Customized Chirality of an Optical Vortex Pair: Helical Dichroism and Enantioselective Force

Abstract

Tailoring the chirality of an optical vortex is crucial for advancing helical chiroptical spectroscopy techniques in various scenarios and attracts great attention. In contrast to the single vortex, the optical vortex pair exhibits richer, fantastic chirality properties due to its additional adjustment parameters. Here, a comprehensive investigation of the chirality for linearly polarized optical vortex pairs based on the vector angular spectrum decomposition method is conducted. The numerical results show that the magnitudes and distributions of local chirality density, helical dichroism, and enantioselective force of the optical vortex pair can be flexibly customized by the position as well as sign combination of vortices, and can vary during free space propagation. The underlying physical mechanism behind these phenomena is ascribed to the interplay of two vortices. Our work can deepen the understanding of the chirality for multiple vortices and open-up the prospect for relevant applications in chiral recognition and manipulation.

1. Introduction

Chiral objects cannot be superimposed on their mirror images by any combination of rotations or translations; they come in left- and right-handed forms known as enantiomers [1,2]. They are ubiquitous in nature, appearing in DNA, viruses, carbohydrates, biomolecules, and proteins [3]. Discrimination of the chirality for some enantiomers plays a key role in the pharmaceutical industry, as their handedness directly influences the potency and toxicity [4]. In addition, light beams also have the chirality [5]. The most representative examples are the left- and right-handed circularly polarized beams whose chirality arises from the circular polarizations, i.e., spin angular momentum [6]. The enantiomers will exhibit differential absorption rates called circular dichroism [7,8,9,10,11,12] and enantioselective force [13,14,15,16] to the left- and right-handed circularly polarized beams, which become powerful tools in chiral recognition and manipulation. Recent years have witnessed the rapidly growing research on the vortex beams due to their fantastic characteristics and extensive applications [17,18,19,20]. Even without the spin angular momentum, the vortex beams are chiral as well, and their chirality comes from the orbital angular momentum $l\hslash$ in each photon with topological charge $l$ and reduced Plank constant $\hslash$ [21]. Specifically, such beams are left- and right-handed for the $l>0$ and $l<0$, respectively [22]. Similarly, the helical dichroism [23,24,25,26,27] and enantioselective force [28] will occur in the interaction between the enantiomers and vortex beams. Such a chiroptical spectroscopy technique exhibits stronger capability in sensitive and precise chiral recognition [29,30,31,32] and has sparked a surge in research.

To optimize the application of the chiroptical spectroscopy technique of the vortex beam in various scenarios, it is necessary to actively customize their chirality for purpose. There have been many works devoted to this issue [33,34,35]. Dale et al. analyzed the chirality of an optical vortex at nanoscale under different focusing conditions [33]. Forbes et al. demonstrated the influence of the polarization degree on the chirality for Bessel vortex beams [34]. Subsequently, Forbes studied the optical chirality of Poincaré vector vortex beams for different topological charges [35]. However, the existing studies are all concentrated on the beams carrying single vortices, and the beams carrying multiple vortices remain unexplored. In fact, the generations and investigations on beams carrying multiple vortices can date back to the last century [36,37,38,39]. Among them, the optical vortex pair is a typical subject of study. In contrast to the single vortex, it has more fascinating properties due to the vortex–vortex interaction and has attracted much attention as of late [40,41,42]. For instance, Dan et al. performed the abrupt auto-focusing of a circular Airy-prime beam with a vortex pair [40]. Successively, Zhen et al. controlled the symmetry of the photonic spin Hall effect in the optical vortex pair [41]. Sun et al. studied the fusion of fractional vortex pairs and their transition to integer vortices [42]. These advancements suggest that the multiple vortices could enrich and customize the optical chirality of vortex beams, a topic yet to be thoroughly investigated.

