Selective trapping of chiral nanoparticles via vector Lissajous beams

Hao Wu; Ping Zhang; Xuejing Zhang; Yi Hu; Zhigang Chen; Jingjun Xu

doi:10.1364/OE.448987

1. Introduction

Polarization is an intrinsic and fundamental vectorial feature of light. It plays a crucial role in designing novel optical fields and realizing exotic light-matter interactions. Different polarizations can be imparted to an optical field in a position-dependent configuration. The resulting so-called vector light, represented by the well-known cylindrical vector beams (CVBs), manifests unique properties in the application of optical manipulation [1–6], optical communication [7], optical microscopy [8,9], material processing [10,11], etc. These progresses have triggered increasing attention in recent years on tailoring the polarization of light.

Quite recently, a new type of CVBs, namely vector Lissajous beams (VLBs), was proposed [12]. Their transverse components have an angular relationship that is described by Lissajous curves. The polarization vectors of these VLBs are defined by two integer parameters (or orders) denoted as (p,q). Compared with traditional CVBs characterized by a single order p, the extra degree of freedom allows for engineering different types (real, imaginary, or complex) of electromagnetic field components. In particular, one can employ the imaginary part of the longitudinal component to control the local distribution of spin angular momentum [12]. Featured with circularly/elliptically-polarizations, traditional CVBs were found useful in the separation and selective trapping of chiral objects that have structures of opposite handedness (also called as enantiomers: S and R enantiomers are defined to stand for left- and right-handed chiral compounds, respectively [13]). Thus one may expect that VLBs, whose polarizations are more controllable, can offer additional functions in such light manipulations.

In this paper, we study the chiral properties of VLBs and their use in manipulating chiral particles. We find that VLBs can exhibit optical chirality for a certain combination of the beam parameters (p,q). Among these combinations, the VLB with (p,q) = (2,1) is found preferable for the separation of S and R enantiomers, as it has two well-separated intensity spots that carry strong chirality. The forces of this optical field exerting on particles with opposite chiral parameters have a relationship of odd and even parity in the transverse and longitudinal dimensions, respectively. The two light spots are capable of separately trapping particles of inverted chirality transversely while applying longitudinal forces of the same direction on these particles. Thus such optical beams may find applications in simultaneous identification and selective delivery of opposite enantiomer molecules.

2. Chiral vector Lissajous beams

To begin with, we consider the generation of VLBs via a high numerical aperture (NA) objective lens with a focal length f. Using the Richards-Wolf vectorial diffraction theory [14,15] and Bessel identities [1], the electric and magnetic components of an optical field in the vicinity of the focus have the expressions of

(1a)$${\textbf E}(r,\varphi ,z) ={-} ikf\int_0^{{\theta _m }} {T(\theta )L(\theta ){{\textbf Q}_E}(r,\varphi ,\theta )} \textrm{exp} ({ikz\cos \theta } )\sin \theta d\theta ,$$

(1b)$${\textbf H}(r,\varphi ,z) ={-} \frac{{ikf}}{Z}\int_0^{{\theta _m }} {T(\theta )L(\theta ){{\textbf Q}_H}(r,\varphi ,\theta )} \textrm{exp} ({ikz\cos \theta } )\sin \theta d\theta ,$$

where $(r,\varphi ,z)$ are the cylindrical coordinates with $r\textrm{ = }\sqrt {{x^2} + {y^2}}$ and $\varphi \textrm{ = arctan}({{y / x}} )$, $k = {{2\pi {n_b}} / \lambda }$ represents the wave number with ${n_b}$ being the refractive index of surrounding medium and $\lambda$ denoting the wavelength in vacuum, Z is the wave impedance, $\theta$ is the polar angle with its maximum value ${\theta _m }$ given by the NA of an objective lens, $T(\theta )$ is the pupil’s apodization function, $L(\theta )$ is the complex amplitude of an optical field illuminating the objective lens, and ${{\textbf Q}_E}$ and ${{\textbf Q}_H}$ introduce the orders $(p,q)$ mentioned in the introduction to control the profiles of VLBs (see Eqs. (10) and (11) in Ref. [12] or Eqs. (8) and (9) in Ref. [16]). In the focal plane (i.e., z = 0), Eq. (1) can be written as:

(2)$$\left[ {\begin{array}{*{20}{c}} {{E_x}}\\ {{E_y}}\\ {{E_z}} \end{array}} \right](r,\varphi ,0) = \left[ {\begin{array}{*{20}{c}} {{i^{p + 1}}{M_{x1}} + {i^{q + 1}}{M_{x2}}}\\ {{i^{p + 1}}{M_{y1}} + {i^{q + 1}}{M_{y2}}}\\ {{i^p}{M_{z1}} + {i^q}{M_{z2}}} \end{array}} \right],\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} {{H_x}}\\ {{H_y}}\\ {{H_z}} \end{array}} \right](r,\varphi ,0) = \left[ {\begin{array}{*{20}{c}} {{i^{p + 1}}{N_{x1}} + {i^{q + 1}}{N_{x2}}}\\ {{i^{p + 1}}{N_{y1}} + {i^{q + 1}}{N_{y2}}}\\ {{i^p}{N_{z1}} + {i^q}{N_{z2}}} \end{array}} \right],$$

where ${M_{tl}}$ and ${N_{tl}}$ ($t = x,y,z$, $l = 1,2$) are obtained by integrating Eq. (1) and they are all real. p and q are used to produce circular (elliptical) polarization by properly engineering the imaginary part of the longitudinal components (i.e., ${E_z}$ and ${H_z}$) that are in charge of the local distribution of spin angular momentum density [12].

Optical fields having circular (elliptical) polarizations are often loosely called as chiral fields [17]. The chirality of light can be characterized by a conservative quantity of an electromagnetic field, namely optical chirality density (OCD), which was introduced to quantify the strength of the coupling between a field and chirality of an object [18]. In general, the OCD is expressed as [18,19]:

(3)$$C = \frac{{{k^2}}}{{2\omega }}{\mathop{\rm Im}\nolimits} ({{\textbf E} \cdot {{\textbf H}^ \ast }} ),$$

where $\omega$ is the angular frequency of light. Positive and negative values of C represent left- and right-handedness chirality densities, respectively. In this framework, the OCD of the VLBs in Eq. (2) reads:

(4)$$\begin{array}{c} C = \frac{{{k^2}}}{{2\omega }}{\mathop{\rm Im}\nolimits} [{{{({ - 1} )}^q}{i^{p + q}}({{M_{x1}}{N_{x2}} + {M_{y1}}{N_{y2}} + {M_{z1}}{N_{z2}}} )} \\ { + {{({ - 1} )}^p}{i^{p + q}}({{M_{x2}}{N_{x1}} + {M_{y2}}{N_{y1}} + {M_{z2}}{N_{z1}}} )} ]. \end{array}$$

Apparently, for $p = q$ (reducing to the case of the classical CVBs), the OCD is null. The VLBs have non-zero chirality densities if $(p,q)$ satisfy certain relations, i.e., $p + q = \textrm{odd}$ and $q \ne 0$. Without loss of generality, only zero and positive integer orders are considered in this paper. To generate these VLBs, we take the function $L(\theta )$ in a form of Bessel–Gaussian profile [15]:

(5)$$L(\theta )= {E_0}{J_1}({2{\beta_0}t} )\textrm{exp} ({ - \beta_0^2{t^2}} ),$$

where ${E_0}$ is the amplitude of the input light, ${\beta _0}$ is the ratio of the pupil radius and the beam waist, $t = {{\sin \theta } / {\sin {\theta _m}}}$, and ${J_1}({\cdot} )$ is the first-order Bessel function of the first kind. In what follows, unless otherwise stated, the parameters used in our study are chosen as ${n_b} = 1$, $\lambda = 532.8\;\textrm{nm}$, $f = 1\;\textrm{mm}$, $\textrm{NA} = 0.95$, ${E_0} = 1$ and ${\beta _0} = 1.6$. Figures 1(a) and 1(b) illustrate the intensity profiles of two VLBs with the lowest orders of $(p,q)$ that can introduce optical chirality in the focal plane. Their non-zero OCDs, anti-symmetrical about the x-axis, mainly distribute in two regions along the y-axis [Figs. 1(c) and 1(d)]. The two regions have a smaller spacing for the case of $({p,q} )= ({0,1} )$; while for the other case, the dominant OCDs exhibit a similar distribution of the light pattern. The former field has a stronger chirality density comparing to the latter one. Figure 1(e) summarizes the OCD of the VLB by examining its maximum value for various combinations of $(p,q)$. A large OCD tends to appear for a small value of $(p,q)$. Note that the average OCDs for all the cases are zero, since the incident beam entering the objective is linearly polarized and thus carries no chirality. Albeit being local, these chiral distributions are useful in manipulating enantiomers.

Fig. 1. (a,b) Intensity profiles of two chiral VLBs in the focal plane. (c,d) OCDs of the VLBs in (a,b). (e) Maximum OCD of a VLB as a function of $({p,q} )$. Panels (a,b) and (c-e) are normalized by the maximum value in (a) and (c), respectively.

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3. Optical force on dipolar chiral particles

3.1 Theoretical model

We consider that a dipolar chiral particle located in vacuum is illuminated by an optical beam. Within the small particle limit [20,21], the polarizability elements of this particle read [21–26]

(6)$${\alpha _{ee}} = \frac{{i6\pi {\varepsilon _0}{\varepsilon _b}}}{{{k^3}}}a_1^{(1 )},\;\;{\alpha _{mm}} = \frac{{i6\pi {\mu _b}}}{{{\mu _0}{k^3}}}b_1^{(1 )},\;\;{\alpha _{em}} ={-} \frac{{6\pi {n_b}}}{{{Z_0}{k^3}}}a_1^{(2 )},$$

where ${\varepsilon _0}$ and ${\mu _0}$ are the permittivity and permeability in vacuum, ${\varepsilon _b}$ and ${\mu _b}$ are the relative permittivity and permeability of the surrounding medium, and ${Z_0} = \sqrt {{{{\mu _0}} / {{\varepsilon _0}}}}$ is the wave impedance in vacuum. The Mie coefficients in Eq. (6) can be expressed in terms of the spherical wave functions [27–29]:

(7)$$\begin{array}{l} a_n^{(1 )} = [{A_n^{(2 )}V_n^{(1 )} + A_n^{(1 )}V_n^{(2 )}} ]{Q_n},\\ a_n^{(2 )} = [{A_n^{(1 )}W_n^{(2 )} - A_n^{(2 )}W_n^{(1 )}} ]{Q_n},\\ b_n^{(1 )} = [{B_n^{(1 )}W_n^{(2 )} + B_n^{(2 )}W_n^{(1 )}} ]{Q_n}, \end{array}$$

with

(8)$$\begin{array}{l} A_n^{(j )} = {Z_s}D_n^{(1 )}({{x_j}} )- D_n^{(1 )}({{x_0}} ),\;\;\;\;B_n^{(j )} = D_n^{(1 )}({{x_j}} )- {Z_s}D_n^{(1 )}({{x_0}} ),\\ W_n^{(j )} = {Z_s}D_n^{(1 )}({{x_j}} )- D_n^{(3 )}({{x_0}} ),\;\;\;\;V_n^{(j )} = D_n^{(1 )}({{x_j}} )- {Z_s}D_n^{(3 )}({{x_0}} ),\\ {Q_n} = {{[{{\psi_n}({{x_0}} )/{\xi_n}({{x_0}} )} ]} / {({V_n^{(1 )}W_n^{(2 )} + V_n^{(2 )}W_n^{(1 )}} )}}. \end{array}$$

Here ${x_0} = {k_0}{r_s}$, ${x_1} = {k_1}{r_s}$ and ${x_2} = {k_2}{r_s}$, where ${k_0}$ is the wave number in vacuum, and ${k_1} = {k_0}\left( {\sqrt {{\varepsilon_s}{\mu_s}} + \kappa } \right)$ and ${k_2} = {k_0}\left( {\sqrt {{\varepsilon_s}{\mu_s}} - \kappa } \right)$ are the wave numbers in the chiral medium with ${\varepsilon _s}$, ${\mu _s}$ and $\kappa$ being the relative permittivity, the relative permeability and the chirality parameter of the chiral particle, respectively. ${Z_s} = \sqrt {{{{\mu _s}} / {{\varepsilon _s}}}}$ is the wave impedance of the particle. ${\psi _n}(x )$ and ${\xi _n}(x )$ are the Riccati–Bessel functions of the first and third kinds, respectively, while $D_n^{(1 )}(x )= {{{{\psi ^{\prime}}_n}(x )} / {{\psi _n}(x )}}$ and $D_n^{(3 )}(x )= {{{{\xi ^{\prime}}_n}(x )} / {{\xi _n}(x )}}$ are the corresponding logarithmic derivatives.

The expression of the time-averaged optical force acting on a dipolar chiral particle can be expressed as [21,22,24–26,30]:

(9)$$\left\langle {\textbf F} \right\rangle = {{\textbf F}_g} + {{\textbf F}_r} + {{\textbf F}_v} + {{\textbf F}_c} + {{\textbf F}_s} + {{\textbf F}_f},$$

with

(10)$$\begin{array}{*{20}{c}} {{{\textbf F}_g} ={-} \nabla \left\langle U \right\rangle ,}&{{{\textbf F}_r} = {{({{C_{ext}} + {C_{recoil}}} )\left\langle {\textbf S} \right\rangle } / c},}\\ {{{\textbf F}_v} = {\mu _0}{\textrm{Re}} ({{\alpha_{em}}} )\left[ {\nabla \times \left\langle {\textbf S} \right\rangle } \right],}&{{{\textbf F}_c} = {C_p}c\left[ {\nabla \times \left\langle {{\textbf L}_s^p} \right\rangle } \right] + {C_m}c\left[ {\nabla \times \left\langle {{\textbf L}_s^m} \right\rangle } \right],}\\ \begin{array}{l} {{\textbf F}_s} = \left[ {2{\omega^2}{\mu_0}{\textrm{Re}} ({{\alpha_{em}}} )- \frac{{k_0^5}}{{3\pi \varepsilon_0^2}}{\mathop{\rm Im}\nolimits} ({{\alpha_{ee}}\alpha_{em}^ \ast } )} \right]\left\langle {{\textbf L}_s^p} \right\rangle \\ + \left[ {2{\omega^2}{\mu_0}{\textrm{Re}} ({{\alpha_{em}}} )- \frac{{k_0^5{\mu_0}}}{{3\pi {\varepsilon_0}}}{\mathop{\rm Im}\nolimits} ({{\alpha_{mm}}\alpha_{em}^ \ast } )} \right]\left\langle {{\textbf L}_s^m} \right\rangle , \end{array}&{{{\textbf F}_f} = \frac{{ck_0^4\mu _0^2}}{{12\pi }}{\mathop{\rm Im}\nolimits} ({{\alpha_{ee}}\alpha_{em}^ \ast } ){\mathop{\rm Im}\nolimits} ({{\textbf E} \times {{\textbf H}^ \ast }} ),} \end{array}$$

where ${{\textbf F}_g}$ is the gradient force related to the optical potential $\left\langle U \right\rangle = {{ - {\textrm{Re}} ({{\alpha_{ee}}} ){{|{\textbf E} |}^2}} / 4} - {{{\textrm{Re}} ({{\alpha_{mm}}} ){{|{\textbf B} |}^2}} / 4} + {{{\mathop{\rm Im}\nolimits} ({{\alpha_{em}}} ){\mathop{\rm Im}\nolimits} ({{\textbf B} \cdot {{\textbf E}^ \ast }} )} / 2}$; ${{\textbf F}_r}$ denotes the radiation pressure that is proportional to the time-averaged Poynting vector $\left\langle {\textbf S} \right\rangle = {{{\textrm{Re}} ({{\textbf E} \times {{\textbf H}^ \ast }} )} / 2}$; ${{\textbf F}_v}$ represents the “vortex” force determined by the energy flow vortex and particle chirality [21]; ${{\textbf F}_c}$ describes the curl spin force associated with the curl of the time-averaged spin angular momentum (SAM) densities [31], with $\left\langle {{\textbf L}_s^p} \right\rangle = {\textrm{Re}} [{{{{\varepsilon_0}({{\textbf E} \times {{\textbf E}^ \ast }} )} / {4i\omega }}} ]$ and $\left\langle {{\textbf L}_s^m} \right\rangle = {\textrm{Re}} [{{{{\mu_0}({{\textbf H} \times {{\textbf H}^ \ast }} )} / {4i\omega }}} ]$; ${{\textbf F}_s}$ describes the spin density force that comes only from the coupling of the particle chirality with SAM densities; and ${{\textbf F}_f}$ is due to the alternating flow of the “stored energy” [32]. The cross section ${C_{ext}} = {C_p} + {C_m}$ is a sum of contributions from the electric dipole channel ${C_p} = {{{k_0}{\mathop{\rm Im}\nolimits} ({{\alpha_{ee}}} )} / {{\varepsilon _0}}}$ and the magnetic dipole channel ${C_m} = {k_0}{\mu _0}{\mathop{\rm Im}\nolimits} ({{\alpha_{mm}}} )$, and the term ${C_{recoil}} = {{ - k_0^4{\mu _0}[{{\textrm{Re}} ({{\alpha_{ee}}\alpha_{mm}^ \ast } )+ {{|{{\alpha_{em}}} |}^2}} ]} / {6\pi {\varepsilon _0}}}$ describes the recoil force related to an asymmetry parameter [30,33].

3.2 Results and discussion

In this section, we investigate the optomechanical behavior of a dipolar chiral particle having a spherical shape illuminated by a chiral VLB. In calculations, the parameters are set as: $P = \;200\;\textrm{mW}$ (beam power), ${r_s} = \;30\;\textrm{nm}$ (particle’s radius), ${\varepsilon _s} = \;2.0 + 0.1i$, and ${\mu _s} = \;1$. The chirality parameter of the particle has a plural form where the real and imaginary (also called as chiral absorption) parts correspond to rotatory power and circular dichroism, respectively [19]. The other parameters are the same with those used in Fig. 1. The VLB with $({p,q} )= ({2,1} )$ is under concern in the following analysis, since it is featured with a high OCD and the regions of inverted chirality densities have a large spacing, which are both beneficial to the separation of enantiomers having opposite handedness.

We first examine the transverse trapping of a dipolar chiral particle, whose chirality parameter is chosen as $\kappa = 0.5 + 0.1i$ (S enantiomer) as an example (other parameters will be discussed later). This large chirality parameter is frequently concerned in literatures, such as Refs. [25,34–38]. Figure 2(a) presents the calculated transverse force ${{\textbf F}_t}$ ($|{{{\textbf F}_t}} |= \sqrt {F_x^2 + F_y^2}$). The upper light spot, subject to left-handedness OCD, exerts relatively stronger forces on this particle, although the bottom light spot has the same intensity pattern. These strong forces are centripetal, thus allowing for a transverse trapping. In contrast, the bottom light spot, featured with right-handedness OCD, tends to push the S enantiomer away. To explore the physical origin of these optical forces, their contributions in ${F_y}$ are analyzed, as shown in Fig. 2(b). Apparently, the components excluding the gradient force ${F_g}$ almost do not make a contribution. For further analysis, the gradient force is decomposed into three parts: two achiral optical forces formulated as ${{\textbf F}_{ach1}} = {{{\textrm{Re}} ({{\alpha_{ee}}} )\nabla {{|{\textbf E} |}^2}} / 4}$ and ${{\textbf F}_{ach2}} = {{{\textrm{Re}} ({{\alpha_{mm}}} )\nabla {{|{\textbf B} |}^2}} / 4}$, and a chiral optical force ${{\textbf F}_{ch}} = {{ - {\mathop{\rm Im}\nolimits} ({{\alpha_{em}}} )\nabla {\mathop{\rm Im}\nolimits} ({{\textbf B} \cdot {{\textbf E}^ \ast }} )} / 2}$. Their distributions along the y-axis are presented in Fig. 2(c). ${F_{ch}}$ and ${F_{ach1}}$ exhibit a quite larger magnitude compared with ${F_{ach2}}$. In the region of $y > 0$ ($y < 0$), where the upper (bottom) light spot locates, the two main contributors in the gradient force tend to strengthen (contradict) each other. Furthermore, the parity of the gradient force is considered. For doing this, we start from the analysis of Eq. (8), and find the spherical wave functions satisfy the following relationships:

(11)$$\begin{array}{l} A_1^{(1 )}[{{k_1}({ - \kappa } )} ]= A_1^{(2 )}[{{k_2}(\kappa )} ],\;\;B_1^{(1 )}[{{k_1}({ - \kappa } )} ]= B_1^{(2 )}[{{k_2}(\kappa )} ],\;\\ W_1^{(1 )}[{{k_1}({ - \kappa } )} ]= W_1^{(2 )}[{{k_2}(\kappa )} ],\;V_1^{(1 )}[{{k_1}({ - \kappa } )} ]= V_1^{(2 )}[{{k_2}(\kappa )} ],\;\\ {Q_1}({ - \kappa } )= {Q_1}(\kappa ). \end{array}$$

Substituting Eq. (11) first into Eq. (7) and then into Eq. (6), one can obtain that the polarizabilities of the dipolar particles with opposite chiral parameters meet the following relations:

(12)$${a_{ee}}({ - \kappa } )= {a_{ee}}(\kappa ),\;{a_{mm}}({ - \kappa } )= {a_{mm}}(\kappa ),\;{a_{em}}({ - \kappa } )={-} {a_{em}}(\kappa ).$$

Using the above formula together with the parity of the optical field shown in Eq. (1):

(13)$$\begin{array}{*{20}{c}} {{{\partial {{|{{\textbf E}({ - x, - y} )} |}^2}} / {\partial j}} = {{ - \partial {{|{{\textbf E}({x,y} )} |}^2}} / {\partial j}},}\\ {{{\partial {{|{{\textbf B}({ - x, - y} )} |}^2}} / {\partial j}} = {{ - \partial {{|{{\textbf B}({x,y} )} |}^2}} / {\partial j}},}\\ {{{\partial {\mathop{\rm Im}\nolimits} [{{\textbf B}({ - x, - y} )\cdot {{\textbf E}^ \ast }({ - x, - y} )} ]} / {\partial j}} = {{\partial {\mathop{\rm Im}\nolimits} [{{\textbf B}({x,y} )\cdot {{\textbf E}^ \ast }({x,y} )} ]} / {\partial j}},} \end{array}({j = x,y} )$$

the symmetrical property of the gradient force is obtained as shown in the following:

(14)$${F_{grad}}({ - \kappa ; - x, - y} )={-} {F_{grad}}({\kappa ;x,y} ).$$

The transverse optical force ${{\textbf F}_t}$, almost contributed by the gradient force, shares the same symmetry relationship described by Eq. (14). Thus for a particle with an opposite chiral parameter (i.e., $\kappa ={-} 0.5 - 0.1i$), the forces experienced by this R enantiomer are centrosymmetric to those for the S enantiomer. In this regard, the two intense spots of the VLB can be used to selectively trap opposite enantiomers in the transverse plane.

Fig. 2. (a) Transverse optical forces (black arrows) experienced by a spherical particle (with chirality parameter $\kappa = 0.5 + 0.1i$) illuminated by a VLB (background). The length and direction of the arrow present the magnitude and direction of the force, respectively. (b,c) Force distribution of (a) along the y-axis (x = 0) together with all of its components (b) and the components consisting of the gradient force (c). The black dots in (b,c) mark the transverse trapping positions.

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Next, we examine the selective trapping of enantiomers with other chirality parameters. To this end, the transverse trapping potential along the y-axis, defined as ${V_y} ={-} \int {{F_y}\; dy}$ [39], is calculated. Its distributions for different S enantiomers are presented in Figs. 3(a) and 3(b). The potential offered by the upper light spot is always a well, indicating a trapping effect. The trapping strength is more influenced by the real part of $\kappa$. In order to achieve a selective trapping for opposite enantiomers, the potential induced by the bottom light spot should be a barrier for an S enantiomer by considering the symmetry property (i.e., Eq. (14)) of the transverse force. This condition is yet not satisfied for all the chirality parameters. For instance, it is only met for ${\textrm{Re}} (\kappa )\ge 0.483$ and $- 0.192 \le {\mathop{\rm Im}\nolimits} (\kappa )\le 0.275$ in Fig. 3(a) and Fig. 3(b), respectively. Other more parameters meeting this condition are summarized in Fig. 3(c). They locate above the curve of ${\textrm{Re}} ({{\kappa_0}} )$ versus ${\mathop{\rm Im}\nolimits} ({{\kappa_0}} )$ (${\kappa _0}$ is hence defined as critical chiral parameter). In particular, as along as the value of ${\textrm{Re}} (\kappa )$ is sufficiently high (say, $\ge 0.945$), a selective trapping always exists regardless of the chiral absorption. For particles of other sizes, the value of ${\kappa _0}$ shows similar distributions [Fig. 3(c)].

