Synthesis of multiple longitudinal polarization vortex structures and its application in sorting chiral nanoparticles

Xinyuan Ying; Guanghao Rui; Guanghao Rui; Shuting Zou; Bing Gu; Bing Gu; Qiwen Zhan; Qiwen Zhan; Yiping Cui

doi:10.1364/OE.427482

1. Introduction

As reflected in the properties of the electric field, a light beam may possess not only linear momentum but also angular momentum (AM), the latter of which can be further classified as spin angular momentum (SAM) and orbital angular momentum (OAM). The intrinsic SAM is associated with the polarization state of light beam, where σ_± = ±1 correspond to the right- and left-hand (RH and LH) circular polarizations respectively [1], while OAM arises from the spatial structure of the optical field which can take arbitrary integer values. It can be characterized by the intrinsic OAM associated with inhomogeneous transversal phase distributions, commonly referring to the spiral phase gradient [2,3], and the extrinsic one relating to the beam trajectory [4–6]. Usually, it is considered that OAM and SAM are independent in a uniform and isotropic transparent medium. However, if an incident spin photon encounters a structure with anisotropic inhomogeneous boundaries, there would be spin-orbit interaction and leads to interesting phenomena such as optical spin-Hall effect [7,8]. In addition, the mutual conversion between SAM and OAM of photons can be expected in a high numerical aperture (NA) focusing system [9–11]. For example, when a circularly polarized vortex beam is tightly focused by a high NA lens, the spin-to-orbit conversion (SOC) would result in a strong longitudinal component of electric field carrying spiral phase at the focal region [12], which has plenty of attractive applications in fluorescent imaging, Raman spectroscopy and optical tweezers [13,14].

The phenomenon of SOC can be observed through the momentum transfer between optical field and nanoparticles [15]. When interacting with a light beam, the nanoparticle would inherit the SAM or OAM of the light, giving rise to the generation of optical forces that drag the nanoparticle to rotate around its own axis [16] or the optical axis [13,17] (namely spin or orbital motion). Since Ashkin and colleagues reported the first stable three-dimensional (3D) optical trapping by using the radiation pressure from a single focused laser beam [18], the continuous development of optical tweezers has become an important tool for research in biology, physical chemistry and soft matter physics [19–21]. Usually, a tightly focused light beam is necessary in an optical tweezers, since the large gradient force would stably confine the nanoparticles in the location that having the largest intensity. In order to manipulate various nanoparticles in separated regions simultaneously, focal fields with multiple foci are necessary, which can be realized by tailoring the spatial distribution of an incident vector light. Most of the previous works focus on the generation of identical multiple focal spots with specific spatial intensity distribution (such as spherical solid/hollow spot [22–25], optical bubble [26], optical chain [27–30], optical needle [31], etc.). Recently, dual coaxial focal spots with different AMs have been reported to rotate nanoparticles located on the optical axis [32]. However, the full control of the focal field in terms of number of focal spots, SAM, OAM, and location in 3D space is still a challenging task. Due to the noncontact and holding nature of optical tweezers, the optical sorting of chiral materials has attracted increasing attentions in recent years. Chirality refers to the geometric property that a substance cannot overlap with its mirror image through translation and rotation operations [33]. The mirror images of a chiral structure are enantiomers, which are widely existed in various macro- and micro- structures. The chiral detection of substances has important applications in pharmacology, toxicology and pharmacokinetics [34–36], since the change of the original chirality of a biomolecule may lead many serious diseases [37,38]. Different from conventional materials, the optical response of chiral nanoparticle not only depends on the electric and magnetic polarizability, but also has close connection to the electromagnetic/magnetoelectric polarizability (also known as chiral polarizability). The interaction between light field and chiral nanoparticles leads to many novel optical force effects, such as attractive/repulsive force [39,40], lateral optical force [41–43], azimuthal/longitudinal optical torque [44,45], which are capable of sorting chiral nanoparticles. In this work, we firstly demonstrate an effective method to generate dual longitudinal polarization vortex (LPV) structure in the transverse plane of the focal region. By sculpting the spatial distribution of a light field at the pupil plane of a high NA lens, the focal field would be split into two focal spots with arbitrary OAM and opposite SAMs. Then this design principle is extended to the creation and manipulation of multiple pairs of dual LPV structures with prescribed locations in 3D space. Furthermore, the feasibility of sorting enantiomers with the use of LPV structure has been validated. The numerical calculation results show that the chiral nanoparticles would be separated by their handedness, while the dynamic behavior of each type of enantiomers can be adjusted individually, providing a feasible route toward all-optical enantiopure chemical syntheses and enantiomers separation in pharmaceuticals.

