In a Pound-Drever-Hall lock, an electronic signal is produced of the form:
My question is: how can we filter to get exactly this term (listed above) without any time-dependent terms?
I know that I can use a bandpass filter around the frequency $\Omega$ will isolate this term:
We don't want the time-dependent terms in the signal and instead we want the imaginary term without frequency:
In the literature, it says that we can combine this signal (using a mixer) with a frequency \$cos(\Omega t)\$ - and a low-pass filter.. but how exactly does this work?
Mixing the signals will produce a term \$cos(\Omega)^2\$. I understand that the average of the signal will have a DC-offset. But if my electronics has a high bandwidth, then this isn't really helping eliminate the time-dependence in these terms, right?
So, in summary, can I use a mixer and a low-pass filter to extract out that particular term? And, if so, how exactly does it work?
(This isn't that important for answering the question, but if you are interested) the function F(omega) is: And we are interested in isolating this particular term because it produces a good looking "error signal" that looks like this picture: