2.1 Radio Interferometry Techniques
2.1.2 Two Element Interferometry
The basic measuring device in synthesis mapping is the interferometer. A two element interferometer has two identical antennas separated by a distanceb. Let us suppose the two antennas are directed towards a point source of some flux density, in a direction indicated by the unit vectors. Since the source is in the far field of the interferometer, the incoming wave front can be considered to be a plane wave.
The plane wave incident on the interferometer arrives at the right antenna first then at the left antenna (see Fig. 2.1). The time delay between two identical antennas with separationb, is given by, tg =b·s/c and is called the geometrical delay. Below we show the schematic diagram for the two element interferometer.
Figure 2.1: Diagrammatic representation of a two-element interferometer Taylor et al. (1999).
The signals from the antennas pass through amplifiers and filters which select the frequency of interest with bandwidthDn. The voltage signal response produced at the two antennas due to the electric field from this point source, are multiplied together and time-averaged in a device called the correlator (Taylor et al., 1999). That is, for input voltages from two antennas,V1(t)andV2(t), the correlator output is proportional to
<V1(t)V2(t)>. Below we represent the two voltage signals as,
V1(t) =v1cos2pn(t tg); V2(t) =v2cos2pnt, the correlator output is then,
r(tg) = 1 T
Z t+T/2
t T/2 v1v2cos(2pnt)cos(2pn(t tg))dt
= 1
2v1v2cos2pntg. (2.7)
We have assumed that the averaging timeT is larger compared to 1/n (Taylor et al., 1999). The result is the interferometer fringe pattern. The correlator output can be recast in terms of the radio brightness integrated over the sky. Let us suppose we haveI(s), representing the sky brightness in the directionsat the frequency n. However, ifA(s)is the effective collecting area of an antenna in the directions, the signal power received by each antenna over a bandwidthDn in a solid angle elementdWisA(s)I(s)DndW. So the correlator signal per solid angle elementdWis,
dr=A(s)I(s)DndWcos2pntg. Integrating over the celestial sphere, we obtain,
r=DnZ A(s)I(s)cos2pb·s
c dW. (2.8)
In practice, the angular response of the antenna element falls rapidly to small values outside a narrow angular width defined by their diameter i.e. for large dishes. It is usually more convenient to refer measurements to a reference positions0, commonly referred to as the phase tracking centre (Taylor et al., 1999). Then we haves
=s0+s and,
r=Dncos
✓
2pnb·s0
c
◆Z
A(s)I(s)cos
✓2pnb·s c
◆
dW (2.9)
Dnsin
✓
2pnb·s0
c
◆Z
A(s)I(s)sin
✓2pnb·s c
◆ dW.
The complex visibility is defined as,
V =|V|eifV =
Z A0(s)I(s)e 2pinb·s/cdW, (2.10)
In eqn. (2.10),A0(s)⌘A(s)/A0 is the normalized antenna beam pattern. Using eqn. (2.10) , we write eqn.
(2.9) as,
r=A0Dn|V|cos
✓2pnb·s
c fV
◆
. (2.11)
An interferometer is a machine for measuring the visibility, which is nothing more than the spatial coher- ence function with a different normalization. The amplitude and phase of the visibility (see eqn. (2.12)) is determined after application of suitable calibration.
In practice, in order to make use of eqn. (2.10), we introduce a convenient coordinate system. A commonly used system is the one where the baseline vector is specified in a coordinate system represented by(u,v,w), wherew points in the direction of interest, herew is chosen to point towards the phase tracking centre (s0).
The coordinates (u,v) are components projected onto the plane perpendicular tow, whereupoints toward the East andvpoints toward the North (Taylor et al., 1999). However, for phase centres not at the zenith, they do not necessarily point to the usual North and East directions.
The coordinates(u,v,w)are measured in wavelengths. However, the positions on the sky are defined byl andmi.e. the direction cosines measured with respect tou andvaxes. Thus a synthesized image in thel-m plane represents a projection of the celestial sphere onto a plane tangent to thel-morigin (Taylor et al., 1999).
In these coordinates, we have,
nb·s
c =ul+vm+wn, nb·s0
c =w, dW= dldm
n = dldm
p1 l2 m2, so that eqn. (2.10) can be written as,
V(u,v,w) =Z Z A0(l,m)I(l,m)e 2pi[ul+vm+w(p1 l2 m2 1)] dldm
p1 l2 m2. (2.12)
Two assumptions are required for eqn. (2.12) to reduce to a two dimensional Fourier transform. First when|l| and|m|are sufficiently small, we have,
w⇣p
1 l2 m2 1⌘
⇡ 1
2 l2+m2 w⇡0,
for|l|,|m|small, i.e. small field imaging, the dependence of the visibility uponw is very small and can be omitted (Taylor et al., 1999). Another assumption, if the baselines are coplanar,wlie in the direction of the celestial pole so,w⇡0 (Taylor et al., 1999). With the above two assumptions, eqn. (2.12) can be written as,
V(u,v) =Z Z A0(l,m)I(l,m)e 2pi(ul+vm)dldm. (2.13) In principle in order to invert the visibilities, eqn. (2.13), we need to have measured the fulluv-plane, which we never do. However, eqn. (2.13) can be inverted to give,
A0(l,m)I(l,m) =Z Z V(u,v)e2pi(ul+vm)dudv. (2.14) V(u,v)is a wave and so consist of the amplitude and phase. To summarize, an interferometer is a device for measuring the amplitude and phase of the complex visibility function. By Van Cittert-Zernike Theorem, the visibility is related to the sky brightness. If the measurement of the visibility is confined to a plane, or if only a small region of the sky is considered,V(u,v)and I(l,m)reduce to a Fourier transform pair. And since the Fourier theory states that any well behaved signal (including images) can be decomposed into its sinusoidal components, we use this theory to retrieve the amplitude and phase of the complex visibility function.