Fig. 3: Evolution of the Bloch wave propagating in the array of rods with varying radii along the propagation direction with ƞ = 0.23 Z.

For the incident Bloch wave with a wave vector k equal to 0.95K in the lower-energy state, a shows the calculated two mode coefficients by solving Eq. 6a. b shows the exact centre-of-energy transverse shift (Hall shift, denoted by circles) obtained by solving Eq. 6b, and the calculated centre-of-energy transverse shifts induced by the combined contribution of \(\langle {{\mathbf{\Lambda }}^{({\boldsymbol{k}})}} \rangle\) and anomalous velocity and V_PT (denoted by upper triangles), and only the contribution of the anomalous velocity (denoted by squares), and only the contribution of the displacement \(\langle {{\mathbf{\Lambda }}^{({\boldsymbol{k}})}} \rangle\) (denoted by lower triangles), respectively. c Spatial trajectory of the centre of the energy of the wave packet. d–f Corresponding results of the incident Bloch wave with the wave vector k equal to 0.8 K