x / y / -20 -10 0 10 20 -20 -10 0 10 20 Z=0 (a) x / y / -20 -10 0 10 20 -20 -10 0 10 20 Z=0 (d) x / y / -20 -10 0 10 20 -20 -10 0 10 20 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 (c) Z=1.0Z$_{R}$ [W] x / y / -20 -10 0 10 20 -20 -10 0 10 20 (b) Z=0.5Z$_{R}$ [W ] x / y / -20 -10 0 10 20 -20 -10 0 10 20 Z=0.5Z$_{R}$ (e) [ W ] x / y / -20 -10 0 10 20 -20 -10 0 10 20 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Z=1.0Z$_{R}$ (f) [W] FIG. 4: (color online) The polarization-dependent rotations of wavepacket centroid (green point) for the model [W]. First row: The wavepacket centroid undergoes a clockwise rotation for left circular polarization σ = +1. Second row: The wavepacket centroid presents an anticlockwise rotation for right circular polarization σ = − 1. Images are labeled by corresponding values of propagation length z and rotation angle ϕ $^{[W]}$. The intensity distributions are plotted in the plane [(a), (d)] z = 0, [(b), (e)] z = 0 . 5 z$_{R}$ , and [(c), (f)] z = z$_{R}$ . Other parameters are the same as those in Fig. 2. We are currently investigating the polarization mode [I]. On substituting Eqs. (2) and (13) into Eq. (19) we have There exists an inherent transverse shift on propagation. This result coincides with that obtained by Li [11] with different methods. According to Eq. (24) the rotation angle is given by In this case, the rotation characteristics of wavepacket centroid vanish, i.e., Ω $^{[I]}$( z ) = 0. We are now in a position to consider the polarization model [II]. On substituting Eqs. (2) and (14) into Eq. (19) we can determine A further important point should be noted is that Eq. (26) can be obtained from Eq. (24) 〈 x 〉 [I] = 0 〈 y 〉 [I] = − σ cot θ k . (24) ϕ [I] = σ arctan π 2 . (25) 〈 x 〉 [II] = 0 , 〈 y 〉 [II] = 0 . (26) 11