Enhanced transverse optical gradient force on Rayleigh particles in two plane waves

Yusen Zhang; Ziheng Xiu; Xiangsuo Fan; Ruxue Li; Huajin Chen; Huajin Chen; Huajin Chen; Huajin Chen; Hongxia Zheng; Hongxia Zheng; Hongxia Zheng; Wanli Lu; Zhifang Lin; Zhifang Lin

doi:10.1364/OE.448458

1. Introduction

Optical forces due to momentum exchange between light and matter are widely used for handling and manipulating microparticles in scientific fields such as physics, chemistry, biology, and medicine [1–14]. Generally, optical forces can be decomposed into gradient and scattering forces (also known as conservative and nonconservative forces). These two forces play quite different roles in optical manipulations [15–24]. The optical gradient force is typically applied to trap particles, known as optical tweezers which are first proposed by Ashkin and coworkers [2]. The scattering force tends to transport particles through pushing particles along the direction of light propagation [25,26] or pulling the particles towards the light source [27,28], and thus it is unfavorable to the optical trapping. To realize the maximum possible efficiency of optical trapping, researchers have tried to reduce the scattering force and enhance the gradient force via shaping the light beam based on various structures and materials [5,6,29,30]. Plasmonic structures such as a bowtie aperture [31], split ring resonators [32,33], plasmonic substrates [34–36] and double-disk metastructure [37], are usually used to tailor local optical fields that can give rise to an enhanced optical gradient force. It was found that the gradient force can be enhanced by about two orders of magnitude in the waveguides composed of hyperbolic metamaterials [38], double negative or single negative refractive index metamaterials [39], compared with the traditional waveguides made of silicon materials. The optical gradient force in paired graphene nanoribbons can also be remarkably enhanced compared to the hybrid plasmon waveguide [40,41]. It should be noted that the enhancement of these optical gradient forces depends on the strong local optical field generated by various structures. However, there has been little work focusing on customizing the trapped object [42–44]. In addition, most of these optical gradient forces mentioned above are still limited to the dipole approximation in tracing the physical orgins [32–36]. It is commonly believed that analytical expression for optical forces based on the dipole approximation is valid when the particle size is much smaller than the illuminating wavelength. But it may be unavailable for some special cases where the higher order multipoles are excited in the particle and significant contribution to the optical forces, even in the Rayleigh regime.

In this paper, we will numerically and analytically show a mechanism that can significantly enhance the transverse optical gradient force acting on a Rayleigh particle through exciting the electric quadrupole on the particle immersed in a simple optical field. The illuminating field is simply composed of two plane waves with linear polarization. The optical gradient force exerted on a traditional dielectric particle can be enhanced by two orders of magnitude by coating an extremely thin silver shell. It is uncovered that the enhanced gradient force is dominated by the contribution from the electric quadrupole excited in the dielectric-silver core-shell particle. For our case in the Rayleigh regime, it is noted that the calculation of optical forces based on the dipole approximation is not available and the optical force from the higher-order multipoles should be taken account into. Besides, the optical potential energy and the optical trapping stiffness for the dielectric-silver core-shell particle can be enlarged two orders of magnitude compared with the dielectric particle. These results enrich the understanding of light-matter interaction and may find applications in optical trapping.

2. Numerical and analytical methods

In this section, we briefly recapitulate both the numerical and analytical methods for calculating optical forces exerted on a particle when the monochramatic optical field illuminate on it. The numerical method based on the full wave simulation (FWS) gives us the exact results to analyze the phenomena while the analytical method based on the multipole expansion theory help us to trace the physics.

2.1 Numerical method: optical forces based on the full wave simulation

Based on the generalized Lorenz-Mie theory [45], the incident optical field can be expanded in terms of vector spherical wave functions $\mathbf {M}_{m,n}^{(1)}(k,\mathbf {r})$ and $\mathbf {N}_{m,n}^{(1)}(k,\mathbf {r})$, which is given explicitly by [46]

(1)$$\begin{aligned} \mathbf{E}_\text{inc} & ={-}E_0\displaystyle\sum_{n,m} iC_{m,n} \Bigl[\,p_{m,n} \mathbf{N}_{m,n}^{(1)}(k,\mathbf{r}) + q_{m,n} \mathbf{M}_{m,n}^{(1)}(k,\mathbf{r}) \,\Bigr], \\ \mathbf{H}_\text{inc} & ={-}\dfrac{1}{Z_0}E_0\displaystyle\sum_{n,m} C_{m,n} \Bigl[\,q_{m,n} \mathbf{N}_{m,n}^{(1)}(k,\mathbf{r}) + p_{m,n} \mathbf{M}_{m,n}^{(1)}(k,\mathbf{r}) \,\Bigr], \end{aligned}$$

where $p_{m,n}$ and $q_{m,n}$ are known as the partial wave expansion coefficients, $Z_0$ is the wave impedance of the background medium, $E_0>0$ characterizes the amplitude of the optical field, $k$ is the wave vector in the background medium, and

(2)$$C_{m,n}=i^{n}\left[\dfrac{(2n+1)}{n(n+1)}\dfrac{(n-m)!}{(n+m)!}\right]^{1/2}.$$

The scattered field can also be expanded in terms of vector spherical wave functions $\mathbf {M}_{m,n}^{(3)}(k,\mathbf {r})$ and $\mathbf {N}_{m,n}^{(3)}(k,\mathbf {r})$, namely,

(3)$$\begin{aligned} \mathbf{E}_\text{sca} & =E_0\displaystyle\sum_{n,m} iC_{m,n} \Bigl[\,a_{m,n} \mathbf{N}_{m,n}^{(3)}(k,\mathbf{r}) + b_{m,n} \mathbf{M}_{m,n}^{(1)}(k,\mathbf{r}) \,\Bigr], \\ \mathbf{H}_\text{sca} & =\dfrac{1}{Z_0}E_0\displaystyle\sum_{n,m} C_{m,n} \Bigl[\,a_{m,n} \mathbf{N}_{m,n}^{(3)}(k,\mathbf{r}) + b_{m,n} \mathbf{M}_{m,n}^{(1)}(k,\mathbf{r}) \,\Bigr]. \end{aligned}$$

The partial wave expansion coefficients of the scattered field $a_{m,n}$ and $b_{m,n}$ can be obtained via the Mie coefficients $a_n$ and $b_n$, namely,

(4)$$a_{mn}=a_{n} p_{mn}, \qquad \qquad b_{mn}=b_{n} q_{mn}.$$

For a core-shell sphere considered in this paper, the Mie coefficients are given by [47]

(5)$$\begin{aligned} a_n&=\dfrac{ \psi_n(y) [\psi'_n(m_s y)-A_n\chi'_n(m_s y)] -m_s \psi'_n(y)[\psi_n(m_s y) -A_n\chi_n(m_s y)]} { \xi_n(y) [\psi'_n(m_s y)-A_n\chi'_n(m_s y)] -m_s \xi'_n(y) [\psi_n(m_s y) -A_n\chi_n(m_s y)]}, \\ b_n&=\dfrac{m_s \psi_n(y) [\psi'_n(m_s y)-B_n\chi'_n(m_s y)] - \psi'_n(y)[\psi_n(m_s y) -B_n\chi_n(m_s y)]} {m_s \xi_n(y) [\psi'_n(m_s y)-B_n\chi'_n(m_s y)] - \xi'_n(y) [\psi_n(m_s y) -B_n\chi_n(m_s y)]}, \end{aligned}$$

with

\begin{aligned} A_n&=\dfrac{m_s \psi_n(m_s x) \psi'_n(m_c x) - m_c\psi'_n(m_s x) \psi_n(m_c x)} {m_s \chi_n(m_s x) \psi'_n(m_c x) - m_c\chi'_n(m_s x) \psi_n(m_c x)}, \\ B_n&=\dfrac{m_s \psi_n(m_c x) \psi'_n(m_s x) - m_c\psi_n(m_s x) \psi'_n(m_c x)} {m_s \psi_n(m_c x) \chi'_n(m_s x) - m_c\chi_n(m_s x) \psi'_n(m_c x)}, \end{aligned}

where $x=kR_{c}$ and $y = kR_{s}$ with $R_{c}$ and $R_{s}$ being the radius of the core and the whole sphere, respectively. $m_{c}$, $m_{s}$ are the refractive indices of the core and shell relative to the surrounding medium. $\psi _{n}(z)$, $\xi _{n}(z)$, $\chi _{n}(z)$ are the Ricatti-Bessel functions given by

(6)$$\begin{array}{c} \psi_{n}(z)=z j_{n}(z), \quad \xi_{n}(z)={-}z h_{n}^{(1)}(z), \quad \chi_{n}(z)=z y_{n}(z), \end{array}$$

in which $j_{n}(z)$, $h_{n}(z)$, and $y_{n}(z)$ are, respectively, the spherical Bessel functions, Hankel function of the first kind, and Neumann function.

