What are even and odd modes and impedances? How they related to differential and common mode impedances?

What are even and odd modes and impedances? How they related to differential and common mode impedances?

Even and odd modes are the two main modes of propagation of the signal through a coupled transmission line pair. Odd mode impedance is defined as impedance of a single transmission line when the two lines in a pair are driven differentially (with signals of the same amplitude and opposite polarity). Even mode impedance is defined as impedance of a single transmission line when the two lines in a pair are driven with a common mode signal (the same amplitude and the same polarity).

Differential impedance is defined as impedance between the two lines when the line pair is driven differentially. This definition effectively makes it equal to twice the odd mode impedance. Common mode impedance is defined as impedance between the two lines when the line pair is driven with common mode stimulus. This makes the common mode impedance to be equal to half the even mode impedance. As a result, even though in presence of coupling between the lines the even mode impedance is always higher than the odd mode impedance, the common mode impedance can be lower than the differential impedance.