2. Generation of multiple pairs of dual foci with arbitrary LPV structure in 3D space

Considering a structured optical field with inhomogeneous distribution of states of polarization (SoP), its local electric field can be expressed in Cartesian coordinate system as:

(1)$${{\boldsymbol E}_0}\textrm{(}x\textrm{,}y\textrm{)} = {A_0}\textrm{(}x\textrm{,}y\textrm{)}[\cos {\xi _0}\textrm{(}x\textrm{,}y\textrm{)}{{\boldsymbol e}_x} + {e^{ - i\Delta \theta \textrm{(}x\textrm{,}y\textrm{)}}}\sin {\xi _0}\textrm{(}x\textrm{,}y\textrm{)}{{\boldsymbol e}_y}] = {A_0}[{e^{i{\xi _0}\textrm{(}x\textrm{,}y\textrm{)}}}{{\boldsymbol e}_1} + {e^{ - i{\xi _0}\textrm{(}x\textrm{,}y\textrm{)}}}{{\boldsymbol e}_2}]/2,$$

where A₀ represents the amplitude and ξ₀ denotes the direction of the polarization orientation, while Δθ indicates the phase difference between x and y components of the electric field. Furthermore, a pair of orthogonal base vector (e₁, e₂) can be applied to analyze the polarization structure of the optical field with e₁ = e_x− eⁱ^{(π/2−Δθ)} e_y and e₂ = e_x + eⁱ^{(π/2−Δθ)} e_y. From Eq. (1), one can easily find that a complex optical field can be decomposed into two orthogonal polarizations with opposite phase terms, which are also related to the spatial coordinate. For example, a locally linear polarization (Δθ = 0) is a combination of both the RH and LH circular polarizations. In addition, it is known that the spatial phase difference of ξ₀(x, y) between e₁ and e₂ would lead to the splitting of the optical field under the tight focusing condition, and two axial separated focal spots would be obtained in the focal volume with orthogonal polarization states. Here, we will study the effects of spatial phase distribution of the illumination on the focal field splitting phenomenon and demonstrate the feasibility of creating multiple pairs of dual foci with controllable AM in 3D space.

Figure 1 illustrates the conceptual scheme of generation of multiple focal spots with arbitrary LPV structure. The incident light is firstly modulated by a phase mask then focused by an objective lens with NA of 0.95. Without loss of generality, we consider a generalized linearly polarized Laguerre-Gaussian (LG) vortex beam with λ = 532 nm and field function l₀(θ) = (2^1/2sinθ/sinα)^|l|exp(−sin²θ/sin²α)exp^ilϕ, where l denotes the topological charge (TC), α is the maximum focusing angle determined by the NA of lens, and ϕ is the azimuthal angle. According to the Richards-Wolf vector diffraction theory, the electric field in the vicinity of the focus of the lens can be calculated by [46,47]:

(2)$${\boldsymbol E}(x,y,z) ={-} \frac{{ikf}}{{2\pi }}\int\limits_0^\alpha {\int\limits_0^{2\pi } {{{\boldsymbol E}_1}} (\theta ,\phi ){e^{ik({\boldsymbol s}\cdot {\boldsymbol r})}}} \sin \theta d\phi d\theta .$$

Here k = n₁k₀ with k₀ = 2π/λ being the wave vector in free space and n₁ being the refractive index of the surrounding medium, f is the focal distance, s is the unit vector along the propagation direction of the ray in image space and r is the position vector from the focal spot to the observation point. Usually, the incident light would be focused by the lens with focal point O(0, 0, 0). However, if we assume that the focal field is split into two separated foci centered at O'(Δx, 0, 0) and O''(−Δx, 0, 0), it has:

(3)$${\boldsymbol s} \cdot ({\boldsymbol r} - {{\boldsymbol r}_{o{o^{\prime}}}}) = {\boldsymbol s} \cdot {\boldsymbol r} - {\boldsymbol s} \cdot {{\boldsymbol r}_{o{o^{\prime}}}} ={-} \rho \sin \theta \cos (\phi - \varphi ) + z\cos \theta \mp \Delta x\sin \theta \cos \phi .$$

Fig. 1. Schematic diagram of setup for creating dual LPV structure. The spatial phase distribution of an incident LG mode light is modulated by a phase mask in the pupil plane. Under the tight focusing condition, components of the RH and LH vibrations of the illumination are focused into separated focal spots along x axis, enabling the arbitrary TC for the longitudinal field of each focal spot.

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Consequently, the required spatial phase modulation applied in the pupil plane can be expressed as:

(4)$$\xi (\theta ,\phi ) ={-} \Delta x\sin \theta \cos \phi + m\phi ,$$

where θ is the focusing angle, mϕ is the introduced Pancharatnam-Berry (PB) phase [48,49]. In this case, apodized field E₁ after refraction can be expressed as:

(5)$${{\boldsymbol E}_1}(\theta ,\phi ) = {l_0}(\theta )[{e^{i{\kern 1pt} \xi (\theta ,{\kern 1pt} \phi )}}{{\boldsymbol e}_r} + {e^{ - i{\kern 1pt} \xi (\theta ,{\kern 1pt} \phi )}}{{\boldsymbol e}_l}].$$

where e_r and e_l are the RH and LH circular polarization basis vector, respectively. It can be seen that the refracted light field can be decomposed into the RH and LH circularly polarized components with opposite phase modulation functions, leading to the creation of dual foci with different spins and locations that are symmetrical with respect to the origin. In this case, the focal field given in Eq. (2) can be rewritten as:

(6)$$\begin{aligned}{{\boldsymbol E}_x} &={-} \frac{{ikf}}{{2\pi }}\int\limits_0^\alpha {\int\limits_0^{2\pi } {{l_0}} (\theta )\sin \theta \sqrt {\cos \theta } {e^{ik[ - \rho \sin \theta \cos (\phi - \varphi ) + z\cos \theta ]}}} \\ & \quad \times [\sin (\phi - \varPsi )\sin \phi + \cos (\phi - \varPsi )\cos \theta \cos \phi ]d\phi d\theta ,\end{aligned}$$

(7)$$\begin{aligned}{{\boldsymbol E}_y} &={-} \frac{{ikf}}{{2\pi }}\int\limits_0^\alpha {\int\limits_0^{2\pi } {{l_0}} (\theta )\sin \theta \sqrt {\cos \theta } {e^{ik[ - \rho \sin \theta \cos (\phi - \varphi ) + z\cos \theta ]}}} \\ & \quad \times [ - \sin (\phi - \varPsi )\cos \phi + \cos (\phi - \varPsi )\cos \theta \sin \phi ]d\phi d\theta ,\end{aligned}$$

(8)$${{\boldsymbol E}_z} ={-} \frac{{ikf}}{{2\pi }}\int\limits_0^\alpha {\int\limits_0^{2\pi } {{l_0}} (\theta ){{\sin }^2}\theta \sqrt {\cos \theta } {e^{ik[ - \rho \sin \theta \cos (\phi - \varphi ) + z\cos \theta ]}}} \cos (\phi - \varPsi )d\phi d\theta ,$$

where Ψ = −kΔxsinθ cosϕ+mϕ. By using identity involving Bessel function, Eq. (8) can be simplified to:

(9)$$\begin{array}{l} {{\boldsymbol E}_z} ={-} \frac{{ikf}}{{2\pi }}\int\limits_0^\alpha {\sqrt {\cos \theta } {{\left( {\frac{{\sqrt 2 \beta \sin \theta }}{{\sin \alpha }}} \right)}^{|l|}}{e^{\left( { - \frac{{{\beta^2}{{\sin }^2}\theta }}{{{{\sin }^2}\alpha }}} \right)}}} [{i^{l + m - 1}}{e^{i(l + m - 1)\varphi }}{J_{l + m - 1}}( - k(\rho \cos \varphi - \Delta x)\sin \theta )\\ \;\;\;\;\;\;\; + {i^{l - m + 1}}{e^{i(l - m + 1)\varphi }}{J_{l - m + 1}}( - k(\rho \cos \varphi - \Delta x)\sin \theta )]{e^{ikz\cos \theta }}{\sin ^2}\theta d\theta , \end{array}$$

where J_n(x) represents the n^th order Bessel function of the first kind. It is clearly shown that the longitudinal electric field E_z has two different phase vortices located at ±Dx with TC of l+m−1 and l − m+1, respectively. By changing the TC of illumination (l) and PB phase applied at the input pupil (m), OAM of the longitudinal component of the two foci can be tuned. In addition, the spacing distance 2Δx between them can be adjusted by tailoring the phase modulation in Eq. (4).

To verify the theoretical prediction, we employ LG beam with l = 1 and set phase modulation according to Eq. (4) with (m, Δx) = (−2, 3λ). Based on the above analysis, it is expecting that two foci with TC l+m–1 = −2 and l − m+1 = 4 in the E_z component at x = −3λ and 3λ, respectively. The required spatial phase distribution modulated in the pupil plane is shown in Fig. 2(a), and the corresponding total, transverse and longitudinal intensity distributions of the focal field are presented in Fig. 2(b)–2(d), showing donut shape with zero amplitude on the optical axis. As shown in Fig. 2(b), the left and right foci are located at r₁ = (−3λ, 0, 0) and r₂ = (3λ, 0, 0) respectively in the focal region. In addition, polarization map superimposed on a zoom-in intensity distribution is also presented as the inset of Fig. 2(b) to demonstrate that the transverse electric field possesses circular polarization with opposite handedness within the main lobe, where blue and orange colors represent the RH and LH, respectively. Consequently, it is demonstrated that left and right focal fields are contributed by the input component of the RH and LH vibration respectively, leading to the different energy flow directions. Specially, the phase distributions of the longitudinal component of the focal field is calculated and presented in the inset of Fig. 2(d). The TC of E_z can be visualized from the phase profile, which are found to be −2 and 4 for left and right focal spots respectively, showing good agreement between theoretical predications and numerical calculations.

