question on even mode and odd mode

In a paper, it says ?Condition for hybrid performance: The transmitted voltages for the even and odd modes must differ from each other by 90 degrees. This may be restated as L * (1/lambda_g_e ? 1/lambda_g_o) + Phi_r = 1/4. L is the length of the coupling section, lambda_g_e and lambda_g_o are the guide wavelengths for the even and odd modes, and Phi_r is the phase shift in the even mode contributed by the reflections form the ends of the coupling section.?



My question is what are even and odd modes here in a rectangular waveguide. And what are lambda_g_e and lambda_g_o (equations)? Are they the guided wavelength according to the cutoff frequencies of the even and odd modes?



Thanks!

I may be wrong, but I did not thing the concept of even/odd mode applied to rectangular waveguide. For a four port, even mode means you apply two in-phase signals (+ and +) to two inputs, and odd mode means you apply two out of phase signals (+ and -) to two inputs. Its just a mathematical concept that people use to calculate things like static capacitance/length to determine impedance. Two adjacent microstrips in "odd mode" will act like there is an "imaginary" ground plane right between them, for instance. If you put +1 volt on one strip's input, and -1 volt on the other strips input, there kind of has to be "0" volts right in the middle of the two strips.

In a waveguide there is no "voltage", because there is no center conductor or ground plane. So you can not say there is "zero voltage" at some point in the waveguide, because to even think of a voltage in a waveguide you have to integrate an Efield along a length. So you could not have an "odd mode" mathematical ground plane in the middle of a rectangular waveguide, because you are integrating over zero length (the thickness of the imaginary ground plane).

I know I am not saying this very clearly, but it my best try.

You have to have 2 waveguides joined somehow. Let?s say they are joined along the narrow sidewall. This wall has to be removed over a certain length ? coupling region.
In that region you have a waveguide that is twice as wide as the original ones.
Even mode is when you launch equal TE10 signals in phase in the two original small waveguides. They become a TE10 mode in the wide waveguide ? ie coupling section.
Odd mode is when the original TE10 signals are opposite in phase ? leading to a TE20 mode in the coupling section.
So the lambda_g_o corresponds to the guided wavelength of the TE20 mode in the coupling section.
I don?t know the details of your structure but this should help.