Optomechanical effects caused by non-zero field quantities in multiple evanescent waves

Yaxin Li; Xinning Yu; Xinning Yu; Tiantao Qu; Jack Ng; Zhifang Lin; Zhifang Lin; Lei Zhang; Lei Zhang; Lei Zhang; Jun Chen; Jun Chen; Jun Chen

doi:10.1364/OE.506758

1. Introduction

Evanescent waves can be generated at an interface through total internal reflection [1]. These near-field waves exhibit strong localization, high energy density, more intricate local momentum, and spatial distribution of spins [2–6], making them valuable for various applications in optics [2,3,7–11], nanotechnology [12,13], and biophotonics [14–17]. For instance, evanescent wave fluorescence biosensors have been developed to improve biometrics [15,16], total internal reflection fluorescence microscopy utilizes the properties of evanescent waves to excite fluorescent molecules for fluorescence calibration samples [14,17], optical skyrmion lattice can be generated in photonic systems with evanescent waves [10], and photonic spin skyrmion, and meron can form in evanescent optical vortices through spin–orbit coupling [18,19]. Even some intriguing mechanical effects arise when investigating the forces exerted on particles immersed in evanescent waves [2,4,9,20–22]. For instance, an achiral particle can experience an anomalous lateral optical force in an evanescent wave [2,22].

The proposal of gradient and scattering forces has been an important step in optical micromanipulation [23–28]. The gradient force is conservative, fulfilling $\nabla \times \boldsymbol {F}_g=0$ by Stokes’ theorem. It is always possible to establish a scalar potential $U$, leading to $\boldsymbol {F}_g=-\nabla U$. In contrast, the scattering force is nonconservative and conforms to $\nabla \cdot \boldsymbol {F}_s=0$. In practice, the gradient force is mainly responsible for optical trapping [23,29–35], whereas the scattering force can be used for pushing or pulling objects for particle transportation [36–43]. Therefore, the analytical decomposition of these forces is crucial for understanding the physical mechanism of light-matter interaction, as demonstrated in previous studies [44–47]. However, analytical expressions to decompose the optical force into scattering and gradient forces in multiple evanescent waves are still lacking. It is our purpose to fill this gap. By employing the Cartesian multipole expansion theory [44], we can effectively decompose the optical force into the gradient (curl-free) and scattering (divergence-free) parts. And this approach proves instrumental in comprehending the relationship between the optical force and field quantities of explicit physical meanings in evanescent waves [22].

In this article, we present the analytical expressions for the scattering and gradient forces exerted on an isotropic sphere of arbitrary size and composition in evanescent waves. By examining these forces, we uncover some fascinating phenomena. In contrast to homogeneous plane waves, the scattering force can be non-zero in counter-propagating evanescent waves. The scattering force results from orbital momentum (OM) density and/or the curl part of the imaginary Poynting momentum (IPM) density under different polarizations. Furthermore, it is found that as the radius of a particle increases, the gradient force within the interference field of evanescent waves becomes independent of the spatial coordinates of the propagating plane. We will elucidate the physical mechanisms underlying these phenomena with the help of analytical expressions of the scattering and gradient forces. Our study provides a rigorous and detailed theoretical foundation for developing and optimizing optical manipulation techniques based on evanescent waves.

2. Results and discussions

2.1 Analytical expressions of scattering and gradient forces in multiple evanescent waves

We consider an optical field composed of $n_p$ evanescent waves [such as Fig. 1(a)]

(1)$$\boldsymbol{E}=\displaystyle\sum_{i=1}^{n_p}\boldsymbol{E}_{i}=\sum_{i=1}^{n_p}\boldsymbol{\mathcal{E}}_{i}e^{ \mathbb{i}\boldsymbol{k}_{i}\cdot\boldsymbol{r}},\quad \boldsymbol{B}=\displaystyle\sum_{i=1}^{n_p}\boldsymbol{B}_{i}=\sum_{i=1}^{n_p}\boldsymbol{\mathcal{B}}_{i}e^{ \mathbb{i}\boldsymbol{k}_{i}\cdot\boldsymbol{r}},$$

where $\mathbb i$ denotes the imaginary unit, and the time dependence $e^{-\mathbb i \omega t}$ has been assumed and omitted for simplicity. Without loss of generality, we assume that the $i\mathrm {th}$ evanescent wave propagates on the $x$-$y$ plane and decays in the $z>0$ half-space, i.e., $\boldsymbol {k}_i=\boldsymbol k_{i,\parallel }+\boldsymbol k_{i,\bot }$, where $\boldsymbol k_{i,\parallel }=k_{ix} \hat {\boldsymbol {e}}_{x}+k_{iy}\hat {\boldsymbol {e}}_{y}$, $\boldsymbol k_{i,\bot }=k_{iz} \hat {\boldsymbol {e}}_{z}=\mathbb {i}\kappa _z\hat {\boldsymbol {e}}_{z}$. The complex amplitude of the electric field $\boldsymbol {\mathcal {E}}_i$ is given by [22]

(2)$$\begin{aligned}\dfrac{\boldsymbol{\mathcal{E}}_i}{E_{0}}=\left(p_i \dfrac{k_{ix} k_{iz}}{k_{i\rho}k_i}-q_i\dfrac{k_{iy}}{k_{i\rho}}\right)\hat{\boldsymbol{e}}_{x}+\left(p_i \dfrac{k_{iy} k_{iz}}{k_{i\rho}k_i}+q_i\dfrac{k_{ix}}{k_{i\rho}}\right)\hat{\boldsymbol{e}}_{y}-p_i \dfrac{k_{i\rho}}{k_i}\hat{\boldsymbol{e}}_{z}. \end{aligned}$$

Here, $E_0$ is the electric field amplitude at $\boldsymbol {r}=0$, and $k_{i\rho }= (k_{ix}^2+k_{iy}^2)^{1/2}$ satisfies $k_{i\rho }^2+k_{iz}^2=k_{i}^2$, where the wave number $k_i=2\pi n_d/\mathrm{\lambda} _i$, $n_d$ is the refractive index of the background medium, and $\mathrm{\lambda} _i$ denotes the wavelength in vacuum. The complex numbers $p_{i}$ and $q_{i}$ characterize the wave’s polarization, and satisfy the relation $|p_{i}|^2+|q_{i}|^2=1$. Their relations to the normalized Stokes parameters in the Poincaré sphere are given by Eqs. (S24) and (S25) in the Supplement 1. The complex amplitude of the magnetic field is expressed as $\boldsymbol {\mathcal {B}}_i={\omega }^{-1} \boldsymbol {k}_i \times \boldsymbol {\mathcal {E}}_i$, following from Maxwell equations.

Fig. 1. Scattering forces exerted on a polystyrene sphere in counter-propagating evanescent waves ($\kappa _z/k_0=0.1$ with $k_0$ denoting the wave number in vacuum). All the beams are transverse magnetic (TM, $H_z=0$) linearly polarized with the polarization parameters $(p,q)=(1,0)$, and the wavelength $\mathrm{\lambda} =1064 \mathrm {nm}$. The polystyrene sphere of radius $R$ is centered arbitrarily at $\{0.1\mathrm {\mu} \textrm{m}, 0.2\mathrm {\mu} \textrm{m}, R+0.5\mathrm{\lambda} \}$ with the permittivity $\varepsilon _s=1.59^2$ and the permeability $\mu =1$. Three Cartesian components of the normalized scattering force $F_{sj}/F_0$ ($j=x, y, z$, and $F_0=\varepsilon |E_0|^2/k^2$ with $\varepsilon$ being the permittivity in the background medium) versus the angle $\theta$ are plotted for two pairs of counter-propagating evanescent waves [see the configuration in (a)], with (b)-(d) corresponding to $R = 0.05\mathrm{\lambda}$ and (e)-(g) corresponding to $R = 0.5\mathrm{\lambda}$. (i)-(k) display the scattering forces versus $R/\mathrm{\lambda}$ for a single pair of counter-propagating evanescent waves, as depicted in configuration (h). The total scattering force $\boldsymbol {F}_s$ is depicted by red spheres, and its three parts $\boldsymbol {F}_\mathrm {P}$, $\boldsymbol {F}_\mathrm {OM}$, and $\boldsymbol {F}_\mathrm {IPM}$ contributed by different physical quantities are marked by orange diamonds, green crosses, and blue triangles, respectively.

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Here, we derive $\boldsymbol {F}_g$ and $\boldsymbol {F}_s$ acting on an isotropic spherical particle of any size and composition in generic evanescent fields, using the Cartesian multipole expansion theory. All the details can be found in Secs. 1 and 2 of the Supplement 1.

When considering a dipole particle, $\boldsymbol {F}_g$ and $\boldsymbol {F}_s$ can be expressed using field quantities of the incident field, including the gradient of electromagnetic energy density [48], momentum density [48], OM density [2,48], and the curl part of IPM density [2,4,49,50]

(3a)$$\begin{aligned} \boldsymbol {F}_g=A_{1}\nabla w^{\mathrm{e}}+B_{1} \nabla w^{\mathrm{m}}, \end{aligned}$$

(3b)$$\begin{aligned} \boldsymbol {F}_s=\boldsymbol {F}_{\mathrm {P}}+\boldsymbol {F}_{\mathrm {OM}}+\boldsymbol {F}_{\mathrm {IPM}}, \end{aligned}$$

where $w^{\mathrm {e}}$ and $w^{\mathrm {m}}$ are the electric and magnetic components of the electromagnetic energy density, respectively. It is evident that $\boldsymbol {F}_g$ is solely related to the gradient of electromagnetic energy density, while $\boldsymbol {F}_s$ consists of three distinct terms

(4a)$$\begin{aligned}\boldsymbol {F}_{\mathrm {P}}=C_{1} \boldsymbol {p}, \end{aligned}$$

(4b)$$\begin{aligned}\boldsymbol {F}_{\mathrm {OM}}=D_{1}\boldsymbol {p}_{\mathrm{o}}^{\mathrm{e}}+E_{1}\boldsymbol {p}_{\mathrm{o}}^{\mathrm{m}}, \end{aligned}$$

(4c)$$\begin{aligned}\boldsymbol {F}_{\mathrm {IPM}}=G_{1}\rm{Im}\widetilde{\boldsymbol p}^{\,\rm{curl}}. \end{aligned}$$

The first term $\boldsymbol {F}_{\mathrm {P}}$ is the generalized radiation pressure, with ${\boldsymbol p}$ denoting the momentum density [48]. The second term $\boldsymbol {F}_{\mathrm {OM}}$ is related to the electric and magnetic parts of OM density $\boldsymbol {p}_{\mathrm {o}}$ [51]. The final term corresponds to the solenoidal (curl) part of the generalized IPM force [2,52,53], where $\rm{Im}\widetilde {\boldsymbol p}^{\,\rm{curl}}$ is the curl part of IPM density [2]. The coefficients $A_{1}\sim E_{1}$, $G_{1}$ are determined by the parameters of particles and the wave vectors of the incident lights.

While for an isotropic spherical particle of any size and composition interacting with multiple evanescent waves, $\boldsymbol {F}_g$ and $\boldsymbol {F}_s$ can be expressed as

(5a)$$\begin{aligned} \boldsymbol {F}_g=\sum_{l=1}^{\infty}\langle\boldsymbol{F}^{(l)}\rangle_{\rm{grad}}=\sum_{l=1}^{\infty}\sum_{i,j} ({A_{l,ij}}\nabla w_{ij}^{\mathrm{e}}+{B_{l,ij}} \nabla w_{ij}^{\mathrm{m}}), \end{aligned}$$

(5b)$$\begin{aligned} \boldsymbol {F}_s =\sum_{l=1}^{\infty}\langle\boldsymbol{F}^{(l)}\rangle_{\rm{curl}}=\boldsymbol {F}_{\mathrm {P}}+\boldsymbol {F}_{\mathrm {OM}}+\boldsymbol {F}_{\mathrm {IPM}}, \end{aligned}$$

where

(6a)$$\begin{aligned}\boldsymbol {F}_{\mathrm {P}}=\sum_{l=1}^{\infty}\sum_{i,j}{C_{l,ij}} \boldsymbol {p}_{ij}, \end{aligned}$$

(6b)$$\begin{aligned}\boldsymbol {F}_{\mathrm {OM}} =\sum_{l=1}^{\infty}\sum_{i,j}({D_{l,ij}}\boldsymbol {p}_{\mathrm{o},ij}^{\mathrm{e}}+{E_{l,ij}}\boldsymbol {p}_{\mathrm{o},ij}^{\mathrm{m}} ), \end{aligned}$$

(6c)$$\begin{aligned}\boldsymbol {F}_{\mathrm {IPM}}=\sum_{l=1}^{\infty}\sum_{i,j} {G_{l,ij}}{\rm{Im}\widetilde{\boldsymbol p}}_{ij}^{\,\rm{curl}}. \end{aligned}$$

In the summations, $i$ and $j$ both run from 1 to the total numbers of beams $n_p$, and $l$ runs over all orders of multipoles. The formulas return to the dipole case in Eqs. (3)–(4) when $l=1$.

For simplicity, all the derivation details, coefficients, and definitions of physical quantities, such as $w_{ij}^{\mathrm {e}}$, are tabulated in Sec. 2 of the Supplement 1.

2.2 Scattering force for arbitrary particle in multiple counter-propagating evanescent waves

The use of counter-propagating evanescent waves can generate various optical fields, such as optical skyrmions and merons [10,19]. Therefore, understanding the optical forces within these optical fields becomes intriguing and important. It is known that the optical force exerted on an isotropic spherical particle in arbitrary pairs of counter-propagating homogeneous plane waves was previously proven to be conservative, resulting in zero scattering force due to symmetry [25]. However, we have discovered that the particle can acquire scattering forces in two pairs of counter-propagating evanescent waves with an included angle $\theta$ [refer to Figs. 1(a)-(g) for example]. Moreover, if the angle $\theta \neq 90 ^{\circ}$, the scattering force can even manifest in the decay direction of the evanescent waves [as in Figs. 1(d) and (g)]. Intriguingly, the scattering forces cannot exist in a single pair of counter-propagating evanescent waves [as shown in Figs. 1(h)-(k)]. Scattering forces can only appear when the number of pairs of counter-propagating evanescent waves $N_p\geq 2$, as also illustrated in Fig. S1 for $N_p=3$.

The presence of scattering forces has also been observed to be dependent on the polarization of the incident light. For example, in the depicted configuration in Fig. 2(g), scattering forces occur for both the transverse magnetic (TM, $H_z=0$) polarization presented in Figs. 2(a)-(b) and the diagonal linear polarization (DLP) shown in Figs. 2(d)-(e). While in the case of circular polarization (CP) displayed in Figs. 2(g)-(i), no scattering forces are observed. Further, we will demonstrate that the mechanisms underlying the generation of scattering forces differ even for the DLP and TM polarizations.

Fig. 2. Scattering forces exerted on a polystyrene sphere when subjected to two pairs of counter-propagating evanescent waves [see the configuration in (g) at $\theta = 90^{\circ}$]. The evanescent waves have TM linear polarization in (a)-(c), diagonal linear polarization (DLP) in (d)-(f), and circular polarization (CP) in (g)-(i) [depicted in the insets of (c), (f), and (i)]. The polarization parameters $(p, q)$ for these waves are $(1, 0)$, $(1, 1)/\sqrt 2$, and $(1, {\mathbb {i}})/\sqrt 2$, respectively. The remaining parameters are consistent with those presented in Figs. 1(i)-(k). The total scattering force ($\boldsymbol {F}_s$) and its three components ($\boldsymbol {F}_{\mathrm {P}}$, $\boldsymbol {F}_{\mathrm {OM}}$, and $\boldsymbol {F}_{\mathrm {IPM}}$) are denoted by the same symbols used in Fig. 1.

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In the following sections, we will analytically and numerically illustrate the underlying physical mechanism of aforementioned phenomena. The optical scattering force generation mechanism in counter-propagating evanescent waves can be understood by studying the scattering force exerted on a particle in the presence of two evanescent waves, i.e., $\boldsymbol {k}_{1}$ and $\boldsymbol {k}_{2}$ in Fig. 1(a) [see also the schematic plot in Fig. 3(a)]. The complex wave vectors are written as

(7)$$\begin{aligned} \boldsymbol{k}_{1}=k_{\rho}\hat{\boldsymbol{e}}_{x}+\mathbb i\kappa_z\hat{\boldsymbol{e}}_{z},\qquad\boldsymbol{k}_{2}=k_{\rho}\cos \theta \hat{\boldsymbol{e}}_{x}+k_{\rho}\sin \theta \hat{\boldsymbol{e}}_{y}+\mathbb i\kappa_z \hat{\boldsymbol{e}}_{z}, \end{aligned}$$

and $\boldsymbol {F}_s$ can be derived from Eqs. (5b) and (6) as

(8)$$\boldsymbol {F}_s=\sum_{\beta=a,b}\boldsymbol {F}_{s,\beta}=\sum_{\beta=a,b}\boldsymbol {F}_{\mathrm {P},\beta}+\boldsymbol {F}_{\mathrm {OM},\beta}+\boldsymbol {F}_{\mathrm {IPM},\beta},$$

with

(9a)$$\begin{aligned}\boldsymbol {F}_{\mathrm {P}}=\sum_{\beta=a,b}\boldsymbol {F}_{\mathrm {P},\beta}=\sum_{l=1}^{\infty}\sum_{\beta=a,b}C_{l,\beta} \boldsymbol {p}_\beta, \end{aligned}$$

(9b)$$\begin{aligned}\boldsymbol {F}_{\mathrm {OM}}=\sum_{\beta=a,b}\boldsymbol {F}_{\mathrm {OM},\beta} =\sum_{l=1}^{\infty}\sum_{\beta=a,b}(D_{l,\beta}\boldsymbol {p}_{\mathrm{o},\beta}^{\mathrm{e}}+ E_{l,\beta}\boldsymbol {p}_{\mathrm{o},\beta} ^{\mathrm{m}}), \end{aligned}$$

(9c)$$\begin{aligned}\boldsymbol {F}_{\mathrm {IPM}}=\sum_{\beta=a,b}\boldsymbol {F}_{\mathrm {IPM},\beta}=\sum_{l=1}^{\infty}\sum_{\beta=a,b}G_{l,\beta} \rm{Im} \widetilde{\boldsymbol p}^{\,\rm{curl}}_\beta. \end{aligned}$$

The double summation over $i$ and $j$ (both indices range from 1 to 2) in Eqs. (5) and (6) can be separated into two distinct groups here, labeled as $a$ and $b$. The first group, denoted as $a$, corresponds to the summation of the contribution of each beam to $\boldsymbol {F}_s$ when it is incident alone ($\sum _{i,j (i=j)}$). The second group, $b$, represents the interference contribution of the two evanescent waves ($\sum _{i,j (i\neq j)}$).

Fig. 3. Scattering forces exerted on a polystyrene sphere in two evanescent waves (TM polarization). (a) Schematic plot of the system configuration. The three Cartesian components of the normalized scattering force versus the angle $\theta$ with a radius of $R = 0.05\mathrm{\lambda}$ in (b) and $R = 0.5\mathrm{\lambda}$ in (c). The remaining parameters employed are identical to those utilized in Figs. 1(b)-(g).

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In the dipole case ($l=1$), the expression for the scattering force $\boldsymbol {F}_s$ in Eq. (9) can be simplified to the form of Eq. (4), which are directly proportional to the field quantities. The relationships between field quantities and the angle $\theta$ formed by the two evanescent waves are presented in Table 1, demonstrating that all the field quantities become zero when $\theta = 180^{\circ}$. As a result, $\boldsymbol {F}_s$ disappears at $\theta = 180^{\circ}$ (i.e., a single pair of counter-propagating evanescent waves), as illustrated in Fig. 3(b). Thus, when there is no momentum, OM, and IPM densities, the optical scattering force can not be induced physically. In general, for an isotropic spherical particle of any size and composition, $\boldsymbol {F}_s$ also disappears when $\theta = 180^{\circ}$ [as seen in Fig. 3(c) for a $R=0.5\mathrm{\lambda}$ particle] as the physical quantities in Eq. (9) become zero at $\theta = 180^{\circ}$ (refer to Table 2), as depicted in Figs. 1(i)-(k) and 3(c).

Table 1. The relationship between field quantities and the angle $\boldsymbol {\theta }$ in the field of two evanescent waves under different polarizations.

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Table 2. The relationship between physical quantities and the angle $\boldsymbol {\theta }$ in the field of two evanescent waves under different polarizations.

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Then a natural question arises: why does the scattering force occur in two pairs of counter-propagating evanescent waves, as depicted in Figs. 1(a)-(g)? For simplicity, we consider the configuration shown in Fig. 2(g) with $\theta =90^{\circ}$, the complex wave vectors of the four evanescent waves can be expressed as

(10)$$\boldsymbol{k}_{i}=k_{\rho}\sin\dfrac{i\pi}{2}\hat{\boldsymbol{e}}_{x}-k_{\rho}\cos\dfrac{i\pi}{2} \hat{\boldsymbol{e}}_{y}+\mathbb{i}\kappa_z\hat{\boldsymbol{e}}_{z},$$

where $i$ runs from $1$ to $4$, and $\boldsymbol {F}_s$ can be derived from Eqs. (5b) and (6) as

(11)$$\boldsymbol {F}_s=\sum_{\beta=a',b',c'}\boldsymbol {F}_{s,\beta}=\sum_{\beta=a',b',c'}\boldsymbol {F}_{\mathrm {P},\beta}+\boldsymbol {F}_{\mathrm {OM},\beta}+\boldsymbol {F}_{\mathrm {IPM},\beta}.$$

The double summation over $i$ and $j$ (both runs from 1 to 4) in Eqs. (5) and (6) can be divided into three categories here, denoted by $a'$, $b'$, and $c'$. Part $a'$ of $\boldsymbol {F}_s$ represents the summation of the contribution of each beam to $\boldsymbol {F}_s$ ($\sum _{i,j (i=j)}$), part $b'$ denotes the interference contributions of counter-propagating evanescent waves ($\sum _{\stackrel {i,j (i\neq j)}{\theta =180^{\circ} }}$), and the last part $c'$ corresponds to the interference contributions of any two adjacent evanescent waves ($\sum _{\stackrel {i,j (i\neq j)}{\theta =90^{\circ} }}$). The contributions of parts $a'$ and $b'$ to the scattering force are zero, i.e., $\boldsymbol {F}_{s,a'}=\boldsymbol {F}_{s,b'}=0$, for the same reason as in the case of a single pair of counter-propagating evanescent waves discussed above. Furthermore, $\boldsymbol {F}_{\mathrm {P},c'}$ is also zero due to the complete cancellation of momentum caused by counter-propagating interference fields. For instance, in the configuration shown in Fig. 2(g), the momentum of the interference field created by beams 1 and 2 counterbalances the momentum of the interference field generated by beams 3 and 4. Nevertheless, Fig. 3 indicates that the scattering force can exist when the angle between two evanescent waves is not $180^{\circ}$ due to non-zero physical quantities listed in Tables 1 and 2. Consequently, for two pairs of counter-propagating evanescent waves with $\theta =90^{\circ}$, the interference of adjacent evanescent waves can generate non-zero OM and IPM densities. In addition, different from momentum density, OM and IPM densities can not be canceled by counter-propagating interference evanescent waves, i.e.,

(12a)$$\begin{aligned}\boldsymbol {p}_{\mathrm{o},{c^{\prime}}} ^{\mathrm{e}}={-}\boldsymbol {p}_{\mathrm{o},{c^{\prime}}} ^{\mathrm{m}}={\rm{Re}} (p^*q)\alpha_1 \begin{bmatrix} -\sin (k_\rho y) \cos (k_\rho x)\\ \sin (k_\rho x) \cos (k_\rho y)\\ 0 \end{bmatrix}, \end{aligned}$$

(12b)$$\begin{aligned}{\rm{Im} \widetilde{\boldsymbol p}_{{\textit{c}}^\prime}^{\text{curl}}} = {({|p| ^2-|q|} ^2)\alpha_2} \begin{bmatrix} \sin (k_\rho x) \cos (k_\rho y)\\ \sin (k_\rho y) \cos (k_\rho x)\\ 0 \end{bmatrix}, \end{aligned}$$

where $\alpha _1={2\varepsilon |E_0|^2k k_\rho \kappa _z e^{-2 \kappa _z z}}/{(\omega \alpha ^2)}$, and $\alpha _2={4\varepsilon |E_0|^2k^2 k_\rho \kappa ^2_z e^{-2 \kappa _z z}}/[\omega \alpha ^2(k_\rho ^2-2 \kappa ^2)]$ with $\alpha ^2={k^2\left | q\right |^2+\left | p\right |^2(\kappa _z ^2+k_{\rho }^2)}$. This, in turn, leads to the emergence of a scattering force acting upon the particle [as illustrated in Figs. 2(a), (b), (d), and (e)]

(13)$$\begin{aligned} \boldsymbol {F}_s =\boldsymbol {F}_{\mathrm {OM},c'}+\boldsymbol {F}_{\mathrm {IPM},c'}=\sum_{l=1}^{\infty}(D_{l,c'} \boldsymbol {p}_{\mathrm{o},c'}^{\mathrm{e}}+ E_{l,c'}\boldsymbol {p}_{\mathrm{o},c'} ^{\mathrm{m}}+G_{l,c'} \rm{Im} \widetilde{\boldsymbol p}_{{\textit{c}}^\prime}^{\text{curl}}) \\ =\sum_{l=1}^{\infty}(D_{l,c'} \boldsymbol {p}_{\mathrm{o}}^{\mathrm{e}}+ E_{l,c'}\boldsymbol {p}_{\mathrm{o}} ^{\mathrm{m}}+G_{l,c'} \rm{Im} \widetilde{\boldsymbol p}^{\,\rm{curl}}), \end{aligned}$$

where the OM and IPM densities are exclusively contributed by the interference of adjacent beams. Thus, physically speaking, constructive interference gives rise to non-zero OM and IPM densities, resulting in the emergence of scattering forces. Further details can be found in Sec. 2.2 of the Supplement 1. One can reach a similar conclusion for two pairs of counter-propagating evanescent waves with $\theta \neq 90^{\circ}$ or an even larger number of pairs of counter-propagating evanescent waves [refer to Figs. 1(b)-(g), S1 and S2].

Especially, the optical scattering force in the decay direction ($z$) solely depends on the IPM density. This can be understood by considering a dipolar particle as an example. Since OM density resides in the $x$-$y$ propagation plane [2], it can not contribute to the $z$ component of the scattering force. When $\theta =90^{\circ}$, the IPM density proportional to $\cos \theta$ becomes zero [refer to Eq. (S56) in the Supplement 1], thus there is no scattering force along the $z$ direction. A similar conclusion holds for a particle of arbitrary size, which can be found in the Sec. 2.3 of the Supplement 1.

Besides the included angle between evanescent waves, the OM and IPM densities also depend on the polarization, as shown in Eq. (12). In the following, we shall present how to switch on and off the optical scattering force via polarizations in multiple counter-propagating evanescent waves. Here, we take the configuration in Fig. 2(g) as an illustration, the conclusion applies to other arbitrary pairs of counter-propagating evanescent waves ($N_p\geq 2$, see Figs. S1, S2 in the Supplement 1). From Eq. (13), we know that both OM and IPM densities will contribute to the scattering force in general. However, the values of these two field quantities are highly polarization dependent [see Eq. (12)], e.g. $\boldsymbol {p}_{\mathrm {o}} ^{\mathrm {e}}$ and $\boldsymbol {p}_{\mathrm {o}} ^{\mathrm {m}}$ are both $\propto {\rm{Re}} (p^*q)$, while ${\rm{Im} \widetilde {\boldsymbol p}^{\,{curl}}\propto} (|p| ^2-|q| ^2)$. Consequently, the field quantities contributing to the scattering force vary under different polarization conditions. The mechanisms of generating scattering force (the last column) under several typical polarizations are listed in Table 3, with the polarization parameters, the corresponding stokes parameters, and the contributed field quantities listed in the 2-5 columns of the table for reference. To make it clear, we also visualize the contributions of a diversity of field quantities to the scattering force in Figs. 2(a)-(c) (TM), Figs. 2(d)-(f) (DLP), and Figs. 2(g)-(i) (CP). From Table 3, we can know that the optical scattering force can be completely switched off when the evanescent wave is circularly polarized. Under other polarizations, the optical scattering force can be either contributed from OM and/or IPM densities. As a result, we can adjust the conservative properties of the optical force by simply tuning the polarization.

Table 3. The contributing physical mechanisms of $ {F}_s$ in two pairs of counter-propagating evanescent waves (${\theta=90^{\circ}}$) under different polarizations.