Fig. 2. Lateral dual foci created by a tightly focused LG illumination with (l, m, Δx) = (1, −1, 3λ). (a) Spatial phase modulation of the phase mask in the pupil plane. (b) Total, (c) transverse and (d) longitudinal components of the intensity distributions of the focal field. The insets in (b) and (d) illustrate SoP and phase map within the main lobe, respectively.

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Furthermore, this strategy can be easily adapted to the separation of multiple pairs of LPV structure in 3D space. Assuming the dual foci are located at O'(Δx, Δy, Δz) and O''(−Δx, −Δy, −Δz), Eq. (3) would be rewritten as:

(10)$${\boldsymbol s} \cdot ({\boldsymbol r} - {{\boldsymbol r}_{o{o^{\prime}}}}){\kern 1pt} ={-} \rho \sin \theta \cos (\phi - \varphi ) + z\cos \theta - ({\pm} \Delta x\sin \theta \cos \phi \pm \Delta y\sin \theta \sin \phi \pm \Delta z\cos \theta ),$$

which allows us to move the foci freely in the image space. In addition, if n pairs of dual foci are required to be generated simultaneously, the corresponding phase modulation function P(θ, ϕ) is expressed as [22]

(11)$$P(\theta ,\phi ) = \sum\limits_{j = 1}^n {\textrm{exp} [ - ik(\Delta {x_j}\sin \theta \cos \phi + \Delta {y_j}\sin \theta \sin \phi + \Delta {z_j}\cos \theta )]} \textrm{,}$$

Then, the electric field can be obtained from Eq. (6–8) as:

(12)$$\begin{aligned}\left[ {\begin{array}{{c}} {{{\boldsymbol E}_x}}\\ {{{\boldsymbol E}_y}}\\ {{{\boldsymbol E}_z}} \end{array}} \right] &={-} \frac{{ikf}}{{2\pi }}\int\limits_0^\alpha {\int\limits_0^{2\pi } {\sin \theta \sqrt {\cos \theta } {e^{ik[ - \rho \sin \theta \cos (\phi - \varphi ) + z\cos \theta ]}} \sum\limits_{j = 1}^n {{{\left( {\frac{{\sqrt 2 \beta \sin \theta }}{{\sin \alpha }}} \right)}^{|{l_j}|}}{e^{\left( { - \frac{{{\beta^2}{{\sin }^2}\theta }}{{{{\sin }^2}\alpha }}} \right)}}} } } {e^{i{l_j}\phi }}\\ & \quad \times\left[ {\begin{array}{{c}} {{{\boldsymbol V}_{{E_x}_{_j}}}}\\ {{{\boldsymbol V}_{{E_y}_{_j}}}}\\ {{{\boldsymbol V}_{{E_z}_{_j}}}} \end{array}} \right]d\phi d\theta \end{aligned}$$

The electric polarization vectors V_E with three components can be expressed as

(13)$$\left\{ {\begin{array}{{c}} {{{\boldsymbol V}_{{E_x}_{_j}}} = \sin (\phi - {\Psi_j})\sin \phi + \cos (\phi - {\Psi_j})\cos \theta \cos \phi \textrm{,}}\\ {{{\boldsymbol V}_{{E_y}_{_j}}} ={-} \sin (\phi - {\Psi_j})\cos \phi + \cos (\phi - {\Psi_j})\cos \theta \sin \phi ,}\\ {{{\boldsymbol V}_{{E_z}_{_j}}} = \cos (\phi - {\Psi_j})\sin \theta ,} \end{array}} \right.$$

where Ψ_j = −k(Δx_j sinθ cosϕ+Δy_j sinθ sinϕ+Δz_j cosθ)+m_jϕ.

Consequently, the theoretical derivations indicate that when addition degree of freedom (Δx and Δy) are introduced, arbitrary number of paired LPV structure with independent TC can be achieved at arbitrary location in the focal volume. To prove it, we consider a complex input beam with parameters given in Tab. 1, and the corresponding spatial phase modulation is presented in Fig. 3(a). The simulated electric field intensities in the focal plane show six focal spots for the total (Fig. 3(b)) and longitudinal (Fig. 3(d)) fields, in which the locations of each pair of dual foci is symmetrical about the origin. It can be seen that the longitudinal component is quite different from the total field, which exhibit solid or donut-shaped profiles but different sizes. In this case, the six foci should have different TCs l_i −m_i +1 = (3, 2, 4) and l_i + m_i −1 = (−1, −2, 0), which can be found out by the phase map in the inset of Fig. 3(d). As the Stokes image shown in Fig. 3(c), each pair of foci still hold opposite handedness. Moreover, the tunable separation between foci in 3D space is considered. With the spatial phase modulation given in Tab. 2 and Fig. 4(a), the focal field consists of two pairs of foci with TC of (3, −1) and (2, −2) which are located at (±3λ, ±3λ, ±3λ) and (±3λ, ${\mp}$ 3λ, ±3λ), respectively. Consequently, we demonstrate that this simple method is capable to generate and separate multiple LPV structures.

Fig. 3. Three pairs of dual foci created by a tightly focused LG illumination in 2D space. (a) Spatial phase modulation of the phase mask in the pupil plane. (b) Total, (c) stokes parameter S₃ and (d) longitudinal components of the intensity distributions of the focal field. The insets in (d) illustrates the phase distribution of the main lobe. The intensities of several foci have been multiplied by a factor of 3 in order to obtain a clear image.

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Fig. 4. Two pairs of dual foci created by a tightly focused LG illumination in 3D space. (a) Spatial phase modulation of the phase mask in the pupil plane. (b) Total intensity distributions of the focal field.

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Table 1. Parameters of multiple pairs of dual foci in transverse plane

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Table 2. Parameters of multiple pairs of dual foci in 3D space

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3. Optical force and torque under dipole approximation

Next, we would like to discuss the strategy to sort chiral nanoparticles by using these LPV structures. Assuming the size of chiral nanoparticles are much smaller than the wavelength of the illumination, dipole approximation can be applied to study the optical force exerted on the nanoparticle. By introducing chiral nanoparticles into the focal region shown in Fig. 1, its polarizability elements can be expressed as [44,45,50]:

(14)$${\alpha _{ee}} = \frac{{i6\pi {\varepsilon _0}{\varepsilon _m}}}{{{k^3}}}{a_1},{\alpha _{mm}} = \frac{{i6\pi {\mu _m}}}{{{\mu _0}{k^3}}}{b_1},{\alpha _{em}} ={-} \frac{{6\pi {n_1}}}{{{Z_0}{k^3}}}{c_1},$$

where ɛ₀ and μ₀ are the permittivity and permeability in vacuum, ɛ_m and μ_m are the relative permittivity and permeability of the surrounding medium. Z₀ = (m₀/e₀)^1/2 is the wave impedance in free space. The corresponding Mie scattering coefficients (a₁, b₁, c₁) can be expressed in terms of the vector spherical wave functions:

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

with

(16)$$\begin{array}{l} A_n^{(j )} = {Z_S}D_n^{(1 )}({{x_j}} )- D_n^{(1 )}({{x_0}} ),B_n^{(j )} = D_n^{(1 )}({{x_j}} )- {Z_S}D_n^{(1 )}({{x_0}} ),\\ W_n^{(j )} = {Z_S}D_n^{(1 )}({{x_j}} )- D_n^{(3 )}({{x_0}} ),V_n^{(j )} = D_n^{(1 )}({{x_j}} )- {Z_S}D_n^{(3 )}({{x_0}} ),\\ {Q_n} = \frac{{{{{\psi _n}({{x_0}} )} / {{\xi _n}({{x_0}} )}}}}{{V_n^{(1 )}W_n^{(2 )} + V_n^{(2 )}W_n^{(1 )}}}. \end{array}$$

Here x₀ = k₀r, x₁ = k₁r, x₂ = k₂r with r being the nanoparticle radius, ${k_1} = {k_0}(\sqrt {{\varepsilon _r}{\mu _r}} + \kappa )/\sqrt {{\varepsilon _m}{\mu _m}}$ and ${k_2} = {k_0}(\sqrt {{\varepsilon _r}{\mu _r}} - \kappa )/\sqrt {{\varepsilon _m}{\mu _m}} $ being the wave numbers in chiral medium and κ denotes the chirality parameter. Z_s = (m_re_m/e_r)^1/2 is the wave impedance of the nanoparticle. ξ_n(x) and ψ_n(x) are the Riccati-Bessel functions of the first and third kinds, respectively. D_n⁽¹⁾(x) = ψ'_n(x)/ψ_n(x) and D_n⁽³⁾(x) = ξ'_n(x)/ξ_n(x) are the corresponding logarithmic derivatives.

Based on the dipole approximation, the time-averaged optical force < F > exerted on a chiral nanoparticle is given by [45,50]:

(17)$$\left\langle {\boldsymbol F} \right\rangle = {F_{chiral}} + {F_{achiral}} = (F_{grad}^c + {F_{rad}} + {F_{vor}} + {F_{spin}}) + (F_{grad}^a + {F_{curl}} + {F_{flow}}),$$

where the expressions of the force components can be found in Tab. 3. Note that c is the speed of light in vacuum and ω = ck₀ is the angular frequency of light, 〈S〉 = 1/2Re[E × H*] denotes the time-averaged Poynting vector, 〈L_e〉 = Re[ɛ₀ɛ_m/(4iω)E × E*] and 〈L_m〉 = Re[μ₀μ_m/(4iω)H × H*] represent the electric and magnetic parts of the time-averaged SAM densities. C_ext = C_e+C_m is a sum of contribution from the electric and magnetic dipole channel with C_e = kIm[α_ee]/ɛ₀ɛ_m and C_m = kμ₀Im[α_ee]/μ_m, respectively. C_recoil = −k⁴μ₀/(6πɛ₀n₁²) (Re[α_eeα*_mm]+|α_em|²) describes the recoil force related to the asymmetry parameter. F_grad and F_rad are the optical force correspond to the gradient force and the radiation pressure; F_curl is referred to the curl-spin force associated with the curl of the SAM densities [51]; F_vor represents the vortex force determined by the energy flow vortex and nanoparticle chirality [52]; F_spin describe the spin density force, which designate a direct coupling of the nanoparticle to the SAM density of the incident wave; F_flow is due to an alternating flow of the “stored energy” [53].

Table 3. Concrete expressions of the optical force

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Besides optical force, optical torque is another important quantity that directly linked to the movement of the nanoparticle [44]:

(18)$$\begin{array}{l} \left\langle {\boldsymbol T} \right\rangle = \left[ { - 2{\mu_0}Re ({\alpha_{em}}) + \frac{{{\mu_0}{k^3}}}{{3\pi {\varepsilon_0}{\varepsilon_m}}}{\mathop{\rm Im}\nolimits} ({\alpha_{ee}}\alpha_{em}^\ast ) + \frac{{\mu_0^2{k^3}}}{{3\pi }}{\mathop{\rm Im}\nolimits} ({\alpha_{mm}}\alpha_{em}^ \ast )} \right]\left\langle {\boldsymbol S} \right\rangle \\ \;\;\;\;\;\;\;\; + \left[ {\frac{{{\mu_0}{k^3}}}{{6\pi {\varepsilon_0}{\varepsilon_m}}}Re ({\alpha_{ee}}\alpha_{em}^\ast ) - \frac{{\mu_0^2{k^3}}}{{6\pi }}Re ({\alpha_{mm}}\alpha_{em}^ \ast )} \right]{\mathop{\rm Im}\nolimits} ({{\boldsymbol E} \times {{\boldsymbol H}^ \ast }} )\\ \;\;\;\;\;\;\;\; - \left[ {\frac{{2\omega }}{{{\varepsilon_0}{\varepsilon_m}}}{\mathop{\rm Im}\nolimits} ({\alpha_{ee}}) - \frac{{\omega {k^3}}}{{3\pi \varepsilon_0^2\varepsilon_m^2}}{\alpha_{ee}}\alpha_{ee}^\ast{-} \frac{{\omega {\mu_0}{k^3}}}{{3\pi {\varepsilon_0}{\varepsilon_m}}}{\alpha_{em}}\alpha_{em}^\ast } \right]\left\langle {{{\boldsymbol L}_e}} \right\rangle \\ \;\;\;\;\;\;\;\; - \left[ {\frac{{2\omega {\mu_0}}}{{{\mu_m}}}{\mathop{\rm Im}\nolimits} ({\alpha_{mm}}) - \frac{{\omega \mu_0^2{k^3}}}{{3\pi {\mu_m}}}{\alpha_{mm}}\alpha_{mm}^\ast{-} \frac{{\omega {\mu_0}{k^3}}}{{3\pi {\varepsilon_0}{\varepsilon_m}{\mu_m}}}{\alpha_{em}}\alpha_{em}^\ast } \right]\left\langle {{{\boldsymbol L}_m}} \right\rangle . \end{array}$$

With Eq. (17) and Eq. (18), the dynamic behavior of the chiral nanoparticles can be precisely simulated. Considering enantiomers with (r, ɛ_r, κ) = (50 nm, 2.5 + 0.1i, ±0.5 ± 0.01i) immersed in water (n₁ = 1.33) are placed in the focal volume presented in Fig. 2, the distributions of the transverse optical force within the dual longitudinal vortex are indicated by the arrows superimposed in Figs. 5(a) and 6(a). In this case, the input power of the illumination is assumed to be 100 mW. It can be clearly seen that chiral nanoparticles with κ = 0.5 + 0.01i and κ = −0.5 − 0.01i are confined within the main lobe of the right and left focal spots with RH and LH spin, respectively. In order to explore more details about the optical force effect, the line-scans of the components of transverse force through the center of the foci are shown in Figs. 5(b) and 6(b). In this case, the achiral force is dominated by the gradient force term ${\boldsymbol F}_{grad}^a$, which tends to trap nanoparticles at positions that have the largest intensity. As for chiral objects, the contribution from chiral force terms becomes nonnegligible, and the equilibrium position can only be formed when the achiral and chiral forces possess the similar trend. Since the direction of chiral force changes with the handedness of both the enantiomers and the focal field, the chiral nanoparticles would be separated and sorted by their chirality. The trapping stability can be demonstrated by the potential depth U_x as the orange curves in Figs. 5(c) and 6(c), which is estimated as the work done by the optical forces along x axis. It can reach 784.2 k_BT and 1732 k_BT for the equilibrium positions marked by asterisk, 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 larger than k_BT can be considered as stable [23]. It is worthy of noting that from the force distribution one may conclude that chiral nanoparticles with κ = 0.5 + 0.01i and κ = −0.5 − 0.01i seem to be confined at the dark center of the left and right focal spots respectively, however, the surrounding high potential barrier would prevent these nanoparticles from entering the central region. Although the lateral sorting is used as an example, it is worthy of noting that the chiral nanoparticles can be captured and rotated in any position of the three-dimensional plane by using this strategy.

