I've got two questions:
- What does a sync filter in a lock-in amplifier?
- What are pros and cons of using it?
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I think you should take a good look at a thorough description of a lock-in amplifier. Here is one
Short summary: A lock in amplifier synchronizes its own output to the phase of the input signal. It requires a reference signal at the frequency you want to amplify. The output level of the amplifier is derived from the output of the phase detector - at phaselock, this output is a DC signal proportional to the amplitude of the input signal. There is an AC signal at twice the reference frequency (2F) riding on the DC which must be filtered out else it will cause harmonics in the output.
This is easy at relatively high frequencies. At very low frequencies, the filters become very long - long delays and large time constants. This is where the synchronous filter comes in. The synchronous filter is a method of removing the 2F signal without using long filters. It has the advantage of providing better (faster) response to changes in the input.
Advantage is already mentioned. Disadvantage is that a synchronous filter can't remove other noise from the DC signal - it only takes out multiples of the reference frequency. If there is a lot of other noise at frequencies near the reference signal, then you will need a normal filter on the DC signal to get rid of the junk.