Radio interferometry

In radio interferometry, multiple radio antennas or apertures (elements) are used for observation in order to increase angular resolution. Without interferometry, radio telescopes would have resolutions comparable to the size of the sun and the moon. Let us begin our discussion by understanding a standard two-element interferometer in one dimension.

Consider a simple source in the far field generating plane waves at a single frequency $\nu$, two identical antennas receiving only one polarization at $\nu$ without any distorting effects either along the intervening media or in the receiving system. The geometric configuration is shown below.

The baseline vector $\mathbf{b}$ between the phase centers of the antennas is written $\mathbf{b}_\lambda$ when expressed in terms of wavelength $\lambda$ (b_\lambda=b/\lambda). The unit vector $\mathbf{s}$ indicates the direction toward the source, and the projected baseline is perpendicular to it. The signal arrives at the reference antenna to the right first and then at the other antenna after a time delay $\tau_g$ related to the physical path length difference $\Delta l_g$ because

$$ \tau_g = \frac{\Delta l_g}{c} = \frac{b\sin\theta}{c} $$

in seconds, which is related to the number of cycles ($\Delta l_g/\lambda = b_\lambda\sin\theta$) and the phase difference

$$ \phi = 2\pi\nu\tau_g = 2\pi b_\lambda \sin\theta $$

which are all time-dependent because $\theta$ changes with time. The number of cycles is, effectively, the number of fringe spacings (the gap between interference fringes) per radian.

The signals from the two antennas are combined in the combiner: either added or multiplied.

The observation of a single point source by an adding interferometer, made of two antennas of diameter $D$ and spacing $d$, is shown in this polar diagram. An interference pattern with fringe spacing $\lambda/d$ is visible within a diffraction pattern of a single antenna having a width of $\sim \lambda/D$. The system noise is generated by the receiver itself. The receiver output created by the measured voltages from the two antennas $V_A$ and $V_B$ will be

$$ R_{A+B}(\tau_g) = \langle [V_A \cos\omega t + V_B\cos\omega(t-\tau_g) + V_r]^2 \rangle $$

where $\omega=2\pi\nu$ is the angular frequency, $V_r$ is the receiver noise, and $\langle\rangle$ denotes time-averaging. Expanding the right hand side, we get

$$ R_{A+B}(\tau_g) \approx \langle [V_A^2 \cos^2 \omega t + V_B^2 \cos^2 \omega(t-\tau_g) + 2 V_AV_B \cos\omega t \cos\omega(t-\tau_g) + V_r^2] \rangle $$

where the cross-terms between the signals and the receiver noise has been ignored because they do not have any long-term phase relationship. The third term on the right hand side can be written as $\langle [\cos\omega\tau_g + \cos(2\omega t-\omega\tau_g)] \rangle$ using the relevant trigonometric identity.¹⁾

The second term $\cos(2\omega t - \omega \tau_g)$ oscillates with RF frequency (as opposed to IF frequency) and, therefore, reduces to zero due to time-averaging. And we know that $\langle \cos^2\omega t \rangle=1/2$ which leads to

$$ R_{A+B}(\tau_g) = \frac{1}{2} [(V_A^2+V_B^2)+V_r^2] + \langle V_AV_B\cos\omega\tau_g \rangle $$

where the first part have the total power terms dominated by the system noise and the second part varying cosinusoidally with time delay represents the interference effects.

Adding interferometers are dominated by gain fluctuations in the total-power terms which might be okay for strong sources, but very bad for weak, i. e. most, sources. In a correlation interferometer, the total-power terms vanish.

According to the definition of cross-correlation, the correlation between two voltage signals can be written as

$$ R_{AB}(\tau) = \langle V_A \cos\omega t V_B \cos\omega(t-\tau) \rangle $$

which can be reduced to only a fringe (interferential) term

$$ R_{AB}(\tau) = \frac{1}{2} \langle V_AV_B\cos\omega\tau \rangle $$

where the total-power terms are absent, unlike the equation for the adding interferometer, and this is plotted below.

The dotted and dashed curves represent two signals (detected by two antennas) phase-shifted with respect to each other, the solid curve shows their multiplication and the solid line the corresponding time-average. As the direction toward the source changes (due to the rotation of the earth), $\tau$ changes and so does the relative phase. At the top phase shift is close to zero, in the middle it becomes $90^\circ$, and at the bottom $180^\circ$. The fringe amplitude varies from $+0.5$ to $-0.5$. One cycle of the RF phase shift corresponds to one fringe cycle.

Here you see the fringe oscillations (cycles) for a point source located on the celestial equator observed by a two-element interferometer lying along the terrestrial equator creating a 15-$\lambda$ long east-west baseline.

The fringes oscillate around a zero mean and there are exactly 15 maxima on each side of the center, where $\theta=0^\circ$. The fringes near the center are more sinusoidal because there the small-angle approximation ($\sin\theta=\theta$) is more applicable. As $b_\lambda=15$, $\lambda/b=1/15 \sim 4^\circ$ which is the angular separation between successive lobes. The fringe spacing increases away from the center because the projected baseline $b\cos\theta$ decreases.