A two-element radio interferometer is the simplest building block of aperture-synthesis radio astronomy. It consists of two spatially separated antennas whose voltage outputs are multiplied and time-averaged by a *correlator*. Even very large arrays with \(N \gg 2\) antennas can be understood as a collection of \(N(N-1)/2\) independent two-element interferometers, each sampling one Fourier component of the sky brightness distribution.

Consider two identical antennas separated by a baseline vector \(\vec{b}\) of length \(b\), pointing toward a distant source in the direction of the unit vector \(\hat{s}\). If the angle between \(\vec{b}\) and \(\hat{s}\) is \(\theta\), then the plane wave from the source arrives at the two antennas at slightly different times. The extra path the wavefront travels to reach antenna 1 is

$$ \vec{b}\cdot\hat{s} = b\cos\theta, $$

which introduces a geometric delay

$$ \tau_g = \frac{\vec{b}\cdot\hat{s}}{c}. $$

If the interferometer is quasi-monochromatic, responding only to a very narrow bandwidth \(\Delta \nu \ll 2\pi/\tau_g\), then the signals received by the two antennas can be written as

$$ V_1(t) = V \cos[\omega(t - \tau_g)], \qquad V_2(t) = V \cos(\omega t), $$

where \(\omega = 2\pi\nu\) is the angular frequency of observation and \(V\) is the voltage amplitude.

The correlator multiplies the signals:

$$ V_1 V_2 = V^2 \cos[\omega(t - \tau_g)]\cos(\omega t) = \frac{V^2}{2}\left[\cos(2\omega t - \omega\tau_g) + \cos(\omega\tau_g)\right]. $$

A time average over many cycles removes the rapid oscillation at \(2\omega t\), leaving only the slowly varying term:

$$ R = \langle V_1 V_2 \rangle = \frac{V^2}{2}\cos(\omega\tau_g). $$

This quantity \(R\) is the correlator output. Its amplitude is proportional to the source flux density \(S\) multiplied by the geometric mean of the antenna collecting areas \((A_1 A_2)^{1/2}\). Its phase encodes precise information about the source direction.

Because the average of the product of uncorrelated noise is zero, the interferometer has no DC output. This suppresses:

* receiver-gain fluctuations * very extended sky signals such as the CMB * short impulsive interference that does not reach both antennas at the same time

This behaviour is fundamentally different from an adding interferometer (e.g., optical Michelson), which sums power rather than multiplying voltages.

As the Earth rotates, the relative angle \(\theta\) changes, and the correlator output traces a sinusoidal fringe:

$$ R(t) = \frac{V^2}{2}\cos(\omega\tau_g(t)). $$

The fringe phase is

$$ \phi = \omega\tau_g = \frac{\omega}{c} b \cos\theta, $$

and its sensitivity to angle is

$$ \frac{d\phi}{d\theta} = -\frac{\omega b}{c}\sin\theta = -2\pi \left(\frac{b\sin\theta}{\lambda}\right). $$

One full fringe cycle \(\Delta\phi = 2\pi\) corresponds to an angular shift

$$ \Delta\theta = \frac{\lambda}{b\sin\theta}, $$

which becomes extremely small when the projected baseline \(b\sin\theta\) spans many wavelengths. This makes interferometers exceptionally powerful for astrometry, routinely achieving absolute positional accuracies \(\sim10^{-3}\,\mathrm{arcsec}\) and differential accuracies \(\sim10^{-5}\,\mathrm{arcsec}\).

If the antennas were isotropic, the interferometer would simply measure a pure sinusoid across the sky—one Fourier component of the brightness distribution. With directive antennas, this sinusoid is multiplied by the individual antenna voltage beams, producing a primary beam envelope (often Gaussian). The Fourier transform of this product corresponds to a finite range of angular spatial frequencies around \(b\sin\theta/\lambda\), set by the antenna diameter \(D\). Because \(D < b\), the interferometer cannot measure the lowest spatial frequencies and therefore cannot detect an isotropic sky component such as the 3-K cosmic microwave background. These missing short spacings can be filled by a sufficiently large single-dish telescope.

Improving instantaneous imaging performance requires more baselines. An array of \(N\) antennas produces \(N(N-1)/2\) simultaneous baselines. The average of the point-source responses of all baselines produces the dirty beam, which approaches a Gaussian with full width \(\theta \simeq \lambda/b\) as the number of antennas grows. Missing short spacings produce negative sidelobe structures such as the characteristic “bowl.”

Because sky brightness distributions usually do not change on observation timescales, one can use Earth-rotation synthesis or movable antennas to accumulate many baselines over time. A single two-element system with movable antennas could reproduce the baselines of a larger array by observing multiple spacings sequentially.

* A two-element interferometer measures the Fourier component of the sky brightness corresponding to its projected baseline. * The correlator output is proportional to \(\cos(\omega\tau_g)\); its phase encodes source position with extremely high precision. * No DC output means large-scale emission, receiver gain drifts, and uncorrelated noise are suppressed. * The fringe spacing \(\Delta\theta = \lambda/(b\sin\theta)\) sets the interferometer’s angular sensitivity. * Directive antennas impose a primary beam envelope on the fringes, limiting the range of spatial frequencies sampled. * Large arrays combine many baselines to synthesize a beam whose width is roughly \(\lambda/b\). * Missing short baselines cause negative bowls and sidelobes in the dirty beam; single-dish data can fill in these spacings.

- Explain how the geometric delay \(\tau_g\) arises from the baseline geometry of a two-element interferometer. - Derive the expression \(R = \frac{V^2}{2}\cos(\omega\tau_g)\) starting from the input voltages. - How does the Earth’s rotation generate interferometric fringes? - Why does a multiplying interferometer have no DC response, and why is this advantageous? - What spatial frequency of the sky brightness distribution is sampled by a baseline of length \(b\)? - Why can’t a two-element interferometer detect an isotropic signal such as the CMB? - How do missing short spacings influence interferometric imaging, and how can they be recovered? - Explain why interferometric astrometry can exceed the angular precision of individual telescope tracking.