Enantioselective optical trapping of chiral nanoparticles using a transverse optical needle field with a transverse spin

Ying Li; Guanghao Rui; Guanghao Rui; Sichao Zhou; Bing Gu; Bing Gu; Yanzhong Yu; Yiping Cui; Qiwen Zhan; Qiwen Zhan

doi:10.1364/OE.403556

1. Introduction

Chirality is a geometrical property where an object cannot be superposed onto its mirror image via either a translational or a rotational operation [1]. The mirror images of a chiral structure are enantiomers, and individual enantiomer are often designated as either right or left-handedness. In fact, this type of symmetry is much harder to be maintained than to be broken, consequently chirality exists widely in various macroscopic and microscopic structures. For example, proteins and nucleic acids are built of chiral amino acids and chiral sugar. In addition, DNA double helix, sugar, quartz, cholesteric liquid crystals and biomolecules are also chiral structures. Although molecules with different handedness have the same chemical construction, usually they would possess distinct chemical behaviors. A chiral biomolecule can be inactive or toxic to cells if its original handedness is varied, which causes many diseases such as phocomelia, Parkinson’s, Alzheimer’s, type II diabetes and Huntington’s [2]. Consequently, the sensitive detection and separation of substances by chirality is therefore of high demand in the fields of pharmacology, toxicology and pharmacodynamics.

The optical field can also be chiral. When illuminated by circularly polarized light, most molecules behave distinct optical responses in terms of refractive index, absorption and Raman scattering [3], which are named circular dichroism, optical rotation and Raman optical activity, providing noninvasive spectroscopic and efficient tool to unscramble the structural, kinetic, and thermodynamic information of molecules [4–7]. However, the chiroptical response in natural chiral materials is relatively weak due to the small electromagnetic interaction volume. Besides, the current spectroscopy is based on the measurement of the average far-field radiation, which suffers from low resolution structural details and considerable amount of sample, making it impossible to detect the structured chirality at nanoscale with high precision.

Different from conventional materials, the chirality of the electromagnetic material induces cross-polarization between electric and magnetic fields, leading to unique optical force effects, which derives from the momentum transfer associated with bending light when tightly focused laser beam interacts with the particles. Since Ashkin and his colleagues reported the first stable three-dimensional optical trapping (or optical tweezers) created using radiation pressure from a single focused laser beam, this technique has evolved from simple manipulation to the application of calibrated forces on, and the measurement of nanometer-level displacement of optically trapped objects [8–10]. In the early days, optical trapping has been mainly implemented in two size regimes: the sub-nanometer (e.g., cooling of atoms, ions and molecules) and micrometer scale (such as cells). Recently, new approaches have been developed to stably trap and manipulate mesoscopic objects, including metallic nanoparticles [11,12], carbon nanotubes [13,14] and quantum dots [15,16]. The continuous development of optical tweezers has revolutionized the experimental study of small particles and become an important tool for research in biology, physical chemistry and soft matter physics. Different from achiral metallic and dielectric materials, the optical response of chiral particle depends on not only the electric and magnetic polarizability, but also has close connection to the electromagnetic/magnetoelectric polarizability (also known as chiral polarizability), which provides additional degree of freedom to tailor the optical force and develop novel optical tweezers techniques. In recent years, plenty of chirality induced optical force effects have been reported (e.g. lateral force [17,18], optical pulling/repulsive force [19–21] and azimuthal/longitudinal optical torque [22,23]), enabling the effective manipulation of chiral particle with controllable dynamic behavior [24–28]. It has been reported that the chiral optical force would be generated along the direction of the photonic spin. In particular, the transverse component of spin angular momentum of light would give rise to a lateral optical force on chiral particles placed on top of a surface, and the direction of the force is determined by the handedness of the specimens [1,29–33]. Brasselet firstly anticipated and realized the optical sorting of particles with different handedness using chiral discriminatory force [34,35]. However, with the decrease of the particle size, the chiral optical force would be gradually suppressed by the achiral gradient force, making it difficult to realize the sorting of chiral nanoparticle.

With the rapid development of the optical engineering, various laser beam shaping system have been developed to tailor the spatial distribution of the light field at will [32,33]. Besides, the inverse design of the complete shaping of the optical focal field with the prescribed features has also been proposed [36–40]. These great progress in optical engineering enable us to synthesize optical focal field with prescribed intensity and three-dimensional polarization state, which are feasible to realize the optical force with unconventional features and are helpful to enhance the performance of the optical tweezers system. For example, the dynamic behaviors of metallic/dielectric nanoparticles with different shapes can be flexibly manipulated by adjusting the spatial distribution of the optical field in terms of phase, amplitude and polarization [41–45]. In this work, we proposed a novel strategy to discriminate and sort chiral nanoparticles by using the lateral optical force. The required incident field is calculated by reversing the radiated field from an array of paired orthogonal electric dipoles located in the focal plane of a 4Pi microscopy. When strongly focused by a high numerical aperture (NA) objective lens, a transverse optical needle field with transverse spin (TONFTS) is obtained in the focal volume, which is different from the previous optical needle fields that has long depth of focus along the optical axis. Deriving from uniform intensity distribution of the optical needle field and the transverse orientation of the photonic spin, this sculptural vectorial optical focal field offers both diminished achiral gradient force and handedness-dependent chiral gradient force. Consequently, the nanoparticles with different handedness can be spatially separated and trapped at different locations along the optical needled field, leading to the lateral deflection of chiral particles in opposite directions.

