content_image/334298.jpg content_image/334299.jpg
〈 x 〉 [W] can be regarded as a small angle inclining from the propagation axis [31, 32]. However, the shift 〈 y 〉 [W] does not change on propagation. [W] 

The angular positions of the wavepacket centroid indicates the rotation angle tan ϕ = 〈 x 〉 $^{[W]}$/ 〈 y 〉 $^{[W]}$, which is significantly different from the previous definition [33]. According to Eq. (19) the rotation angle is given by 

<img src='content_image/334298.jpg'>

Very surprisingly, the rotation angle is proportional to the Gouy phase for circular polar- ization σ = ± 1. In all cross sections denoted by the distance from the z = 0 plane, the transverse patterns are quite similar. However the overall scale changes and the pattern rotates as a whole. The former is caused by diffraction, and the latter is seen due to the existence of transverse momentum currents. 

Since the rotating wavepacket passes the distance between different cross-sections in time t = z/c , the instant angular velocity can be given by Ω $^{[W]}$( z ) = dϕ $^{[W]}$/dt = cdϕ $^{[W]}$/dz , so that 

<img src='content_image/334299.jpg'>

The rotation velocity of the wavepacket centroid decreases as the propagation distance in- creases. When the condition σ = 0 is satisfied and the rotation characteristics of wavepacket centroid vanish, i.e., Ω $^{[W]}$( z ) = 0. This is why the linear polarized wavepacket cannot present the optical Magnus effect. However the linear polarization can be represented as a superpo- sition of two circularly polarized components [7]. As the result, the polarization-dependent split of the wavepacket intensity distribution arises. Thus, the same mechanism also leads to other effects such as the SHEL. 

Figure 4 shows the polarization-dependent rotations of wavepacket centroid. At the plane z = const. , the intensity of the wavepacket can be regarded as I $^{[W]}$( r ) = c $^{2}$p [W] z ( r ). The wavepacket centroid (green spots) whose angular positions indicate the rotation angle ϕ . For the left circular polarization σ = +1, the wavepacket centroid exhibits a clockwise ro- tation [Figs. 4(a)-4(c)]. For the right circular polarization σ = − 1, however, the wavepacket centroid presents an anticlockwise rotation [Figs. 4(d)-4(f)]. During the wavepacket prop- agation from z = 0 to z = + z$_{R}$ , the rotation angle amounts to ϕ [W] = σπ/ 4. The physics underlying this intriguing effect is the combined contributions of transverse spin and orbital currents. The novel polarization-dependent rotations differs from the conventional optical Magnus effect, in that light-matter interaction is not required. 