Here, we develop a theoretical description of how the optical chirality can be tailored by a linearly polarized optical vortex pair. The analytical expressions of the electromagnetic fields for optical vortex pairs are first derived by the vector angular spectrum decomposition method under the paraxial condition. Then, the effects of the vortices’ positions, sign combinations, and propagation distance on the magnitude and spatial distributions of intensity, phase, local chirality density, helical dichroism, and enantioselective force for the optical vortex pairs are examined numerically and the physical mechanisms behind this are discussed. This work provides a foundation for further investigations on the dynamics of optical vortex pairs and the improvement of the existing approaches in chiroptical spectroscopy techniques of chiral recognition and manipulation.

2. Theoretical Model and Formulae

In the Cartesian coordinates $(x,y,z)$, the scalar field of the optical vortex pair on the initial plane $z=0$ can be expressed as [41]

$$u\left(x,y,0\right)={\left({\displaystyle \frac{x-x_1}{w_0}}+i{\displaystyle \frac{y-y_1}{w_0}}\right)}^{\left|l\right|}{\left({\displaystyle \frac{x-x_2}{w_0}}+i\mathrm{sign}(l){\displaystyle \frac{y-y_2}{w_0}}\right)}^{\left|l\right|}{e}^{-\left(x^2+y^2\right)/w_0^2},$$

(1)

where $w_0$ is the beam waist and $(x_1,y_1)$ and $(x_2,y_2)$ are the positions of two vortices with the same magnitude $\left|l\right|$ of the topological charges. Here, the sign of topological charge for the first vortex is always positive, whereas that for the second vortex depends on the sign function $\mathrm{sign}(\cdot )$.

Next, we will derive the electromagnetic fields of such an optical vortex pair. For simplicity without loss of generality, $\left|l\right|=1$ is assumed throughout this paper. Employing the following three formulae,

$${\displaystyle {\int }_{-\infty }^{+\infty }{e}^{-a{x}^2+\mathrm{i}bx}\mathrm{d}x=\sqrt{\frac{\pi }{a}}{e}^{-b^2/(4a)}},$$

(2)

$${\displaystyle {\int }_{-\infty }^{+\infty }x{e}^{-a{x}^2+\mathrm{i}bx}\mathrm{d}x=\frac{\mathrm{i}b}{2a}\sqrt{\frac{\pi }{a}}{e}^{-b^2/(4a)}},$$

(3)

$${\displaystyle {\int }_{-\infty }^{+\infty }{x}^2{e}^{-a{x}^2+\mathrm{i}bx}\mathrm{d}x=\sqrt{\frac{\pi }{a}}\left(\frac{1}{2a}-\frac{b^2}{4a^2}\right)}{e}^{-b^2/(4a)},$$

(4)

the angular spectrum of Equation (1) with $\left|l\right|=1$ is written as

$$\begin{array}{cc}\hfill \tilde{u}\left(k_x,k_y\right)& ={\displaystyle \frac{1}{4{\pi }^2}}{\displaystyle {\int }_{-\infty }^{+\infty }u\left(x,y,0\right)\exp \left[-\mathrm{i}\left(k_{x}x+k_{y}y\right)\right]\mathrm{d}{k}_x\mathrm{d}{k}_y}\hfill \\ & ={\displaystyle \frac{w_0^2}{8\pi }}\left\{\begin{array}{l}{\displaystyle \frac{w_0^2}{2}}\left(\mathrm{sign}(l)k_y^2-k_x^2\right)-\mathrm{i}{w}_0^2{k}_x{k}_y\left(\mathrm{sign}(l)+1\right)+2AB\\ +w_0\left[\mathrm{i}\left(A+B\right)k_x-\left(\mathrm{sign}(l)A+B\right)k_y\right]+1-\mathrm{sign}(l)\end{array}\right\}{e}^{-w_0^2\left(k_x^2+k_y^2\right)/4},\hfill \end{array}$$