Fig. 5. Chiral detection with the use of dual LPV structure illustrated in Fig. 2. For chiral nanoparticles with κ = 0.5 + 0.01i, (a) Intensity pattern superimposed with the distribution of the optical force, (b) line-scan of the total, chiral and achiral force along x-axis, (c) line-scan of the longitudinal optical torque and potential depth along x-axis, (d) trajectories of the nanoparticles in the transverse plane of the focal field. The initial positions of the nanoparticles are indicated by the coordinates.

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Fig. 6. Chiral detection with the use of dual LPV structure illustrated in Fig. 2. For chiral nanoparticles with κ = −0.5 − 0.01i, (a) Intensity pattern superimposed with the distribution of the optical force, (b) line-scan of the total, chiral and achiral force along x-axis, (c) line-scan of the longitudinal optical torque and potential depth along x-axis, (d) trajectories of the nanoparticles in the transverse plane of the focal field. The initial positions of the nanoparticles are indicated by the coordinates.

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So far, we demonstrate that the chiral nanoparticles can be well restricted along radial direction of the foci. Due to the axial symmetry of the focal spot, the optical force would be constant along the azimuthal direction, giving rise to the longitudinal optical torque (shown in Figs. 5(c) and 6(c)) that drags the nanoparticle rotating around the beam center. Considering the conservation law of AM in this closed physical system, the nanoparticle would inherit the OAM of the LPV structure, therefore its rotation direction is determined by the sign of TC. In this example, chiral nanoparticles with κ = 0.5 + 0.01i and κ = −0.5 − 0.01i would rotate counterclockwise and clockwise around the right and left focal spots, respectively. In order to elaborately describe the dynamic motion of the enantiomers affected by the longitudinal vortex structure, Langevin dynamics (LD) equation is adopted for each chiral nanoparticle [54,55]:

(19)$$m\frac{{{d^2}{\boldsymbol R}}}{{d{t^2}}} = {\boldsymbol F}({\boldsymbol R},t) - \chi \frac{{d{\boldsymbol R}}}{{dt}} + {\boldsymbol \eta }\textrm{,}$$

where R is center position of the nanoparticle with mass m, F is the optical force, the friction coefficient χ = 6πvr is given by Stokes’ law with v = 0.89 mPa·s being the dynamic viscosity of water. η is the stochastic Brownian force satisfying a Gaussian probability distribution <η_i²> = 2k_BTχ, where i is the coordinate index.

To solve the LD equation, BAOAB method is applied due to its excellent computation speed and accuracy [56,57]. Figure 5(d) shows the trajectory of several chiral nanoparticles with κ = 0.5 + 0.01i and initial position randomly distributed in the focal plane. The total duration of the simulation is 50 μs (corresponding to 0.5 μs LD time steps for a LD time step of 100). It is clearly seen that the chiral nanoparticle would be bounced off by the left focal spot but be confined and rotating around the right focal spot with RH polarization. Similarly, for chiral nanoparticles with opposite handedness, the left vortex field with LH polarization is capable to sustain a stable trapping and rotating (shown in Fig. 6(d)). Note that the influence of Brownian motion to the trajectory of the nanoparticle is negligible due to the strong optical force, which can also be demonstrated by the calculated potential depth. Consequently, we demonstrate that the significantly different dynamic behaviors of chiral nanoparticles provide a feasible way to easily distinguish the type of enantiomers.

4. Conclusions

In conclusion, we proposed a method to realize dual foci with arbitrary LPV structures in the transverse plane, which is realized by modulating the PB phase of the incident LG beam at the pupil plane of a high NA lens. We also demonstrated that the strategy can be applied to create multiple pairs of dual foci in 3D space, both the location and the TC of each LPV structure can be independently controlled. Furthermore, the induced optical force and torque from the interaction between chiral nanoparticles and the dual LPV structures are calculated using dipole approximation. Numerical results demonstrated that the nanoparticle would experience a transverse chiral optical force with direction determined by the chirality of the material, consequently the enantiomers with different handedness would be trapped and separated by the dual LPV structures with opposite energy flow directions. Moreover, we demonstrated that there is distinct difference in the dynamic behaviors between the RH and LH chiral nanoparticles. It is worthy of noting that the working wavelength of the proposed sorting strategy can be conveniently adjusted. In addition, the high focusing efficiency of the optical trapping system enables to sustain the force stability with relatively low input power. Consequently, the thermal damage can be avoided by tuning the working wavelength and reducing the laser power. The application of this method may open new avenues for all-optical enantiopure chemical syntheses and enantiomers separation in pharmaceuticals.

Funding

National Natural Science Foundation of China (11504049, 11774055, 12074066, 92050202).