Based on the Maxwell stress tensor method, the time-averaged optical force on a sphere can be exactly calculated by the integral over a closed surface of the particle and reads [48,49]

(7)$$\langle\mathbf{F}\rangle=\oint_S\,\, \mathbf{\hat{r}}\cdot \overset{{\boldsymbol{\leftrightarrow}}}{\langle\mathbf{T}\rangle} \,\mbox{d} S,$$

where $\hat {\mathbf {r}}$ is related to the outward unit normal vector on the closed surface $S$, and the time-averaged Maxwell’s stress tensor reads [48,49],

(8)$$\begin{aligned} \overset{{\boldsymbol{\leftrightarrow}}}{\langle\mathbf{T}\rangle}=\frac{1}{2} \text{Re}\bigl[\varepsilon_0 \mathbf{E} \mathbf{E}^{{\ast}}+\mu_0 \mathbf{H} \mathbf{H}^{{\ast}}-\frac{1}{2}(\varepsilon_0 \mathbf{E}\cdot\mathbf{E}^{{\ast}}+\mu_0 \mathbf{H}\cdot\mathbf{H}^{{\ast}}) \overset{{\boldsymbol{\leftrightarrow}}}{\mathbf{I}} \bigr]. \end{aligned}$$

In Eq. (8), $\varepsilon _{0}$ ($\mu _{0}$) is the permittivity (permeability) in vacuum and $\stackrel {\leftrightarrow }{\mathbf {I}}$ is the unit tensor and the superscript ${*}$ represents the complex conjugate. $\mathbf {E}$ and $\mathbf {H}$ are the total field after the particle is positioned, viz. $\mathbf {E}=\mathbf {E}_{\text {inc}}+\mathbf {E}_{\text {sca}}$ and $\mathbf {H}=\mathbf {H}_{\text {inc}}+\mathbf {H}_{\text {sca}}$. Substituting Eqs. (1) and (3) into Eq. (8), one can derive the three components of the optical force $F_{x}$, $F_{y}$, and $F_{z}$ in the lossless background medium [50–52]

(9)$$\begin{aligned} F_{x}=\mbox{Re}\, \bigl[ { F}_{1} \bigr], \;\; F_{y}=\mbox{Im}\, \bigl[ { F}_{1} \bigr], \;\; F_{z}=\mbox{Re}\, \bigl[ { F}_{2} \bigr]. \end{aligned}$$

In Eq. (9), $F_{1}$ and $F_{2}$ are given by

(10)$$\begin{aligned} {F}_1&=\dfrac{2 \pi \varepsilon_0 {E}_{0}^{2}} {k^{2}}\,\displaystyle\sum_{n,m}\,\bigl[c_{11}F_{1}^{(1)}-c_{12}F_{1}^{(2)}+c_{13}F_{1}^{(3)}\bigr], \\ {F}_2&={-}\dfrac{4 \pi \varepsilon_0 {E}_{0}^{2}} {k^{2}}\,\displaystyle\sum_{n,m}\,\bigl[c_{21}F_{2}^{(1)}+c_{22}F_{2}^{(2)}\bigr], \end{aligned}$$

and

\begin{aligned} F_{1}^{(1)} &= \tilde{a}_{m n}\tilde{b}_{m_1 n}^{*}+\tilde{b}_{m n}\tilde{a}_{m_1 n}^{*}-\tilde{p}_{m n}\tilde{q}_{m_1 n}^{*}-\tilde{q}_{mn}\tilde{p}_{m_1 n}^{*}, \\ F_{1}^{(2)} &= \tilde{a}_{m n}\tilde{a}_{m_1 n_1}^{*}+\tilde{b}_{m n}\tilde{b}_{m_1n_1}^{*}-\tilde{p}_{mn}\tilde{p}_{m_1 n_1}^{*}-\tilde{q}_{m n}\tilde{q}_{m_1 n_1}^{*}, \\ F_{1}^{(3)} &= \tilde{a}_{m n_1}\tilde{a}_{m_1 n}^{*}+\tilde{b}_{m n_1}\tilde{b}_{m_1 n}^{*}-\tilde{p}_{m n_1}\tilde{p}_{m_1 n}^{*}-\tilde{q}_{mn_1}\tilde{q}_{m_1n}^{*}, \\ F_{2}^{(1)} &= \tilde{a}_{m n}\tilde{a}_{m n_1}^{*}+\tilde{b}_{m n}\tilde{b}_{m n_1}^{*}-\tilde{p}_{mn}\tilde{p}_{mn_1}^{*}-\tilde{q}_{m n}\tilde{q}_{m n_1}^{*}, \\ F_{2}^{(2)}&= \tilde{a}_{m n}\tilde{b}_{m n}^{*}-\tilde{p}_{m n}\tilde{q}_{m n}^{*}, \end{aligned}

where the coefficients are

\begin{aligned} c_{11} & =\left[\dfrac{(n-m)(n+m+1)}{n^{2}(n+1)^{2}}\right]^{1/2},\\ c_{12} & = \left[\dfrac{n(n+2)(n+m+1)(n+m+2)}{(n+1)^{2}(2n+1)(2n+3)}\right]^{1/2},\\ c_{13} & = \left[\dfrac{n(n+2)(n-m)(n-m+1)}{(n+1)^{2}(2n+1)(2n+3)}\right]^{1/2},\\ c_{21} & = \left[\dfrac{n(n+2)(n-m+1)(n+m+1)}{(n+1)^{2}(2n+1)(2n+3)}\right]^{1/2},\\ c_{22} & = \dfrac{m}{n(n+1)}, \end{aligned}

with $m_{1} = m + 1, n_{1} = n + 1$, and the coefficients are

(11)$$\begin{aligned} \tilde a_{mn}&=a_{mn}-\frac{1}{2}\, p_{mn}, & & \tilde p_{mn}=\frac{1}{2}\, p_{mn}, \\ \tilde b_{mn}&=b_{mn}-\frac{1}{2}\, q_{mn}, & & \tilde q_{mn}=\frac{1}{2}\, q_{mn},\\ a_{mn}&=a_{n} p_{mn}, & & b_{mn}=b_{n} q_{mn}. \end{aligned}$$

2.2 Analytical method: optical forces based on the multipole expansion theory

Based on the multipole expansion technique, the optical forces given in Eq. (7) can also be decomposed into the interception and the recoil forces [53]

(12)$$\langle\mspace{1mu}{\mathbf{F}_\text{{ME}}}\mspace{1mu}\rangle=\displaystyle\sum_{n=1}^{\infty} \langle\mspace{1mu}{\mathbf{F}^{(n)}_{\text{int}}}\mspace{1mu}\rangle+\displaystyle\sum_{n=1}^{\infty} \langle\mspace{1mu}{\mathbf{F}^{(n)}_{\text{rec}}}\mspace{1mu}\rangle,$$

where the interception force reads

(13)$$\begin{aligned} &\langle\mspace{1mu}{\mathbf{F}^{(n)}_{\text{int}}}\mspace{1mu}\rangle=\langle\mspace{1mu}{\mathbf{F}^{\text{e}\,(n)}_{\text{int}}}\mspace{1mu}\rangle+\langle\mspace{1mu}{\mathbf{F}^{\text{m}\,(n)}_{\text{int}}}\mspace{1mu}\rangle, \\ &\langle\mspace{1mu}{\mathbf{F}^{\text{e}\,(n)}_{\text{int}}}\mspace{1mu}\rangle=\frac{1}{2 n !} \textrm{Re}\left[\left(\nabla^{(n)} {\boldsymbol{E}}^{*}\right) \stackrel{(n)}{\text{ .. }} \ \stackrel{\leftrightarrow}{\mathbb{O}}{_{\operatorname{elec}}^{(n)}}\right], \\ &\langle\mspace{1mu}{\mathbf{F}^{\text{m}\,(n)}_{\text{int}}}\mspace{1mu}\rangle=\frac{1}{2 n !} \textrm{Re}\left[\left(\nabla^{(n)} {\boldsymbol{B}}^{*}\right) \stackrel{(n)}{\text{ .. }} \ \stackrel{\leftrightarrow}{\mathbb{O}}{_{\operatorname{mag}}^{(n)}}\right], \end{aligned}$$

and the recoil force reads

(14)$$\begin{aligned} &\langle\mspace{1mu}{\mathbf{F}^{(n)}_{\text{rec}}}\mspace{1mu}\rangle=\langle\mspace{1mu}{\mathbf{F}^{\text{e}\,(n)}_{\text{rec}}}\mspace{1mu}\rangle+\langle\mspace{1mu}{\mathbf{F}^{\text{m}\,(n)}_{\text{rec}}}\mspace{1mu}\rangle+\langle\mspace{1mu}{\mathbf{F}^{\text{x}\,(n)}_{\text{rec}}}\mspace{1mu}\rangle, \\ &\langle\mspace{1mu}{\mathbf{F}^{\text{e}\,(n)}_{\text{rec}}}\mspace{1mu}\rangle={-}\frac{1}{4 \pi \varepsilon_{0}} \frac{(n+2) k^{2 n+3}}{(n+1) !(2 n+3) ! !} \operatorname{Im}\left[\stackrel{\leftrightarrow}{\mathbb{O}}{_{\text{elec }}^{(n) *}} \stackrel{(n)}{\text{ .. }} \ \stackrel{\leftrightarrow}{\mathbb{O}}{_{\text{elec }}^{(n+1)}}\right], \\ &\langle\mspace{1mu}{\mathbf{F}^{\text{m}\,(n)}_{\text{rec}}}\mspace{1mu}\rangle={-}\frac{\mu_{0}}{4 \pi} \frac{(n+2) k^{2 n+3}}{(n+1) !(2 n+3) ! !} \operatorname{Im}\left[\stackrel{\leftrightarrow}{\mathbb{O}}{_{\operatorname{mag}}^{(n) *}} \stackrel{(n)}{\text{ .. }} \ \stackrel{\leftrightarrow}{\mathbb{O}}{_{\operatorname{mag}}^{(n+1)}}\right], \\ &\langle\mspace{1mu}{\mathbf{F}^{\text{x}\,(n)}_{\text{rec}}}\mspace{1mu}\rangle=\frac{Z_{0}}{4 \pi} \frac{k^{2 n+2}}{n n !(2 n+1) ! !} \textrm{Re}\left[\stackrel{\leftrightarrow}{\mathbb{O}}{_{\text{elec }}^{(n)}} \stackrel{(n-1)}{\text{ .. }} \ \stackrel{\leftrightarrow}{\mathbb{O}}{_{\operatorname{mag}}^{(n) *}}\right] \stackrel{(2)}{. .} \ \stackrel{\leftrightarrow}{\epsilon}. \end{aligned}$$

In Eqs. (13) and (14), $\langle \mspace {1mu}{\mathbf {F}^{(n)}_{\text {int}}}\mspace {1mu}\rangle$ and $\langle \mspace {1mu}{\mathbf {F}^{(n)}_{\text {rec}}}\mspace {1mu}\rangle$ denote, respectively, the interception force and recoil force of $2^{n}$-pole. Physically, they can be understood as originating from the two different process: 1, light is intercepted by the particle and then an interception force is induced; 2, light is re-emitted by the coupling between multipoles induced in the particle and then a recoil force is exerted on the particle. ${{\stackrel{\text{\tiny$\boldsymbol{\leftrightarrow}$}}{\boldsymbol{\mathbb{\epsilon}}}}}$ is the Levi-Civita tensor, and the superscripts “e”, “m” and “x” represent, respectively, the contribution arising from the electric multipoles, magnetic multipoles and hybrid term. The electric and magnetic multipoles $2^{n}$-pole of order $n$ are described by rank-$n$ tensors ${\stackrel{\text{\scriptsize$\boldsymbol{\leftrightarrow}$}}{\boldsymbol{\mathbb{O}}}_{\text{elec}}^{\text{\raisebox{-1.8ex}{${(n)}$}}}}$ and ${\stackrel{\text{\scriptsize$\boldsymbol{\leftrightarrow}$}}{\boldsymbol{\mathbb{O}}}_{\text{mag}}^{\text{\raisebox{-1.8ex}{${(n)}$}}}}$, whose mathematical definition can be found in Ref. [53] for details. Considering the lower order cases with $n = 1$ and 2 corresponding to the dipole and quadrupole, the interception force can be written as

(15)$$\mathbf{F}_\text{int}=\mathbf{F}_{p}+\mathbf{F}_{m}+\mathbf{F}_{Q^{e}}+\mathbf{F}_{Q^{m}},$$

with

(16)$$\begin{aligned} &\mathbf{F}_{p} =\frac{1}{2} \textrm{Re}\left[\left(\nabla \mathbf{E}_\text{inc}^{*}\right) \cdot \mathbf{p}\right], & & \mathbf{F}_{m} =\frac{1}{2} \textrm{Re}\left[\left(\nabla \mathbf{B}_\text{inc}^{*}\right) \cdot \mathbf{m}\right], \\ &\mathbf{F}_{Q^{e}} =\frac{1}{4} \textrm{Re}\left[\left(\nabla \nabla \mathbf{E}_\text{inc}^{*}\right): \stackrel{\leftrightarrow}{\boldsymbol{Q}}^{e}\right], & & \mathbf{F}_{Q^{m}} =\frac{1}{4} \textrm{Re}\left[\left(\nabla \nabla \mathbf{B}_\text{inc}^{*}\right): \stackrel{\leftrightarrow}{\boldsymbol{Q}}^{m}\right], \end{aligned}$$

and the recoil force is given by

(17)$$\mathbf{F}_\text{rec}=\mathbf{F}_{p m}+\mathbf{F}_{Q^{e} p}+\mathbf{F}_{Q^{m} m}+\mathbf{F}_{Q^{e} Q^{m}},$$

with

(18)$$\begin{aligned} & \mathbf{F}_{p m} ={-}\frac{k^{4}}{12 \pi \varepsilon_{0} c_{0}} \textrm{Re}\left(\mathbf{p} \times \mathbf{m}^{*}\right),\\ & \mathbf{F}_{Q^{e} p} ={-}\frac{k^{5}}{40 \pi \varepsilon_{0}} \operatorname{Im}\left[\stackrel{\leftrightarrow}{\boldsymbol{Q}}^{e}\cdot \mathbf{p}^{*}\right],\\ & \mathbf{F}_{Q^{m} m} ={-}\frac{k^{5}}{40 \pi \varepsilon_{0} c_{0}^{2}} \operatorname{Im}\left[\stackrel{\leftrightarrow}{\boldsymbol{Q}}^{m} \cdot \mathbf{m}^{*}\right], \\ & \mathbf{F}_{Q^{e} Q^{m}}={-}\frac{k^{6}}{240 \pi c_{0} \varepsilon_{0}} \textrm{Re}\left[\mathbf{Q}_{x}^{e} \times \mathbf{Q}_{x}^{m *}\right.\left.+\mathbf{Q}_{y}^{e} \times \mathbf{Q}_{y}^{m *}+\mathbf{Q}_{z}^{e} \times \mathbf{Q}_{z}^{m *}\right], \end{aligned}$$

where $c_{0}$ denotes the speed of light in vacuum. The electric dipole and quadrupole, together with the magnetic dipole and quadrupole, can be described by

