## 1. Introduction

One important task in biology is the separation of enantiomers, which can be achieved by using different techniques such as gas chromatography, high performance liquid chromatography, or capillary electrophoresis [

1]. Different groups [

2,

3,

4,

5] have recently analyzed the use of optical fields to generate enantiomeric separation. In this sense, a study has been made of lateral optical forces (LOF) generated by interfering plane waves [

6] in order to obtain chirality sorting, and the LOF effect on paired chiral particles has also been discussed [

7]. Furthermore, another way of generating LOF is given by evanescent waves, which transport spin angular momentum and could be used for chirality sorting [

3]. Thus, micro-metric particles have been sorted in a fluidic medium by using the spin-dependent optical forces of chiral particles [

8]. Moreover, our group has demonstrated that it is possible to obtain chiral resolution of nanoparticles by using an optical conveyor belt [

9,

10,

11,

12] through the superposition of two orthogonal counter-propagating temporally-de-phased beams [

13,

14]. In this paper, we are going to analyze the effects of the temporal de-phasing parameter on the chiral resolution of the optical conveyor belt, showing that the maximum separation distance between enantiomers can be obtained by selecting the correct range of the de-phased parameter. Finally, the dynamics of enantiomers are studied in their final trap region.

## 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.

It is interesting to note that when enantiomers are finally trapped at different

z positions (see

Figure 2 and

Figure 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

$\delta >0$. One explanation for this result could be that, at the final time, particles describe a limit cycle in which

$z(t)$ reaches a limit value

${z}_{\pm \delta}^{\tau}$ (each enantiomer goes to a final z position that depends on their value

$\pm \delta $ and de-phase

$\tau $) at the fixed value

$r={r}_{p}$ (see

Figure 2). In this case (when the limit cycle is reached

$z={z}_{\pm \delta}^{\tau},\phantom{\rule{0.166667em}{0ex}}r={r}_{p}$), Equations (

28) and (

30) can be approximated to:

Introducing Equation (

33) into (

32), we finally obtain that:

As can be observed in

Figure 5, the approximated function (

34) qualitatively explains the final frequency values

$\omega $, although quantitatively, the predictions are overestimated be approximately 1 Hz.

A possible explanation for this observed overestimation is that we have used the approximations:

in order to obtain the analytical approximated Equation (

34).

Finally,

Figure 6 shows the particle separation of particles with different chiral parameter

$\delta $ when we use the de-phasing parameter

$\tau =10$ Hz, taking in all cases a final time of 50 s.

As can be observed, the particle separation is null for low values of $\delta $, increasing for higher values of $\delta $ until reaching a saturation zone for a fixed time of 50 s. It is important to note that at this time, particles with $\delta >0.7$ have not reached their limit cycle.