# Fringe Stopping

## 1 Fringe Rates

In our discussion of earth-rotation synthesis, we examined the idea of how, as the earth rotates, the projection of a baseline toward a source changes. This changing baseline projection versus time causes the phase measured by the baseline relative to that source to oscillate as a sinusoid that gradually changes period, depending on where the source is on the sky. Recall that in our discussion of the measurement equation, we wrote down an expression for phase of a visibility as a function of the delay between when the wavefront from a source hit antennas i and j:

$V_{ij}(\nu ) \propto e^{-2\pi i\tau _{ij}\nu } = e^{-2\pi i\frac{\vec b\cdot \hat s}{\lambda }} \,\!$

Since, as shown above, the delay $\tau =\frac{\vec b_{ij}\cdot \hat s}{c}$ is related to the projection of the baseline toward a source, we can write down the maximum rate at which τ could be changing if the only thing that is moving the antennas is the earth rotating:

$\frac{d\tau }{dt}\le \Omega _\oplus \frac{b}{c}. \,\!$

That is to say, the rate of change of delay must be less than the maximum delay across the baseline (b / c), rotated by the angular frequency of the earth’s rotation, $\Omega _\oplus$. Now we can write down the maximum fringe rate, which is the rate at which a fringe goes through a full oscillation:

$f_{max}=\nu \frac{d\tau }{dt}=\Omega _\oplus \frac{b}{\lambda } \,\!$

So the upshot is that the longer you baseline is, in units of wavelengths, the higher your fringe rate.

## 2 Fringe Stopping

For very long baselines (or high frequencies) fringe rates can get quite large. If you are observing at 100 GHz with a 100-m baseline, the maximum fringe rate is 2.4 Hz. This means that your correlator integration time needs to be significantly shorter than this — say 100 ms – to avoid washing out your fringes by integrating for a substantial fraction of the period. You can see that if you have a large number of antennas, this can set a data rate that is quite annoyingly high.

The solution to this problem is to stop the fringe at the center of your field of view. This is done by mixing the natural fringe rate of the measurement with an oscillating sine wave that corresponds to your prediction of the fringe rate of the phase center, given your known baseline length and orientation. Thus, instead of the full fringe-rate, you get the beat frequency between the actual fringe-rate and your predicted fringe-rate. If you’ve done a good job predicting your fringe rate, this can allow you to integrate over much longer timescales. Fringe stopping is often, though not always, by changing the phase of the local oscillator that mixes the IF signal directly from the antenna feed down to a frequency that can be digitized.

The hard part about fringe stopping is that you need to know your antenna positions pretty well (though not perfectly) before you observe, which means you can’t rely on self-calibration for that. Another annoying aspect of fringe stopping is that your fringe-rate depends on where you are looking. Fringe rates near the horizon can be nearly zero; fringe rates near zenith approach a maximum (the exact location of the maximum fringe rate depends on the latitude you are observing from). Hence, it is impossible to stop fringes over a wide field of view. While you may be stopping the fringe at your phase center, the beating of the natural fringe rate with your predicted fringe rate will increase away from phase center. If you are observing with antennas that have wide fields of view, and you integrate for too long, then you will find that your effective field of view will have been narrowed as you begin to wash out sources with higher fringe rates away from phase center. Consider yourself warned.