Selective transport of chiral particles by optical pulling forces

Hongxia Zheng; Hongxia Zheng; Hongxia Zheng; Xiao Li; Huajin Chen; Huajin Chen; Huajin Chen; Zhifang Lin; Zhifang Lin; Zhifang Lin

doi:10.1364/OE.444627

1. Introduction

Chirality describes the geometry property of a substance, whose mirror image is non-superimposable with itself no matter how to rotate or translate it [1]. Passive enantiomer sorting has become a focus of extensive research in material science, chemistry, biology, and particularly, the pharmaceutical industry, since many drugs are single enantiomer [2], which is different from the situation that both handedness are widespread in nature. Compared to the complex chemical means [3], the optical methodology on the basis of optical force is less invasive and hopefully efficient, and it has developed into a research topic which attracts broad interest [4–22].

The optical force originates from the mechanical effect of light [23,24]. Light carries momentum, which can be exchanged with particles during the light scattering process, leading to a force exerted on the particles. Optical manipulation based on the optical force has become an indispensable tool in a diversity of research fields [23–32] after the first experimental demonstration in 1970 by Ashkin [33]. Recently, the optical chirality discrimination and/or sorting based on the optical manipulation techniques, has developed increasingly both in the theoretical proposals [4,5,7,8,10–14,16,18–22] and experimental implementations [6,9,15,17]. This outstanding development is attributed to the extremely different mechanical response regarding to the chirality of particles. To be specific, particles with certain chirality can be selectively trapped via optical tweezers [4,7,9,15,16,20,21], sorted by the longitudinal optical force [5,6,13,17–19,22] or the lateral optical force of which the force direction is perpendicular to the direction of light propagation and intensity gradient [8,10–12,14]. The chirality sorting employing the optical pulling force [34–37] is rarely studied except in the case of two-plane-wave field [5,22]. Two-plane-wave field [5,22] is the first scheme taking advantage of the counter-intuitive optical pulling force to realize chirality sorting, but its experimental feasibility is limited by the relatively narrow tunable range of the directions of wave vetors. In this paper, we design an interferential optical field which is composed of three plane waves with the same circular polarization, to sort chiral particles, which obviously improves the experimental feasibility attributing to the relatively wide tunable range of the diretions of wave vectors. Based on the multipole expansion theory of optical forces [38], we numerically present that such a simple optical field can stably trap a particle with certain handedness in the transverse plane while pull it along the longitudinal direction to the light source, which finally realize the selective transport of chiral particles with different handedness. Our scheme may find applications in transporting and separating chiral objects.

2. Theory and formulations

In this section, we briefly recapitulate the multipole expansion theory for calculating optical forces exerted on a chiral particle when the interferential plane waves illuminate on it. The optical force is also decomposed into conservative gradient force and nonconservative scattering force.

The chiral particle is characterized by the following constitutive relations [39]

(1)$$\boldsymbol D= \varepsilon_0\varepsilon_p \boldsymbol E+{\boldsymbol{i}}\kappa\sqrt{\varepsilon_0\mu_0}\, \boldsymbol H , \qquad \boldsymbol B={-}{\boldsymbol{i}}\kappa\sqrt{\varepsilon_0\mu_0}\, \boldsymbol E+ \mu_0\mu_p \boldsymbol H ,$$

where i represents the imaginary unit, $\kappa$ is the chiral parameter describing the chirality of the object, while $\varepsilon _0$ ($\mu _0$) and $\varepsilon _p$ ($\mu _p$) denote the vacuum permittivity (permeability) and relative permittivity (permeability) of the particle, respectively.

The electric and magnetic fields composed of multiple plane waves can be written as

(2)$$\boldsymbol E=\displaystyle\sum_{i=1}^{n_p}\boldsymbol E_i =\displaystyle\sum_{i=1}^{n_p} E_0 \boldsymbol{\mathcal{E}}_i e^{\boldsymbol{i}\, k\,\boldsymbol{\hat{k}}_i.\boldsymbol{r}}, \qquad \boldsymbol B=\displaystyle\sum_{i=1}^{n_p}\boldsymbol B_i =\displaystyle\sum_{i=1}^{n_p} B_0\,\boldsymbol{\mathcal{B}}_i e^{\boldsymbol{i}\, k\,\boldsymbol{\hat{k}}_i.\boldsymbol{r}},$$

in which $k$ is the wave number of constituent plane waves in background medium, $\boldsymbol {\hat {k}}_i$ is the unit vector denoting the direction of the $i$th wave vector $\boldsymbol {k}_i=k\,\boldsymbol {\hat {k}}_i$, $B_0=E_0/c$ with $c$ being the light speed in background, $E_0$ is the amplitude of each plane wave, $\boldsymbol {\mathcal {E}}_i$ and $\boldsymbol {\mathcal {B}}_i=\boldsymbol {k}_i \boldsymbol {\times } \boldsymbol {\mathcal {E}}_i$ are complex amplitude vectors of the $i$th plane wave, and $n_p$ is the number of the plane waves making up the optical fields, respectively.

Within the framework of multipole expansion theory, the time-averaged optical force $\boldsymbol F$ is generally expressed as [38,40]

(3)$$\boldsymbol F=\displaystyle\sum_{l=1}^{\infty} \boldsymbol F^{(l)}_{\textrm{int}}+\displaystyle\sum_{l=1}^{\infty} \boldsymbol F^{(l)}_{\textrm{rec}},$$

where $\boldsymbol F^{(l)}_{\textrm {int}}$ and $\boldsymbol F^{(l)}_{\textrm {rec}}$ are, respectively, termed as interception force and recoil force on multipoles of various orders, which can be understood as two physical processes: 1, light is intercepted by the particle and then an extinction force will be exerted on the particle; 2, light is re-emitted by the oscillating multipoles excited in the particle and then an recoil force is applied. To be specific, the interception and recoil forces on multipoles of various orders can be further written as

