Response to Quasi-Monochromatic Radiation

Till now we had assumed that the radiation from the source was monochromatic. Let us now consider the more realistic case of quasi-monochromatic radiation, i.e. the radiation spectrum^4.4 contains all frequencies in a band $\Delta \nu$ around $\nu$, with $\Delta \nu$ small compared to $\nu$. If the radiation at some frequency $\nu$ arrives in phase at the two antennas in the interferometer, the radiation at some adjacent frequencies will arrive out of phase, and if $\Delta \nu$ is large enough, there will be frequencies at which the radiation is actually 180 degrees out of phase. Intuitively hence one would expect that averaging over all these frequencies would decrease the amplitude of the fringe. More rigorously, we have

$$r(\tau_g) = {1\over \Delta\nu}\int_{\nu - {\Delta\nu \over 2}}^{\nu + {\Delta\nu \over 2}} \cos(2\pi \nu \tau_g) d\nu$$ (4.3.3)

$$= {1\over \Delta\nu} Re \left[ \int_{\nu - {\Delta\nu \over 2}}^{\nu + {\Delta\nu\over 2}} e^{i2\pi \nu \tau_g}d\nu \right]$$

$$= \cos(2\pi\nu\tau_g) \left[ {\sin(\pi\Delta\nu\tau_g) \over \pi\Delta\nu\tau_g} \right]$$

The quantity in square brackets, the sinc function, decreases rapidly with increasing bandwidth. Hence as one increases the bandwidth that is accepted by the telescope, the fringe amplitude decreases sharply. This is called fringe washing. However, since in order to achieve reasonable signal to noise ratio one would require to accept as wide a bandwidth as possible^4.5, it is necessary to find a way to average over bandwidth without losing fringe amplitude. To understand how this could be done, it is instructive to first look at what the fringe would be for a spatially extended source.

Let the direction vector to some reference point on the source be ${\mathbf s_0}$, and further assume that the source is small that it lies entirely on the tangent plane to the sky at ${\bf s_0}$, i.e. that the direction to any point on the source can be written as ${\bf s = s_0 + }$ $\sigma$,   ${\bf s_0}$.$\sigma$$= 0$,   $\tau_g =$ ${\bf s_0}$.${\bf b}$. Then, from the van Cittert-Zernike theorem we have^4.6:

$$r(\tau_g) = \mathit{ Re} \left[\int \, I({\bf s}) e^{-i2\pi{\bf s}.{\bf b} \over \lambda} d{\bf s} \right]$$

$$= {\mathit Re} \left[ e^{-i2\pi{\bf s_0}.{\bf b} \over \lambda} \int I({\bf s}) e^{-i2\pi{\sigma}.{\bf b} \over \lambda} d{\bf s} \right]$$

$$= \vert\mathcal{V}\vert\cos(2\pi\nu\tau_g +\Phi_{\mathcal{V}})$$ (4.3.4)

where $\mathcal{V}$, the complex visibility is defined as:

$$\mathcal{V} = \vert\mathcal{V}\vert e^{-i\Phi_\mathcal{V}} = \int I({\bf s}) e^{2\pi{\sigma}.{\bf b} \over \lambda}$$ (4.3.5)

The information on the source size and structure is contained entirely in $\mathcal{V}$, the factor $\cos(2\pi\nu\tau_g)$ in eqn. (4.3.4) only contains the information that the source rises and sets as the earth rotates. Since this is trivial and uninteresting, it can safely be suppressed. Conceptually, the way one could suppress this information is to introduce along the electrical signal path of antenna $1$ an instrumental delay $\tau_i$ which is equal to $\tau_g$. Then we will have $r(\tau_g) = \vert\mathcal{V}\vert\cos(\Phi_\mathcal{V})$, i.e. the fast fringe oscillation has been suppressed. One can then average over frequency and not suffer from fringe washing. Since $\tau_g$ changes with time as the source rises and sets, $\tau_i$ will also have to be continuously adjusted. This adjustment of $\tau_i$ is called delay tracking. In most existing interferometers however, the process of preventing fringe washing is slightly more complicated than the conceptual scheme described above. The complication arises because delay tracking is usually done digitally in the baseband, i.e. after the whole chain of frequency translation operations described in Chapter 3. The geometric delay is however suffered by the incoming radiation, which is at the RF frequency.

Figure 4.3: A two element interferometer with fringe stopping and delay tracking (see text).

Footnotes

... spectrum^4.4

Radiation from astrophysical sources is inherently broadband. Radio telescopes however have narrow band filters which accept only a small part of the spectrum of the infalling radiation.

... possible^4.5

See Chapter 5

... have^4.6

apart from some constant factor related to the gain of the antennas which we have been ignoring throughout.