2.4 Fringe Stopping and Complex Correlator

With the Earth rotation, the cosine term of Eq.2.22 modulates the correlator output quasi-sinusoidaly with a natural fringe rate of

$$\nu_{LO} d\tau_g/dt \simeq \Omega_{earth} {\rm b} \nu_{LO} /c$$ (2.27)

which is of order of 10 Hz for b$= 300$ m baselines and $\nu_{LO} = 100$ GHz. Note that the fringe rate only depends on the effective angular resolution ( $b\nu_{LO}/c \simeq b/\lambda$ is the angular resolution, $2''$ in the above example).

The fringe rate is somewhat too high for simple digital sampling of the visibility. An exception is VLBI (because there is no other choice), although the resolutions are $< 1$mas. The usual technique is to modulate the phase of the local oscillator $\Phi_{LO}$ such that $\Phi_{LO}(t) = 2\pi\nu_{LO}\tau_g(t)$ at any given time. Then Eq.2.25 is reduced to

$$r_r = A_o \vert V\vert {\rm cos}(\pm 2 \pi \nu_{IF}\Delta\tau - \Phi_V)$$ (2.28)

(with the + sign for USB conversion, and the - sign for LSB conversion), is a slowly varying output, which would be constant for a point source at the reference position (or delay tracking center). This process is called Fringe Stopping, since it stops the fringe pattern modulation. After fringe stopping, we can no longer measure the amplitude $\vert V\vert$ and the phase $\Phi_V$ separately, since $r_r$ is now a constant for a point source. A modulation of the delay tracking could be used to separate $\vert V\vert$ and $\Phi_V$. Instead, it is more convenient and effective to use a second correlator, with one signal phase shifted by $\pi/2$. Its output is

$$r_i = A_o \vert V\vert {\rm sin}(\pm 2 \pi \nu_{IF}\Delta\tau - \Phi_V)$$ (2.29)

With both correlators, we measure directly the real $r_r$ and imaginary $r_i$ parts of the complex visibility $r$. The device is thus called a ``complex'' correlator.

Note:

From Eq.2.28, a delay tracking error $\Delta \tau$ appears as a phase slope as a function of frequency, with

$$\Phi(\nu_{IF}) = \pm 2 \pi \nu_{IF} \Delta\tau$$ (2.30)