Acknowledgment

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

Disclosures

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

Data availability

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

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$⟨ F ⟩$	component	expression
$F_{c h i r a l}$	$F_{g r a d}^{c}$	$- \nabla \frac{1}{2} Im (α_{e m}) Im (B \cdot E^{*})$
	$F_{r a d}^{c}$	$\frac{n_{1}}{c} C_{r e c o i l} ⟨ S ⟩$
	$F_{v o r}$	$μ_{0} R e (α_{e m}) \nabla \times ⟨ S ⟩$
	$F_{s p i n}$	$\begin{array}{l} [2 ω^{2} μ_{0} R e (α_{e m}) - \frac{k^{5}}{3 π ε_{0}^{2} ε_{m}^{2}} Im (α_{e e} α_{e m}^{})] ⟨ L_{e} ⟩ \\ + [2 ω^{2} μ_{0} R e (α_{e m}) - \frac{k^{5} μ_{0}}{3 π ε_{0} ε_{m} μ_{m}} Im (α_{m m} α_{e m}^{})] ⟨ L_{m} ⟩ \end{array}$
$F_{a c h i r a l}$	$F_{g r a d}^{a}$	$\nabla [\frac{1}{4} R e (α_{e e}) \| E \|^{2} + \frac{1}{4} R e (α_{m m}) \| B \|^{2}]$
	$F_{r a d}^{a}$	$\frac{n_{1}}{c} C_{e x t} ⟨ S ⟩$
	$F_{c u r l}$	$\frac{C_{e} c}{n_{1}} \nabla \times ⟨ L_{e} ⟩ + \frac{C_{m} c}{n_{1}} \nabla \times ⟨ L_{m} ⟩$
	$F_{f l o w}$	$\frac{c k^{4} μ_{0}^{2}}{12 π n_{1}} Im (α_{e e} α_{m m}^{}) Im (E \times H^{})$

$⟨ F ⟩$	component	expression
$F_{c h i r a l}$	$F_{g r a d}^{c}$	$- \nabla \frac{1}{2} Im (α_{e m}) Im (B \cdot E^{*})$
	$F_{r a d}^{c}$	$\frac{n_{1}}{c} C_{r e c o i l} ⟨ S ⟩$
	$F_{v o r}$	$μ_{0} R e (α_{e m}) \nabla \times ⟨ S ⟩$
	$F_{s p i n}$	$\begin{array}{l} [2 ω^{2} μ_{0} R e (α_{e m}) - \frac{k^{5}}{3 π ε_{0}^{2} ε_{m}^{2}} Im (α_{e e} α_{e m}^{})] ⟨ L_{e} ⟩ \\ + [2 ω^{2} μ_{0} R e (α_{e m}) - \frac{k^{5} μ_{0}}{3 π ε_{0} ε_{m} μ_{m}} Im (α_{m m} α_{e m}^{})] ⟨ L_{m} ⟩ \end{array}$
$F_{a c h i r a l}$	$F_{g r a d}^{a}$	$\nabla [\frac{1}{4} R e (α_{e e}) \| E \|^{2} + \frac{1}{4} R e (α_{m m}) \| B \|^{2}]$
	$F_{r a d}^{a}$	$\frac{n_{1}}{c} C_{e x t} ⟨ S ⟩$
	$F_{c u r l}$	$\frac{C_{e} c}{n_{1}} \nabla \times ⟨ L_{e} ⟩ + \frac{C_{m} c}{n_{1}} \nabla \times ⟨ L_{m} ⟩$
	$F_{f l o w}$	$\frac{c k^{4} μ_{0}^{2}}{12 π n_{1}} Im (α_{e e} α_{m m}^{}) Im (E \times H^{})$

Synthesis of multiple longitudinal polarization vortex structures and its application in sorting chiral nanoparticles

Abstract

1. Introduction

2. Generation of multiple pairs of dual foci with arbitrary LPV structure in 3D space

3. Optical force and torque under dipole approximation

4. Conclusions

Funding

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (3)

Equations (19)

Optics Express

index (i)	l_i	m_i	location (Δx_i, Δy_i)	TC of the LPV structure	handedness
1	1	−1	5λcos(π/6), 5λsin(π/6)	3	LH
2	0	−1	5λcos(3π/6), 5λsin(3π/6)	2	LH
3	2	−1	5λcos(5π/6), 5λsin(5π/6)	4	LH
4	1	−1	−5λcos(π/6), −5λsin(π/6)	−1	RH
5	0	−1	−5λcos(3π/6), −5λsin(3π/6)	−2	RH
6	2	−1	−5λcos(5π/6), −5λsin(5π/6)	0	RH

index (i)	l_i	m_i	location (Δx_i, Δy_i, Δz_i)	TC of the LPV structure	handedness
1	1	−1	(3λ, 3λ, 3λ)	3	LH
2	0	−1	(3λ, −3λ, 3λ)	2	LH
3	1	−1	(−3λ, −3λ, −3λ)	−1	RH
4	0	−1	(−3λ, 3λ, −3λ)	−2	RH