(4)$$\boldsymbol F^{(l)}_{\textrm{int}}=\boldsymbol F^{\textrm{e} (l)}_{\textrm{int}}+\boldsymbol F^{\textrm{m} (l)}_{\textrm{int}}, \qquad \boldsymbol F^{(l)}_{\textrm{rec}}=\boldsymbol F^{\textrm{e} (l)}_{\textrm{rec}}+\boldsymbol F^{\textrm{m} (l)}_{\textrm{rec}}+\boldsymbol F^{\textrm{x} (l)}_{\textrm{rec}}.$$

Substituting the optical field Eq. (2) into Eq. (3), after lengthy algebra (detailed derivation process in Ref. [41]), one can arrive at

(5)$$\begin{aligned} \langle\mspace{1mu}{\boldsymbol F^{{\boldsymbol{e}}(l)}_{\textrm{int}}}\mspace{1mu}\rangle &= \dfrac{\pi\,(2l+1)}{l\,(l+1)}\;\displaystyle\sum_{i,j} \Bigl\{\textrm{Im} \bigl[a_l \,\boldsymbol Y_{l,ij}^{\,(1)}\bigr] +\textrm{Re} \bigl[c_l \, \boldsymbol Y_{l,ij}^{\,(2)}\bigr]\Bigr\},\\ \langle\mspace{1mu}{\boldsymbol F^{{\boldsymbol{}m}(l)}_{\textrm{int}}}\mspace{1mu}\rangle &= -\dfrac{\pi\,(2l+1)}{l\,(l+1)}\;\displaystyle\sum_{i,j} \Bigl\{\textrm{Im} \bigl[b_l \,\boldsymbol Y_{l,ij}^{\,(3)}\bigr] -\textrm{Re} \bigl[c_l \, \boldsymbol Y_{l,ij}^{\,(4)}\bigr]\Bigr\},\\ \langle\mspace{1mu}{\boldsymbol F^{{\boldsymbol{}e}(l)}_{\textrm{rec}}}\mspace{1mu}\rangle &= -\dfrac{\pi}{2\,(l+1)^{2}}\;\displaystyle\sum_{i,j} \textrm{Im} \bigl[a^{*}_l\boldsymbol{\cdot}a_{l+1} \,\boldsymbol Y_{l,ij}^{\,(5)}+\boldsymbol{i}\boldsymbol{\cdot}a^{*}_l\boldsymbol{\cdot}c_{l+1} \,\boldsymbol Y_{l,ij}^{\,(6)} -\boldsymbol{i}\boldsymbol{\cdot}c^{*}_l\boldsymbol{\cdot}a_{l+1} \,\boldsymbol Y_{l,ij}^{\,(7)}+c^{*}_l\boldsymbol{\cdot}c_{l+1} \,\boldsymbol Y_{l,ij}^{\,(8)}\bigr],\\ \langle\mspace{1mu}{\boldsymbol F^{{\boldsymbol{}m}(l)}_{\textrm{rec}}}\mspace{1mu}\rangle &= -\dfrac{\pi}{2\,(l+1)^{2}}\;\displaystyle\sum_{i,j} \textrm{Im} \bigl[b^{*}_l\boldsymbol{\cdot}b_{l+1} \,\boldsymbol Y_{l,ij}^{\,(8)}-\boldsymbol{i}\boldsymbol{\cdot}b^{*}_l\boldsymbol{\cdot}c_{l+1} \,\boldsymbol Y_{l,ij}^{\,(7)} +\boldsymbol{i}\boldsymbol{\cdot}c^{*}_l\boldsymbol{\cdot}b_{l+1} \,\boldsymbol Y_{l,ij}^{\,(6)}+c^{*}_l\boldsymbol{\cdot}c_{l+1} \,\boldsymbol Y_{l,ij}^{\,(5)}\bigr],\\ \langle\mspace{1mu}{\boldsymbol F^{{\boldsymbol{}x}(l)}_{\textrm{rec}}}\mspace{1mu}\rangle &= \dfrac{\pi\,(2l+1)}{2l^{2}\,(l+1)^{2}}\;\displaystyle\sum_{i,j} \textrm{Re} \bigl[a^{*}_l\boldsymbol{\cdot}b_l \,\boldsymbol Y_{l,ij}^{\,(9)}+c^{*}_l\boldsymbol{\cdot}c_l \,\boldsymbol Y_{l,ij}^{\,(9)} -c^{*}_l\boldsymbol{\cdot}a_l \,\boldsymbol Y_{l,ij}^{\,(10)}+c^{*}_l\boldsymbol{\cdot}b_l \,\boldsymbol Y_{l,ij}^{\,(11)}\bigr]. \end{aligned}$$

In Eq. (5), $a_l$, $b_l$, and $c_l$ are Mie coefficients [21,39] and $\boldsymbol Y$ vectors are defined as

(6)$$\begin{aligned} \boldsymbol Y_{l,ij}^{\,(1)} &= Q_{l,ij}^{\,(1)} \boldsymbol Z_{\textrm{ee},ij}^{(1)}-Q_{l,ij}^{\,(2)} \boldsymbol Z_{\textrm{mm},ij}^{(1)}, \qquad \qquad \boldsymbol Y_{l,ij}^{\,(2)} = Q_{l,ij}^{\,(1)} \boldsymbol Z_{\textrm{me},ij}^{(1)}+Q_{l,ij}^{\,(2)} \boldsymbol Z_{\textrm{em},ij}^{(1)},\\ \boldsymbol Y_{l,ij}^{\,(3)} &= Q_{l,ij}^{\,(1)} \boldsymbol Z_{\textrm{mm},ij}^{(1)}-Q_{l,ij}^{\,(2)} \boldsymbol Z_{\textrm{ee},ij}^{(1)}, \qquad \qquad \boldsymbol Y_{l,ij}^{\,(4)} = Q_{l,ij}^{\,(1)} \boldsymbol Z_{\textrm{em},ij}^{(1)}+Q_{l,ij}^{\,(2)} \boldsymbol Z_{\textrm{me},ij}^{(1)},\\ \boldsymbol Y_{l,ij}^{\,(5)} &= R_{l,ij}^{\,(1)} \boldsymbol Z_{\textrm{ee},ij}^{(1)\,*}-R_{l,ij}^{\,(2)} \boldsymbol Z_{\textrm{mm},ij}^{(1)\,*}-4\boldsymbol{i}\, R_{l,ij}^{\,(3)} \boldsymbol S_{\textrm{em},ij}^{(1)\,*}+ R_{l,ij}^{\,(4)} \boldsymbol Z_{\textrm{ee},ij}^{(1)}-R_{l,ij}^{\,(5)} \boldsymbol Z_{\textrm{mm},ij}^{(1)}+4\boldsymbol{i}\, R_{l,ij}^{\,(6)} \boldsymbol S_{\textrm{em},ij}^{(1)},\\ \boldsymbol Y_{l,ij}^{\,(6)} &= R_{l,ij}^{\,(1)} \boldsymbol Z_{\textrm{em},ij}^{(1)\,*}+R_{l,ij}^{\,(2)} \boldsymbol Z_{\textrm{me},ij}^{(1)\,*}+4\boldsymbol{i}\, R_{l,ij}^{\,(3)} \boldsymbol S_{\textrm{ee},ij}^{(1)\,*}+ R_{l,ij}^{\,(4)} \boldsymbol Z_{\textrm{me},ij}^{(1)}+R_{l,ij}^{\,(5)} \boldsymbol Z_{\textrm{em},ij}^{(1)}+4\boldsymbol{i}\, R_{l,ij}^{\,(6)} \boldsymbol S_{\textrm{mm},ij}^{(1)},\\ \boldsymbol Y_{l,ij}^{\,(7)} &= R_{l,ij}^{\,(1)} \boldsymbol Z_{\textrm{me},ij}^{(1)\,*}+R_{l,ij}^{\,(2)} \boldsymbol Z_{\textrm{em},ij}^{(1)\,*}-4\boldsymbol{i}\, R_{l,ij}^{\,(3)} \boldsymbol S_{\textrm{mm},ij}^{(1)\,*}+ R_{l,ij}^{\,(4)} \boldsymbol Z_{\textrm{em},ij}^{(1)}+R_{l,ij}^{\,(5)} \boldsymbol Z_{\textrm{me},ij}^{(1)}-4\boldsymbol{i}\, R_{l,ij}^{\,(6)} \boldsymbol S_{\textrm{ee},ij}^{(1)},\\ \boldsymbol Y_{l,ij}^{\,(8)} &= R_{l,ij}^{\,(1)} \boldsymbol Z_{\textrm{mm},ij}^{(1)\,*}-R_{l,ij}^{\,(2)} \boldsymbol Z_{\textrm{ee},ij}^{(1)\,*}-4\boldsymbol{i}\, R_{l,ij}^{\,(3)} \boldsymbol S_{\textrm{em},ij}^{(1)}+ R_{l,ij}^{\,(4)} \boldsymbol Z_{\textrm{mm},ij}^{(1)}-R_{l,ij}^{\,(5)} \boldsymbol Z_{\textrm{ee},ij}^{(1)}+4\boldsymbol{i}\, R_{l,ij}^{\,(6)} \boldsymbol S_{\textrm{em},ij}^{(1)\,*},\\ \boldsymbol Y_{l,ij}^{\,(9)} &= \boldsymbol{i}\boldsymbol{\cdot}R_{l,ij}^{\,(4)} \bigr[\boldsymbol Z_{\textrm{ee},ij}^{(1)}-\boldsymbol Z_{\textrm{mm},ij}^{(1)\,*}\bigl]+\boldsymbol{i}\boldsymbol{\cdot}R_{l,ij}^{\,(5)} \bigr[\boldsymbol Z_{\textrm{ee},ij}^{(1)\,*}-\boldsymbol Z_{\textrm{mm},ij}^{(1)}\bigl] -4\, R_{l,ij}^{\,(7)} \boldsymbol S_{\textrm{em},ij}^{(1)\,*}-4\, R_{l,ij}^{\,(6)} \boldsymbol S_{\textrm{em},ij}^{(1)},\\ \boldsymbol Y_{l,ij}^{\,(10)} &= R_{l,ij}^{\,(4)} \bigr[\boldsymbol Z_{\textrm{me},ij}^{(1)}+\boldsymbol Z_{\textrm{me},ij}^{(1)\,*}\bigl]+R_{l,ij}^{\,(5)} \bigr[\boldsymbol Z_{\textrm{em},ij}^{(1)\,*}+\boldsymbol Z_{\textrm{em},ij}^{(1)}\bigl] +4\boldsymbol{i}\, R_{l,ij}^{\,(7)} \boldsymbol S_{\textrm{ee},ij}^{(1)}+4\boldsymbol{i}\, R_{l,ij}^{\,(6)} \boldsymbol S_{\textrm{mm},ij}^{(1)},\\ \boldsymbol Y_{l,ij}^{\,(11)} &= R_{l,ij}^{\,(4)} \bigr[\boldsymbol Z_{\textrm{em},ij}^{(1)}+\boldsymbol Z_{\textrm{em},ij}^{(1)\,*}\bigl]+R_{l,ij}^{\,(5)} \bigr[\boldsymbol Z_{\textrm{me},ij}^{(1)\,*}+\boldsymbol Z_{\textrm{me},ij}^{(1)}\bigl] -4\boldsymbol{i}\, R_{l,ij}^{\,(7)} \boldsymbol S_{\textrm{mm},ij}^{(1)}-4\boldsymbol{i}\, R_{l,ij}^{\,(6)} \boldsymbol S_{\textrm{ee},ij}^{(1)}, \end{aligned}$$

where the coefficients $Q$ and $R$, dependent on $x_{ij}$ with $x_{ij}=\boldsymbol {\hat {k}}_i.\boldsymbol {\hat {k}}_j$, are all defined in terms of Legendre polynomials by

(7a)$$\begin{aligned}Q_{l,ij}^{(1)}&= \displaystyle\sum^{l}_{m=1}\!\!^{(2)} m(2l+1-m)(2l+1-2m)P_{l-m}(x_{ij}),\\ Q_{l,ij}^{(2)} &= \displaystyle\sum^{l}_{m=2}\!\!^{(2)} m(2l+1-m)(2l+1-2m)P_{l-m}(x_{ij}),\\ R^{(1)}_{l,ij}&=\displaystyle\sum^{l}_{m=1}\!\!^{(2)} (m+1)(2l+2-m)(2l+1-2m)[2(m+1)l-(m^{2}-m-4)]P_{l-m}(x_{ij}),\\ R^{(2)}_{l,ij}&=\displaystyle\sum^{l}_{m=2}\!\!^{(2)} m(m+2)(2l+1-m)(2l+1-2m)(2l+3-m)P_{l-m}(x_{ij}), \\ R^{(3)}_{l,ij}&=\displaystyle\sum^{l}_{m=1}\!\!^{(2)} (m+1)(2l+2-m)(2l+1-2m)P_{l-m}(x_{ij}), \end{aligned}$$

(7b)$$\begin{aligned}R^{(4)}_{l,ij}&= \displaystyle\sum^{l}_{m=2}\!\!^{(2)} (2l+1-m)(2l+1-2m)[2m^{2} l-m(m+1)(m-2)]P_{l-m}(x_{ij}),\\ R^{(5)}_{l,ij}&= \displaystyle\sum^{l}_{m=1}\!\!^{(2)} (m+1)(m-1)(2l-m)(2l+2-m)(2l+1-2m)P_{l-m}(x_{ij}), \\ R^{(6)}_{l,ij}&= \displaystyle\sum^{l}_{m=2}\!\!^{(2)} m(2l+1-m)(2l+1-2m)P_{l-m}(x_{ij}),\\ R^{(7)}_{l,ij}&= \displaystyle\sum^{l}_{m=1}\!\!^{(2)} (2l+1-2m)(2l^{2}-2(m-1)l+m^{2}-m)P_{l-m}(x_{ij}), \end{aligned}$$

in which the summation $\displaystyle\sum^{l}_{m=1}\!\!^{(2)}$ and $\displaystyle\sum^{l}_{m=2}\!\!^{(2)}$ above represent the index $m$ odd and even positive integers satisfying $0<m\leq l$, and $P_{n}(x)$ is the Legendre polynomial. The field quantities in Eq. (6) are defined as

(8)$$\begin{aligned} \boldsymbol Z_{\textrm{ee},ij}^{(1)} &= \dfrac 12\,\Bigl[\nabla D_{\textrm{ee},ij}^{(1)}- \nabla\boldsymbol{\times}\boldsymbol S_{\textrm{ee},ij}^{(1)}-2 \boldsymbol{i}\,\textrm{Re}\,\boldsymbol S_{\textrm{em},ij}^{(1)}\Bigr], \\ \boldsymbol Z_{\textrm{mm},ij}^{(1)} &= \dfrac 12\,\Bigl[\nabla D_{\textrm{mm},ij}^{(1)}- \nabla\boldsymbol{\times}\boldsymbol S_{\textrm{mm},ij}^{(1)}-2 \boldsymbol{i}\,\textrm{Re}\,\boldsymbol S_{\textrm{em},ij}^{(1)}\Bigr],\\ \boldsymbol Z_{\textrm{me},ij}^{(1)} &= \dfrac 12\,\Bigl[\nabla D_{\textrm{me},ij}^{(1)}-\nabla\boldsymbol{\times}\boldsymbol S_{\textrm{me},ij}^{(1)}-\boldsymbol{i}\,\bigl(\boldsymbol S_{\textrm{ee},ij}^{(1)}+\boldsymbol S_{\textrm{mm},ij}^{(1)}\bigr)\Bigr], \\ \boldsymbol Z_{\textrm{em},ij}^{(1)} &= \dfrac 12\,\Bigl[\nabla D_{\textrm{em},ij}^{(1)}-\nabla\boldsymbol{\times}\boldsymbol S_{\textrm{em},ij}^{(1)}+\boldsymbol{i}\,\bigl(\boldsymbol S_{\textrm{ee},ij}^{(1)}+\boldsymbol S_{\textrm{mm},ij}^{(1)}\bigr)\Bigr], \end{aligned}$$

where the field moments for any pair $(i,j)$ of the plane waves are given by

(9)$$\begin{array}{llll} D_{\textrm{ee},ij}^{(1)}= (\boldsymbol{\mathcal{E}}_i.\boldsymbol{\mathcal{E}}_j^{*}) \boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, & \quad & D_{\textrm{mm},ij}^{(1)}= (\boldsymbol{\mathcal{B}}_i.\boldsymbol{\mathcal{B}}_j^{*}) \boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, \\ D_{\textrm{em},ij}^{(1)}= (\boldsymbol{\mathcal{E}}_i.\boldsymbol{\mathcal{B}}_j^{*}) \boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, & & D_{\textrm{me},ij}^{(1)}= (\boldsymbol{\mathcal{B}}_i.\boldsymbol{\mathcal{E}}_j^{*}) \boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, \\ \boldsymbol S_{\textrm{ee},ij}^{(1)}= (\boldsymbol{\mathcal{E}}_i\boldsymbol{\times}\boldsymbol{\mathcal{E}}_j^{*}) \boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, & & \boldsymbol S_{\textrm{mm},ij}^{(1)}= (\boldsymbol{\mathcal{B}}_i\boldsymbol{\times}\boldsymbol{\mathcal{B}}_j^{*}) \boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, \\ \boldsymbol S_{\textrm{em},ij}^{(1)}= (\boldsymbol{\mathcal{E}}_i\boldsymbol{\times}\boldsymbol{\mathcal{B}}_j^{*}) \boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, & & \boldsymbol S_{\textrm{me},ij}^{(1)}= (\boldsymbol{\mathcal{B}}_i\boldsymbol{\times}\boldsymbol{\mathcal{E}}_j^{*}) \boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}. \end{array}$$

To calculate the decomposed gradient and scattering forces, the following separated field vectors $\boldsymbol Z^{g}$ ($\boldsymbol Z^{s}$) and $\boldsymbol S^{g}$ ($\boldsymbol S^{s}$) instead of $\boldsymbol Z$ and $\boldsymbol S$ in Eq. (8) should be substituted into the $\boldsymbol Y$ vectors in Eq. (6). The $\boldsymbol Z$ vectors can be separated as

(10)$$\begin{aligned} \boldsymbol Z_{\textrm{ee},ij}^{(1)\,\textrm{g}} &= \dfrac 12\,\nabla D_{\textrm{ee},ij}^{(1)}, \qquad \boldsymbol Z_{\textrm{mm},ij}^{(1)\,\textrm{g}} = \dfrac 12\,\nabla D_{\textrm{mm},ij}^{(1)},\\ \boldsymbol Z_{\textrm{me},ij}^{(1)\,\textrm{g}}&= \dfrac 12\,\nabla D_{\textrm{me},ij}^{(1)}, \qquad \boldsymbol Z_{\textrm{em},ij}^{(1)\,\textrm{g}}= \dfrac 12\,\nabla D_{\textrm{em},ij}^{(1)}, \end{aligned}$$

and

(11a)\begin{align}\boldsymbol Z_{\textrm{ee},ij}^{(1)\,\textrm{s}}={-}\dfrac 12\,\Bigl[\nabla\boldsymbol{\times}\boldsymbol S_{\textrm{ee},ij}^{\,(1)}+2 \boldsymbol{i}\,\textrm{Re}\,\boldsymbol S_{\textrm{em},ij}^{\,(1)}\Bigr], \end{align}

(11b)\begin{align}\boldsymbol Z_{\textrm{mm},ij}^{(1)\,\textrm{s}}={-}\dfrac 12\,\Bigl[\nabla\boldsymbol{\times}\boldsymbol S_{\textrm{mm},ij}^{\,(1)}+2 \boldsymbol{i}\,\textrm{Re}\,\boldsymbol S_{\textrm{em},ij}^{\,(1)}\Bigr], \end{align}

(11c)\begin{align}\boldsymbol Z_{\textrm{me},ij}^{(1)\,\textrm{s}}={-}\dfrac 12\,\Bigl[\nabla\boldsymbol{\times}\boldsymbol S_{\textrm{me},ij}^{\,(1)}+\boldsymbol{i}\,\bigl(\boldsymbol S_{\textrm{ee},ij}^{\,(1)}+\boldsymbol S_{\textrm{mm},ij}^{\,(1)}\bigr)\Bigr], \end{align}

(11d)\begin{align}\boldsymbol Z_{\textrm{em},ij}^{(1)\,\textrm{s}}={-}\dfrac 12\,\Bigl[\nabla\boldsymbol{\times}\boldsymbol S_{\textrm{em},ij}^{\,(1)}-\boldsymbol{i}\,\bigl(\boldsymbol S_{\textrm{ee},ij}^{\,(1)}+\boldsymbol S_{\textrm{mm},ij}^{\,(1)}\bigr)\Bigr]. \end{align}

The partition form of $\boldsymbol S$ vectors are

(12)$$\begin{array}{lll} \boldsymbol S_{\textrm{ee},i j}^{\,(1)\,\textrm{g}} ={-}\dfrac{\boldsymbol{i}}{1-x_{i j}}\textrm{Re} \nabla D_{\textrm{em},i j}^{(1)}, \\ \boldsymbol S_{\textrm{mm},i j}^{\,(1)\,\textrm{g}} =\dfrac{\boldsymbol{i}}{1-x_{i j}}\textrm{Re} \nabla D_{\textrm{em},i j}^{(1)},\\ \boldsymbol S_{\textrm{em},i j}^{\,(1)\,\textrm{g}} =\dfrac{\boldsymbol{i}}{2(1-x_{ij})}\, \textrm{Re}\,\Big[\nabla D_{\textrm{ee},i j}^{(1)}-\nabla D_{\textrm{mm},i j}^{(1)}\Big], \end{array}$$

and

(13)$$\begin{aligned} & \boldsymbol S_{\textrm{ee},i j}^{\,(1)\,\textrm{s}} ={-}\dfrac{\boldsymbol{i}}{1-x_{i j}}\textrm{Im} \nabla\!\boldsymbol{\times}\!\boldsymbol G_{\textrm{ee},i j}^{(1)}+\boldsymbol S_{\textrm{ee},i j}^{\,(1)}\delta_{i,j}, \\ & \boldsymbol S_{\textrm{mm},i j}^{\,(1)\,\textrm{s}} ={-}\dfrac{\boldsymbol{i}}{1-x_{ij}} \textrm{Im} \nabla\!\boldsymbol{\times}\! \boldsymbol G_{\textrm{mm},i j}^{(1)}+\boldsymbol S_{\textrm{mm},i j}^{\,(1)}\delta_{i,j}, \\ & \textrm{Im}\, \boldsymbol S_{\textrm{em},i j}^{\,(1)\,\textrm{s}} =\dfrac{1}{2(1-x_{ij})} \textrm{Im}\Big[\nabla\!\boldsymbol{\times}\!\boldsymbol G_{\textrm{me},i j}^{(1)*}-\nabla\!\boldsymbol{\times}\!\boldsymbol G_{\textrm{em},i j}^{(1)}\Big]+\boldsymbol S_{\textrm{em},i j}^{\,(1)}\delta_{i,j}, \end{aligned}$$

where $\delta _{i,j}$ represents the Kronecker delta and $\boldsymbol G$ vectors are defined as

(14)$$\begin{array}{llll} \boldsymbol G_{\textrm{ee},ij}^{(1)}={-}i\, (\boldsymbol{k}_j.\boldsymbol{\mathcal{E}}_i)\,\boldsymbol{\mathcal{E}}_j^{*}\boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, \qquad \boldsymbol G_{\textrm{mm},ij}^{(1)}={-}i\, (\boldsymbol{k}_j.\boldsymbol{\mathcal{B}}_i)\,\boldsymbol{\mathcal{B}}_j^{*}\boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, \\ \boldsymbol G_{\textrm{em},ij}^{(1)}={-}i\, (\boldsymbol{k}_j.\boldsymbol{\mathcal{E}}_i)\,\boldsymbol{\mathcal{B}}_j^{*}\boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}, \qquad \boldsymbol G_{\textrm{me},ij}^{(1)}={-}i\, (\boldsymbol{k}_j.\boldsymbol{\mathcal{B}}_i)\,\boldsymbol{\mathcal{E}}_j^{*}\boldsymbol{\cdot}e^{\boldsymbol{i}\,(\boldsymbol{k}_i-\boldsymbol{k}_j).\boldsymbol{r}}. \end{array}$$

Re $\boldsymbol S_{\textrm {em},ij}^{\,(1)}$ in Eq. (11) only contributes to the scattering force. It is noted that Eq. (5) gives the numerical value of the optical forces in unit of ${\varepsilon _b E_0^{2}}/k^{2}$ with $\varepsilon _b$ being the permittivity of background medium since we have assumed $k=\varepsilon _b=\mu _b=c=1$ in the force expression for simplicity.

3. Results and discussions

To illustrate the selective transporting and/or sorting of chiral particles, we consider that a chiral spherical particle is illuminated by an optical field composed of three interferential plane waves, viz. $n_p=3$ in Eq. (2), as is schematically shown in Fig. 1. Assuming that their wave vectors have the same polar angle with respect to $z$ axis and show three-fold rotational symmetry around $z$ axis, the incident electric field is given by

(15)$$\boldsymbol E=\boldsymbol E_1+\boldsymbol E_2+\boldsymbol E_3, \qquad \boldsymbol E_i= E_0 \boldsymbol{\mathcal{E}}_i e^{\boldsymbol{i}\, k\,\boldsymbol{\hat{k}}_i.\boldsymbol{r}}, \qquad \boldsymbol{\mathcal{E}}_i=p \, \boldsymbol{\hat \theta}_{k_i}+q\,\boldsymbol{\hat \phi}_{k_i}, \qquad i=1,2,3,$$

where $\boldsymbol {\hat \theta }_{k_i}$ and $\boldsymbol {\hat \phi }_{k_i}$ denote, respectively, the directions of increasing polar angle and azimuthal angle in spherical coordinate system for the $i$th wave vector. The three waves share the same polarization vector $(p,q)$ satisfying the normalization condition $|p|^{2}+|q|^{2}=1$, which describes the polarized state of the plane wave, e.g., $(p,q)=(1,\pm \boldsymbol{i})/\sqrt {2}$ describes left/right circular polarization, respectively. The direction of the $i$th plane wave is denoted by $\boldsymbol {\hat k}_i=\cos \phi _i \sin \theta \,\boldsymbol {\hat x}+\sin \phi _i \sin \theta \,\boldsymbol {\hat y}+\cos \theta \,\boldsymbol {\hat z}$, with $\theta$ and $\phi _i=(i-1)\boldsymbol {\times } 120^{\circ }$ being the polar angle and azimuthal angle of the $i$th plane wave $\boldsymbol {\hat k}_i$ in spherical coordinate system. Substituting Eq. (15) into Eq. (9), one can readily derive that the scalar field quantities $D$ are independent on $z$, which results in the zeros of the irrotational parts of vectors $\boldsymbol Z$ and $\boldsymbol S$ in Eqs. (10) and (12) and finally the vanishing gradient force in $F_z$, leaving us with the scattering force in the $z$ direction only. For $\theta <90^{\circ }$ in our system, the optical field represents a composed wave propagating along the positive $z$ direction. $F_z>0$ indicates the pushing force along the positive $z$ direction while $F_z<0$ denotes the pulling force along the negative $z$ direction, as shown by the blue arrow in Fig. 1(a). In all our simulation, the background is vacuum, the wavelength is $\lambda =1.064~\mu m$, the field amplitude $E_0$ is set such that the irradiance of a single plane wave is $I_0=|E_0|^{2}/{(2\boldsymbol {\cdot }Z_0)}=1.0~mW/\mu m^{2}$ with $Z_0$ being the wave impedance of vacuum, and the three beams are all left circularly polarized with $(p,q)=(1,\boldsymbol{i}/{\sqrt {2}})$ except otherwise stated.

Fig. 1. (a) Schematic illustration of a spherical chiral particle immersed in an optical field constituted by three plane waves with wave vectors $\boldsymbol {\hat k}_i=\cos \phi _i \sin \theta \,\boldsymbol {\hat x}+\sin \phi _i \sin \theta \,\boldsymbol {\hat y}+\cos \theta \,\boldsymbol {\hat z}$ with $\theta <\pi /2$ and $\phi _i=2(i-1)\pi /3$, $i=1,2,3$. $F_{\rho }$ is the radial force exerted on the particle on $x$-$o$-$y$ plane, where $F_{\rho }<0$ and $F_{\rho }>0$ denote, respectively, the particle can be draw to and away from the coordinate origin. $F_z>0$ and $F_z<0$ are known as pushing and pulling forces (shown by the blue arrow). (b) Spatial profiles of the lateral force $F_z$ acting on a conventional dielectric particle with $\varepsilon =2.5$ and radius $r=0.4~\mu m$ in vacuum. The three constituent plane waves share the same wavelength $\lambda =1.064~\mu m$, $\theta =90^{\circ }$, and left circular polarization with $(p,q)=(1,\boldsymbol{i})/{\sqrt {2}}$. The periodic distribution of $F_z>0$ and $F_z<0$ suggests that an introduction of $z$-component of $\boldsymbol {k}_i$ will result in both pushing and pulling forces. (c) Spatial profiles of the longitudinal force $F_z$ exerted on a chiral particle ($\kappa =0.45$). The polar angle is set as $\theta =75^{\circ }$ while other parameters are the same as (b). (d) Spatial profiles of the light intensity, in which the shaded regions denote the pulling forces are exerted to the chiral particle ($\kappa =0.45$) wherein. The region surrounded by the dashed black circle represents roughly a periodic unit in the $x$-$o$-$y$ plane which will be focused in the following text. Parameters are the same as (c).

