s11 s12
Sij=Vi-/Vj+, where I is the output port and J is the input port, - is incident onto the port from the network and + is incident from the port to the network. The port has a reference plane from which we relate everything to.
But how is this valid in non-TEM waveguides?
Problem 1: We can't measure V or I in RF. (Remember we redefined voltage and current).
Solution: Measure it using power. (Sij=sqrt(Pi-)/sqrt(Pj+), where P+=0.5|V+|2/Z, P-=0.5|V-|2/Z, so we defined the a=V+/sqrt(Z), b=V-/sqrt(Z). and the powers become P+=0.5|a|2, P-=0.5|b|2. Therefore Sij=b/a.
Problem 2: This wll cause the S parameters to be dependant on Zc.
Solution: We'll use Zc=50 Ohms as our standard, relate everything to it.
Problem 3: We need a matched load over the RF spectrum to be able to measure the S parameters.
Solution: Use the standard 50 Ohms load.
Solution 3: Futher discussions
Some people will say the standard load termination will still cause reflections into the port, thats correct, but it'll cause no input wave from the terminated port. i.e. The reflected wave is reflected from the inside of the port, not incident from outside the refrence plane (remember V+ is defined as incident from outside the refrence plane), so it won't contradict out definition of the S-parameters.
You can check Wikipedia's link on S parameters for more info