(19)$$\begin{aligned} &\mathbf{p}=\alpha_{e} \mathbf{E}, & & \stackrel{\leftrightarrow}{{\mathbf{Q}}}^{{e}}=\frac{\beta_{e}}{2}\left(\nabla \mathbf{E}+\nabla \mathbf{E}^{\mathrm{T}}\right), \\ &\mathbf{m}=\alpha_{m} \mathbf{B}, & & \stackrel{\leftrightarrow}{\mathbf{Q}}^{m}=\frac{\beta_{m}}{2}\left(\nabla \mathbf{B}+\nabla \mathbf{B}^{T}\right), \end{aligned}$$

with the polarizabilities [17]

(20)$$\begin{array}{l} \alpha_{e}=\frac{i 6 \pi \varepsilon_{0}}{k^{3}} a_{1}, \quad \quad \alpha_{m}=\frac{i 6 \pi}{\mu_{0} k^{3}} b_{1},\quad \quad \beta_{e}=\frac{i 40 \pi \varepsilon_{0}}{k^{5}} a_{2}, \quad \quad\beta_{m}=\frac{i 40 \pi}{\mu_{0} k^{5}} b_{2}. \end{array}$$

Other quantities in Eq. (18) are defined as

(21)$$\begin{array}{lll} \mathbf{Q}_{x}^{e}=\mathbf{e}_{x} \cdot \overleftrightarrow{\mathbf{Q}}^{e}, & \mathbf{Q}_{y}^{e}=\mathbf{e}_{y} \cdot \overleftrightarrow{\mathbf{Q}}^{e}, & \mathbf{Q}_{z}^{e}=\mathbf{e}_{z} \cdot \overleftrightarrow{\mathbf{Q}^{e}} \\ \mathbf{Q}_{x}^{m}=\mathbf{e}_{x} \cdot \overleftrightarrow{\mathbf{Q}}^{m},& \mathbf{Q}_{y}^{m}=\mathbf{e}_{y} \cdot \overleftrightarrow{\mathbf{Q}}^{m},& \mathbf{Q}_{z}^{m}=\mathbf{e}_{z} \cdot \overleftrightarrow{\mathbf{Q}}^{m} \end{array} $$

To understand the physics of the optical force, the interception and recoil forces in Eqs. (13) and (14) can be divided into the gradient and scattering parts [17,20,21,23]

(22)$$\langle\mspace{1mu}{\mathbf{F}_\text{int}}\mspace{1mu}\rangle =\left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{grad}}+\left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{curl}}+\left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{grad}}+\left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{curl}},\\$$

with

(23)$$\begin{aligned} & \left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{grad}} =u_{n}^{(1)} \operatorname{Im} \sum_{i, j} a_{n}\left[Q_{n, i j}^{(1)} \mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{gr}}-Q_{n, i j}^{(2)} \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{gr}}\right], \\ & \left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{curl}} =u_{n}^{(1)} \operatorname{Im} \sum_{i, j} a_{n}\left[Q_{n, i j}^{(1)} \mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{cr}}-Q_{n, i j}^{(2)} \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{cr}}\right], \\ & \left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{grad}} =u_{n}^{(1)} \mathrm{Im} \sum_{i, j} b_{n}\left[Q_{n, i j}^{(1)} \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{gr}}-Q_{n, i j}^{(2)} \mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{gr}}\right], \\ & \left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{curl}} =u_{n}^{(1)} \operatorname{Im} \sum_{i, j} b_{n}\left[Q_{n, i j}^{(1)} \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{cr}}-Q_{n, i j}^{(2)} \mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{cr}}\right],\\ \end{aligned}$$

and

(24)$$\begin{aligned} \quad\quad & \langle\mspace{1mu}{\mathbf{F}_\text{rec}}\mspace{1mu}\rangle=\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{grad}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{curl}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{grad}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{curl}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{x}(n)}\right\rangle_{\mathrm{grad}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{x}(n)}\right\rangle_{\mathrm{curl}},\\ \end{aligned}$$

with

\begin{aligned} \quad\quad&\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{grad}}= u_{n}^{(2)} \operatorname{Im} \sum_{i, j} a_{n+1} a_{n}^{*}\left[R_{n, i j}^{(1)}\left(\mathbf{Z}_{\mathrm{ee}, i j}^{(1) g r}\right)^{*}-R_{n, i j}^{(2)}\left(\mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{gr}}\right)^{*}-4 i R_{n, i j}^{(3)}\left(\mathbf{S}_{\mathrm{em}, i j}^{(1) \text{ grad }}\right)^{*}\right.\\ &\quad\quad\quad\quad\quad \left.+R_{n, i j}^{(4)} \mathbf{Z}_{e e, i j}^{(1) \mathrm{gr}}-R_{n, i j}^{(5)} \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{gr}}+4 i R_{n, i j}^{(6)} \mathbf{S}_{\mathrm{em}, i j}^{(1)\mathrm{grad}} \right], \\ &\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{curl}}= u_{n}^{(2)} \operatorname{Im} \sum_{i, j} a_{n+1} a_{n}^{*}\left[R_{n, i j}^{(1)}\left(\mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{cr}}\right)^{*}-R_{n, i j}^{(2)}\left(\mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{cr}}\right)^{*}-4 i R_{n, i j}^{(3)}\left(\mathbf{S}_{\mathrm{em}, i j}^{(1) \mathrm{curl}}\right)^{*}\right.\\ &\quad\quad\quad\quad\quad\left.+R_{n, i j}^{(4)} \mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{cr}}-R_{n, i j}^{(5)} \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{cr}}+4 i R_{n, i j}^{(6)} \mathbf{S}_{\mathrm{em}, i j}^{(1) \mathrm{curl}}\right],\\ &\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{grad}}= u_{n}^{(2)} \operatorname{Im} \sum_{i, j} b_{n+1} b_{n}^{*}\left[R_{n, i j}^{(1)}\left(\mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{gr}}\right)^{*}-R_{n, i j}^{(2)}\left(\mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{gr}}\right)^{*}-4 i R_{n, i j}^{(3)} \mathbf{S}_{\mathrm{em}, i j}^{(1) \mathrm{grad}}\right.\\ &\quad\quad\quad\quad\quad\left.+R_{n, i j}^{(4)} \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{gr}}-R_{n, i j}^{(5)} \mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{gr}}+4 i R_{n, i j}^{(6)}\left(\mathbf{S}_{\mathrm{em}, i j}^{(1) \mathrm{grad}}\right)^{*}\right], \\ \end{aligned}

