A Two Element Interferometer

Consider a two element interferometer shown in Figure 4.1. Two antennas $1,2$ whose (vector) separation is b, are directed towards a point source of flux density S. The angle between the direction to the point source and the normal to the antenna separation vector is $\theta$. The voltages that are produced at the two antennas due to the electric field from this point source are $v_1(t)$ and $v_2(t)$ respectively. These two voltages are multiplied together, and then averaged. Let us start by assuming that the radiation emitted by the source is monochromatic and has frequency $\nu$. Let the voltage at antenna $1$ be $v_1(t) = \cos(2\pi\nu t)$. Since the radio waves from the source have to travel an extra distance $b\sin\theta$ to reach antenna $2$, the voltage there is delayed by the amount $b\sin\theta /c$. This is called the geometric delay, $\tau_g$. The voltage at antenna $2$ is hence $v_2(t) = \cos(2\pi\nu (t -\tau_g) )$, where we have assumed that the antennas have identical gain. $r(\tau_g)$, the averaged output of the multiplier is hence:

$$r(\tau_g) = {1 \over {\rm T}} \int_{t-T/2}^{t+T/2} \cos(2\pi \nu t)\cos(2\pi \nu (t-\tau_g)) dt$$ (4.2.1)

$$= {1 \over {\rm T}} \int_{t-T/2}^{t+T/2} (\cos(4\pi \nu t - 2\pi\tau_g) +\cos(2\pi \nu \tau_g)) dt$$

$$= \cos(2\pi \nu \tau_g)$$

where we have assumed that the averaging time T is long compared to $1/\nu$. The $\cos(4\pi\nu t)$ factor hence averages out to 0. As the source rises and sets, the angle $\theta$ changes. If we assume that the antenna separation vector, (usually called the baseline vector or just the baseline) is exactly east west, and that the source's declination $\delta_0 =0$, then $\theta = \Omega_E t$, ( where $\Omega_E$ is the angular frequency of the earth's rotation) we have:

$$r(\tau_g) = \cos(2\pi \nu\ \times\ b/c\ \times\ \sin(\Omega_E (t-t_z)))$$ (4.2.2)

where $t_z$ is the time at which the source is at the zenith. The output $r(\tau_g)$, (also called the fringe), hence varies in a quasi-sinusoidal form, with its instantaneous frequency being maximum when the source is at zenith and minimum when the source is either rising or setting (Figure 4.2).

Figure 4.2: The output of a two element interferometer as a function of time. The solid line is the observed qausi-sinosoidal output (the fringe), the dotted line is a pure sinusoid whose frequency is equal to the peak instantaneous frequency of the fringe. The instantaneous fringe frequency is maximum when the source is at the zenith (the center of the plot) and is minimum when the source is rising (left extreme) or setting (right extreme).

Now if the source's right ascension was known, then one could compute the time at which the source would be at zenith, and hence the time at which the instantaneous fringe frequency would be maximum. If the fringe frequency peaks at some slightly different time, then one knows that assumed right ascension of the source was slightly in error. Thus, in principle at least, from the difference between the actual observed peak time and the expected peak time one could determine the true right ascension of the source. Similarly, if the source were slightly extended, then when the waves from a given point on the source arrive in phase at the two ends of the interferometer, waves arising from adjacent points on the source will arrive slightly out of phase. The observed amplitude of the fringe will hence be less than what would be obtained for a point source of the same total flux. The more extended the source, the lower the fringe amplitude^4.2. For a sufficiently large source with smooth brigtness distribution, the fringe amplitude will be essentially zero^4.3. In such circumstances, the interferometer is said to have resolved out the source.

Further, two element interferometers cannot distinguish between sources whose sizes are small compared to the fringe spacing, all such sources will appear as point sources. Equivalently when the source size is such that waves from different parts of the source give rise to the same phase lags (within a factor that is small compared to $\pi$), then the source will appear as a point source. This condition can be translated into a limit on $\Delta\theta$, the minimum source size that can be resolved by the interferometer, viz.,

$$\pi \nu \Delta \theta b /c~~ \lesssim ~~\pi \hskip 0.25in \L... ...ightarrow \hskip 0.25in \Delta \theta ~~\lesssim ~~\lambda / b$$

i.e., the resolution of a two element interferometer is $\sim \lambda / b$. The longer the baseline, the higher the resolution.

Observations with a two element interferometer hence give one information on both the source position and the source size. Interferometers with different baseline lengths and orientations will place different constraints on the source brightness, and the Fourier transform in the van Cittert-Zernike theorem can be viewed as a way to put all this information together to obtain the correct source brightness distribution.

Footnotes

... amplitude^4.2

assuming that the source has a uniform brightness distribution

... zero^4.3

This is related to the fact that in the double slit experiment, the interference pattern becomes less distinct and then eventually disappears as the source size is increased (see e.g. Born & Wolf, `Principles of Optics', Sixth Edition, Section 7.3.4). In fact the double slit setup is mathematically equivalent to the two element interferometer, and much of the terminology in radio interferometry is borrowed from earlier optical terminology.