2. Construction of a transverse optical needle field with a transverse spin

In order to create an optical field that possesses the capability to laterally sort chiral nanoparticles, two requirements need to be satisfied: Firstly, the trapping light must have a transverse component of the spin angular momentum, which would product chiral optical force normal to the propagation direction of the light beam. Secondly, the achiral optical force caused by the interaction between the chiral nanoparticle and the optical field must be conquered by its chiral counterpart. As for the transverse spin angular momentum (e.g. focal field with photonic spin along y axis), it can be synthesized by two electric dipoles oscillating along x and z axis in the x-z plane with the same strength and phase difference of ±π/2, which are located at the focal point of a high NA lens. By changing the oscillating direction and the sign of the phase difference of the dipoles, both the orientation and the handedness of the photonic spin can be adjusted accordingly [39,43]. However, the optical field generated with this method is highly limited to a focal volume that is smaller than diffraction limit, leading to the dominating achiral gradient force that would pull the enantiomer towards the center of the focus. To sort enantiomers with chiral optical force, an optical needle field with transverse spin angular momentum and long uniform intensity along the spin axis is required to alleviate the dominating effect of the achiral gradient force. Consequently, TONFTS has great potentials in chiral optical tweezers, which can be constructed by pattering an array of paired orthogonal dipoles in the focal volume. As shown in Fig. 1(a), two pairs of orthogonal electric dipoles with the same oscillating direction are located mirror-symmetric with respect to the x-z plane. With the antenna pattern synthesis method for discrete linear dipole array, the radiation pattern of the dipoles collected by a high NA lens in the pupil plane can be calculated by the linear superposition of the emitted field of these infinitesimal dipoles:

(1)$${\boldsymbol{E}_{\boldsymbol{0}}}(\theta ,\varphi ) = ({E_\theta }{\boldsymbol{a}_{\boldsymbol{\theta}}} + {E_\varphi }{\boldsymbol{a}_{\boldsymbol{\varphi}}}){F_N},$$

where E_θ and E_ϕ are the radial and azimuthal components of the incident field, respectively. Similar methods have been demonstrated to generate optical needle field carrying different polarizations [46–49]. However, the optical needle fields reported previously have extended depth of focus along the optical axis, in which cases the radiation field from longitudinal discrete dipole array would have a rotational symmetry with respect to the pupil plane of the high NA lens. As for the linear dipole array in the transverse plane, the array factor F_N relates to not only the phase delay caused by the spacing distance d_n and initial phase difference β_n for each pair of the dipoles, but also depends on the spatial position of the observation point P(θ, ϕ):

(2)$${F_N} = \sum\limits_{n = 1}^N {{A_n}({e^{j\frac{{k{d_n}\sin \theta \sin \varphi + {\beta _n}}}{2}}}} + {e^{ - j\frac{{k{d_n}\sin \theta \sin \varphi + {\beta _n}}}{2}}}),$$

where k is the wave-vector in the medium, N is the total number of paired orthogonal dipole elements, A_n is the ratio of the radiation amplitude between the n-th dipole pair and the standard dipole pair with normalized amplitude.

Fig. 1. Diagram of the proposed optical tweezers setup. (a) The incident light is tailored by the vectorial optical field generator (VOF-Gen) to vectorial light with complex spatial distribution and then highly focused by the 4Pi focusing system to generate TONFTS. The orientation of the optical needle and the photonic spin is obtained through coherent superposition of the radiation patterns from an array of paired orthogonal electric dipoles. The intensity distribution superimposed with the polarization map of the (b) right-propagating and (c) left-propagating illumination.

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For a conventional objective lens obeys sine condition, the required incident pupil field is:

(3)$${E_{ri}}({r,\varphi } )= \frac{1}{{\sqrt {\cos \theta } }}{F_N}({X_{ri}} \cdot {\boldsymbol{e}_{\boldsymbol{x}}} + {Y_{ri}} \cdot {\boldsymbol{e}_{\boldsymbol{y}}}),$$

(4)$${X_{ri}} = {e^{i\frac{\pi }{2}}}\sin \theta \cos \varphi - \cos \theta {\cos ^2}\varphi - {\sin ^2}\varphi ,$$

(5)$${Y_{ri}} = {e^{i\frac{\pi }{2}}}\sin \theta \sin \varphi - \cos \theta \cos \varphi \sin \varphi + \sin \varphi \cos \varphi ,$$

where r = f·sinθ, f is the focal length of the objective lens, e_x and e_y are the unit vectors along x and y axis, respectively. After being focused by a high NA lens, the electric field in the vicinity of the focus can be calculated by using Richard–Wolf vectorial diffraction theory:

(6)$${\boldsymbol{E}_{rf}}({r_p},\phi ,{z_p}) = {C_s}\int_0^{{\theta _{\max }}} {\int_0^{2\pi } {({X_{rf}} \cdot {\boldsymbol{e}_{\boldsymbol{x}}} + {Y_{rf}} \cdot {\boldsymbol{e}_{\boldsymbol{y}}} + {Z_{rf}} \cdot {\boldsymbol{e}_{\boldsymbol{z}}})} } \times {e^{jk{r_p}\sin \theta \cos (\varphi - \phi ) + j{z_p}\cos \theta }}\sin \theta {F_N}d\theta d\varphi ,$$

where ${C_s} = \pi {\lambda ^{ - 1}}\sqrt {{{2P} / {({{n_m}{\varepsilon_0}c\pi N{A^2}} )}}}$, P is the power of the illumination, n_m is the refractive index of the surrounding medium, λ is the wavelength in medium, θ_max is the maximal focusing angle determined by NA of the lens, r_p = (x² + y²)^1/2 and ϕ = tan⁻¹(y/x) are the polar coordinates in the focal volume. The electric field components X_rf, Y_rf, and Z_rf can be expressed as:

(7)$${X_{rf}} = {e^{i\phi }}\sin \theta \cos \theta \cos \varphi - {\cos ^2}\theta {\cos ^2}\varphi - {\sin ^2}\varphi ,$$

(8)$${Y_{rf}} = {e^{i\phi }}\sin \theta \cos \theta \sin \varphi + {\sin ^2}\theta \sin \varphi \cos \varphi ,$$

(9)$${Z_{rf}} = {e^{i\phi }}{\sin ^2}\theta - \cos \theta \sin \theta \cos \varphi .$$

A 4Pi focusing system is applied to correct the asymmetric radiation patterns from the off-axis electric dipoles, which is consisted of two lens with effective NA of 0.74 and two counter-propagating beams, and the space in-between the lenses is filled with water (n_m = 1.33) [33]. Compared with the right-propagating light beam that described by Eqs. (3)–(5), the E_x component of the left propagating field E_li would be out of phase, which is necessary to maintain the symmetry of the optical needle field about x axis:

(10)$${E_{li}}({r,\varphi } )= \frac{1}{{\sqrt {\cos \theta } }}{F_N}({X_{li}} \cdot {\boldsymbol{e}_{\boldsymbol{x}}} + {Y_{li}} \cdot {\boldsymbol{e}_{\boldsymbol{y}}}),$$

(11)$${X_{li}} ={-} {e^{i\frac{\pi }{2}}}\sin \theta \cos \varphi - \cos \theta {\cos ^2}\varphi - {\sin ^2}\varphi ,$$

(12)$${Y_{li}} = {e^{i\frac{\pi }{2}}}\sin \theta \sin \varphi + \cos \theta \cos \varphi \sin \varphi - \sin \varphi \cos \varphi .$$

After being focused by a 4Pi focusing system, the electric field in the vicinity of the focus can be described as:

\boldsymbol{E}({r_p},\phi ,{z_p}) = {\boldsymbol{E}_{rf}}({r_p},\phi ,{z_p})\textrm{ + }{\boldsymbol{E}_{lf}}({r_p}, - \phi , - {z_p})\textrm{ = }{C_s}\int_0^{{\theta _{\max }}} {\int_0^{2\pi } {({X_f} \cdot {\boldsymbol{e}_{\boldsymbol{x}}} + {Y_f} \cdot {\boldsymbol{e}_{\boldsymbol{y}}} + {Z_f} \cdot {\boldsymbol{e}_{\boldsymbol{z}}})} } \times

(13)$$({e^{jk{r_p}\sin \theta \cos (\varphi - \phi ) + j{z_p}\cos \theta }} + {e^{jk{r_p}\sin \theta \cos (\varphi + \phi ) - j{z_p}\cos \theta }})\sin \theta {F_N}d\theta d\varphi ,$$

(14)$${X_f} = {e^{i\phi }}\sin \theta \cos \theta \cos \varphi - {\cos ^2}\theta {\cos ^2}\varphi - {\sin ^2}\varphi ,$$

(15)$${Y_f} = {e^{i\phi }}\sin \theta \cos \theta \sin \varphi + {\sin ^2}\theta \sin \varphi \cos \varphi ,$$

(16)$${Z_f} = {e^{i\phi }}{\sin ^2}\theta - \cos \theta \sin \theta \cos \varphi .$$

Besides, in order to achieve a TONFTS along y axis, the parameters of each paired orthogonal dipoles in the linear array in terms of spacing distance d_n, radiation amplitude A_n and phase difference β_n need to be carefully chosen. Generally, large dipole spacing d_n brings large modulation of the interacted dipole field which makes it impossible to realize uniform intensity, while small d_n leads to optical needle field with limited length. Radiation amplitude A_n controls the intensity near the positions of the dipole elements, making sure the peak intensities around each dipole position are approximately equal. Phase difference β_n is used to control the overall flatness profile of the transversal intensity distribution. In this work, particle swarm algorithm is utilized with multi-objective optimization (length and flatness of the optical needle field). Particle swarm optimization is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. In this case, the radiation amplitude A_n, spacing distance d_n and phase difference β_n were chosen as the parameter group. Based on the reasonable value range, one hundred groups of parameters were randomly selected at the initialization stage. Besides, length and flatness of the optical needle in the y-direction and the intensity symmetry in the x-z plane are set as the figure of merit. The globally optimized parameters are summarized in Table 1. In addition, the corresponding distribution of the required incident field for generating TONFTS is calculated with Eqs. (3)–(5) and presented in Figs. 1(b) and 1(c). Besides amplitude, the state of polarization (SOP) at the beam cross-section is also indicated by the polarization ellipses, where green and blue colors represent left- and right-handedness respectively. It can be clearly seen that both the amplitude and the polarization distributions of the illumination are neither uniform nor symmetric. For the right-propagating illumination, most of the input light is in the lower half, the polarization of which have opposite handedness compared with that of the top half. Besides, the SOP of the incident light would change its ellipticity from circular to linear along the vertical direction, while its elevation angle would also vary along the horizontal direction. As for the left-propagating illumination, the SOP distribution is the mirror image of the case shown in Fig. 1(b) due to the phase difference between the E_x components.

Table 1. Optimized parameters of the TONFTS

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To illustrate the feasibility of the proposed strategy to generate TONFTS, the characteristics of the optical focal field are numerically explored with Eqs. (6)–(9) and presented in Fig. 2, with the assumption that the optical wavelength of the illumination is 632 nm and incident power is 150 mW. From the three-dimensional intensity distribution (Fig. 2(a)) and the corresponding line-scans (Fig. 2(b)), it can be clearly seen that a uniform and symmetric optical needle field along y axis is obtained. The length of the optical needle is measured to be 1.61λ, which is defined as the transversal full width of above 80% maximum intensity. Besides, the full-width-half-maximum of the focal field along x and z axes are found to be 0.3985λ and 0.3377λ, respectively. It is worthy of noting that the needle length can be further extended by adding more elements in the dipole array. To examine the polarization of the optical needle field, intensity distribution superimposed with SOP in three longitudinal slices (y = −0.8λ, 0, and 0.8λ) are shown in Figs. 2(c)–2(e). One may find from the polarization ellipse that the ellipticity within the main lobe keeps close to 1, indicating the optical needle field maintains circular polarization in the x-z plane. To better study the polarization characteristics of the focal field, both the full Stokes parameters S and the spin density D (D ∝ Im(E*×E)) of focused beam in the x-z plane (y = 0) are calculated and illustrated in Figs. 2(f)–2(k). It can be clearly that both the normalized S₃ and D_y are nearly unit in the main lobe, indicating that the spin axis of the optical needle field is mainly in the positive y axis. To quantitively evaluate the quality of the TONFTS, the beam purity η = Φ_RHC/(Φ_RHC+Φ_LHC) is calculated to be 0.8359, which is defined as intensity percentage of the right-hand circular polarization component in the x-z plane throughout the optical needle field: ${\Phi _{RHC({LHC} )}} = \int\limits_{ - {x_l}}^{{x_h}} {\int\limits_{ - {y_l}}^{{y_h}} {\int\limits_{ - {z_l}}^{{z_h}} {{{|{{E_{RHC({LHC} )}}({x,y,z} )} |}^2}dxdydz} } } ,$ where the integration area is restricted by x_l/y_l/z_l and x_h/y_h/z_h (marked on Fig. 2(b)).

Fig. 2. Synthesized TONFTS in the focal volume of the 4Pi focusing system. (a) Intensity distribution of the TONFTS in three-dimensional space. (b) Intensity line-scans of the TONFTS along different axes. Intensity distribution superimposed with SOP in x-z plane at (c) y = −0.8λ, (d) 0 and (e) 0.8λ. (f-h) Stokes images and (i-k) spin density distribution in the vicinity of the focus.

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To demonstrate that the complexity of the required incident light is still within the current optical engineering technology, the vectorial optical field is experimentally generated and compared with the theoretical design. Figure 3(a) illustrates the diagram of the vectorial optical field generator (VOF-Gen) shown in Fig. 1, which is the key system to create an arbitrary optical field with independent control of phase, amplitude, and polarization distribution on the pixel level. Taking advantage of the high definition television format of the Holoeye HEO 1080P reflective phase-only liquid crystal spatial light modulator, full control of all the degrees of freedom to create an arbitrary complex optical field is enabled [37]. The experimental results of the input beam designed for generating TONFTS is captured by a charge coupled device camera and shown in Fig. 3(f). Besides, the full Stokes parameter measurement (shown in Figs. 3(f)–3(i)) is also performed, showing very good agreement between the theoretical predications (shown in Figs. 3(b)–3(e)) and the experimental behavior. In order to quantitatively evaluate the synthesized input light in terms of the overall quality of the SOP, the cumulative normalized Stokes parameters are introduced:

(17)$${P_i} = \sqrt {{{\sum {S_i^2({{x_0},{y_0}} )} } / {\sum {S_0^2({{x_0},{y_0}} )} }}} ,\quad \quad \quad i = 1,2,3$$

where (x₀, y₀) are the indexes of the pixels of the Stokes image. Table 2 shows the theoretical and experimental values of P₁, P₂ and P₃ of the input light presented in Figs. 3(b)–3(e) and 3(f)–3(i). The average error is calculated to be about 15%, demonstrating that the VOF-Gen is capable of sculpturing the complex light field and realizing TONFTS with high quality.

Fig. 3. Experimentally generated complex optical field for synthesizing TONFTS. (a) Diagram of the experimental setup. HWP, half-wave plate; P, polarizer; BS, beam splitter; L, lens; M, mirror; SF, spatial filter. (b-e) Theoretical and (f-i) experimental Stokes images of the incident light.

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Table 2. Theoretical and experimental P_i values for the incident light presented in Fig. 3(b) and Fig. 3(f).

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3. Optical force and torque under dipole approximation

To demonstrate the unique mechanical response using this TONFTS, we then analyze the optical interaction by introducing enantiomers into the focal volume of the 4Pi focusing system. The scattering behavior of the particle can be rigorously solved by the Mie theory, and the corresponding Mie scattering coefficients ($a_n^{(1)},a_n^{(2)},b_n^{(1)},b_n^{(2)}$) govern the relationship between the incident and scattering fields [23,50–53]:

(18)$$\begin{aligned} &a_n^{(1)} = [A_n^{(2)}V_n^{(1)} + A_n^{(1)}V_n^{(2)}]{Q_n}, \;\;\; a_n^{(2)} = [A_n^{(1)}W_n^{(2)} - A_n^{(2)}W_n^{(1)}]{Q_n},\\ &b_n^{(1)} = [B_n^{(1)}W_n^{(2)} + B_n^{(2)}W_n^{(1)}]{Q_n},\;\; b_n^{(2)} = a_n^{(2)}, \end{aligned}$$

with

(19)$$\begin{aligned} &A_n^{(j)} = {Z_S}D_n^{(1)}({x_j}) - D_n^{(1)}({x_0}), \;\;\; B_n^{(j)} = D_n^{(1)}({x_j}) - {Z_S}D_n^{(1)}({x_0}),\\ &W_n^{(j)} = {Z_S}D_n^{(1)}({x_j}) - D_n^{(3)}({x_0}), \;\; V_n^{(j)} = D_n^{(1)}({x_j}) - {Z_S}D_n^{(3)}({x_0}),\\ &{Q_n} = \frac{{{{{\psi _n}({x_0})} / {{\xi _n}({x_0})}}}}{{V_n^{(1)}W_n^{(2)} + V_n^{(2)}W_n^{(1)}}}, \end{aligned}$$

where x₀= kr_s, x₁ = k₁r_s, and x₂ = k₂r_s with r_s represents the radius of particle and wave number ${k_1} = k(\sqrt {{\varepsilon _r}{\mu _r}} - \kappa )\textrm{/}\sqrt {{\varepsilon _m}} , {k_2} = k(\sqrt {{\varepsilon _r}{\mu _r}} + \kappa )\textrm{/}\sqrt {{\varepsilon _m}} ,$ with ε_r and μ_r represents the relative permittivity and permeability of the chiral medium, and ε_m represents the relative permittivity of the immersion medium. ${Z_S} = \sqrt {{{{\mu _r}{\varepsilon _m}} / {{\varepsilon _r}}}}$, while ψ_n(x) and ξ_n(x) are the Riccati-Bessel functions of the first and third kinds and $D_n^{(1)}(x) = {\psi ^{\prime}_n}(x)/{\psi _n}(x),$ $D_n^{(3)}(x) = {\xi ^{\prime}_n}(x)/{\xi _n}(x)$ are the corresponding derivatives.

Considering a spherical chiral particle that is much smaller than the incident wavelength, it can be treated in quasi-static limit and represented by point polarizability. Within the dipole approximation, the induced electric dipole moment p and magnetic dipole moment m can be expressed as p = α_eeE+α_emB and m = −α_emE+α_mmB, where E and B are the electric field and magnetic induction of the incident optical fields, α_ee, α_mm and α_em describe the electric, magnetic and electromagnetic polarizability of the sample. Note that α_em is related to the chirality parameter κ of the material that the particle is made of, and the imaginary part of α_em will change its sign if the handedness of the chiral sample changes. The polarizability elements of the dipolar chiral particle relate to the scattering coefficients [23,47]:

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

where ${Z_0} = \sqrt {{\mu _0}{\varepsilon _m}/{\varepsilon _0}}$ is the wave impedance in vacuum and ε₀ and μ₀ are the vacuum permittivity and permeability. It can be proved that the Mie coefficients satisfy a₁⁽¹⁾(−κ) = a₁⁽¹⁾(κ), b₁⁽¹⁾(−κ) = b₁⁽¹⁾(κ), but a₁⁽²⁾(−κ) = −a₁⁽²⁾(κ), consequently only the electromagnetic polarizability will change its sign if the handedness of the chiral sample changes.

With the counter-propagating incident fields with expressions given in Eqs. (3)–(6) and Eqs. (10)–(12), the optical force exerted on a chiral nanoparticle in the TONFTS can be derived in terms of the electric and magnetic dipoles [32,54,55]:

(21)$$\left\langle \boldsymbol{F} \right\rangle = \frac{1}{2}{\rm{Re}} [(\nabla {\boldsymbol{E}^{\ast }}) \cdot \boldsymbol{p} + (\nabla {\boldsymbol{B}^{\ast}}) \cdot \boldsymbol{m} - \frac{{{Z_0}{k^4}}}{{6\pi }}(\boldsymbol{p} \times {\boldsymbol{m}^{\ast}})].$$

The force expression can be written as follows:

(22)$$\begin{aligned} \left\langle \boldsymbol{F} \right\rangle ={-} &\nabla \left\langle U \right\rangle + \frac{{n{}_m}}{c}({{C_{ext}} + {C_{recoil}}} )\left\langle \boldsymbol{S} \right\rangle + {\mu _0}\nabla \times {\rm{Re}} [{{\alpha_{em}}} ]\left\langle \boldsymbol{S} \right\rangle \textrm{ + }\nabla \times \left\{ {{C_p}\frac{c}{{{n_m}}}\left\langle {{\boldsymbol{L}_{\boldsymbol{p}}}} \right\rangle + {C_m}\frac{c}{{{n_m}}}\left\langle {{\boldsymbol{L}_{\boldsymbol{m}}}} \right\rangle } \right\}\\ &+ \left\{ {2{\omega^2}{\mu_0}{\rm{Re}} [{{\alpha_{em}}} ]- \frac{{{k^5}}}{{3\pi \varepsilon_0^2\varepsilon_m^2}}{\mathop{\rm Im}\nolimits} [{{\alpha_{ee}}\alpha_{em}^ \ast } ]} \right\}\left\langle {{\boldsymbol{L}_{\boldsymbol{p}}}} \right\rangle + \left\{ {2{\omega^2}{\mu_0}{\rm{Re}} [{{\alpha_{em}}} ]- \frac{{{k^5}{\mu_0}}}{{3\pi {\varepsilon_0}{\varepsilon_m}}}{\mathop{\rm Im}\nolimits} [{{\alpha_{mm}}\alpha_{em}^ \ast } ]} \right\}\left\langle {{\boldsymbol{L}_{\boldsymbol{m}}}} \right\rangle \\ &+ \frac{{c{k^4}\mu _0^2}}{{12\pi {n_m}}}{\mathop{\rm Im}\nolimits} [{{\alpha_{ee}}\alpha_{mm}^ \ast } ]{\mathop{\rm Im}\nolimits} [{\boldsymbol{E} \times {\boldsymbol{H}^{\ast}}} ]. \end{aligned}$$

where c is the speed of light in vacuum, 〈S〉 = Re[E × H*]/2 denotes the time-averaged Poynting vector, 〈U〉 = −1/4(Re[α_ee]|E|² + Re[α_mm]|B|² − 2Im[α_em]Im[B·E*]) is an energy term due to the interaction of the dipolar chiral particle with the TONFTS, 〈L_p〉 = Re[ε₀ε_m/(4iω)E × E*] and 〈L_m〉 = Re[μ₀/(4iω)H × H*] represent the electric and magnetic part of the time-averaged spin angular momentum densities respectively, ω is the angular frequency. C_ex_t = C_p + C_m = k(Im[α_ee]/ (ε₀ε_m) + μ₀Im[α_mm]) is a sum of contribution from the electric and magnetic dipole channel. C_recoil = −k⁴μ₀/(6πε₀ε_m)Re[α_eeα*_mm] − k⁴μ₀/(6πε₀ε_m)|α_em|² describes the recoil force and is related to the asymmetry parameter. The first and second terms in Eq. (22) correspond to the gradient force and the radiation pressure, respectively. The third term is a vortex force determined by the energy flow vortex around the particle and the optical activity, while the fourth term represents the scattering force associated with the curl of spin angular momentum densities. The fifth and sixth terms are referred to the spin density force that is related to the spin angular momentum densities. The particle chirality makes no explicit contribution to the last term, which is due to the alternating flow of the stored energy. Consequently, the movement of a chiral particle immersed in the optical focal field is subject to the induced time-averaged optical force, arising from the transfer of the linear momentum between the light and the material.

Assuming a lossy chiral nanoparticle with radius of 5 nm located near the center of the TONFTS, the optical force is calculated with Eq. (22) and the corresponding force distributions in different planes are illustrated in Fig. 4. Note that the magnetic field B is calculated from the spatial distributions of the electric fields in the focal volume as B = 1/(iω)∇×E. The parameters of the chiral material are set to be (ε, κ) = (1.6²+0.04i, −0.5 + 0.02i). The chiral particles can be a conventional electromagnetic material with chiral shape, or a spherical particle made of chiral material. By controlling the concentration of the lossy constituent added, the chiral particle with controllable absorption can be precisely fabricated. Figures 4(a) and 4(b) shows the x- and z-component of the optical force in the x-z plane (y = 0), indicating that the particles in the focal area would be trapped at x = 0 and z = 0. Besides, from the distribution of the y-component of the optical force F_y in the x-y plane (shown in Fig. 4(c)) it can be clearly seen that there is an equilibrium position at y = 0. To explore the physical mechanism of the induced force, the total optical force has been divided into different parts and the corresponding line-scans are presented in Figs. 4(d)–4(f). Firstly, it can be seen that the scattering force which contains the second, third, fourth, fifth and sixth terms in Eq. (22) is maximized along the direction of the photonic spin, but still is difficult to compete with the gradient force. Secondly, the gradient force can be decomposed into the combination of chiral and achiral gradient force, which would change/not change its sign for particles with opposite chirality. Specifically, achiral gradient force refer to F_achiral= −1/4(Re[α_ee]|E|² + Re[α_mm]|B|²) and chiral gradient force refer to F_chiral= 1/2Im[α_em]Im[B·E*]. Thirdly, the achiral gradient force is dominating in the plane perpendicular to the orientation of the photonic spin, which confines the enantiomers at the center of the focus. Lastly, due to the uniform intensity distribution of the optical needle field, the achiral gradient force would become negligible along y-axis in the vicinity of the focus, while the movement of the particle is primarily determined by the chiral gradient force. As a particle with refractive index larger than the ambient environment, it would be trapped at the location possessing zero net optical force and negative slope of the optical force distribution. The stability of the optical trapping can be demonstrated by the potential depth U, which is estimated as the work done by optical forces along the axes ${U_n} = - \int {F_n} \cdot dn (n = x,y,z)$. It can reach 1.658 k_BT and 7.5 k_BT for the equilibrium positions at the center and the ends of the optical needle field respectively, where k_B is the Boltzmann constant and T = 300 K is the absolute temperature of the ambient. Traditionally an optical trap with potential depth U larger than k_BT can be considered as stable [11]. Consequently, a chiral nanoparticle with chirality parameter (κ = −0.5 + 0.02i) would be stably trapped at the central position (x = y = z = 0) of the TONFTS. In addition, a chiral nanoparticle with opposite handedness (κ = 0.5 + 0.02i) is also considered and the corresponding optical force distributions are presented in Figs. 4(g)–4(l). In this case, the particle is still centrally confined in the x-z plane by the achiral gradient force, while it will be pushed out of the focus by the chiral gradient force along y axis and finally be stably trapped at the ends of the TONFTS, and the separation distance between the enantiomers should be around 0.5λ or 0.6λ. Therefore, we demonstrated that the TONFTS is capable to realize laterally sorting of chiral nanoparticle, in which the enantiomers with different handedness would be trapped at the center and two ends of the optical needle field, respectively. It is worthy of noting that the equilibrium positions of the enantiomers can be easily switched, which is realized by changing the handedness of the orthogonal dipoles (Eqs. (4) and (5)).