\begin{aligned} &\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{curl}}= u_{n}^{(2)} \mathrm{Im} \sum_{i, j} b_{n+1} b_{n}^{*}\left[R_{n, i j}^{(1)}\left(\mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{cr}}\right)^{*}-R_{n, i j}^{(2)}\left(\mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{cr}}\right)^{*}-4 i R_{n, i j}^{(3)} \mathbf{S}_{\mathrm{em}, i j}^{(1) \mathrm{curl}}\right.\\ &\quad\quad\quad\quad\quad\left.+R_{n, i j}^{(4)} \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{cr}}-R_{n, i j}^{(5)} \mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{cr}}+4 i R_{n, i j}^{(6)}\left(\mathbf{S}_{\mathrm{em}, i j}^{(1) \mathrm{curl}}\right)^{*}\right],\\ &\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{x}(n)}\right\rangle_{\mathrm{grad}}= u_{n}^{(3)} \operatorname{Im} \sum_{i, j} a_{n} b_{n}^{*}\left\{R_{n, i j}^{(4)}\left[\mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{gr}}-\left(\mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{gr}}\right)^{*}\right]+4 i R_{n, i j}^{(6)}\left(\mathbf{S}_{\mathrm{em}, i j}^{(1) \mathrm{grad}}\right)^{*}\right.\\ &\quad\quad\quad\quad\quad\left.+R_{n, i j}^{(5)}\left[\left(\mathbf{Z}_{\mathrm{mm}, i j}^{(1) g \mathrm{r}}\right)^{*}-\mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{gr}}\right]+4 i R_{n, i j}^{(7)} \mathbf{S}_{\mathrm{em}, i j}^{(1)\mathrm{grad}}\right\}, \\ &\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{x}(n)}\right\rangle_{\mathrm{curl}}= u_{n}^{(3)} \operatorname{Im} \sum_{i, j} a_{n} b_{n}^{*}\left\{R_{n, i j}^{(4)}\left[\mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{cr}}-\left(\mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{cr}}\right)^{*}\right]+4 i R_{n, i j}^{(6)}\left(\mathbf{S}_{\mathrm{em}, i j}^{(1) \mathrm{curl}}\right)^{*}\right.\\ &\quad\quad\quad\quad\quad\left.+R_{n, i j}^{(5)}\left[\left(\mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{cr}}\right)^{*}-\mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{cr}}\right]+4 i R_{n, i j}^{(7)} \mathbf{S}_{\mathrm{em}, i j}^{(1) \mathrm{curl}}\right\},\\ \end{aligned}

where the field quantities in Eqs. (23) and (24) are defined as

(25)$$\begin{aligned} & \mathbf{S}_{\mathrm{em}, i j}^{(1) \operatorname{grad}}={-}\frac{i}{2\left(1-x_{i j}\right)} \nabla\left[D_{\mathrm{mm}, i j}^{(1)}-D_{\mathrm{ee}, i j}^{(1)}\right],\quad\quad\quad\quad \mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{gr}}=\frac{1}{2} \nabla D_{\mathrm{ee}, i j}^{(1)},\\ & \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{gr}}=\frac{1}{2} \nabla D_{\mathrm{mm}, i j}^{(1)},\quad\quad \quad\quad \mathbf{S}_{\mathrm{em}, i j}^{(1) \text{ curl }}={-}\frac{1}{2\left(1-x_{i j}\right)} \nabla \times\left[\mathbf{G}_{\mathrm{em}, i j}^{(1)}-\mathbf{G}_{\mathrm{me}, j i}^{(1) *}\right], \\ & \mathbf{Z}_{\mathrm{mm}, i j}^{(1) \mathrm{cr}}={-}\frac{1}{2} \nabla \times \mathbf{S}_{\mathrm{mm}, i j}^{(1)}-i \textrm{Re} \mathbf{S}_{\mathrm{em}, i j}^{(1)}, \quad\quad\mathbf{Z}_{\mathrm{ee}, i j}^{(1) \mathrm{cr}}={-}\frac{1}{2} \nabla \times \mathbf{S}_{\mathrm{ee}, i j}^{(1)}-i \textrm{Re} \mathbf{S}_{\mathrm{em}, i j}^{(1)},\\ \quad \quad & Q_{n, i j}^{(1)}=\sum_{m=1}^{n}{}^{(2)} m(2 n+1-m)(2 n+1-2 m) P_{n-m}\left(x_{i j}\right),\\ & Q_{n, i j}^{(2)}=\sum_{m=2}^{n}{}^{(2)} m(2 n+1-m)(2 n+1-2 m) P_{n-m}\left(x_{i j}\right),\\ & R_{n, i j}^{(1)}=\sum_{m=1}^{n}{}^{(2)}(m+1)(2 n+2-m)(2 n+1-2 m)\left[2(m+1) n-\left(m^{2}-m-4\right)\right] P_{n-m}\left(x_{i j}\right),\\ & R_{n, i j}^{(2)}=\sum_{m=2}^{n}{ }^{(2)} m(m+2)(2 n+1-m)(2 n+1-2 m)(2 n+3-m) P_{n-m}\left(x_{i j}\right),\\ & R_{n, i j}^{(3)}=\sum_{m=1}^{n}{}^{(2)}(m+1)(2 n+2-m)(2 n+1-2 m) P_{n-m}\left(x_{i j}\right),\\ & R_{n, i j}^{(4)}=\sum_{m=2}^{n}{}^{(2)}(2 n+1-m)(2 n+1-2 m)\left[2 m^{2} n-m(m+1)(m-2)\right] P_{n-m}\left(x_{i j}\right),\\ & R_{n, i j}^{(5)}=\sum_{m=1}^{n}{}^{(2)}(m+1)(m-1)(2 n-m)(2 n+2-m)(2 n+1-2 m) P_{n-m}\left(x_{i j}\right),\\ & R_{n, i j}^{(6)}=\sum_{m=2}^{n} {}^{(2)}m(2 n+1-m)(2 n+1-2 m) P_{n-m}\left(x_{i j}\right),\\ & R_{n, i j}^{(7)}=\sum_{m=1}^{n}{}^{(2)}(2 n+1-2 m)\left[2 n^{2}-2(m-1) n+m^{2}-m\right] P_{n-m}\left(x_{i j}\right).\\ \end{aligned}$$

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

(26)$$\begin{aligned} &D_{\mathrm{ee}, i j}^{(1)}=\left(\boldsymbol{\mathcal{E}}_{i} \cdot \boldsymbol{\mathcal{E}}_{j}^{*}\right) e^{{\boldsymbol{i}\left(\boldsymbol{k}_{i}-\boldsymbol{k}_{j}\right)\cdot\boldsymbol{r}}}, & & & D_{\mathrm{mm}, i j}^{(1)}=\left(\boldsymbol{\mathcal{B}}_{i} \cdot \boldsymbol{\mathcal{B}}_{j}^{*}\right) e^{{\boldsymbol{i}\left(\boldsymbol{k}_{i}-\boldsymbol{k}_{j}\right) \cdot \boldsymbol{r}}} \\ &\mathbf{S}_{\mathrm{ee}, i j}^{(1)}=\left(\boldsymbol{\mathcal{E}}_{i} \times \boldsymbol{\mathcal{E}}_{j}^{*}\right) e^{{\boldsymbol{i}\left(\boldsymbol{k}_{i}-\boldsymbol{k}_{j}\right) \cdot \boldsymbol{r}}}, & & & \mathbf{S}_{\mathrm{mm}, i j}^{(1)}=\left(\boldsymbol{\mathcal{B}}_{i} \times \boldsymbol{\mathcal{B}}_{j}^{*}\right) e^{{\boldsymbol{i}\left(\boldsymbol{k}_{i}-\boldsymbol{k}_{j}\right) \cdot \boldsymbol{r}}} \\ &\mathbf{S}_{\mathrm{em}, i j}^{(1)}=\left(\boldsymbol{\mathcal{E}}_{i} \times \boldsymbol{\mathcal{B}}_{j}^{*}\right) e^{{\boldsymbol{i}\left(\boldsymbol{k}_{i}-\boldsymbol{k}_{j}\right) \cdot \boldsymbol{r}}}, & & & \mathbf{G}_{\mathrm{em}, i j}^{(1)}={-}i\left(\boldsymbol{k}_{j} \cdot \boldsymbol{\mathcal{E}}_{i}\right) \boldsymbol{\mathcal{B}}_{j}^{*} e^{{\boldsymbol{i}\left(\boldsymbol{k}_{i}-\boldsymbol{k}_{j}\right) \cdot \boldsymbol{r}}} \\ &\mathbf{G}_{\mathrm{me}, i j}^{(1)}={-}{i}\left(\boldsymbol{k}_{j} \cdot \boldsymbol {\mathcal{B}}_{i}\right) \boldsymbol {\mathcal{E}}_{j}^{*} e^{{\boldsymbol{i}\left(\boldsymbol{k}_{i}-\boldsymbol{k}_{j}\right) \cdot \boldsymbol{r}}}, \end{aligned}$$

$\boldsymbol {\hat {k}}_i$ is the unit vector denoting the direction of the $i$-th wave vector ${\boldsymbol k_i}=k\,\boldsymbol {\hat {k}}_i$, $\mathbf {\mathcal {E}}_i$ and $\mathbf {\mathcal {B}}_i={\boldsymbol k_i }\times \mathbf {\mathcal {E}}_i$ are complex amplitude vectors of the $i$-th plane wave, respectively.

One can finally write the optical force in Eq. (12) in terms of the gradient force and the scattering force

(27)$$\langle\mspace{1mu}{\mathbf{F}_\text{{ME}}}\mspace{1mu}\rangle=\displaystyle\sum_{n=1}^{\infty} \langle\mspace{1mu}{\mathbf{F}^{(n)}_g}\mspace{1mu}\rangle+\displaystyle\sum_{n=1}^{\infty} \langle\mspace{1mu}{\mathbf{F}^{(n)}_c}\mspace{1mu}\rangle,$$

with

(28)$$\begin{aligned} & \langle\mspace{1mu}{\mathbf{F}^{(n)}_g}\mspace{1mu}\rangle =\left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{grad}}+\left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{grad}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{grad}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{grad}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{x}(n)}\right\rangle_{\mathrm{grad}},\\ & \langle\mspace{1mu}{\mathbf{F}^{(n)}_c}\mspace{1mu}\rangle =\left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{curl}}+\left\langle\mathbf{F}_{\mathrm{int}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{curl}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{e}(n)}\right\rangle_{\mathrm{curl}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{m}(n)}\right\rangle_{\mathrm{curl}}+\left\langle\mathbf{F}_{\mathrm{rec}}^{\mathrm{x}(n)}\right\rangle_{\mathrm{curl}}.\\ \end{aligned}$$

3. Results and discussions

To demonstrate the enhanced transverse optical gradient force on the Rayleigh particle, we consider that a dielectric-silver core-shell particle is illuminated by an optical field formed by two linearly polarized plane waves, as illustrated by Fig. 1. The two plane waves have the same wavelength. Their wave vectors all lie in the $xoz$ plane and electric fields are given by [54]

(29)$$\begin{aligned} \mathbf{E}_{1,2}= & \frac{E_0}{\sqrt{1+\left|m_{1,2}\right|^{2}}}\left(\cos \gamma \mathbf{e}_{x}+m_{1,2} \mathbf{e}_{y}\mp \sin \gamma \mathbf{e}_{z}\right) e^{i k(z \cos \gamma \pm x \sin \gamma)} \\ \end{aligned}$$

where the upper and lower signs correspond to indices 1 and 2, and $\gamma$ is the incident angle between the wave vector and the $z$ axis. The wave polarizations are characterized by the complex parameters $m_{1,2}$. $\mathbf {e}_{x}$, $\mathbf {e}_{y}$, and $\mathbf {e}_{z}$ are the unit vectors in the Cartesian coordinate. The time dependence $e^{-i{\omega }t}$ is assumed and suppressed. Here the polarization parameter $m_{1} = m_{2} = 1$, the incident angle $\gamma = 45^{\circ}$, and the incident wavelength $\lambda = 532$ nm are considered throughout the paper. $E_0=8.68\times 10^{5}V/m$ is set such that the irradiance of a single plane wave $I_0=E_0^{2}/(2 Z_0)=1.0 mW/\mu {m}^{2}$. The dielectric-silver core-shell nanoparticle can be synthesized by a simple chemical reduction technique [55]. The relative permittivity of the dielectric core is $\varepsilon _{c} = 2.53$, and the one of silver shell is $\varepsilon _{s} =-11.4486+0.137561i$ considering the incident wavelength $\lambda = 532$ nm. The permittivity of silver can derived by the Drude model [55] $\varepsilon (\omega )=\varepsilon _{\infty }-\omega _{p}^{2} /\left (\omega ^{2}+i \omega \eta \right )$, where $\varepsilon _{\infty }$ is the high-frequency limit dielectric constant, $\omega {_p}$ is the plasma frequency, and $\eta$ is the damping constant.

Fig. 1. Schematic illustration of a dielectric-silver core-shell particle illuminated by an interference field composed of two linearly polarized plane waves in vacuum. The core and shell radii are denoted by $R_c$ and $R_s$, respectively.

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Based on the Eq. (27) within the multipole expansion theory, we calculate the $x$ component of the transverse optical force $F^{x}_\text {{ME}}$ together with its decomposed parts $F^{x}_g$ and $F^{x}_c$ acting on two different spherical nanoparticles versus their position in the $xoz$ plane, as shown in Figs. 2(a)-(c) for the dielectric particle and Figs. 2(d)-(f) for the dielectric-silver core-shell particle. The dielectric particle has the radius $R = 20$ nm and the core-shell one has the core radius $R_{c} = 17.75$ nm and the shell radius $R_{s} =20$ nm, where the thickness of the silver shell 2.25 nm demonstrates the strongest enhancement of transverse optical forces for the particle with radius 20 nm. Here, the order $n=3$ is taken to guarantee the computing accuracy. The calculating results show that the transverse optical forces $F^{x}_\text {{ME}}$ on both dielectric and dielectric-silver core-shell particle are almost entirely determined by the gradient forces, since the scattering forces are much smaller than the gradient forces and thus can be ignored. In addition, one can see that all spatial profiles of the optical forces exhibit a period distribution. This is because the transverse gradient force arising from the gradient of the light intensity and thus determined by the intensity distribution formed by the two interfering plane waves. In particular, one can find that the magnitude of transverse gradient force for the core-shell particle is over 100 times greater than the one of the conventional dielectric particle, as verified by Fig. 2(g). Besides, the transverse gradient force acting on the core-shell particle can also be strengthened by 5 times in magnitude comparing with a homogeneous silver particle with the same radius of $R =20$ nm. As a consequence, the transverse gradient force on a conventional dielectric particle can be remarkably enhanced via coating an ultra-thin silver shell, quite different from those based on the strong optical field induced by various plasmonic structures [31–37].

Fig. 2. The spatial distribution of the $x$ component of the transverse optical force $F^{x}_\text {{ME}}$ as well as its decomposed parts $F^{x}_g$ and $F^{x}_c$ acting on a dielectric nanoparticle (a)-(c) and a dielectric-silver core-shell nanoparticle (d)-(f). The panels (a) and (d), (b) and (e), (c) and (f), respectively, correspond to the total force $F^{x}_\text {{ME}}$, the gradient force $F^{x}_g$, and the scattering force $F^{x}_c$. (g) The ratio of the magnitude of transverse optical force $F^{x}_\text {{ME}}$ of the dielectric-silver core-shell particle to one of the dielectric particle. The dielectric particle has the radius $R = 20$ nm and the core-shell one has the core radius $R_{c} = 17.75$ nm as well as the shell radius $R_{s} =20$ nm. The optical forces are in the unit of $\mathrm {pN} \mathrm {mW}^{-1} \mu \mathrm {m}^{2}$.

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To reveal the physical origin of the enhanced transverse gradient force, we employ the analytical expressions of optical forces given in the Eq. (12) to compute the contribution of each term to the gradient force. The results for the dielectric and core-shell particles are shown in Figs. 3(a) and (b) , where all the parameters are the same as those in Fig. 2 except that the particle is positioned at $(x, 0, 0)$ in the Cartesian coordinate. We can clearly see that the transverse gradient force $F^{x}_\text {ME}$ for the both cases originate overwhelmingly from the interception force $F^{x}_\text {int}$ while the recoil force $F^{x}_\text {rec}$ exhibits a negligible contribution. The transverse optical force $F^{x}_\text {FWS}$ is also calculated with the numerical method based on the FWS for comparison, which is in excellent accordance with the results from the analytical theory, verifying the Eq. (12). It can be further traced to the contribution from each multipole to the transverse optical force $F_x$ via the Eq. (15). Figure 3(c) for the dielectric particle shows that the transverse gradient force comes from the terms of the magnetic dipole $F^{x}_m$ as well as electric quadrupole $F^{x}_{Q^{e}}$, whereas the contribution from the term of the electric dipole $F^{x}_p$ is substantially small. The results imply that in this case the magnetic dipole and electric quadrupole are excited in the dielectric particle. Generally, it is hard to excite the higher-order multipole such as electric quadrupole in the Rayleigh dielectric particle with the permeability $\mu =1$. The interesting phenomenon is also found in the case of the core-shell particle by coating a much thin silver shell on the dielectric particle, as illuminated in Fig. 3(d). One can see that the transverse gradient force for the core-shell particle is mostly dominated by the electric quadrupole while the term of magnetic dipole has a small contribution. The results are much different from the case of the dielectric particle.

Fig. 3. The $x$ component of the transverse optical forces and the relative errors for the dielectric particle (a) (c) (e) and the core-shell particle (b) (d) (f). The transverse optical force together with its decomposed interception and recoil forces are calculated with the Eq. (12) based on the multipole expansion method as shown in (a) and (b), where the transverse optical force $F^{x}_\text {FWS}$ based on the FWS is plotted for comparison. The interception force and its decomposed terms are computed with the Eq. (15) as presented in (c) and (d). The relative errors between the dipole approximation $F^{x}_D$ and the accurate FWS $F^{x}_\text {FWS}$ are given in (e) and (f). All the parameters are the same as those in Fig. 2 except that the particle is fixed at $(x, 0, 0)$ in the Cartesian coordinate.

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In general, when the size of a dielectric particle is significantly smaller than the wavelength of light, the particle can commonly be treated as an electric dipole and thus the optical force on the particle is studied within the dipole approximation. In contrast, both the magnetic dipole and electric quadrupole can be excited in our cases, even the particle is in the Rayleigh regime. As a result, the analytical theory for the optical force is not valid for our cases. An intuitive result can see the relative error between transverse gradient forces obtained from the dipole approximation and the FWS, as displayed in Figs. 3(e) and (f) for the dielectric and core-shell particles. The relative error is defined as

(30)$$\text{Error}_\text{D}=\frac{\left|\mathbf{F}_{D}-\mathbf{F}_\text{FWS}\right|}{\mathbf{F}_\text{FWS}} \times 100 \%,$$

where $\mathbf {F}_{D}=\mathbf {F}_{p}+\mathbf {F}_{m}+\mathbf {F}_{pm}$. The calculating results clearly show that the relative error for the dielectric particle is about $\pm 160\%$ and the one for the core-shell particle is close to $\pm 100\%$, manifesting that the optical force based on the dipole approximation is not applicable any more for the two Rayleigh particles.

As shown in both cases for the dielectric and core-shell particles above, the transverse optical gradient force is overwhelmingly dominant over the scattering force, and hence it can be used to trap particles. The trapping phenomenon can be visualized by the optical potential energy displayed in Figs. 4(a) and (b), corresponding to Figs. 2(a) and (d). The landscapes of the optical potential energy for both cases exhibit a periodic change from pits to humps as the $x$, corresponding to the stable and unstable trapping positions, and thus it can serve as a comprehensive picture for the trapping to detrapping transition. Furthermore, the optical potential energy for the core-shell particle is over 100 times the magnitude outweighing that of the dielectric particle, in agreement with the results in Fig. 2(g). Accordingly, two orders of magnitude enhancement of the optical trap stiffness for the core-shell particle is also achieved compared with one of the dielectric particle, facilitating the realization of stable trapping in experiments. It is noted that both the potential energy as well as the trap stiffness are proportional to the incident irradiance.

Fig. 4. (a) and (b) Optical potential energy $\langle U\rangle$ corresponding to Figs. 2(a) and (d), showing the trapping phenomena. The optical potential energy is in the unit of $k_BT$.

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4. Conclusion

In summary, based on the full wave simulation and the multipole expansion method, we have shown that an enhanced transverse optical gradient force can be induced on a dielectric-silver core-shell Rayleigh particles illuminated by an interfering field composed of two plane waves with linear polarization. The enhanced gradient force is two orders of magnitude stronger than that of the conventional dielectric particle through coating an ultra-thin silver shell on the dielectric particle. It should be noted that such enhanced force can be achieved by a relative long range wavelength from $530~nm$ to $535~nm$ while the incident angle $\gamma$ should keep approach $45^{\circ }$. In particular, the enhanced gradient force is attributed to the contribution from the electric quadrupole that usually can not be excited in conventional Rayleigh. As a consequence, the calculation of optical forces within the dipole approximation is invalid for Rayleigh particles in our case since the higher-order multipole is excited and shows a significant contribution to the optical force. Such enhanced transverse gradient force can be used to trap particles with strengthened trapping stiffiness. Specifically, the optical potential energy as well as the trapping stiffness for the dielectric-silver core-shell particle are both two orders of magnitude compared with the dielectric particle, in favor of realizing a stable trapping.

Funding

National Natural Science Foundation of China (12074084, 12174076, 62001129, 62104052); Natural Science Foundation of Guangxi Province (2021GXNSFBA075029, 2021GXNSFDA196001, 2021JJB170012); Scientific Base and Talent Special Project of Guangxi Province (AD19110095, AD19245130); Open Project of State Key Laboratory of Surface Physics in Fudan University (KF2019_11); Innovation Project of Guangxi Graduate Education (YCSW2021308).

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.

References

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]

2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef]

3. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]

4. A. Ashkin, “Optical trapping and manipulation of neutral particles using lasers,” Proc. Natl. Acad. Sci. U. S. A. 94(10), 4853–4860 (1997). [CrossRef]

5. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011). [CrossRef]

6. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]

7. M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011). [CrossRef]

8. R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76(2), 026401 (2013). [CrossRef]

9. Y. M. Li and K. Yao, Optical Tweezers Technique (in Chinese), (Beijing Science, Beijing, 2015).

10. P. H. Jones, O. M. Maragó, and G. Volpe, Optical tweezers: Principles and applications, (Cambridge University, Cambridge, 2015).

11. D. L. Gao, W. Q. Ding, M. Nieto-Vesperinas, X. M. Ding, M. Rahman, T. H. Zhang, C. Lim, and C. W. Qiu, “Optical manipulation from the microscale to the nanoscale: fundamentals, advances and prospects,” Light: Sci. Appl. 6(9), e17039 (2017). [CrossRef]

12. H. B. Xin, Y. C. Li, Y. Zhang, and B. J. Li, “Optical Forces: From Fundamental to Biological Applications,” Adv. Mater. 32(37), 2001994 (2020). [CrossRef]

13. Y. Z. Shi, T. T. Zhu, T. H. Zhang, A. Mazzulla, D. P. Tsai, W. Q. Ding, A. Q. Liu, G. Cipparrone, J. J. Sáenz, and C. W. Qiu, “Chirality-assisted lateral momentum transfer for bidirectional enantioselective separation,” Light: Sci. Appl. 9(1), 62 (2020). [CrossRef]

14. T. T. Zhu, Y. Z. Shi, W. Q. Ding, D. P. Tsai, T. Cao, A. Q. Liu, M. Nieto-Vesperinas, J. J. Sáenz, P. C. Wu, and C. W. Qiu, “Extraordinary multipole modes and ultra-enhanced optical lateral force by chirality,” Phys. Rev. Lett. 125(4), 043901 (2020). [CrossRef]

15. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61(2), 569–582 (1992). [CrossRef]

16. P. Wu, R. Huang, C. Tischer, A. Jonas, and E. L. Florin, “Direct measurement of the nonconservative force field generated by optical tweezers,” Phys. Rev. Lett. 103(10), 108101 (2009). [CrossRef]

17. Y. K. Jiang, J. Chen, J. Ng, and Z. F. Lin, “Decomposition of optical force into conservative and nonconservative components,” arXiv 1604:05138 (2016).

