Complex visibility
= Complex visibility
{title2=$V(u,v)$}
A complex visibility is a calibrated complex correlation of the signals received at a particular <interferometric baseline>. In a narrow field, with scalar radiation, the <far-field approximation for an outgoing source>, a narrow frequency channel, negligible noncoplanar phase curvature and a steady sky of mutually incoherent emitters,
$$
V(u,v)=\iint A(l,m)I(l,m)e^{-2\pi i(ul+vm)}\,dl\,dm.
$$
Here $l,m$ are small dimensionless sky-direction offsets, $I$ is brightness in the chosen channel, and $A$ is the <primary beam>. This is the <Fourier transform> of beam-weighted brightness. The two-dimensional formula requires those approximations: a wide field generally needs the third baseline coordinate and curved sky geometry.
For an explicit one-dimensional derivation, set the <primary beam> to one over the small field, and place independent emitters at angular offsets $\theta_j$. Choose a <wave phase> convention in which, after removing the <phase centre>'s delay, their received <complex amplitudes> are
$$
E_1(t)=\sum_j a_j(t),\qquad
E_2(t)=\sum_j a_j(t)e^{2\pi i u\theta_j},\qquad
\langle a_j a_k^*\rangle=I_j\delta_{jk}.
$$
Here $\delta_{jk}$ is the <Kronecker delta> and angle brackets denote an <expected value> estimated by time averaging a steady signal. The last equality states mutual incoherence: different emitters have zero averaged cross term, although each emitter's two received copies remain correlated. Multiplying and averaging gives
$$
V(u)=\langle E_1E_2^*\rangle
=\sum_{j,k}\langle a_j a_k^*\rangle e^{-2\pi i u\theta_k}
=\sum_j I_j e^{-2\pi i u\theta_j}.
$$
For continuous brightness this becomes $V(u)=\int I(\theta)e^{-2\pi i u\theta}\,d\theta$. Real brightness implies the <Hermitian symmetry> $V(-u)=V(u)^*$; it does not require brightness to be even. In particular, an asymmetric real source usually has a nonzero imaginary <complex visibility>. At zero baseline, $V(0)=\sum_j I_j$ is the total beam-weighted source strength. Ordinary separated-element correlations need not measure that zero-baseline value.