Please explain in detail how to take an S21 measurement for an antenna? I know what it is... but how do I physically connect the antenna to the VNA to take the measurement? Thank you in advance!
In a two-port network, S21 is the complex number that describes the linear power gain of the network. (it's kinda backward - S2 is output, S1 is input).
It's a complex number because it is used to characterize both magnitude and phase changes to the input. There is a different gain number for every frequency of interest, because, in general, gain and phase vary with frequency.
A single antenna is in general a one-port network. It therefore does not have a Port 2. So, S21 does not make sense.
However, I think your confusion stems from the fact that many instruments can ONLY measure S21. (forward) In that case, a reflection bridge is often used to measure reflected power. It actually uses the forward power measurement S21 (yes, it connects to both ports 1 and 2) to determine the value of "S11" at a THIRD port (so, actually S33).
These bridges cleverly route all the power from S1 to S3. It also cleverly routes all the reflected power coming back from S3 to S2. So, S33, which is really the "S11" of port S3, is, due to the clever routing, just S21. And S21 is the one thing we can easily measure!
In short: A reflection bridge lets you use S21 to find "S11" on its third port.
An antenna is single port device so there is only S11 parameter. If there are more that one antenna ( like two different TX and RX Antenna) then only S12 and S21 will come in the picture, that is antenna coupling or antenna isolation matrix. This can be measured using two port of VNA.