On Forward and Backward Modes in 1D Periodic Bounded Structures

We discuss the issue of identifying the forward/backward nature of modes in bounded one-dimensional periodic structures. This identification is based on the possibility of adequately and uniquely defining the phase velocity in these types of structure. We propose a general definition of phase velocity for one-dimensional scalar waves and show that, according to that general definition, the voltage and current waves in nonhomogeneous lossless transmission lines with positive per-unit-length capacitance and inductance are necessarily forward waves. We analyze in detail the particular case of periodic transmission lines and question the conclusions about the forward/backward nature of their modal solutions that are traditionally drawn from the inspection of the Brillouin diagrams. Numerical results for the case of corrugated parallel-plate waveguides support the idea that all modes can be considered forward-like as long as a periodic transmission line model remains a sensible and reliable description of the problem. In more general scenarios, we show that an appropriate definition of the phase velocity can still be found for electromagnetic waves with at least one linearly polarized field and that they are also necessarily forward waves if they propagate through media with positive $\varepsilon$ and $\mu$ parameters. Finally, we relate our discussion to the effective refractive index of periodic structures, highlighting that although its definition is not valid for a general periodic structure, it can be useful in many practical cases.