(5)

where $A=\left(x_1+\mathrm{i}{y}_1\right)/w_0$ and $B=\left(x_2+\mathrm{isign}(l)y_2\right)/w_0$. According to the vector angular spectrum decomposition method under the paraxial condition, the electric fields of the optical vortex pair with polarization parameters $(p_x,p_y)$ can be expressed as

$$E\left(x,y,z\right)=e^{\mathrm{i}kz}{\displaystyle {\int }_{-\infty }^{+\infty }\tilde{u}\left(k_x,k_y\right)e^{\mathrm{i}\left[k_{x}x+k_{y}y-z\left(k_x^2+k_y^2\right)/2k\right]}\mathrm{d}{k}_x\mathrm{d}{k}_y},$$

(6)

where $\tilde{u}={\tilde{u}}_x\widehat{x}+{\tilde{u}}_y\widehat{y}+{\tilde{u}}_z\widehat{z}$ is the vector angular spectrum with ${\tilde{u}}_{x,y}=p_{x,y}\tilde{u}(k_x,k_y)$ and ${\tilde{u}}_z=\mathrm{i}(p_x{k}_x+p_y{k}_y)\tilde{u}(k_x,k_y)/k$ that can be derived from the divergence equation $\nabla \cdot E=0$. Correspondingly, the magnetic fields can be calculated in a similar manner, as

$$H\left(x,y,z\right)=\exp \left(\mathrm{i}kz\right){\displaystyle {\int }_{-\infty }^{+\infty }\tilde{h}\left(k_x,k_y\right)e^{\mathrm{i}\left[k_{x}x+k_{y}y-z\left(k_x^2+k_y^2\right)/2k\right]}\mathrm{d}{k}_x\mathrm{d}{k}_y},$$

(7)

where $\tilde{h}=h_x\widehat{x}+h_y\widehat{y}+h_z\widehat{z}$. Based on the relationships $\nabla \times E=\mathrm{i}\omega \mu H$ and $\nabla \cdot H=0$, it can be derived that $h_x=-p_y\tilde{u}(k_x,k_y)/Z$, $h_y=p_x\tilde{u}(k_x,k_y)/Z$, and ${\tilde{u}}_z=\mathrm{i}(p_x{k}_y-p_y{k}_x)\tilde{u}(k_x,k_y)/k/Z$, with Z being the wave impedance in a vacuum. Then, using the following relationships

$$\left[\begin{array}{l}u\left(x,y,z\right)\\ \partial u\left(x,y,z\right)/\partial x\\ \partial u\left(x,y,z\right)/\partial y\end{array}\right]=e^{\mathrm{i}kz}{\displaystyle {\int }_{-\infty }^{+\infty }\left[\begin{array}{c}1\\ k_x\\ k_y\end{array}\right]\tilde{u}\left(k_x,k_y\right)e^{\mathrm{i}\left[k_{x}x+k_{y}y-z\left(k_x^2+k_y^2\right)/2k\right]}\mathrm{d}{k}_x\mathrm{d}{k}_y},$$

(8)

The electromagnetic fields of the optical vortex pair can be finally presented as

$$E=\left(p_x\widehat{x}+p_y\widehat{y}\right)u\left(x,y,z\right)+{\displaystyle \frac{\mathrm{i}}{k}}\left(p_x{\displaystyle \frac{\partial u\left(x,y,z\right)}{\partial x}}+p_y{\displaystyle \frac{\partial u\left(x,y,z\right)}{\partial y}}\right)\widehat{z},$$

(9)

$$H={\displaystyle \frac{1}{Z}}\left[\left(-p_y\widehat{x}+p_x\widehat{y}\right)u\left(x,y,z\right)-{\displaystyle \frac{\mathrm{i}}{k}}\left(p_y{\displaystyle \frac{\partial u\left(x,y,z\right)}{\partial x}}-p_x{\displaystyle \frac{\partial u\left(x,y,z\right)}{\partial y}}\right)\widehat{z}\right],$$

(10)

where

$$u=\left[{\displaystyle \frac{w_0^4}{w^2}}\left({\displaystyle \frac{x+\mathrm{i}y}{w^2}}-{\displaystyle \frac{A}{w_0}}\right)\left({\displaystyle \frac{x+\mathrm{isign}(l)y}{w^2}}-{\displaystyle \frac{B}{w_0}}\right)+\left({\displaystyle \frac{w_0^4}{2w^4}}-{\displaystyle \frac{w_0^2}{2w^2}}\right)\left(v-1\right)\right]e^{-\left(x^2+y^2\right)/w^2+\mathrm{i}kz},$$

(11)

$${\displaystyle \frac{\partial u}{\partial x}}=-{\displaystyle \frac{2x}{w^2}}u+{\displaystyle \frac{w_0^4}{w^4}}\left[{\displaystyle \frac{2x+\mathrm{i}(1+\mathrm{sign}(l))y}{w^2}}-{\displaystyle \frac{A+B}{w_0}}\right]e^{-\left(x^2+y^2\right)/w^2+\mathrm{i}kz},$$

(12)

$${\displaystyle \frac{\partial u}{\partial y}}=-{\displaystyle \frac{2y}{w^2}}u+{\displaystyle \frac{w_0^4}{w^4}}\left[{\displaystyle \frac{\mathrm{i}(1+\mathrm{sign}(l))x-2\mathrm{sign}(l)y}{w^2}}-\mathrm{i}{\displaystyle \frac{A\mathrm{sign}(l)+B}{w_0}}\right]e^{-\left(x^2+y^2\right)/w^2+\mathrm{i}kz},$$

(13)

with $w=\sqrt{2\mathrm{i}z/k+w_0^2}$.

Physically speaking, the inherent chiral nature of the local electromagnetic fields can be quantified by the local chirality density defined as

$$C={\displaystyle \frac{\omega }{2c^2}}\mathrm{Im}\left(E\cdot H^{\ast }\right),$$

(14)

where $\omega$ is the angular frequency of the beam, c is the speed of light in a vacuum, and superscript “*” represents the complex conjugate. Then, we will explore the relationships between the local chirality density, helical dichroism, and enantioselective force. According to the theory of quantum electrodynamics, the dipole expansion form of the interaction between the particle $\xi$ at location $R_{\xi }$ and the light field can be described by the Hamiltonian operator as [23,26,27]

$${\widehat{H}}_{\mathrm{int}}\left(\xi \right)=-{\epsilon }_0^{-1}{\widehat{\mu }}_j\left(\xi \right){\widehat{d}}_j^{\perp }\left(R_{\xi }\right)-{\widehat{m}}_j\left(\xi \right){\widehat{b}}_j\left(R_{\xi }\right),$$

(15)

in which ${\epsilon }_0$ is the permittivity in vacuum, ${\widehat{\mu }}_j\left(\xi \right)$ and ${\widehat{m}}_j\left(\xi \right)$ are the electric and magnetic dipole operators, respectively, ${\widehat{d}}_j^{\perp }\left(R_{\xi }\right)$ and ${\widehat{b}}_j\left(R_{\xi }\right)$ are the electric displacement and magnetic field mode operators (transverse to the Poynting vector), respectively, and can be expressed as

$${\widehat{d}}_j^{\perp }\left(r\right)={\displaystyle \sum _{k,l}\mathrm{\Omega }\left[E_{\left|l\right|,j}\left(r\right){\widehat{a}}_{\left|l\right|}\left(k{\widehat{z}}_j\right)-\mathrm{H}.\mathrm{c}.\right]},{\widehat{b}}_j\left(r\right)={\displaystyle \sum _{k,l}\mathrm{\Omega }\left[\mathrm{i}{H}_{\left|l\right|,j}\left(r\right){\widehat{a}}_{\left|l\right|}\left(k{\widehat{z}}_j\right)-\mathrm{H}.\mathrm{c}.\right]},$$

(16)

Here, $E_{\left|l\right|,j}\left(r\right)$ and $H_{\left|l\right|,j}\left(r\right)$ are the electromagnetic fields in the location $r$, $\mathrm{\Omega }$ is a normalization constant, ${\widehat{a}}_{\left|l\right|}\left(k{\widehat{z}}_j\right)$ is the annihilation operator, and H.c. stands for Hermitian conjugate. When a chiral particle in the initial state $\left|E_0\right.〉$ absorbs a photon from a single mode $\left(k,l\right)$ input laser with occupation number n, it will reduce the occupation of the mode to n-1 and result in the particle being in the excited state $\left|E_{\alpha }\right.〉$. In this process, the initial and final states of the total light–matter system are denoted by $\left|I\right.〉=\left|E_0\right.〉\left|n\left(k,l\right)\right.〉$ and $\left|F\right.〉=\left|E_{\alpha }\right.〉\left|n-1\left(k,l\right)\right.〉$, respectively. Such a process can be represented by the matrix using the first-order time-dependent perturbation theory as follows

$$M_{FI}=〈F\left|{\widehat{H}}_{\mathrm{int}}\left(\xi \right)\right|I〉\propto {\epsilon }_0{E}_{\left|l\right|,j}{\mu }_j^{\alpha 0}+\mathrm{i}{H}_{\left|l\right|,j}{m}_j^{\alpha 0},$$

(17)

where ${\mu }_j^{\alpha 0}$ and $m_j^{\alpha 0}$ are the pure real polar vector and imaginary axial vector, respectively. According to the Fermi golden rule, the absorption rate of the chiral particle can be calculated as the modulus square of $M_{FI}$. In this process, three distinct terms $\mu \mu$, $mm$, and $\mu m$ will be produced. Only the interference term $\mu m$ between electric and magnetic transition dipole moments are responsible for the helical dichroism, and the pure electric dipole $\mu \mu$ and magnetic dipole $mm$ are equivalent for the enantiomers and are neglected. Thus, the chirality-related absorption rate can be written as

$$〈\mathrm{\Gamma }〉\propto {〈{\left|M_{FI}\right|}^2〉}_{\mathrm{EM}}\propto \mathrm{Im}\left({\epsilon }_{0}E\cdot H^{*}\right){\mu }^{\alpha 0}{m}^{\alpha 0}\propto C.$$

(18)

This equation illustrates that the chirality-related absorption rate is proportional to the local chirality density. The helical dichroism is proportional to the difference between the local chirality density of the beam field for the opposite sign of topological charge, which can be expressed as

$$〈{\mathrm{\Gamma }}^{\left|l\right|}〉-〈{\mathrm{\Gamma }}^{-\left|l\right|}〉\propto C^{\left|l\right|}-C^{-\left|l\right|}.$$

(19)

If the input photon is annihilated and then created again at the same chiral particles, the potential energy of this two-photon interaction can be described by the second-order time-dependent perturbation theory as

$$U=\mathrm{Re}{\displaystyle \sum _F\frac{\left.〈I\right|H_{\mathrm{int}}\left|F\right.〉\left.〈F\right|H_{\mathrm{int}}\left|I\right.〉}{E_F-E_I}},$$

(20)

Similarly, in the expansion form of the potential energy, only the interference term between electric and magnetic transition dipole moments has the chiral effects, which is calculated as

$$U_{\mathrm{EM}}\propto \mathrm{Re}\left[\left(E_i\cdot r_i\right)\times \left(\mathrm{i}{H}_j\cdot r_j\right)\cdot G_{ij}+\mathrm{H}.\mathrm{c}.\right],$$

(21)

with the mixed electric–magnetic polarizability tensor that has opposite signs for enantiomers and is given by

$$G_{ij}\left(\omega ,-\omega \right)={\displaystyle \sum _{\alpha }\left(\frac{{\mu }_i^{0\alpha }{m}_j^{\alpha 0}}{E_{\alpha }-E_0-\hslash \omega }+\frac{m_j^{0\alpha }{\mu }_i^{\alpha 0}}{E_{\alpha }-E_0+\hslash \omega }\right)}.$$

(22)

To account for the randomly oriented nature of chiral particles in the liquid or gas phase, Equation (21) should be rotationally averaged. The enantioselective force is formed as

$$F=\nabla 〈U_{\mathrm{EM}}〉\propto \kappa \nabla \mathrm{Im}\left(E\cdot H^{*}\right)\propto \kappa \nabla C,$$

(23)

with chirality parameter $\kappa =〈G_{ij}〉$ of the chiral particle.

3. Results and Discussion

In this section, we will numerically perform the intensity, phase, local chirality density, helical dichroism, and enantioselective force of the optical vortex pair with different positions and sign combinations of vortices in a vacuum. If not otherwise specified, the propagation distance, Rayleigh distance, polarization parameters, and initial beam waist of the optical vortex pair are assumed as $z=0$, $zR=k{w}_0^2/2$, $(p_x,p_y)=(1,0)$, and $w_0=4\lambda$, with $\lambda$ being the wavelength in a vacuum. In the following figures, the $I_0$, $C_0$, ${\mathrm{\Gamma }}_0$, and $F_0$ denote the max magnitudes of the intensity, local chirality density, helical dichroism, and transverse enantioselective force of the optical vortex pair in a vacuum.

In the first case, the second vortex and topological charges of the optical vortex pair are fixed at $(x_2,y_2)=(0,0)$ and $(l_1,l_2)=(1,1)$, respectively, and the corresponding intensity I, phase $\phi$, and local chirality density C of the optical vortex pair are plotted in Figure 1. As we can see, when the positions of two vortices are overlapped, their topological charges will couple to each other. In this case, the optical vortex pair is degenerated to a single vortex with topological charge $l=2$, and its intensity and local chirality density spatial distribution are both hollow and centrally symmetric. The zero values of intensity and local chirality density at the coordinate origin are induced by the single-phase singularity. If the two vortices are separated from each other, the intensity and local chirality density are shaped as the crescents with broken symmetries, and their magnitudes are obviously amplified. In addition, these distributions will be rotated by 180 degrees when the first vortex is transferred to a centrosymmetric position relative to coordinates origin. It is originated from the co-existences of two-phase singularities. Notably, if the first vortex is far away from the second vortex, the intensity and local chirality density spatial distributions will approach that of the single vortex with topological charge $l=1$, and their magnitudes are enhanced further. At this time, the influence of the first vortex on the second vortex is weak.

Then, the intensity I, phase $\phi$, and local chirality density C of the optical vortex pair with opposite topological charges $(l_1,l_2)=(1,-1)$, i.e., vortex dipole, is examined from Figure 2. The other parameters are the same as those in Figure 1. It shows that the sign change of topological charges has no influence on the intensity, but not for the phase and local chirality density. For the case of superposition of two vortices, the phase singularity will vanish, accompanied by the zero local chirality density. The underlying physical mechanism behind this phenomenon is ascribed to the annihilation of two vortices. That is to say, the orbital angular momentum on the beam vanishes and such a beam is no longer chiral. When the distance between two vortices exists, the phase does not have the spiral structure anymore, in contrast to that in Figure 1, and there exists an abrupt phase jump of $\pi$ across the line linking the two vortices. The spatial distribution of the local chirality density is still asymmetric, and its centroid will vary together with the location rotation of the first vortex. If two vortices are far away from each other, the intensity and local chirality density spatial distribution will also be close to the single vortex with topological charge $l=-1$. In this case, the magnitudes of intensity and local chirality density are also enlarged with the increase in distance between the two vortices.

To display the propagation characteristics of the chirality for the optical vortex pair in the free space, we plot the local chirality density C of the optical vortex pair with the second vortex $(x_2,y_2)=(0,0)$ for different propagation distances z = 0, z = zR/2, and z = zR in Figure 3. As we can see, the single vortex with topological charge $l=2$ will diffract in the propagation and the local chirality density will manifest itself as the expanding doughnut structure with an unchanged distribution pattern. If the two vortices are not overlapped, the diffraction effect still exists but the distribution pattern of the local chirality density will vary versus the propagation distance. Such a phenomenon is associated with the changing position of the phase singularities of the beam field in propagation. In addition, the magnitude of the local chirality density will weaken in the propagation, which stems from the diffraction effect of the amplitude for the electromagnetic fields.