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Substituting Eq. (15) into Eqs. (9) and (14), one can easily get the optical forces, including its gradient and scattering parts, are $z$-independent. In other words, the forces exerted on the particle are all translational invariance with respect to $z$, thus we can focus on the forces analysis within the $x$-$o$-$y$ plane. It has been shown that lateral force field with $F_z<0$ and $F_z>0$ can emerge with rotational symmetries for a conventional dielectric sphere located on the $x$-$o$-$y$ plane, where all plane waves lie on with the polar angle being $\theta =90^{\circ }$ [42]. An illustrative result is shown in Fig. 1(b) as an example, where left circular polarization is used. It is reasonably believed that $F_z<0$ will not vanish immediately with the emerging positive $z$-component of $\boldsymbol {k}_i$ with the polar angle $\theta <90^{\circ }$, at least when the $\theta$ is close enough to $90^{\circ }$, which is exactly the pulling force to transport particles towards light source. The shaded regions in Fig. 1(c) present the pulling forces exerted on a chiral particle illuminated by three left circular polarized plane waves with $\theta =75^{\circ }$ as an example. The color map in Fig. 1(d) demonstrates the rotationally symmetric intensity distribution of the aforementioned light field, which makes up an optical lattice on the $x$-$o$-$y$ plane. Particles illuminated by such an optical field will present a periodically distributed force field in the $x$-$o$-$y$ plane, as can be seen by the pulling forces denoted by shaded regions. The region surrounded by the dashed black circle in Fig. 1(d) denote roughly a periodical unit in the optical lattice. Without loss of generality, we focus on it to study the stable transporting and/or sorting of chiral particles. Other transporting channels with exactly the same property force field can be easily found periodically in the transverse plane.

The three-plane-wave optical field can selectively transport chiral particles with the picture that particles with certain chirality can be drawn to the periodic intensity maxima and then pulled to the light source, while particles with opposite chirality will be de-trapped from these intensity maxima and then disperses in the pushing force regions. To be specific, two different situations are demonstrated in Figs. 2(a)-(b) and Figs. 2(c)-(d), respectively. Spatial distribution of forces for $|\kappa |=0.45$ and $|\kappa |=0.05$ are shown in Fig. 2 as examples. The radial force $F_\rho$ is the radial component of the transverse force $\boldsymbol F_\perp$, see Fig. 1(a) for a schematic illustration. The color maps in Fig. 2 present the radial forces $F_\rho$ exerted on a particle located at corresponding position in the $x$-$o$-$y$ plane. It is obviously shown that the particle with positive chirality shown in Figs. 2(a) and (c) can be trapped stably at the intensity maxima of the optical lattice (the origin of the coordinate system is considered here as an example), since the particle will return to the equilibrium position when displaced slightly, due to the negative radial forces $F_\rho <0$. The trapped particles can then be pulled along the negative $z$ direction via optical pulling forces $F_z<0$ depicted by the shaded region, leading to long-ranged transport of a positive handedness particle towards light source since all forces is invariant with respect to $z$, which is mentioned before. However, the particles with corresponding negative chirality, as shown in Figs. 2(b) and (d), cannot be stably pulled along the negative $z$ direction. As shown in Fig. 2(b), the particle with negative chirality is always subject to pushing forces to be pushed away from the side of the light source, which is completely different from the situation of the positive chirality particle, leading to clean chirality sorting. It is noted the almost same, if not exactly identical, landscapes of pulling forces are found in Fig. 2(c) and Fig. 2(d). However, the de-trapping in transverse plane, as shown in Fig. 2(d), frustrates the application of optical pulling force in long-ranged pulling particles along the negative $z$ direction, since the free-walking particles will be exerted by optical pushing forces along the positive $z$ direction besides the pulling one. Particles with opposite chirality can therefore be distinguished by taking advantage of the long-ranged pulling force on a certain chirality, since the incident optical field is composed of long-ranged propagating plane waves. Based on our three-plane-wave optical field with left circular polarization, the chiral particles with positive handedness can be collected in the side of light source while the opposite one cannot, which enables its application to the practical optical chirality sorting. The collection of the chiral particles with negative chirality can also be achieved by using the right circularly polarized plane waves, as can be seen in the following results.

Fig. 2. Spatial distribution of radial forces (color representation) and pulling forces (shaded region) to show two different situations. $\theta =75^{\circ }$, $r=0.4~\mu m$, and $\kappa =\pm 0.45$ for (a) and (b) while $\theta =89^{\circ }$, $r=0.5~\mu m$, and $\kappa =\pm 0.05$ for (c) and (d), respectively. Other parameters are the same as those in Fig. 1.

