## 3 Discussion  
In the numerical calculation, we have used the values of \(\lambda=1070\) nm, \(a=\lambda/15\), \(n_{p}=1.45\), \(n=1.33\), \(w_{0}=5\lambda\), \(w_{1}=4\lambda\), \(b_{1}=0.9\), \(\delta=\pm 0.06\), and \(A_{0}=2\times 10^{8}\) V/m assuming that the initial position of the enantiomers is \(r_{0}=3\lambda,\theta_{0}=0,z_{0}=10\lambda\). Figure 1 shows the potential energy W(\(r,z\)) given by Equation (20). As can be observed, W shows a maximum at radial position \(r_{p}=1.6\lambda\) (green dashed line), which is the radial coordinate where particles will be initially trapped according to Equation (28). Thus, by using the indicated fixed values, the chiral conveyor is mainly controlled by the de-phase \(\tau\), which governs the conveyor velocity. In fact, if \(\tau=0\), there will not be any temporal dependence in Equations (29) and (30), and there will not be a chiral conveyor; consequently, W will act like a chiral trap. In this case, both enantiomers will be trapped in the same region, as shown in Figure 2a, and as a result, there is no effective enantiomeric separation. Figure 2b-f shows the enantiomeric separation produced by the action of the chiral optical conveyor described by Equations (1), (7), and (25)-(27). As can be observed, the particle trajectories are spiral in all cases, the pitch and velocity of optical conveyors being strongly dependent on de-phase parameter \(\tau\)[19]. In this sense, the behavior of chiral conveyors is very similar to dielectric conveyors, and the main difference from the results obtained in [19] is that the movement of particles in the \(z\) direction is limited due to the axial dependence of parameters \(W_{0}(r,z)\) and \(W_{s}(r,z)\). Furthermore, as can be seen in Figure 2, on increasing \(\tau\), the enantiomeric separation also increases until it reaches a maximum separation that slowly decreases for high values of de-phase \(\tau\). This result can be clarified in Figure 3, where the behavior of enantiomeric separation (\(\Delta\), defined as the axial distance between trapped enantiomers) is represented as a function of \(\tau\). As previously mentioned, there is a fast increment of \(\Delta\) when \(\tau\) rises, obtaining a maximum value of \(18\lambda\) for \(\tau=10\) Hz, which remains nearly constant (see the inset graphics in Figure 3), until reaching a value of \(\tau=100\) Hz to decrease slowly subsequently. Figure 4 shows the particles position at different times for a de-phase parameter \(\tau=10\) Hz. As can be observed in Figure 4a, enantiomers are trapped at \(r=r_{p}\) in different z positions in a short time (\(t=0.01\) s). Figure 4b-d shows that positive enantiomers (\(\delta>0\)) are trapped more quickly than negative ones. It can also be observed in Figure 4b-e that there are two traps, but at the end (\(t=10\) s), only one of them is stable for each enantiomer. Figure 4: Particles’ position at different times: (**a**) \(t=0.01\) s, (**b**) \(\tau=0.1\) s, (**c**) \(t=1\) s, (**d**) \(t=2\) s, (**e**) \(t=5\) s, and (**f**) \(t=10\) s, for \(\tau=10\) Hz (\(\delta>0\) blue and \(\delta<0\) red; green dots show the initial particles’ position). Figure 3: Enantiomeric separation versus \(\tau\). The inset figure shows enantiomeric separation versus \(\tau\) in the logarithmic scale. It is interesting to note that when enantiomers are finally trapped at different \(z\) positions (see Figures 2 and 4f), they describe circular trajectories with angular frequencies \(\omega\), which are shown in Figure 5.  
As can be observed in Figure 5, at low and high \(\tau\) values, the frequency of the circular trajectories \(\omega\) shows similar values for both enantiomers (1 kHz \(<\omega<\) 1 Hz). However, for \(\tau\) values between 1 and 1000 Hz, the frequency \(\omega\) is nearly constant and equal to 2 Hz for enantiomers with \(\delta<\) 0 and 3 Hz in the case of \(\delta>\) 0.