For the second case, the positions of the two vortices of the optical vortex pair are centrosymmetric with respect to the coordinate origin. Figure 4 depicts the intensity I, phase $\phi$, and local chirality density C of the optical vortex pair with two vortices $(x_2,y_2)=(-x_1,-y_1)=(2\lambda ,2\lambda )$ for topological charges $(l_1,l_2)=(1,1)$ or $(l_1,l_2)=(1,-1)$. Correspondingly, the intensity distribution has centrosymmetry as well, along with the near-zero magnitudes at the locations of two-phase singularities. It is worthy to note that the transverse beam fields at the phase singularities are zero, but the longitudinal beam fields are not. The local chirality density for the p-polarized optical vortex pair totally comes from the latter. Therefore, it is not zero near the phase singularities and its sign is opposite to the sign of topological charge. The spatial distribution of the local chirality density is symmetric for $(l_1,l_2)=(1,1)$ but asymmetric for $(l_1,l_2)=(1,-1)$.

Figure 5 illustrates helical dichroism of a propagating optical vortex pair for a chiral particle with different positions of vortices. Analogous to the local chirality density, the spatial distribution of helical dichroism also has high dependence on position and takes on positive and negative values simultaneously. The distribution pattern and diffraction behaviour of helical dichroism for overlapped vortices is identical to that of local chirality density with positive topological charges due to the zero chirality for those with opposite signs of topological charges. The spatial distributions of the helical dichroism for the separated vortices have the axial symmetries on the initial plane z = 0, but such symmetries are broken in the propagation plane z = zR. In fact, the distribution patterns of separated vortices are also more similar to that of local chirality density with positive topological charges.

Finally, we perform the magnitude and direction vector (denoted by the black arrows) of transverse enantioselective force $F$ for a propagating optical vortex pair for a chiral particle with different positions and sign combinations of vortices in Figure 6. When the positions of two vortices are coincident to each other, the magnitude distribution of the transverse enantioselective force possesses the circularly symmetric structure with inner and outer rings for the case of two positive topological charges. Meanwhile, the direction vector of such a force has the axial symmetry, and there exists an enantioselective trapping point at the coordinate origin, i.e., phase singularity. However, if the signs of two topological charges are opposite, there does not exist the transverse enantioselective force due to the non-chiral beam field. Then, the symmetry of the force distribution will not exist as the first vortex transfers to the position $(2\lambda ,2\lambda )$. When the two vortices are centrosymmetric relative to the coordinate origin, the magnitude distribution of force will become symmetric again, but the direction vector is not for the negative topological charge of the second vortex. In addition, the spatial distributions of the transverse enantioselective force will change during the propagation in free space. The vector direction of this enantioselective force will be reversed for a chiral particle with $\kappa <0$. This means the enantiomers will be exerted the discriminatory force and be separated in such field.

4. Conclusions

In conclusion, we have developed a theoretical framework to analyze the chirality for an optical vortex pair. Using the vector angular spectrum decomposition method, the analytical expressions of the electromagnetic fields for an optical vortex pair have been derived. Then, the intensity, phase, local chirality density, helical dichroism, and enantioselective force of the linearly polarized optical vortex pair have been numerically simulated. It has been found that their magnitude can be amplified or suppressed by increasing or decreasing the distance of vortices, respectively. The spatial distributions of these quantities can be tailored by sign combinations of vortices as well, and show strong dependence on the propagation distance. In addition, the transverse enantioselective force directions can be affected by the forms of the vortices. Such phenomena arise from the vortex–vortex collision, coupling, and annihilation. These results may be helpful for the further understanding of chirality of multiple vortices and pave the way for potential applications in chiral recognition and manipulation.