Seminarium: Discussion on Forward and Backward Modes in 1D Periodic Bounded Structures

Periodic structures are typically analysed based on their dispersion (or Brillouin) diagrams, which gives much information about the characteristics of the wave propagation in the structure. In this talk, we discuss one particular characteristic, namely the "forward" or "backward" nature of the waves. Traditionally, waves are identified as "forward" or "backward" based on whether the phase and group velocities have the same or opposite signs. However, because modal solutions in periodic media consist of an infinite combination of spatial harmonics, this conventional definition of phase velocity can lead to incorrect interpretations.

We present a generalized definition of phase velocity applicable to one-dimensional scalar waves. We show that waves in periodic transmission lines are forward as long as the constitutive parameters (per unit length capacitance and inductance) are positive. In other words, modal solutions sometimes classified as "backward" are demonstrated to be forward-like waves dominated by higher-order spatial harmonics. This interpretation is supported by simulation results of 1D photonic crystals and periodically corrugated parallel-plate waveguides. We also discuss vector waves and highlight that in this case, a similar conclusion is not possible unless at least one of the vector fields (the E or H-field) is linearly polarized. In that case, the time-averaged Poynting vector is always locally aligned with the phase flow of the linearly polarized field (for positive material parameters), meaning that, if anything, these waves are always to be classified as forward. However, it can be argued that this type of classification of waves in periodic structures is not appropriate.