Fig. 4. Calculated optical force on 5 nm (radius) chiral absorbing nanoparticle at wavelength of 632 nm using the proposed strategy with TONFTS. Distribution of the (a) x-, (b) z-, and (c) y-component of the optical force exerted on an chiral nanoparticle with (ε, κ) = (1.6²+0.04i, −0.5 + 0.02i) located near the center of the TONFTS, and the corresponding components of the optical force along (d) x- (e) z- and (f) y-axis. Distribution of the (g) x-, (h) z-, and (i) y-component of the optical force exerted on the chiral nanoparticle with opposite handedness (κ = 0.5 + 0.02i) and the components of the optical force along (j) x- (k) z- and (l) y-axis.

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For comparison, the optical force effects of chiral nanoparticle interacting with the optical focal field with pure transverse spin (along y axis) is also considered, which can be designed by reversing the radiation field from two orthogonal electric dipoles with phase difference of π/2 at the focus of a high NA lens. Note that in this case the intensity distribution of the focal field along y axis is not uniform anymore. As shown in Figs. 5(a)–5(c), although the particle can still be trapped at the center of the x-z plane, the difference in optical force which is mainly determined by the chiral gradient force becomes negligible, therefore the enantiomers cannot be separated by the achiral gradient force in three-dimensional space. From Eq. (22) one can find that the chiral gradient force is proportional to the gradient of the optical chirality density Im[B·E*]. Compared with the TONFTS, the optical focal field with purely transverse spin angular momentum carries relatively weak E_y, H_x and H_z, leading to the small optical chirality density. As is shown in Figs. 5(d)–5(g), the optical chirality of purely transverse spin filed in y-z plane and x-y plane is an obviously smaller than TONFTS, and the low chirality gradient leads to negligible chiral gradient force. Consequently, the TONFTS we proposed in this work has the advantage of providing negligible achiral gradient force and large optical chiral density in the transversal direction. Due to the enhanced chiral gradient force provided by the TONFTS, it is feasible to stably trap and laterally sort natural /artificial chiral materials with smaller chirality parameter by increasing the laser power, or constructing complex optical focal field that consists of more pair of electric and magnetic dipoles, which would boost the efficiency and sensitivity of the techniques involving the enantiomers sorting and detecting.

Fig. 5. The optical force effect of optical field with transverse spin exerted on 5 nm (radius) chiral absorbing nanoparticle at the wavelength of 632 nm. Optical force along (a) x-, (b) y-, and (c) z-axis exerted on chiral nanoparticles with (ε, κ) = (1.6²+0.04i, ±0.5 + 0.02i) located near the focus of an optical focal field with transverse spin along y axis. Optical chirality distribution of the optical focal field with pure transverse spin and TONFTS in the (d, f) y-z plane and (e, g) x-y plane, respectively.

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Moreover, due to the spin angular momentum transfer from the TONFTS, the chiral particle would experience an intrinsic torque Γ and spin around its center of mass:

(23)$$\begin{aligned} &\left\langle {\boldsymbol{\varGamma}} \right\rangle = [ - 2{\mu _0}{\rm{Re}} ({\alpha _{em}}) + \frac{{{\mu _0}{k^3}}}{{3\pi {\varepsilon _0}{\varepsilon _m}}}{\mathop{\rm Im}\nolimits} ({\alpha _{ee}}\alpha _{em}^ \ast ) + \frac{{\mu _0^2{k^3}}}{{3\pi }}{\mathop{\rm Im}\nolimits} ({\alpha _{mm}}\alpha _{em}^ \ast )]\left\langle \boldsymbol{S} \right\rangle \\ &+ [\frac{{{\mu _0}{k^3}}}{{6\pi {\varepsilon _0}{\varepsilon _m}}}{\rm{Re}} ({\alpha _{ee}}\alpha _{em}^ \ast ) - \frac{{\mu _0^2{k^3}}}{{6\pi }}{\rm{Re}} ({\alpha _{mm}}\alpha _{em}^ \ast )]{\mathop{\rm Im}\nolimits} (\boldsymbol{E} \times {\boldsymbol{H}^{\ast}})\\ &+ [\frac{{2\omega }}{{{\varepsilon _0}{\varepsilon _m}}}{\mathop{\rm Im}\nolimits} ({\alpha _{ee}}) - \frac{{\omega {k^3}}}{{3\pi \varepsilon _0^2\varepsilon _m^2}}{\alpha _{ee}}\alpha _{ee}^ \ast{-} \frac{{\omega {\mu _0}{k^3}}}{{3\pi {\varepsilon _0}{\varepsilon _m}}}{\alpha _{em}}\alpha _{em}^ \ast ]\left\langle {{\boldsymbol{L}_{\boldsymbol{p}}}} \right\rangle \\ &+ [2\omega {\mu _0}{\mathop{\rm Im}\nolimits} ({\alpha _{mm}}) - \frac{{\omega \mu _0^2{k^3}}}{{3\pi }}{\alpha _{mm}}\alpha _{mm}^ \ast{-} \frac{{\omega {\mu _0}{k^3}}}{{3\pi {\varepsilon _0}{\varepsilon _m}}}{\alpha _{em}}\alpha _{em}^ \ast ]\left\langle {{\boldsymbol{L}_{\boldsymbol{m}}}} \right\rangle . \end{aligned}$$

To intuitively illustrate the activity of the trapped enantiomers, the distributions of the optical torque in three x-z planes containing the equilibrium locations (y = −0.6λ, 0, 0.5λ) are calculated and shown in Fig. 6. It can be clearly seen that the chiral particle would rotate around y axis. The rotation orientation of the particle is determined by that of the photonic spin and it is not related to the chirality of the material. Besides, the rotation frequencies ${\Omega }p = - {\Gamma }/8\pi \xi {r_s}^{3}$ at the equilibrium positions are calculated to be 7.3259×10⁵ Hz (viscosity of water ξ is about 0.001 N⋅s/m² at 20^°C)) for particles with κ = −0.5 + 0.02i and 4.0034×10⁵ for particles with κ = 0.5 + 0.02i. Note that the chiral particle trapped at the center of the TONFTS would experience a faster rotation frequency compared to its counterpart with the opposite handedness trapped at the ends of the optical needle field, providing another feasible way to distinguish the enantiomers by their dynamic behaviors.

Fig. 6. Calculated optical torque distribution at equilibrium positions. Distribution of the y-component of the optical torque in the z-x plane at (a) y = −0.6λ, (b) 0 and (c) 0.5λ. The x- and z-components of the optical torque is indicated by the arrows.

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4. Conclusions

In conclusion, we propose a method to lateral sort chiral nanoparticles with different handedness by using the transverse optical needle field with transverse spin. To achieve controllable orientation of both the photonic spin and the optical needle field, the required pupil field of the illumination is analytically derived through reversing the radiation patterns from an array of paired orthogonal electric dipoles located in the focal plane of a 4Pi microscopy and experimentally generated with a home-built VOF-Gen. Furthermore, the scattering behavior of the chiral particle is evaluated by the Mie scattering coefficients and the induced optical force from the interaction between the chiral nanoparticle and the TONFTS is calculated under dipole approximation. Numerical results demonstrated that the particles experience a transverse chiral optical force with direction determined by the handedness of the chiral material. Besides, the transverse achiral gradient force is negligible along the optical needle field, giving rise the possibility to lateral sort and trap chiral nanoparticle with different handedness at different locations. Moreover, we demonstrated that the intrinsic torque leads to the rotation of the chiral nanoparticle with handedness-dependent rotation frequency around the spin axis. This versatile trapping method may open new avenues for all-optical enantiopure chemical syntheses and enantiomer separations in pharmaceuticals.

Funding

National Natural Science Foundation of China (11504049, 11774055).

Acknowledgements

G. R. acknowledged the support by the Zhishan Young Scholar Program of Southeast University.

Disclosures

The authors declare no conflicts of interest.

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	A_n (radiation amplitude)	d_n (spacing distance)	β_n (phase difference)
Dipole-pair 1	1	0.6524λ	0.1273π
Dipole-pair 2	0.685	1.5038λ	π

	P₁	P₂	P₃
Theoretical values	0.6621	0.1916	0.7245
Experimental values	0.6661	0.1920	0.7296

	A_n (radiation amplitude)	d_n (spacing distance)	β_n (phase difference)
Dipole-pair 1	1	0.6524λ	0.1273π
Dipole-pair 2	0.685	1.5038λ	π

	P₁	P₂	P₃
Theoretical values	0.6621	0.1916	0.7245
Experimental values	0.6661	0.1920	0.7296

Enantioselective optical trapping of chiral nanoparticles using a transverse optical needle field with a transverse spin

Abstract

1. Introduction

2. Construction of a transverse optical needle field with a transverse spin

3. Optical force and torque under dipole approximation

4. Conclusions

Funding

Acknowledgements

Disclosures

References

Cited By

Figures (6)

Tables (2)

Equations (24)

Optics Express