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2.3 Gradient force in the interference field of evanescent waves

In addition to optical scattering force, gradient force plays a significant role in the near-field particle trapping of evanescent waves. Usually, the gradient force in a multi-beam interference field is spatially dependent due to the inhomogeneous distribution of optical field intensity [45]. Here, based on our developed analytical expressions, we uncover that within the interference field of multiple evanescent waves, the gradient force exerted on an isotropic particle becomes spatially constant [independent of position ($x, y$) in the plane of propagation] as the particle radius increases. This observation is illustrated in Figs. 4(a) and (b), taking the configuration depicted in Fig. 3(a) as an example. Notably, this conclusion holds true for not only dielectric but also metallic particles, as seen in Figs. 4(c) and (d), and for an interference field composed of any number of evanescent waves with arbitrary polarization (i.e., each evanescent wave in the interference field can have the same or a different polarization state). Further details and examples are provided in Figs. S3-S6 in the Supplement 1.

Fig. 4. The normalized gradient force ($|\boldsymbol {F}_{g}|/F_0=\sqrt {F_{gx}^2+F_{gy}^2+F_{gz}^2}/F_0$) exerted on an isotropic sphere in the interference field of two evanescent waves ($\kappa _z/k_0=0.4$) with the configuration plotted in Fig. 3(a). All the beams are circularly polarized with the polarization parameters $(p,q)=(1,{\mathbb {i}})/\sqrt 2$. The sphere is centered at $(x, y, R+0.5\mathrm{\lambda} )$, where the radius $R$ is $0.2\mathrm{\lambda}$ and $2.2\mathrm{\lambda}$ in (a), (c) and (b), (d). The sphere’s permittivity is $\varepsilon _s=1.59^2$ (polystyrene) and $\varepsilon _s=-58.85+1.38 \mathbb {i}$ (gold) in (a)-(b) and (c)-(d), respectively. Other parameters used are the same as those in Fig. 3, except for the angle arbitrarily set as $\theta =60^{\circ}$.

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In two evanescent waves shown in Fig. 3(a), the gradient force $\boldsymbol {F}_{g}$ can be deduced from Eq. (5a) as

(14)$$\begin{aligned} \boldsymbol {F}_g=\sum_{\beta=a,b}\boldsymbol {F}_{g,\beta}=\sum_{l=1}^{\infty}\sum_{\beta=a,b}(A_{l,\beta}\nabla w^{\mathrm{e}}_\beta+ B_{l,\beta}\nabla w^{\mathrm{m}}_\beta), \end{aligned}$$

where

(15a)$$\begin{aligned} \nabla w^{\mathrm{e}}_a={-}{\varepsilon}|E_0|^2\kappa_z e^{{-}2 \kappa_z z} \hat{\boldsymbol{e}}_{z}, \end{aligned}$$

(15b)$$\begin{aligned} \nabla w^{\mathrm{m}}_a={-}\dfrac{\varepsilon |E_0|^2 }{\alpha^2}[|q|^2 (k_\rho^2+\kappa_z^2 )+|p|^2 k^2]\kappa_ze^{{-}2 \kappa_ z z} \hat{\boldsymbol{e}}_{z}, \end{aligned}$$

(15c)$$\begin{aligned}\nabla w^{\mathrm{e}}_b\propto \begin{bmatrix} \gamma_1 \sin(k_\rho\gamma)-\gamma_2 \cos (k_\rho\gamma)\\ \gamma_1 \sin(k_\rho\gamma)- \gamma_2\cos (k_\rho\gamma)\\ \gamma_1\cos(k_\rho\gamma)+\gamma_2 \sin (k_\rho\gamma) \end{bmatrix}, \end{aligned}$$

(15d)$$\begin{aligned}\nabla w^{\mathrm{m}}_b\propto \begin{bmatrix} \gamma_3\sin(k_\rho\gamma)+\gamma_2 \cos (k_\rho\gamma)\\ \gamma_3\sin(k_\rho\gamma)+\gamma_2 \cos (k_\rho\gamma)\\ \gamma_3\cos(k_\rho\gamma)-\gamma_2 \sin (k_\rho\gamma) \end{bmatrix}, \end{aligned}$$

with $\gamma =x-x\cos \theta -y\sin \theta$, $\gamma _1=k^2 |q|^2 \cos \theta + |p|^2 (\kappa _z^2 \cos \theta +k_\rho ^2)$, $\gamma _2=2{\rm{Re}}(p^*q)\kappa _z k\sin \theta$, and $\gamma _3=k^2 |p|^2 \cos \theta +|q|^2 (\kappa _z^2 \cos \theta +k_\rho ^2)$. Part $a$ of the summation in Eq. (14) represents the summation of the contribution of each beam to $\boldsymbol {F}_{g}$ when it is incident alone ($\sum _{i,j (i=j)}$), and the other part $b$ shows the interference contribution of the two evanescent waves ($\sum _{i,j (i\neq j)}$). Therefore, as Eq. (15) shows, $\nabla w^\mathrm {e}_b$ and $\nabla w^\mathrm {m}_b$, which are related to the interference, depend on the spatial position. In contrast, $\nabla w^\mathrm {e}_a$ and $\nabla w^\mathrm {m}_a$ are independent of positions.

