2D Relation Between Sky and Aperture Planes

Below, we derive the generalized 2D Fourier transform relation between the visibility and the source brightness distribution in the ($u,v,w$) system. The geometery for this derivation is shown in Fig 10.3.

Let the vector $\bar L_o$ represent the direction of the phase center and the vector $\bar D_\lambda$ represent the location of all antennas of an array with respect to a reference antenna. Then $\tau_g=\bar D_\lambda . \bar L_o$. Note that $2\pi \bar D_\lambda . \bar L_o = 2 \pi w$ is phase by which the visibility should be rotated to stop the fringe. For any source in direction $\bar L=\bar L_o+ \bar \sigma$, the output of an interferometer after fringe stopping will be

$$V(\bar D_\lambda)=\int\limits_{4\pi} I(\bar L)B(\bar L)e^{2\pi\iota \bar D_\lambda.(\bar L- \bar L_o)} d\Omega.$$ (10.2.8)

The vector $\bar L = (l,m,n)$ is in the sky tangent plane, $\bar L_o$ is the unit vector along the $w$ axis and $\bar D_\lambda = (u,v,w)$. The above equation can then be written as

$$V(u,v,w)=\int\int I(l,m)B(l,m)e^{2\pi\iota (ul+vm+w(\sqrt{1-l^2-m^2}-1))} {dldm \over \sqrt{1-l^2-m^2}}.$$ (10.2.9)

Figure 10.3: Relationship between the ($l,m$) co-ordiantes and the ($u,v,w$) co-ordinates

If the array is such that all antennas are exactly located in the ($u,v$) plane, $w$ is exactly zero and the above equation reduces to an exact 2D Fourier transform relation between the source brightness distribution and the visibility. This is true for a perfect east-west array (like WSRT or ATCA). However to maximize the $uv$-coverage arrays like GMRT or VLA are not perfectly east-west. As mentioned earlier, the integrals in the above equation are finite for a small portion of the sky (being limited by the primary beam patter of the antennas). If the field of view being mapped is small (ie. for small $l$ and $m$) $\sqrt{1-l^2+m^2}-1 \approx -{1 \over 2}(l^2 + m ^2)$ and can be neglected. Eq. 14.1.1 becomes

$$V(u,v,w) \approx V(u,v,0) = \int\int I(l,m)B^\prime(l,m)e^{2\pi\iota (ul+vm)} dldm.$$ (10.2.10)

where $B^\prime = B/ \sqrt{1-l^2-m^2}$. Neglecting the $w$-term puts restrictions on the field of view that can be mapped without being effected by the phase error which is approximately equal to $\pi(l^2+m^2)w$. Eq. 10.2.10 shows the 2D Fourier transform relation between the surface brightness and visibility.

Since there are finite number of antennas in an aperture synthesis array, the $uv$-coverage is not continuous. Let

$$S(u,v)=\matrix{ 1,~{\rm for~all~ measured}~ ($u,v$)~ {\rm points} \cr 0,~ {\rm every~ where~ else.}~~~~~~~~~~~~~~~~ }$$ (10.2.11)

Then to represent the real life situation, assuming that $B(l,m)=1$ over the extent of the source, Eq. 10.2.10 becomes

$$V(u,v)S(u,v) = \int\int I(l,m)e^{2\pi\iota (ul+vm)} dldm.$$ (10.2.12)

Inverting the above equation and using the convolution theorem, we get $I^D = I*DB$ where $DB$ is the Fourier transform of $S$. $DB$ is the transfer function of the the telescope for imaging and is referred to as the Dirty Beam. $I^D$ represents the raw image produced by an earth rotation aperture synthesis telescope and is referred to as the Dirty Map. Contribution of Dirty Beam to the map and methods of removing these these effects will be discussed in greater detail in later lectures.

In all the above discussion, we have assumed the observations are monochromatic with negligible frequency bandwidth and that the ($u,v$) measurements are instantaneous measurements. None of these assumptions are true in real life. Observations for continuum mapping are made with as large a frequency bandwidth as possible (to maximize the sensitivity) and the data is recorded after finite integration. Both result into degradation in the map plane and these effects will be discussed in the later chapters.

Neglecting the $w$-term essentially implies that the source brightness distribution is approximated to be restricted to the tangent plane at the phase center in the sky rather than on the surface of the celestial sphere. At low frequencies, where the antenna primary beams are larger and the radio emission from sources is also on a larger scale, this assumption restricts the mappable part of the sky to a fraction of the primary beam. Methods to relax this assumption will also be discussed in a later lecture.