structure.

The next step is to connect the electronic components (Fig. 7). In sequence, the signal from the feedhorn goes to the receiver, then to the A/D converter LabPro, and finally to a computer via USB. We use software which comes with LabPro to control sampling frequency (integration time) and duration of recording.

Telescope pointing adjustment is the final step before the experiment. We prepare a table of the Sun's azimuthal and elevation angles as a function of time using an on-line tool provided by the U.S. Naval Observatory.³⁸ The antenna is set to the single-dish mode (i.e., dish facing toward the Sun). We align the planes of the mount's top plate and ladder parallel to sunlight using their shadows. The azimuth is set to that of the Sun, and we adjust the elevation angle of the dish to maximize the signal from the Sun on a voltage meter. We note that our dish is an off-axis paraboloid antenna, so the direction of the dish looks very offset from the direction of the Sun; visual alignment is misleading. We therefore need to use the voltage meter for alignment. We later installed a foot-long rod on the dish and marked a point (on the dish) at which the shadow of the rod tip falls when pointed toward the Sun. We then rotate the dish by 180° around the metal pole for interferometer measurements.

The signal amplitudes from the two side mirrors need to be balanced. We check the voltage readout from each side mirror separately by blocking the optical path of the other (or by removing the other mirror). We then move the central mirror toward the side of stronger signal to decrease its effective surface area.

Once the mirrors are set and the telescope is pointed toward the Sun, we begin taking interferometer measurements. We should see fringes (as in Fig. 4) as we slew the telescope and sweep across the Sun in the azimuthal direction. We typically spend 10–30 s on each “sweep” observation, and then correct the telescope pointing before the next sweep. The pattern may be seen as variations of the voltage readout, or as a fringe pattern in a plot if the LabPro and computer are in use. The LabPro and computer do not know about telescope pointing, and record only the readout voltage as a function of time. We therefore need to convert the time to azimuthal angle after the measurements. We record the start and end azimuthal angles in sweeping the Sun. Typically, we start from a far-off position, say 10°–20° away in azimuth, and sweep across the Sun. We assume that the telescope slew speed is constant (approximately correct when we record for a long time, e.g., 20–30 s). The projection effect, i.e., the cos(elevation) term, must be accounted for in calculation of arc length in the sky.

We change baseline length by sliding the side mirrors on the ladder and repeat fringe measurements. The baseline length should be determined from the fringe pattern, but for reference we record the side mirror separation using a tape measure fixed to the ladder.

Radio interference was initially a problem. We conducted a site search across the campus using the dish and a commercial receiver (a so-called satellite finder for ∼$10–20, which is used to find commercial television satellites when a dish is installed) and compared the strengths of the Sun and ambient radio signals. We conveniently found that one spot in front of our building was radio quiet.

Geosynchronous satellites are located along a thin belt in the sky so the Sun's sidereal path will be aligned along this belt in some seasons, which hinders the experiment. Thus, the Sun's position should be checked at the planning stage of the experiment.

Unfortunately, our telescope mount structure is slightly wider than a standard doorway. It does not fit on most of our elevators and cannot pass through exit doors of our building. We thus have to carry it out via a loading deck. Anyone contemplating construction of such a telescope would be wise to factor in such size restrictions.

The telescope can be used as a single-dish radio telescope by pointing the dish directly toward the sky. The beam size of our dish is roughly ∼1° in X band, with which we can barely resolve the Sun (∼0.5° diameter). We can compare the profiles of the Sun and a commercial satellite (a point source) to find this experimentally. On the other hand, the Sun's diameter can be easily resolved and determined with the interferometer. The comparison of the single-dish and interferometer measurements permits students to appreciate the superiority of interferometry in terms of spatial resolution.

Figure 8 shows results from a student group's lab report. Panel (a) is an example of a fringe pattern of the Sun from which students determine the baseline length by measuring the interval between peaks and troughs (and from their readings of the side mirror separation). This group repeated fringe measurements three times at each of ten different baseline lengths. Panel (b) shows a fit of Eq. (27), i.e., the Fourier transform of the Sun. The null point at B_λ = 96 in the fit suggests that their measurement of the Sun's diameter is $∼36′$ at ∼11 GHz. [Note that the Sun's reported diameter at ∼10 GHz is about $34′$ with little dependence on solar activity (i.e., sunspot number). This diameter is calculated from the observed radio-to-optical diameter ratios³⁹ and the optical diameter of $∼30′$⁠.] These results demonstrate a proof of concept demonstrated by our students, and a variety of exercises can be developed for a student lab beyond what is described here.

The authors thank Peter Koch, the previous Chair of the Department of Physics and Astronomy at Stony Brook University, for providing funds to develop this experiment, and Munetake Momose for useful discussion. The authors also thank students in the lab course, Kendra Kellogg, Melissa Louie, and Stephanie Zajac, for letting us use plots from their lab report. This work was supported by the NSF through Grant No. AST-1211680. J.K. also acknowledges support from NASA through grants NNX09AF40G, NNX14AF74G, a Herschel Space Observatory grant, and a Hubble Space Telescope grant.