## Abstract

Chiral resolution is a fundamental problem in pharmaceutics or agrochemicals, so a great effort has been made to generate experimental techniques capable of producing mechanical separation of a mixture of enantiomers. Unlike other techniques that are usually employed, such as chiral resolving agents or chiral chromatography, we propose a new technique which is directly applicable in solution and without further processing. This technique is based on optical forces, since we show that with the proper design of the polarization states of the incident beams and temporal dephasing, a chiral sensitive optical conveyor can be obtained that is able to transport enantiomers in opposite directions. The implementation of such an optical conveyor with the required focused optical fields produces a well-defined trapping region for each enantiomer, since theoretical simulations over a large number of chiral particle trajectories show that it is possible to reach values of enantiomeric excess of over 99%.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Chirality or handedness is a property of rigid objects that cannot be superimposed with their mirror image. The two non-superimposable mirror-image forms of chiral bodies are called enantiomers. These objects are abundant in nature, both on macroscopic and microscopic scales. However, notable differences in behavior can be seen between the two scales. The two chiral forms of macroscopic objects can be discriminated in achiral media by motion, but due to the random thermal motion on the microscopic scale, the enantiomers exhibit the same physical and chemical properties when they interact with an achiral environment [1]. Therefore, the only difference between enantiomers on the microscopic scale is observed when they interact with other chiral bodies. This has special relevance in biology, since all enzymes, proteins, carbohydrates, amino acids, nucleosides and many hormones are chiral compounds, and are usually present in nature in only one enantiomeric form. For example, all the natural amino acids are l-enantiomers, while all the natural sugars are d-enantiomers (where the l and d denote the ability to rotate polarized light to the left or right that exhibit the different enantiomers respectively). In the same way, drugs are also chiral molecules and as a result the two enantiomers exhibit marked differences in biological activities such as pharmacology, pharmacokinetics, toxicology or metabolism. Thus, one enantiomer may produce the desired therapeutic activities, while the other may be inactive or, in the worst cases, produce undesired or toxic effects. Therefore, chiral resolution is one of the fundamental problems for pharmaceutical industries [2] and other biochemical related industries like agrochemicals [3]. Usually, chiral resolution is obtained by different techniques such as crystallization of enantiopure crystals, the use of chiral resolving agents or chiral chromatography [4].

Furthermore, in recent decades, a huge advance has been made in the mechanical manipulation of objects on the micro and nanoscale using optical forces. In 1986, Ashkin at al. [5] demonstrated that the forces exerted by a strongly focused laser beam can be used to spatially confine small dielectric particles or atoms, which is usually known as an optical tweezer. Basically, this effect is due to the so-called gradient or dipole force, i.e., the light induces an electric dipole in the particle, which, in an environment with a low refractive index, tends to be located at areas of maximum intensity. Therefore, using diffractive techniques, complex optical tweezers have been developed [6, 7], including multiple trap structures or dynamical effects, making them a powerful tool for the manipulation of micro and nanoparticles. In addition, other sophisticated transport systems based on optical forces have been designed, for example, a micro-conveyor belt for sub-micron particles has been developed using two counter-propagating Bessel beams with controllable phase shift between them [8]. In the same way, in recent years different configurations have been described for the generation of the so-called tractor beams [9–15], which are traveling waves that transport illuminated objects back to their sources.

Recently, a growing interest has emerged in the study of chiral discrimination employing optical forces [16–27]. Therefore, on the basis of the high circular dichroism that exhibit chiral liquid crystal droplets, it has been demonstrated both experimentally and theoretically that it is possible to selective trapping chiral microparticles [17–19], with the radius much longer than light wavelength. However, it is not easy to transfer these results to smaller particles, with the radius smaller than light wavelength, as is the case of the vast majority of biomolecules. The main drawback of this is the fact that chiral forces are usually surpassed by the optical forces that do not depend on chirality, so the force field should be designed properly to generate an efficient chirally selective force. In this sense, Canaguier-Durand et al [16] showed that with two counter-propagating incoherent plane waves with circular polarization it is possible to generate a pure chiral force proportional to the imaginary part of the chiral polarizability, which is able to get the mechanical separation of enantiomers. However, since they use plane waves to avoid any effect from the gradient or dipole force from the standard optical tweezers, the experimental implementation is difficult because the system does not have enough energy to act as a chiral optical tweezer in solution (there are also other drawbacks, such as the temperature increase due to the light absorption at the working conditions, since they work near the resonance). Based on the same principle, Fernandes et al [28] theoretically proposed an optical conveyor belt for chiral metamaterial particles with size of about 200 nanometers, but also employing plane waves and with several restrictions to the particle properties. Furthermore Wang and Chang reported in reference [20] that a lateral force (perpendicular to the propagation direction of light) emerges from the coupling between the particle chirality and the near field structure of the light reflected from a substrate surface. This force pushes chiral particles with opposite handedness in opposite directions. In the same way, Hayat et al. [21] and Alizadeh et al [22] propose the use of the extraordinary spin angular momentum (SAM) of evanescent waves and chiral Surface Plasmon Polaritons respectively to generate chirally sensitive optical forces. Selective trapping at molecular level by circulary polarized light has also been studied by Bradshaw et al [26,27]. Moreover, using the concept of optical helicity [29], Cameron et al. used the interference of two plane waves with perpendicular polarization to generate a chiral selective force proportional to the gradient of the optical helicity profile and the real part of the chiral polarizability [23,30], which enables to work far of the resonance that limits the temperature problems. However, as in the case of reference [16], the use plane waves limits the direct applicability in solution, so they proposed an interesting device to exploit the gradient of optical helicity, using it as the driving force for a chiral Stern-Gerlach deflector, where the sign of the deflection angle depends on the particle chirality.

In this paper we design a chiral sensitive optical conveyor that is able to transport the two enantiomers in opposite directions employing tightly focused optical fields required for optical tweezers. Through the proper design of dipolar forces and optical helicity, we show that chiral resolution for particles on the nanoscale can be achieved. Section 2 introduces the foundation of the system designed, which is based on the optical conveyors for dielectric particles. However, unlike the tractor beams described in the bibliography, which consist in one traveling wave that transports the particles [9, 11], our idea is to use two traveling waves propagating in opposite directions, with each enantiomer being sensitive to one of these waves. Finally, the practical application of the proposed device is discussed in section 3, where the ability of the system for generating enantiomeric excess is evaluated through the numerical simulation of a large number of particle trajectories, including effects such as Brownian motion or the axial attenuation of the optical fields.

## 2. Chiral optical conveyor

The behavior of a small particle in a homogeneous and isotropic medium subjected to a strong force field **F**(**r**(t)) can be determined by the over-damped Langevin equation [31]:

**r**(t) denotes the particle position,

*γ*the friction coefficient and

**W**(t) is a random force that causes the thermal fluctuations (Brownian motion), that usually dominate the movement of small biomolecules in free solution. Therefore, obtaining chiral resolution requires a proper design of the force field, which will cause the two enantiomers to be located in different spatial regions.

Let us consider a chiral nanoparticle, with a much smaller size than the light wavelength, in a source-free, lossless, non-magnetic (*μ _{m}*=1), non-dispersive and isotropic medium with relative dielectric permittivity

*∊*. In this case, the time-averaged optical force exerted by a monochromatic field can be obtained using dipolar approximation [21,32–34]:

_{m}**p**and

**m**are the electric and magnetic dipoles of the particle,

**E**and

**H**are the incident electric and magnetic fields respectively (the asterisk indicates the complex conjugate and ⊗ is the dyadic product), while

*μ*

_{0}is the magnetic permeability of vacuum,

*η*is the vacuum impedance, n the refractive index and k is the wavenumber. For chiral particles (assuming isotropic polarizabilities), the electric

**p**and magnetic

**m**dipoles are defined as: where

*∊*

_{0}denotes the vacuum permittivity,

*α*and

_{e}*α*are the electric and magnetic polarizabilities respectively, while

_{m}*χ*denotes the chiral polarizability, which has opposite sign for the two enantiomeric forms of a chiral compound (note that all the defined polarizabilities have volume dimensions, and are directly comparable). Introducing Eqs. (3) and (4) into Eq. (2) and taking into account that polarizabilities are complex magnitudes, it is possible to express the optical forces in dipolar approximation into different contributions related to the real and imaginary parts of the particle properties and some electromagnetic field quantities, such as SAM density, Poynting vector, energy density or helicity [21] (a detailed description of all the terms is showed in appendix). However, depending on the working conditions, many of these contributions can be neglected. For example, to avoid undesired thermal effects due to high power usually required, the working wavelength is commonly far from the particle resonances, which allows us to neglect all the contributions from the imaginary parts of particle properties. Therefore, the study of the optical forces for chiral biomolecules is usually limited to the contribution of only two terms [23,24,30]:

**k**

*denote the wavevector (k is the wavenumber) and*

_{i}**ê**

*determines the polarization state. Let us consider the interference of two plane waves with a fixed dephase*

_{i}*φ*and the same polarization state

**ê**

_{1}=

**ê**

_{2}= {1, 0, 0} and the following wavevectors, which form an angle +

*θ*and −

*θ*with respect the z-axis respectively:

Introducing the total electric and magnetic fields (**E**=**E**_{1}+Exp[j*φ*]**E**_{2} and **H**=**H**_{1}+Exp[j*φ*]**H**_{2}) into Eq. (5), it could be observed that, for these fields, the unique non-zero component of the force is F* _{z}*, and it is given by:

Therefore, the motion equation for the axial coordinate, without taking into account the thermal fluctuations, is obtained replacing the Eq. (9) into Eq. (1), and it can be written as [9]:

where*σ*=4

_{d}*π∊*

_{0}A

^{2}kcos(

*θ*)

*α*

_{e,R}/

*γ*is an amplitude which depends on the electric field amplitude A together with material and medium properties. The dielectric particles tends to be located in the zeros of the sin function (stationary points with zero velocity [9]), and if we modify the fixed dephase with a electrooptic phase modulator or a piezoelectric mirror in a time scale very much slower than the field frequencies it is possible to transport the particles in the desired direction. So for example, if we introduce a linear variation of the dephase of the form

*φ*=

*ξ*t, second term of Eq. (10) results in a travelling wave that control the particle motion, with a phase velocity given by

*v*=

*ξ*/(2kcos(

*θ*)). Therefore, if the light has enough intensity to fulfill the condition |

*σ*| > |

_{d}*v*|, the particle moves with the phase velocity of the interference pattern, and the direction is controlled with the sign of the phase shift coefficient

*ξ*. In the same way, it is possible to generate an optical conveyor like the one shown by Eq. (10) with the optical helicity term of Eq. (5), by the superposition of two beams with the same wavevectors given by Eq. (8) but with perpendicular polarization states (

**ê**

_{1}= {1, 0, 0} and

**ê**

_{2}= {0, −

*jcos*(

*θ*),

*jsin*(

*θ*)}). As in the previous case, the unique non-vanishing component of the force is the axial coordinate, but in this case it depends on the chiral polarizability as:

Thus, the motion equation for the axial coordinate for this conveyor takes the form:

where*σ*=−8

_{c}*π∊*

_{0}A

^{2}kcos(

*θ*)sin

^{2}(

*θ*)

*χ*is an amplitude with dimensions of velocity which now depends on the chiral polarizability

_{R}/γ*χ*, so only chiral particles will be sensitive to the optical conveyor. However, for a linear variation of the dephase of the form

_{R}*φ*=

*ξ*t (as the previous case), both enantiomers will be moved in the same direction because the motion is only controlled by the phase velocity of the traveling wave, so there is no influence from the sign of

*σ*in the movement, and it will not be used as an efficient chiral resolution device. In order to solve this problem, we propose the use of a more complex polarization state for the two plane waves, which will provide two travelling waves propagating in opposite direction for the motion equation, being each enantiomer only sensitive to one of these waves. Let us consider two planes waves with the wavevectors described by Eq. (8) and the following polarization states:

_{c}*are constants that fix the polarization state of each beam and*

_{i}*φ*describes a fixed dephasing. Therefore, proceeding in the same way that previous configurations, we introduce the total electric and magnetic fields (

**E**=

**E**

_{1}+Exp[j

*φ*]

**E**

_{2}and

**H**=

**H**

_{1}+Exp[j

*φ*]

**H**

_{2}) into Eq. (5), and as in the previous cases, the unique non-zero component of the force is F

*, but in this case takes a more complex form and depends on dielectric and chiral polarizabilities:*

_{z}Therefore, for this case, the motion equation for the particles can be written as:

*σ*=Λ

_{i}*. When we apply a temporal variation for the dephase*

_{i}/γ*φ*=

*ξ*t, the motion equation presents two counter-propagating travelling waves with different amplitudes, and they can be used to efficiently separate enantiomers by the proper selection of the beams polarization, so it will made that each enantiomer be only sensitive to one of this travelling waves. However, for clarify this, it is more convenient to re-write the Eq. (17) in the following form:

*ζ*=

*σ*

_{2}/

*σ*

_{1}is the ratio between the amplitudes

*σ*

_{1}and

*σ*

_{2}. Therefore, the motion Eq. (18) is equivalent to an optical conveyor one, but with a more complex phase velocity, given by:

*v*=

*ξ*/(kcos(

*θ*)). Therefore, the sign of the phase velocity of the resulting interference depends on the amplitude ratio

*ζ*, so it is possible to transport two enantiomers in opposite directions with the appropriate design of the amplitude coefficients

*σ*. Basically, for one enantiomer we need |

_{i}*ζ*| > 1, while for the other |

*ζ*| < 1. To fulfill such conditions, the polarization states of the incident beams must be carefully selected. Let us define

*β*=|

*χ*|/

_{R}*α*

_{e,R}as the ratio between the module of chiral and electric polarizabilities (usually |

*χ*| is about one or more orders of magnitudes lower than

_{R}*α*

_{e,R}), so the chiral polarizability is

*χ*=±

_{R}*βα*

_{e,R}(the sign depends on the enantiomer considered). Therefore, taking the polarization parameters b

_{1}=b

_{2}=2

*β*sin

^{2}(

*θ*), the ratio

*ζ*=Λ

_{2}/Λ

_{1}from Eq. (16) can be written as:

Hence, for the enantiomer with negative *χ _{R}*, the

*ζ*parameter takes a value greater than −2, and therefore, the phase velocity of the traveling wave given by Eq. (20) becomes negative, so the enantiomer will be moved in the negative direction of the propagation coordinate z. On the other hand, for the other enantiomer, which exhibits positive

*χ*, the

_{R}*ζ*parameter is 0, and consequently, the phase velocity of the traveling wave that controls the particle behavior is positive, and it will be moved in the positive direction of the axial coordinate. This behavior is proven by the numerical solution of the trajectory Eq. (17) for the two enantiomers with parameter values:

*σ*

_{1}=10

^{−3}m/s,

*β*=0.01,

*θ*=15°,

*λ*=1070 10

^{−9}m and

*ξ*=10 Hz. The results are shown in Fig. 1, obtaining the expected behavior, since the two enantiomers move in opposite directions, and therefore the desired chiral spatial separation is achieved. Analytic trajectories can be also estimated by directly integrating the phase velocity given by Eq. (20), so the particle’s trajectory can be written as:

Therefore, as can be seen in Fig. 1, with the appropriate values of *ζ* the numerical results matches the trajectory given by Eq. (22), proving that the particles travel with the dynamic interference pattern generated by the two traveling waves. For the enantiomer with positive *χ _{R}*, the ratio

*ζ*is 0, so the particle motion exhibit the expected linear behavior for the positive direction of the axial coordinate. In the case of the negative enantiomer, the

*ζ*value takes a value of 2.3094, and the particle moves to the negative direction of the axial coordinate with slightly ondulatory trajectory due to the arctan function of Eq. (22).

## 3. Chiral resolution

Finally, we are going to show the viability of the conveyor designed for chiral resolution by analyzing a real three dimensional system, including the thermal motion effect. Optical conveyors for dielectric particles were experimentally delivered in bibliography under different configurations. In our case, we are going to use two focused cylindrical vector beams generated by annular apodization (Fig. 2), in a similar way as those employed to generate optical conveyors for dielectric particles in references [11,12], but with two main differences. On one hand, in order to include chiral sensitive conveyors we use elliptical polarization combining radial and azimuthal polarizations instead of pure radial or azimuthal polarizations employed in such references. On the other hand, in this work we use a 4-pi configuration as it is shown in Fig. 2, with two counter-propagating cylindrical beams instead of the corresponding co-propagating structure used in references [11,12], since this configuration will be easier to implement experimentally (however both configurations are allowed and will provide similar results). The system includes two polarization control systems to generate the desired polarization state for both beams; an electrooptic phase modulator to control the temporal dephasing between the two beams; the focusing system and two different ring shaped amplitude mask in order to generate the cylindrical vector beams with different k* _{z}*. Therefore, lets consider two counter-propagating electric fields with the following polarization states in cylindrical coordinates (E

*, E*

_{r}*, E*

_{ϕ}*), obtained by approximations [11,12] of the diffraction integrals of Youngworth et al. [35]:*

_{z}*ϑ*=2A

*δ*c

^{1/2}sSinc(ks

*δ*z/2), c=cos(

*θ*), s=sin(

*θ*), M is the matrix for the inversion of z coordinate and

*δ*is the ring width, while

*ê*={cJ

_{ρ}_{1}(krs),0,jsJ

_{0}(krs)} and

*ê*={0,J

_{ϕ}_{1}(krs),0} denote the radial and azimuthally polarization vectors of the focused cylindrical vector beams. As in the previous section, B

_{1}and B

_{2}control the degree of ellipticity of the two beams. Now a more complex force field is generated, with radial and axial forces (a detailed description is given in appendix), although the axial movement is also controlled by the two traveling waves. So, by taking B

_{1}=

*β*/q and B

_{2}=

*β*, the two enantiomers will be moved in opposite directions without any restriction in the

*β*value (q is an adjustable constant greater than 1). This was analyzed by a numerical solution of Eq. (1) with the proper values of the forces for 1600 particle trajectories of each enantiomer including Brownian motion and without interaction between the particles (3200 individual simulations). The random function force W(t) has a Gaussian probability distribution with correlation function <W

*(t),W*

_{i}*(t)>=2*

_{j}*γ*k

*T*

_{B}*δ*

_{i,j}

*δ*(t–t′), k

*being the Boltzmann’s constant, T the temperature and the friction coefficient*

_{B}*γ*=6

*πν*a depends on the medium viscosity

*ν*and the particle radius a. Therefore, the Brownian motion has been estimated by randomly generated constant forces in time intervals of 10

^{−7}s with the described Gaussian distribution, being the system of coupled differential equations for the particle’s position solved numerically in such intervals.

Numerical simulations were made with realistic parameter values, so the parameters of beams used in simulations were *θ*=0.65, A=2 10^{8} V/m, *λ*=1.070 10^{−6} m, *ξ*=100 Hz, q=1.15 and *δ*=0.06, resulting in a total power per unit area in the focal region of about 2.2 MW/cm^{2} for each beam (total power in the focus region of about 78 mW). It is important to note that for these conditions, the thermal heating due to radiation can be neglected since it will be lower than 1 K as has been reported in bibliography [36] (in this reference it is estimated a temperature increase of 0.8 K in water for a focused beam of 100 mW at 1064 nm). Regarding the particle properties, chiral particles of radius a=*λ*/15 were analyzed, and their polarizabilities were estimated through Mie scattering coefficients [34,37] with values for parameters n* _{p}*=1.5, n=1.33 and the chirality parameter Δ= ±0.1 (similar to others present in bibliography [20,24,28]), resulting in the following values

*α*

_{e,R}=5 10

^{−23}m

^{3}and

*χ*=±1.1 10

_{R}^{−23}m

^{3}(see appendix for a complete description of particle’s polarizabilities). The medium used in the simulations was water at a temperature of T=275 K, for which the viscosity was

*ν*=16.7 10

^{−4}Pa.s. The initial position of each experiment was randomly distributed over a region between −15

*μ*m and 15

*μ*m for the z coordinate and between 0 and 1.5

*μ*m for the radial coordinate.

Figure 3 shows the initial and final particle distribution after 5 seconds of the 3200 chiral particles analyzed (1600 of each enantiomer) without particle-particle interaction. As can be seen, an efficient chiral discrimination is obtained, since chiral particles with positive *χ _{R}* (red points) tends to locate in a region between −19

*μ*m and −9

*μ*m of the z coordinate and between 0 and 0.75

*μ*m for the radial coordinate, which coincides with the first maximum of the Bessel functions that form the radial trap as can be seen in the density plot of the potential energy distribution. Thus, after 1600 simulations, 1181 enantiomers with positive

*χ*were located in this region, while only 4 enantiomers with negative

_{R}*χ*lay in the same region. A second region in the negative range of z coordinate (−10 to 0

_{R}*μ*m) with excess of positive chiral particles was observed for the second maximum of the Bessel functions (radial coordinate between 0.75 and 1.5

*μ*m), where 159 +

*χ*enantiomers were located and zero of the other enantiomer. The enantiomers with negative

_{R}*χ*(blue points) moved in the positive direction of the axial coordinate, being mainly located in a region from 5 to 15

_{R}*μ*m for the z coordinate and between 0 and 0.75

*μ*m for the radial coordinate. After five seconds 1460 −

*χ*enantiomers, more than 91% of the total, were located in this region, while only 3 of the other enantiomer lay in the same region. According to these results, the trapping efficiency for enantiomers with negative

_{R}*χ*is greater than for the positive one, as can be seen in the number of particles that scatter from the traps (260 of 1600 particles scattered for

_{R}*χ*enantiomers and 140 of 1600 particles scattered for −

_{R}*χ*enantiomers). Therefore, chiral separation was achieved, trapping each enantiomer in two clearly distinguishable regions, which were separated by more than 14

_{R}*μ*m.

Figure 4 shows the temporal evolution of the enantiomeric excess for the two main regions where the enantiomers were located. As can be seen, an efficient chiral resolution was achieved in both regions, with final values over 99% for the region with excess of positive *χ _{R}* particles and over 98% for the region with excess of negative

*χ*enantiomers. The temporal evolution follows a saturation curve, so that it reaches a stable value in the two regions from 4.5 seconds, which is the time spent for the enantiomers departing from more distant positions to reach trapping regions, as can be observed in the inset of Fig. 4, where examples of the trajectories obtained for the axial coordinate of 128 particles (64 of each enantiomer) are plotted. Therefore, chiral resolution with the designed optical conveyor was demonstrated for particles with radius of 70 nm. The application of this technique to smaller particles will be possible, but in order to beat the thermal energy, the power of the beams must be increased or maybe it must be explored the use of other structured light beams which will provide better behavior. However, since the particles’ polarizabilities scales with the volume, it would not be expected that the present technique itself will be applied to particles smaller than about 5 nm, but the possibility of using it in combination with other techniques that suppress the thermal motion in molecules can be studied, for example the use in combination with an ABEL trap [38].

_{R}## 4. Conclusion

In conclusion, an efficient new chiral resolution technique based on optical forces has been proposed for particles for which the dipolar approximation is valid. This was achieved by the design of a new class of optical conveyors with the ability to distinguish between enantiomers. For this purpose, we proposed transporting particles by using two travelling waves propagating in opposite directions, despite the single travelling wave used for dielectric particles, each enantiomer is sensitive to a different travelling wave, and consequently transported in opposite directions. This was accomplished with the proper selection of the polarization states of the incident beams, since the beams’ ellipticity must be proportional to the ratio between chiral and dielectric polarizabilities of the chiral particles that will be separated. Finally, through theoretical simulations of a large number of trajectories we proved that an efficient mechanical separation between enantiomers can be obtained with this new class of optical conveyor generated with the focused optical fields required for trapping applications. Therefore, each kind of enantiomer in a clearly distinguishable trapping region was located, the enantiomeric excess reached being 98% and 99% for the main trapping regions.

## Appendix

## Optical forces

Let us consider a chiral nanoparticle, with a much smaller size than the light wavelength, in a source-free, lossless, non-magnetic (*μ _{m}*=1), non-dispersive and isotropic medium with relative dielectric permittivity

*∊*. In this case, the time-averaged optical force exerted by a monochromatic field can be obtained using dipolar approximation [21,32–34]:

_{m}**p**and

**m**are the electric and magnetic dipoles of the particle,

**E**and

**H**are the incident electric and magnetic fields respectively (the asterisk indicates the complex conjugate and ⊗ denote the dyadic product), while

*μ*

_{0}is the magnetic permeability of vacuum,

*η*is the vacuum impedance, n the refractive index and k is the wavenumber. For chiral particles (assuming isotropic polarizabilities), the electric

**p**and magnetic

**m**dipoles are defined as: where

*∊*

_{0}denotes the vacuum permittivity,

*α*and

_{e}*α*are the electric and magnetic polarizabilities respectively, while

_{m}*χ*denotes the chiral polarizability, which has opposite sign for the two enantiomeric forms of a chiral compound (note that all the defined polarizabilities have volume dimensions, and are directly comparable). Introducing Eqs. (26) and (27) into Eq. (25) and taking into account that polarizabilities are complex magnitudes, it is possible to express the optical forces in dipolar approximation into different contributions related to the real and imaginary parts of the particle properties and some electromagnetic field quantities, which can be written as [21]:

## Optical conveyor for cylindrical vector beams

In this section, we realize a more detailed description of the calculation of the optical forces for the proposed set-up for chiral resolution. As it was mentioned in the article, we propose the use of a 4-pi configuration with annular apodization of convergent cylindrical vector beams. Based on the papers by Richards and Wolf [39, 40], Youngworth and Brown [35] demonstrated that the electric field components (using cylindrical coordinates) in the vicinity of the focus of a radially or azimuthally polarized beam can be obtained by using vectorial diffraction theory. In the special case of annular apodization and for sufficiently narrow rings, analytic representation of the electric fields near focus can be obtained through approximations [11,41] of the diffraction integrals. Therefore, let us consider two counter-propagating electric fields with the following polarization states in cylindrical coordinates (E* _{r}*, E

*, E*

_{ϕ}*):*

_{z}*ϑ*=2A

*δ*c

^{1/2}s Sinc(ks

*δ*z/2), c=cos(

*θ*), s=sin(

_{i}*θ*), M is the matrix for the inversion of z coordinate (M=((1,0,0),(0,1,0),(0,0, −1))), while

_{i}*ê*={cJ

_{r}_{1}(krs), 0, jsJ

_{0}(krs) } and

*ê*={0, J

_{ϕ}_{1}(krs), 0} denote the radial and azimuthally polarization vectors of cylindrical beams generated, whose degree of ellipticity is controlled by the parameters B

_{1}and B

_{2}. The corresponding magnetic fields, neglecting the effects of the derivatives of the sinc function, can be expressed as:

The potential that controls the motion of chiral particles can be calculated from

Therefore, for the total fields (**E**=**E**_{1}+**E**_{2} and **H**=**H**_{1}+**H**_{2}) given by Eqs. (41), (42), (43) and (44), the potential can be divided in three different contributions, one describing an optical trap (U* _{t}*), and the other two are related to the travelling waves propagating in opposite directions (U

_{c1}and U

_{c2}), so it can be written as:

*χ*=±

_{R}*βα*

_{e,R}(the sign depends on the considered enantiomer). The values used for B

_{1}and B

_{2}parameters, that fix the ellipticity of the incident beams, were B

_{1}=

*β*/q and B

_{2}=

*β*. The optical forces were obtained calculating the gradient of the potential U, and the particles trajectory were simulated by numerical solution of motion equation including thermal effects, the values used are shown in the main manuscript.

## Particle’s polarizabilities

We consider small chiral particles with radius *a* much lower than the wavelength of the counter-propagating beams, inside on a dielectric media of refractive index *n*. The optical force on the chiral sphere (CS) can be deduced by using the dipole approximation for which (CS) is modeled as a magnetic and electric dipole induced given by [42]:

**E**and

**H**are the incident electric and magnetic fields respectively,

*∊*

_{0}denotes the vacuum permittivity,

*μ*

_{0}is the magnetic permeability of vacuum,

*η*is the vacuum impedance,

*α*and

_{e}*α*are the electric and magnetic polarizabilities respectively, while

_{m}*χ*denotes the chiral polarizability, which has opposite sign for the two enantiomeric forms of a chiral compound (note that all the defined polarizabilities have volume dimensions, and are directly comparable). Using the Bohren [37] notation for scattering coefficient of an optically active particle, the polarizability coefficients of previous equations can be written as [34]:

_{0}is the vacuum wavenumber, while

*a*

_{1},

*b*

_{1}and

*c*

_{1}are the coefficients of the scattered field obtained by Bohren for a spherical particle optically active given by:

*x*=

*k*

_{0}

*n a*,

*J*is

*L*or

*R*and they make reference to the wavenumbers

*k*=

_{L}*k*

_{0}

*N*=

_{L}*k*

_{0}(

*n*+Δ) and

_{p}*k*=

_{R}*k*

_{0}

*N*=

_{R}*k*

_{0}(

*n*− Δ) with

_{p}*n*the refractive index of the particle and Δ the chirality parameter. The auxiliary functions

_{p}*ψ*

_{1}(

*x*) =

*x j*

_{1}(

*x*), ${\xi}_{1}(x)=x{h}_{1}^{1}(x)$ are related with

*j*

_{1}(

*x*) the spherical Bessel functions and ${h}_{1}^{1}(x)$ the spherical Hankel function of the first kind. The relative index

*m*,

_{L}*m*and parameter

_{R}*m*are defined as follows [37]:

The values used in the simulations were n* _{p}*=1.5, n=1.33, a=

*λ*/15,

*λ*=1070 10

^{−9}m and Δ=±0.1, being the real part of the polarizabilities

*α*=5 10

_{R}^{−23}m

^{3}and

*χ*=±1.1 10

_{R}^{−23}m

^{3}and the imaginary part of the polarizabilities

*α*=5 10

_{I}^{−25}m

^{3}and

*χ*=±1 10

_{I}^{−25}m

^{3}, which are two orders of magnitude lower than the corresponding real parts for both cases.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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