Boy, this is getting really bizzare!
Jian: Take a very simple example. Forget using any transmission lines to directly measure incident and reflected waves. Throw away the connecting Zo transmission lines and substitute lumped Zo loads. Exactly equivalent situation. We can still use the equation for Gamma. And to further simplify, we will use nothing but lossless lumped circuits, inductors and capacitors. It is now all just lumped element circuit theory. None of that confusing EM stuff. No transmission line, no loss, so all your problems with certain lossy transmission lines goes completely away!
Now calculate your G for Zin = 0 + jX and Zo = 0 - jX. D is zero and G is infinite. There is NO loss, there is NO transmission line. What we have is the reflection coefficient of an inductor (Zin = 0+jX) in a system normalized to a pure imaginary impedance (Zo=0-jX). The reflection coefficient is infinite. Does not matter if you conjugate the numerator or not.
Now, why is it infinite? Reasonable question. Recall, that we are just using pure lumped circuit theory. If this ridiculous infinite Gamma is an approximation, we have BIG problems. Instead, let's try to understand it.
Remember that S-parameters are measured with each port terminated in the normalizing impedance, in this case Zo = -jX. You put a -jX (=Zo) in parallel with a +jX (= Zin) and you have a perfect open circuit. This is a C in parallel with an L. This is a resonant circuit. You excite this lossless LC with a sinewave at the resonant frequency and the voltage on the LC circuit grows to infinity (force 1.0 A into an open circuit, you calculate the voltage!). This is what the infinite Gamma means, you have a resonant circuit and you are exciting it at the resonant frequency. There is no loss in this case, so discussion of loss is pointless. There is no transmission line either, so discussion of incident and reflected waves is also pointless.
If you calculate the reflection coefficient for the same inductor in a 50 Ohm system, you take out the capacitor and substitute a 50 Ohm resistor (still no transmission line). Now you get a complex Gamma. This is what we are all used to. In a pure real system (all ports same Zo) mag(Gamma) <= 1 for passive structures. This condition no longer holds when normalizing to complex Zo, as shown above. It also no longer holds when normalizing different ports to different Zo's.
All the above is pure lumped circuit theory. No transmission lines at all. No loss at all. But we still get your "ridiculous" Gamma. Me thinks what is needed is understanding the meaning of the result, it is nothing to do with loss and nothing to with transmission lines.
Jian, you continue to state that you can't distinguish between the incident and reflected waves on certain lossy transmission lines. You did not address the simple counter example I suggested (long line, turn off the excitation before the reflected wave arrives). If your ideas are correct, it has to work for that situation too.
Loucy: I don't know how familar you are with MoM. If I go over your head, let me know and I will simplify. Basically, all the planar MoM software out there that I am familar with solves for current in the metal. It does not solve for incident and reflected waves. It solves for the current under the condition that the voltage on the metal must be zero (or proportional to the current if there is loss).
OK, we got the current in the metal (true for all MoM software), now what do we do? In Sonnet's case, we have perfectly conducting sidewalls. We put an infinitely small gap between the sidewalls and the circuit and impress a voltage across that gap. Because the gap is infinitely small, our path of integration is infinitely short and we get one unique value for the voltage. The sidewalls are a perfect short circuit. Thus, the relation between the port voltages and port currents are the Y-parameters of the circuit. For a single given EM solution, there is only one voltage for each port and only one current. This is what I mean by unique. To within numerical precision and to within the accuracy of the underlying EM analysis, we get the exact answer.
For unshielded tools, one way to proceed is to extend a length of open circuit stub on the ground side of the port. Now, the port voltage is across a gap between the stub (on one side of the port) and the circuit (on the other side of the port). You still have to excite it with a voltage across the gap. But now, you can look at the standing wave (by viewing the current) that results and extract an incident and reflected wave. Now you can get S-parameters directly. I have a feeling that this is where Jian encounters his difficulty in certain lossy cases.
I have never explored that because we don't use this technique. This technique does indeed introduce approximations (due to voltage definition ambiguities, which require an integration of some kind from line to "ground"). I wonder if these might be the approximations that Jian is talking about, however, these approximations are for all cases except lossless, homogenous (same dielectric everywhere). Even though these approximations are usually fairly small, we do not tolerate them, so we do not proceed in this manner.
Basically, Loucy, a precise defintion of reflection coefficient can not come from incident and reflected waves (except for lossless, homogenous) because of ambiguity in determining the "line voltage". If you want precision, you have to use perfect short circuit terminations and calculate Y-parameters, and then covert to whatever form is desired by the user (usually S-parameters).
After we are all done in Sonnet, we convert the Y parameters, as desired, to S parameters by pure circuit theory. Couldn't care less (for the analysis output) as to what are incident and reflected waves. It is really quite a beautiful approach. I think gradually even those working in unshielded analysis will come to realize its significance.