18. J. J. Du, C. H. Yuen, X. Li, K. Ding, G. Q. Du, Z. F. Lin, C. T. Chan, and J. Ng, “Tailoring optical gradient force and optical scattering and absorption force,” Sci. Rep. 7(1), 18042–18047 (2017). [CrossRef]

19. S. Sukhov and A. Dogariu, “Non-conservative optical forces,” Rep. Prog. Phys. 80(11), 112001 (2017). [CrossRef]

20. X. N. Yu, Y. K. Jiang, H. J. Chen, S. Y. Liu, and Z. F. Lin, “Approach to fully decomposing an optical force into conservative and nonconservative components,” Phys. Rev. A 100(3), 033821 (2019). [CrossRef]

21. H. X. Zheng, X. N. Yu, W. L. Lu, J. Ng, and Z. F. Lin, “GCforce: Decomposition of optical force into gradient and scattering parts,” Computer Phys. Commun. 237, 188–198 (2019). [CrossRef]

22. Y. K. Jiang, H. Z. Lin, X. Li, J. Chen, J. J. Du, and J. Ng, “Hidden symmetry and invariance in optical forces,” ACS Photonics 6(11), 2749–2756 (2019). [CrossRef]

23. H. X. Zheng, X. Li, Y. K. Jiang, J. Ng, Z. F. Lin, and H. J. Chen, “General formulations for computing the optical gradient and scattering forces on a spherical chiral particle immersed in generic monochromatic optical fields,” Phys. Rev. A 101(5), 053830 (2020). [CrossRef]

24. H. X. Zheng, X. Li, J. Ng, H. J. Chen, and Z. F. Lin, “Tailoring the gradient and scattering forces for longitudinal sorting of generic-size chiral particles,” Opt. Lett. 45(16), 4515–4518 (2020). [CrossRef]

25. J. Baumgartl, M. Mazilu, and Kholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]

26. C. Maher-McWilliams, P. Douglas, and P. F. Barker, “Laser-driven acceleration of neutral particles,” Nat. Photonics 6(6), 386–390 (2012). [CrossRef]

27. J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011). [CrossRef]

28. S. Sukhov and A. Dogariu, “Negative nonconservative forces: optical "tractor beams" for arbitrary objects,” Phys. Rev. Lett. 107(20), 203602 (2011). [CrossRef]

29. M. Woerdemann, C. Alpmann, M. Esseling, and C. Denz, “Advanced optical trapping by complex beam shaping,” Laser Photonics Rev. 7(6), 839–854 (2013). [CrossRef]

30. M. A. Taylor, M. Waleed, A. B. Stilgoe, H. Rubinsztein-Dunlop, and W. P. Bowen, “Enhanced optical trapping via structured scattering,” Nat. Photonics 9(10), 669–673 (2015). [CrossRef]

31. J. Berthelot, S. S. Aćimović, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014). [CrossRef]

32. M. H. Alizadeh and B. M. Reinhard, “Plasmonically enhanced chiral optical fields and forces in achiral split ring resonators,” ACS Photonics 2(3), 361–368 (2015). [CrossRef]

33. T. Cao and Y. M. Qiu, “Lateral sorting of chiral nanoparticles using Fano-enhanced chiral force in visible region,” Nanoscale 10(2), 566–574 (2018). [CrossRef]

34. S. B. Wang and C. T. Chan, “Strong optical force acting on a dipolar particle over a multilayer substrate,” Opt. Express 24(3), 2235–2241 (2016). [CrossRef]

35. M. M. Salary and H. Mosallaei, “Tailoring optical forces for nanoparticle manipulation on layered substrates,” Phys. Rev. B 94(3), 035410 (2016). [CrossRef]

36. A. Ivinskaya, M. I. Petrov, A. A. Bogdanov, I. Shishkin, P. Ginzburg, and A. S. Shalin, “Plasmon-assisted optical trapping and anti-trapping,” Light: Sci. Appl. 6(5), e16258 (2017). [CrossRef]

37. R. C. Jin, J. Li, Y. Wang, M. Zhu, J. Li, and Z. Dong, “Optical force enhancement and annular trapping by plasmonic toroidal resonance in a double-disk metastructure,” Opt. Express 24(24), 27563–27568 (2016). [CrossRef]

38. Y. He, S. He, J. Gao, and X. Yang, “Giant transverse optical forces in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Express 20(20), 22372–22382 (2012). [CrossRef]

39. V. Ginis, P. Tassin, C. M. Soukoulis, and I. Veretennicoff, “Enhancing optical gradient forces with metamaterials,” Phys. Rev. Lett. 110(5), 057401 (2013). [CrossRef]

40. B. Zhu, G. Ren, Y. Gao, Y. Yang, M. J. Cryan, and S. Jian, “Giant gradient force for nanoparticle trapping in coupled graphene strips waveguides,” IEEE Photonics Technol. Lett. 27(8), 891–894 (2015). [CrossRef]

41. W. L. Lu, H. J. Chen, S. Y. Liu, J. Zi, and Z. F. Lin, “Extremely strong bipolar optical interactions in paired graphene nanoribbons,” Phys. Chem. Chem. Phys. 18(12), 8561–8569 (2016). [CrossRef]

42. J. Glückstad, “Optical manipulation: Sculpting the object,” Nat. Photonics 5(1), 7–8 (2011). [CrossRef]

43. A. Jannasch, A. F. Demirörs, P. J. D. Van Oostrum, A. Van Blaaderen, and E. Schäffer, “Nanonewton optical force trap employing anti-reflection coated, high-refractive-index titania microspheres,” Nat. Photonics 6(7), 469–473 (2012). [CrossRef]

44. N. Wang, X. Li, J. Chen, Z. F. Lin, and J. Ng, “Gradient and scattering forces of anti-reflection-coated spheres in an aplanatic beam,” Sci. Rep. 8(1), 17423 (2018). [CrossRef]

45. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011).

46. Y. L. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34(21), 4573 (1995). [CrossRef]

47. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, New York, 1983).

48. J. D. Jackson, Classical Electrodynamics, 3rd edn (John Wiley and Sons, New York, 1999).

49. A. Zangwill, Modern Electrodynamics (Cambridge University, New York, 2012).

50. J. Ng, Z. F. Lin, C. T. Chan, and P. Sheng, “Photonic clusters formed by dielectric microspheres: Numerical simulations,” Phys. Rev. B 72(8), 085130 (2005). [CrossRef]

51. N. Wang, J. Chen, S. Y. Liu, and Z. F. Lin, “Dynamical and phase-diagram study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87(6), 063812 (2013). [CrossRef]

52. H. Chen, N. Wang, W. L. Lu, S. Liu, and Z. Lin, “Tailoring azimuthal optical force on lossy chiral particles in Bessel beams,” Phys. Rev. A 90(4), 043850 (2014). [CrossRef]

53. Y. K. Jiang, H. J. Chen, J. Chen, J. Ng, and Z. F. Lin, “Poynting vector, orbital and spin momentum and angular momentum versus optical force and torque on arbitrary particle in generic optical fields,” arXiv 1511:08546 (2015).

54. A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5(1), 011039 (2015). [CrossRef]

55. O. Pe na-Rodríguez and U. Pal, “Au@Ag core–shell nanoparticles: efficient all-plasmonic Fano-resonance generators,” Nanoscale 3(9), 3609 (2011). [CrossRef]

Enhanced transverse optical gradient force on Rayleigh particles in two plane waves

Abstract

1. Introduction

2. Numerical and analytical methods

2.1 Numerical method: optical forces based on the full wave simulation

2.2 Analytical method: optical forces based on the multipole expansion theory

3. Results and discussions

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (35)

Optics Express