Due to the decay property of the evanescent wave, the $z$ component of gradient force dominate. Let’s focus first on the gradient force’s $z$ component (decay direction, $F_{gz}/F_0$). In Fig. 5(a), we present the numerical simulation results for the relative standard deviation [54] [$\sigma (u)=\sqrt {\frac {\sum _{i=1}^{n}\left (u_{i}-\bar {u}\right )^{2}}{n-1}}/\bar {u}$] of $F_{gz}/F_0$ on the $x$-$y$ plane to reflect its dependence on the spatial position ($x, y$) as the particle radius increases. When the particle radius is small, the relative standard deviation is significant, indicating that the $z$-component of the gradient force depends on the spatial coordinates, as shown in the inset indicated by the blue arrow. However, as the radius increases, the relative standard deviation eventually tends to zero, and the $z$-component of the gradient force gradually becomes independent of the spatial position, as shown in the inset indicated by the black arrow. This phenomenon can be understood by the study of a particle placed at an arbitrary position shown in Fig. 5(b). As the radius increases, the contribution of $F_{gz,b}$ resulting from the interference diminishes gradually, approaching zero. Meanwhile, $F_{gz,a}$ tends towards $F_{gz}$, indicating a ratio of $F_{gz,a}$ to $F_{gz}$ approaching $1$. This implies that the gradient force is gradually equal to the direct addition of the gradient force when each beam incidents alone. Therefore, when the particle radius is small, both $F_{gz,a}$ and $F_{gz,b}$ contribute to $F_{gz}$. Since $F_{gz,a}$ is independent of the space coordinates, the distribution of $F_{gz}$ on the $x$-$y$ plane is solely determined by $F_{gz,b}$, resulting in the force distribution pattern corresponding to the blue arrow in Fig. 5(a). However, as the radius increases sufficiently, the contribution of $F_{gz,b}$ diminishes, resulting in $F_{gz}$ being entirely contributed by $F_{gz,a}$. Consequently, $F_{gz}$ only shows force distribution independent of the spatial position, as shown in the force distribution diagram denoted by the black arrow in Fig. 5(a). Moreover, the $x$ and $y$ components of the gradient force are entirely contributed by $\nabla w^\mathrm {e}_b$ and $\nabla w^\mathrm {m}_b$, and thus, related to the spatial position as seen from Eqs. (14) and (15). However, the comparison between the force distribution in Fig. 4(a) and that denoted by the blue arrow in Fig. 5(a) with all parameters being the same reveals that the gradient force is mainly contributed by its $z$ component, i.e., $|\boldsymbol {F}_{g}| \approx |F_{gz}|$. This is especially true for sufficiently large radii, as the $x$ and $y$ components of the gradient force disappear with diminishing interference effects. Thus, when the radius is large enough, the gradient force is no longer related to the spatial distribution. It should be noted that, this phenomenon is impossible in the interference field composed of homogeneous plane waves. It can be seen from Eqs. (15a) and (15b) that in the case of homogeneous plane waves, $\kappa _z=0$, thus $\nabla w^\mathrm {e}_a=\nabla w^\mathrm {m}_a=0$, contributing no gradient force [46]. Therefore, in the interference field of homogeneous plane waves, the gradient force is only contributed by the interference part, resulting in a pattern related to the spatial position and disappearing as the radius increases and the interference effect weakens.

Fig. 5. The relative standard deviation of the normalized gradient force and the proportion of its two constituent parts. The system used is shown in Fig. 3(a). (a) The relative standard deviation of the $z$ component of the normalized gradient force ($F_{gz}/F_0$) on the $x$-$y$ plane (same range as Fig. 4). The distribution of $|F_{gz}|/F_0$ on the $x$-$y$ plane for particles with radii of $0.2\mathrm{\lambda}$ (blue arrow) and $2.2\mathrm{\lambda}$ (black arrow) are shown in the insets. (b) $F_{gz,a}/F_{gz}$ and $F_{gz,b}/F_{gz}$ versus the dimensionless sphere radius $R/\mathrm{\lambda}$. The sphere’s center is arbitrarily set at $\{0.1\mathrm {\mu} \textrm{m}, 0.2\mathrm {\mu} \textrm{m}, R+0.5\mathrm{\lambda} \}$. $\theta$ in (a)-(b) is set at $60^{\circ}$. (c) The relative standard deviation of $|{\boldsymbol {F}}_{g}|/F_0$ as a function of $R/\mathrm{\lambda}$ and the angle $\theta$. Other parameters used are the same as those in Fig. 4(a).

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Furthermore, we find that the radius needed for achieving the gradient force independent of spatial coordinates is influenced by the angle $\theta$, and this relationship is not monotonic, as demonstrated in Fig. 5(c). On the right side of the relative standard deviation peak of the gradient force, we notice that a larger included angle between the incident beams generally results in a reduced radius for achieving position-independent gradient force. Conversely, on the other side of the peak, we observe that the closer the two incident beams are, the smaller the radius needed. When the angle $\theta$ is $0 ^{\circ}$, the two beams propagate in the same direction without interference, and thus, the required minimum radius approaches 0. This conclusion is also valid for the case of an interference field comprising multiple evanescent waves, where the minimum included angle determines the minimum radius required to achieve a position-independent gradient force (refer to Fig. S4 in the Supplement 1). This interesting observation favors versatile optical trapping with evanescent waves.

3. Conclusions

Based on the multipole expansion theory, we present the analytical expressions of the scattering and gradient forces exerted on an isotropic sphere of arbitrary size and composition in evanescent waves. With the help of these analytical expressions, we find interesting phenomena about the scattering and gradient forces in the interference field composed of multiple evanescent waves. In contrast to the scenario involving homogeneous plane waves, we observe the presence of non-zero scattering forces when particles encounter counter-propagating evanescent waves. This phenomenon arises when there are multiple pairs of counter-propagating evanescent waves ($N_p\geq 2$), as it results in the existence of non-zero OM and/or IPM densities induced by the interference of adjacent evanescent waves with an angle $\theta \neq 180^{\circ}$. Furthermore, in the case of $\theta \neq 90^{\circ}$, the scattering force in the decay direction can also occur due to a non-zero IPM density. We also illustrate the different generation mechanisms of the scattering force under different polarizations, which can be used to switch on and off the scattering force. Moreover, our study reveals an interesting feature of the gradient force. As the particle size increases, the gradient force becomes independent of the spatial coordinates $(x, y)$ in the interference field generated by evanescent waves (assuming beams propagate in the $x$-$y$ plane and decay in the $z>0$ half-space). This is attributed to the presence of the non-interference term of the electromagnetic energy density gradient, which is absent in the case of homogeneous plane waves, leading to a unique phenomenon that cannot occur under plane wave conditions. Our study offers a comprehensive understanding of the optical force characteristics in the presence of multiple evanescent waves, facilitating versatile near-field manipulations.

