Two Element Interferometers in Practice

To see this more clearly, let us consider the interferometer shown in Figure 4.3. The signals from antennas $1,2$ are first converted to a frequency $\nu_{BB}$ using a mixer which is fed using a local oscillator of frequency^4.7 $\nu_{LO}$, i.e. $\nu_{LO} = \nu_{RF} - \nu_{BB}$. Along the signal path for antenna $1$ an additional instrumental delay $\tau_i = \tau_g + \Delta \tau$ is introduced, as is also a time varying phase shift $\Phi_f$. The reasons for introducing this phase shift will be clear shortly. Then (see also equations 4.2.1 and 4.3.4) we have:

$$r(\tau_g) = \vert\mathcal{V}\vert \left<\cos(\Phi_\mathcal{V} + 2\pi\nu_{BB}t-2\pi\nu_{RF}\tau_g)\cos(2\pi\nu_{BB}(t-\tau_i) + \Phi_f)\right>$$ (4.4.6)

$$= \vert\mathcal{V}\vert cos(\Phi_\mathcal{V} + 2\pi(\nu_{RF} -\nu_{BB})\tau_g - \nu_{BB}\Delta\tau -\Phi_f)$$

$$= \vert\mathcal{V}\vert \cos(\Phi_\mathcal{V} + 2\pi\nu_{LO}\tau_g - \nu_{BB}\Delta\tau -\Phi_f)$$ (4.4.7)

So, in order to compensate for all time varying phase factors, it is not sufficient to have $\tau_i = \tau_g$, one also needs to introduce a time varying phase $\Phi_f = 2\pi\nu_{LO}\tau_g$. This additional correction arises because the delay tracking is done at a frequency different from $\nu_{RF}$. The introduction of the time varying phase is called fringe stopping. Fringe stopping can be achieved in a variety of ways. One common practice is to vary the instantaneous phase of the local oscillator signal in arm $1$ of the interferometer by the amount $\Phi_f$. Another possibility (which is the approach taken at the GMRT), is to digitally multiply the signal from antenna $1$ by a sinusoid with the appropriate instantaneous frequency.

Another consequence of doing delay tracking digitally is that the geometric delay can be quantized only upto a step size which is related to the sampling interval with which the signal was digitized. In general therefore $\Delta\tau$ is not zero, and is called the fractional sampling time error. Correction for this error will be discussed in the Chapter 9.

The delay tracking and fringe stopping corrections apply for a specific point in the sky, viz. the position $s_0$. This point is called the phase tracking center^4.8. Signals, such as terrestrial interference, which enter from the far sidelobes of the antennas do not suffer the same geometric delay $\tau_g$ as that suffered by the source. Consequently, delay tracking and fringe stopping introduces a rapidly varying change in the phase of these signals. On long baselines, where the fringe rate is rapid, the terrestrial interference could hence get completely decorrelated. While this may appear to be a terrific added bonus, in principle, terrestrial interference is usually so much stronger than the emission from cosmic sources, that even the residual correlation is sufficient to completely swamp out the desired signal.

We end this chapter by re-establishing the connection between what we have just done and the van Cittert-Zernike theorem. The first issue that we have to appreciate is that the van Cittert-Zernike theorem deals with the complex visibility, $\mathcal{V} =\vert\mathcal{V}\vert e^{-i\Phi_\mathcal{V}}$. However, the quantity that has been measured is $r(\tau_g) =\vert\mathcal{V}\vert\cos({-\Phi_\mathcal{V}})$. If one could also measure $\vert\mathcal{V}\vert\sin({-\Phi_\mathcal{V}})$, then of course one could reconstruct the full complex visibility. This is indeed what is done at interferometers. Conceptually, one has two multipliers instead of the one in Figure 4.3. The second multiplier is fed the same input as that in Figure 4.3, except that an additional phase difference of $\pi/2$ is introduced in each signal path. As can be easily verified, the output of this multiplier is $\vert\mathcal{V}\vert\sin({-\Phi_\mathcal{V}})$. Such an arrangement of two multipliers is called a complex correlator. The two outputs are called the sine and cosine outputs respectively. For quasi-sinsoidal processes, one has to introduce a $\pi/2$ phase difference at each frequency present in the signal. The corresponding transformation is called a Hilbert transform^4.9. How the complex correlator is achieved at the GMRT is described in Chapter 9. The output of the complex correlator is hence a single component of the Fourier transform of the source brightness distribution^4.10. The component measured depends on the antenna separation as viewed from the source, i.e. $({\bf b.s_0})/\lambda$, which is also called the projected baseline length. For a large smooth source, the Fourier transform will be sharply peaked about the origin, and hence the visibility measured on long baselines will be small.

Further Reading

1. Thompson, R. A., Moran, J. M. & Swenson, G. W. Jr., `Interferometry & Synthesis in Radio Astronomy', Wiley Interscience.
2. R. A. Perley, F. R. Schwab, & A. H. Bridle, eds., `Synthesis Imaging in Radio Astronomy', ASP Conf. Series, vol. 6.

Footnotes

... frequency^4.7

Note that it is important that the phase of the local oscillator signal be identical at the two antennas, i.e. the local oscillator signal has to be distributed in a phase coherent way to both antennas in the interferometer. Chapter 23 explains how this is acheived at the GMRT.

... center^4.8

For maximum sensitivity, one would also point the antennas such that their primary beam maxima are also at $s_0$.

... transform^4.9

see Chapter 1

... distribution^4.10

This is true only if the antenna dimensions are neglected. Strictly speaking, the measured visibility is an average over the visibilities in the range ${\bf b +a}$ to ${\bf b - a}$ where $a$ is the diameter of the antennas and b is the separation between their midpoints. As will be seen in Chapter 14 the fact that one has information on visibilities on scales smaller than $b$ is useful when attempting to image large regions of the sky.