Fig. 3. (a,b) Transverse trapping potential along the y-axis (x = 0) for various chirality parameters of particles positioned in the VLB employed in Fig. 2: ${\mathop{\rm Im}\nolimits} (\kappa )= 0.1$ in (a) and ${\textrm{Re}} (\kappa )= 0.5$ in (b). These potentials are normalized by the factor ${k_B}T$ (${k_B}$ is the Boltzmann constant and $T = 300\;\textrm{K}$ is the absolute temperature). (c) Critical chirality parameter ${\kappa _0}$ for particles of different radii.

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Finally, we consider the longitudinal optical force ${F_z}$ of the VLB exerting on enantiomers. ${F_z}$ along the y-axis (x = 0) together with its contributions are displayed in Fig. 4(a) for an S enantiomer ($\kappa = 0.5 + 0.1i$). Pushing (${F_z} > 0$) and pulling (${F_z} < 0$) forces appear in the upper and bottom light spot, respectively. The longitudinal optical force is nearly not contributed by the gradient force ${F_g}$ and the alternating flow force ${F_f}$ [Fig. 4(a)]. This feature holds for the cases of other chirality parameters, as shown in Figs. 4(b) and 4(c). In this regard, we can safely write ${F_z}$ in the form of ${F_z} \cong {F_r} + {F_v} + {F_c} + {F_s}$ based on Eq. (9). Using the result in Eq. (12) together with the symmetry property [Eq. (13)] of the optical field, it is readily to obtain:

(15)$${F_z}({ - \kappa ; - x, - y} )= {F_z}({\kappa ;x,y} ).$$

Thus, for opposite enantiomers that are assumed to be trapped transversely in different light spots, they experience longitudinal forces with the same direction and strength. This non-conserved force is helpful for the delivery of the chiral particles. Its strength is more sensitive to the imaginary part of the chirality parameter [Figs. 4(b) and 4(c)], which is contrary to the case of the transverse force. The sign of the longitudinal force is only determined by the sign of the chiral absorption [Fig. 4(c)].

Fig. 4. Longitudinal forces exerting on S enantiomers and their components for the VLB employed in Fig. 2. (a) Along the y-axis (x = 0) and $\kappa = 0.5 + 0.1i$; (b,c) at the transverse trapping position (as marked by the black dot in (a)): ${\mathop{\rm Im}\nolimits} (\kappa )= 0.1$ in (b) and ${\textrm{Re}} (\kappa )= 0.5$ in (c). The inset in (a) only shows the gradient force ${F_g}$ and the alternating flow force ${F_f}$.

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4. Conclusion

In summary, we have investigated the chiral properties of VLBs in the manipulation of chiral particles. It is found that the OCD is not zero for certain relations of orders $({p,q} )$. We focus on a VLB whose chiral property is more beneficial to the separation of opposite enantiomers. For dipolar particles with opposite chiral parameters, the forces offered by this optical field have a relationship of odd and even parity in the transverse and longitudinal directions, respectively. The two spots of this optical beam can trap particles of inverted chirality transversely, while always exerting a force in the same direction longitudinally. Such a selective trapping effect is discussed for various chiral parameters and radii of dipolar particles. Our results may find potential applications in optical selection and separation of chiral particles at nanometer scales. More intriguing phenomena are expected by employing the VLBs to manipulate chiral particles in nontrivial media [40,41].

Funding

National Key Research and Development Program of China (2017YFA0303800); National Natural Science Foundation of China (12022404, 62075105, 91750204); 111 Project (B07013).

Disclosures

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

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Selective trapping of chiral nanoparticles via vector Lissajous beams

Abstract

1. Introduction

2. Chiral vector Lissajous beams

3. Optical force on dipolar chiral particles

3.1 Theoretical model

3.2 Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (16)

Optics Express