Angular Co-ordinates

As described in Chapter 4, the response of an interferometer to quasi-monochromatic radiation from a point source located at the phase center is given by

$$r(\tau(t)) = cos(2\pi\nu_o \tau),$$ (10.1.1)

where $\tau=\tau_o = (D/c) sin(\theta(t))$ is the geometrical delay, $\theta$ is the direction which the antennas are tracking with respect to the vertical direction, $\lambda$ is the wavelength, $\nu_o$ is the center frequency of the observing band and $D$ is the separation between the antennas. As the antennas track the source, the geometrical delay changes as a function of time. This changing $\tau$ is exactly compensated with a computer controlled delay and for a point source at the phase center, the output of the interferometer is the amplitude of the fringe pattern.

For a source located at an angle $\theta = \theta_o + \Delta\theta$, for small $\Delta\theta$, $\tau=\tau_o + \Delta\theta (D/c)cos(\theta(t))$. Since fringe stopping compensates for $\tau_o$, the response of the interferometer for a source $\Delta\theta$ away from the phase center is $cos(2\pi \Delta\theta D_\lambda cos(\theta))$ where $D_\lambda = D/\lambda$. If the phase center is shifted by equivalent of $\lambda/4$, the interferometer will pick up an extra phase of $\pi/2$ and the response will be sinusoidal instead of co-sinusoidal. Hence, an interferometer responds to both even and odd part of the brightness distribution. Interferometer response can then be written in complex notation as

$$r(\tau(t)) = e^{2\pi\iota\Delta\theta D_\lambda cos(\theta)}.$$ (10.1.2)

Writing $u=D_\lambda cos(\theta)$, which is the projected separation between the antennas in units of wavelength in the direction normal to the phase center and $l=sin(\Delta\theta) \approx \Delta\theta$, we get

$$r(u,l) = e^{2\pi\iota u l} = e^{2\pi\iota u\Delta\theta}$$ (10.1.3)

as the complex response of a two element interferometer for a point source of unit flux located $\Delta\theta$ away from the phase center given by the direction $\theta_o$.

Usually the phase center coincides with the center of the field being tracked by all the antennas. Let the normalized power reception patter of antennas (which are assumed to be identical) at a particular frequency be $B(\Delta\theta)$ and the surface brightness of an extended source be represented by $I(\Delta\theta)$. The response of the interferometer to a point source located $\Delta\theta$ away from the phase center would then be $I(\Delta\theta)B(\Delta\theta)e^{2\pi\iota u \Delta\theta}$. For an extended source with a continuous surface brightness distribution, the response is given by

$$V(u) = \int{B(\Delta\theta)I(\Delta\theta)e^{2\pi\iota u\Delta\theta}d\Delta\theta} = \int{B(l)I(l)e^{2\pi\iota u l} dl}.$$ (10.1.4)

The above equation is a 1D Fourier transform relation between the source brightness distribution and the output of the visibility function $V$. The integral is over the entire sky visible to the antennas but is finite only for a range of $l$ limited by the antenna primary reception pattern $B(l)$. In practice, $u$ is calculated as a function of the source position in the sky, specified in astronomical co-ordinate system, as seen by the observer on the surface of the earth.

$l$ in the above equation is the direction of the elemental source flux relative to the pointing center. $u$ then has the interpretation of spatial frequency and $V(u)$ represents the 1D spatial frequency spectrum of the source.