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Figure 3 presents the phase diagrams with respect to the particle radius $r$ and the chiral parameter $\kappa$, where the particle is illuminated by composed optical field with $\theta =75^{\circ }$ in Figs. 3(a) and (c) while $\theta =89^{\circ }$ in Figs. 3(b) and (d). The regions in color indicate parameter spaces where the optical pulling force $F_z<0$ exerts on the particle locating at the intensity maxima like the coordinate origin, while gray regions represent pushing forces. Furthermore, the color regions covered with green points correspond to the effective pulling that can realize chirality sorting, viz., under such parameters the particles can be stably trapped at the intensity maxima and then pulled along negative $z$ direction, as shown in Figs. 2(a) and (c) for examples. The remained color region represent the situation that pulling forces will act on particles located at the intensity maxima but these positions are not equilibrium positions where particles can be stably trapped, as shown in the case of Fig. 2(d). The criterion for stably trapping at the origin used in this paper is that the radial force satisfies $F_\rho <0$ for $\rho \leq 0.05~\mu m$. It is obviously demonstrated in Figs. 3(a) and (c), also in Figs. 3(b) and (d), that the composed optical field with opposite helicity exerts similar, if not exactly identical, forces on the particles with opposite handedness, one can therefore selectively pull chiral particles by simply tuning the light polarization. It is well-known that discriminating the chiral particles with small chirality is not easy, because of the weak chiral effect. In our system, the chiral particles with weak chirality can also be selectively transported by setting the polar angle $\theta$ large enough, as illustrated by comparing Figs. 2(a) and (c) and Figs. 2(b) and (d). For $\kappa =0.45$ and $\kappa =0.05$, we also show the phase spaces of optical longitudinal forces versus particle radius $r$ and polar angle $\theta$ in Figs. 4(a) and (b), respectively. The tunable range of the polar angle $\theta$ to implement selectively transporting, shown by color regions covered by green points, is demonstrated in Fig. 4 with nearly $24^{\circ }$ for $\kappa =0.45$ and $13^{\circ }$ for $\kappa =0.05$, which provides us a relatively wide range to manipulate the direction of wave vectors $\boldsymbol {\hat {k}}_i$.

Fig. 3. Phase diagrams with respect to the particle radius $r$ and the chiral parameter $\kappa$. Regions where particles can be stably trapped in the intensity maxima of a periodic region like the origin of the coordinate system and then subject to $F_z<0$ (pulling force) are demonstrated by color patches covered with green points. The dashed white lines refer to $\kappa =0.0$. $\theta =75^{\circ }$ and $(p,q)=(1,\pm \boldsymbol{i})/{\sqrt {2}}$ for (a) and (c) while $\theta =89^{\circ }$ and $(p,q)=(1,\pm \boldsymbol{i})/{\sqrt {2}}$ for (b) and (d). Other parameters are the same as those in Fig. 1.

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Fig. 4. Phase diagrams with respect to the particle radius $r$ and the polar angle $\theta$. Three waves are all left circularly polarized with $(p,q)=(1,\boldsymbol{i})/{\sqrt {2}}$ while $\kappa =0.45$ for (a) and $\kappa =0.05$ for (b). The color regions covered with green points denote effective pulling similar to the situations presented in Fig. 2(a) or (c) while the rest color regions are the situations similar to Fig. 2(d). The dashed white line is a reference line for $\theta =66^{\circ }$ for (a) while $\theta =77^{\circ }$ for (b). Other parameters are the same as those in Fig. 1.

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The structured optical field composed of three plane waves limits its applications to Mie particles with radius around $r \approx \lambda /2$, as shown in Fig. 3. To pull micro-particles with size larger than $\lambda$, optical field constituted by multiple plane waves can be employed. Figure 5 presents the examples of $n_p=4, 5, 6, 9$, respectively. Although the multiple-plane-wave fields extend the application of transporting particles to large particle size, it’s noted that their performance in chirality sorting is poor. The effective pulling, which is denoted by the color regions covered by green points, occurs in both sides of white lines in Fig. 5 for both positive and negative chiralities. Compared to the cleanly sorting shown in Fig. 3 where effective pulling only occurs in one side of the white lines for one kind of chirality, the multiple-plane-wave fields will also stably pull the opposite handedness particle to the light source, polluting the chirality sorting scenario. This suggests that the complexity introduced in the composed field will not establish a better configuration to realize chirality sorting. The three-plane-wave optical field presented in our paper, simple but efficient, is the best choice to implement chirality sorting by pulling force compared to other multiple-plane-wave fields.

Fig. 5. Phase diagrams with respect to the particle radius $r$ and the chiral parameter $\kappa$ for multiple plane waves, viz. 4 beams for (a), 5 beams for (b), 6 beams for (c) and 9 beams for (d). All plane waves share the same polar angle $\theta =80^{\circ }$ and left circular polarization $(p,q)=(1,\boldsymbol{i})/{\sqrt {2}}$ while their azimuthal angles are $\phi _i=(i-1)\boldsymbol {\times } 360^{\circ }/i$. The dashed white reference lines correspond to $\kappa =0.0$. Other parameters are the same as those in Fig. 1.

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4. Conclusions

In summary, we have presented a simple yet efficient passive all-optical proposal to selectively transport chiral particles based on the optical pulling force. The proposal takes advantage of an optical field composed of three circular polarized plane waves whose wave vectors show three-fold rotational invariance property around $z$ axis. The composed optical field presents net energy flow along positive $z$ direction. It is shown that a chiral particle can be transversely trapped by the optical radial force $F_\rho <0$ to the intensity maxima, while the particle with opposite chirality will be driven away by the reversed radial force $F_\rho >0$. Meanwhile, the transversely trapped chiral particle is transported by the optical pulling force $F_z<0$, leading to all-optical chirality sorting. Our proposal on the basis of three-plane-wave field presents a relatively wide tunable range in the manipulation of the directions of wave vectors $\boldsymbol {\hat {k}}_i$, which obviously improves the experimental feasibility. Besides, our demonstration allows for transporting in multiple channels simultaneously, which might add an additional hopefully efficient passive method for optical selection and separation of chiral particles.

Funding

National Natural Science Foundation of China (12074084, 12174076, 11804061); Natural Science Foundation of Guangxi Province (2018GXNSFBA281021, 2021GXNSFDA196001); National Key Research and Development Program of China (2016YFA0301103, 2018YFA0306201); Scientific Base and Talent Special Project of Guangxi Province (AD19110095); Open Project of State Key Laboratory of Surface Physics in Fudan University (KF2019_11).

Disclosures

The authors declare no conflicts of interest.

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 transport of chiral particles by optical pulling forces

Abstract

1. Introduction

2. Theory and formulations

3. Results and discussions

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (19)

Optics Express