x / y / -20 -10 0 10 20 -20 -10 0 10 20 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 (c) x / y / -20 -10 0 10 20 -20 -10 0 10 20 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 (f) x / y / -20 -10 0 10 20 -20 -10 0 10 20 (b) x / y / -20 -10 0 10 20 -20 -10 0 10 20 (a) x / y / -20 -10 0 10 20 -20 -10 0 10 20 (e) x / y / -20 -10 0 10 20 -20 -10 0 10 20 (d) FIG. 6: (color online) The distribution of the longitudinal angular momentum density for the model [W]. First row: Left circular polarization σ = +1. Second row: Right circular polarization σ = − 1. [(a), (d)] Orbital angular momentum density j [W] Oz . [(b), (e)] Spin angular momentum density j [W] Sz . [(c), (f)] Total angular momentum density j [W] Oz + j [W] Sz . The cross section is chosen as z = 0 and the intensity is plotted in normalized units. Other parameters are the same as those in Fig. 2. over the whole x − y plane [24] The transverse angular momentum components are given by In the z = 0 plane, we have J [M] x /P [M] z = 〈 y 〉 [M] and J [M] y /P [M] z = −〈 x 〉 $^{[M]}$. Thus, the transverse angular momentum can be obtained by measuring the position of wavepacket centroid [10]. In order to reveal the rotation characteristics, it is necessary for us to know the transverse angular momentum in any cross section. We first consider the model [W], the transverse angular momenta are given by J [W] x = − πσ sin 2 θ 2 k $^{2}$z$_{R}$ , J [W] y = πz sin 2 θ 2 k $^{2}$z 2 R . (40) 15 P [M] = ∫ ∫ p $^{[M]}$( r )d x d y, (36) J [M] = ∫ ∫ j $^{[M]}$( r )d x d y. (37) J [M] x = 〈 y 〉 P [M] z − zP [M] y , (38) J [M] y = zP [M] x − 〈 x 〉 P [M] z . (39)