Funding

National Natural Science Foundation of China (12174231, 12074084, 12074169, 12074230); Fundamental Research Program of Shanxi Province (202103021222001); Stable Support Plan Program of Shenzhen Natural Science Fund (20200925152152003); Shanxi “1331 Project”; Research Project Supported by Shanxi Scholarship Council of China; Program of Education and Teaching Reform in Shanxi Province (J20230003).

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.

Supplemental document

See Supplement 1 for supporting content.

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	TE/TM ^a	DLP ^b and ALP ^c	CP ^d	EP ^e
$p$	$\propto {\cos^{2} \frac{θ}{2}, \sin θ, 0}$		$\propto {\cos \frac{θ}{2}, \cos \frac{θ}{2}, \sin θ}$
$p_{o}^{e}$			$\propto {\cos^{2} \frac{θ}{2}, \sin θ, 0}$
$p_{o}^{m}$			$\propto {\cos \frac{θ}{2}, \sin θ, 0}$
$I m {\tilde{p}}^{c u r l}$	$\propto {\sin^{2} θ, \cos^{2} \frac{θ}{2} \sin θ, \sin^{2} θ}$	$\propto {\sin θ, \cos^{2} \frac{θ}{2}, \sin θ}$	${0, 0, 0}$	$\propto {\sin^{2} θ, \cos^{2} \frac{θ}{2} \sin θ, \sin^{2} θ}$

	$R e (p^{*} q)$	$\| p \|^{2} - \| q \|^{2}$	stokes parameters of Poincaré sphere	field quantities	$F_{s}$
CP	0	0	$τ = 0, χ = 0, σ = \pm 1$	$p_{o}^{e} = p_{o}^{m} = I m {\tilde{p}}^{c u r l} = 0$	$0$
EP	$0$	$\neq 0$	$τ \neq 0, χ = 0, σ \neq 0$	$p_{o}^{e} (p_{o}^{m}) = 0,$	$F_{I P M}$
TE/TM	$0$	$\neq 0$	$τ = \pm 1, χ = 0, σ = 0$	$I m {\tilde{p}}^{c u r l} \neq 0$	$F_{I P M}$
DLP	$\neq 0$	$0$	$τ = 0, χ = \pm 1, σ = 0$	$p_{o}^{e} (p_{o}^{m}) \neq 0,$	$F_{O M}$
DLP	$\neq 0$	$0$	$τ = 0, χ = \pm 1, σ = 0$	$I m {\tilde{p}}^{c u r l} = 0$	$F_{O M}$
ALP	$\neq 0$	$\neq 0$	$τ \neq 0, χ \neq 0, σ = 0$	$p_{o}^{e} (p_{o}^{m}) \neq 0$	$F_{O M} + F_{I P M}$
ALP	$\neq 0$	$\neq 0$	$τ \neq 0, χ \neq 0, σ = 0$	$I m {\tilde{p}}^{c u r l} \neq 0$	$F_{O M} + F_{I P M}$

	TE/TM ^a	DLP ^b and ALP ^c	CP ^d	EP ^e
$p$	$\propto {\cos^{2} \frac{θ}{2}, \sin θ, 0}$		$\propto {\cos \frac{θ}{2}, \cos \frac{θ}{2}, \sin θ}$
$p_{o}^{e}$			$\propto {\cos^{2} \frac{θ}{2}, \sin θ, 0}$
$p_{o}^{m}$			$\propto {\cos \frac{θ}{2}, \sin θ, 0}$
$I m {\tilde{p}}^{c u r l}$	$\propto {\sin^{2} θ, \cos^{2} \frac{θ}{2} \sin θ, \sin^{2} θ}$	$\propto {\sin θ, \cos^{2} \frac{θ}{2}, \sin θ}$	${0, 0, 0}$	$\propto {\sin^{2} θ, \cos^{2} \frac{θ}{2} \sin θ, \sin^{2} θ}$

	$R e (p^{*} q)$	$\| p \|^{2} - \| q \|^{2}$	stokes parameters of Poincaré sphere	field quantities	$F_{s}$
CP	0	0	$τ = 0, χ = 0, σ = \pm 1$	$p_{o}^{e} = p_{o}^{m} = I m {\tilde{p}}^{c u r l} = 0$	$0$
EP	$0$	$\neq 0$	$τ \neq 0, χ = 0, σ \neq 0$	$p_{o}^{e} (p_{o}^{m}) = 0,$	$F_{I P M}$
TE/TM	$0$	$\neq 0$	$τ = \pm 1, χ = 0, σ = 0$	$I m {\tilde{p}}^{c u r l} \neq 0$	$F_{I P M}$
DLP	$\neq 0$	$0$	$τ = 0, χ = \pm 1, σ = 0$	$p_{o}^{e} (p_{o}^{m}) \neq 0,$	$F_{O M}$
DLP	$\neq 0$	$0$	$τ = 0, χ = \pm 1, σ = 0$	$I m {\tilde{p}}^{c u r l} = 0$	$F_{O M}$
ALP	$\neq 0$	$\neq 0$	$τ \neq 0, χ \neq 0, σ = 0$	$p_{o}^{e} (p_{o}^{m}) \neq 0$	$F_{O M} + F_{I P M}$
ALP	$\neq 0$	$\neq 0$	$τ \neq 0, χ \neq 0, σ = 0$	$I m {\tilde{p}}^{c u r l} \neq 0$	$F_{O M} + F_{I P M}$

Optomechanical effects caused by non-zero field quantities in multiple evanescent waves

Abstract

1. Introduction

2. Results and discussions

2.1 Analytical expressions of scattering and gradient forces in multiple evanescent waves

2.2 Scattering force for arbitrary particle in multiple counter-propagating evanescent waves

2.3 Gradient force in the interference field of evanescent waves

3. Conclusions

Funding

Disclosures

Data Availability

Supplemental document

References

Supplementary Material (1)

Data Availability

Cited By

Figures (5)

Tables (3)

Equations (27)

Optics Express