A modal decomposition approach to topological wave propagation

The design of valley-Hall topological insulators is typically informed by analyzing the primitive unit cell, using the Berry curvature of the Bloch-eigenmodes to predict energy localization on the domain boundary, or the dispersion diagram of the semi-infinite lattice to predict the group velocity of topological waves. However, practical systems are finite in size and the propagating topological wave is composed of a finite bandwidth as guaranteed by the Fourier uncertainty principle. Hence, the propagation predicted by the ideally infinite lattice for a given frequency deviate from practical topological systems driven by finite bandwidth wave packets. In this work, we demonstrate that the propagating topological waves in valley Hall lattices can be interpreted using the linear degenerate modal basis. We show that a small subset of closely spaced modes comprise the topological waves, which enables the construction of analytical reduced-order models that accurately capture the topological wave. This result is related to the sparse density of the modal spectrum inside the topological band gap, which allows for the energy to be carried nearly completely by a narrow band of spectrally isolated eigenmodes that are spatially localized along the domain boundary. This result is then employed to refine group velocity predictions of propagating topological waves with respect to input signal bandwidth and frequency by matching the modal spectrum to the supercell dispersion diagram describing the semi-infinite system. Moreover, we utilize the damped modal spectrum to characterize variations in wave group velocity in damped topological lattices and predict edge-to-bulk transitions. Hence, this work establishes a framework for accurately characterizing undamped and damped topological wave propagation in quantum valley Hall systems using the machinery of classical dynamics.