Lateral optical force along the translationally invariant direction of optical fields formed by circularly polarized plane waves

Based on the generalized Lorenz-Mie method and Maxwell stress tensor theory, we demonstrate that a lateral optical force (OF) can be induced on a spherical particle along the translationally invariant direction of incident optical fields formed by multiple circularly polarized plane waves. It is analytically shown that the lateral OF depends on two quantities associated with the illuminating field, regardless of the particle properties such as material composition and particle size. Interestingly, for some special cases like two- or three-wave interference field, the lateral OF is rigorously proportional to the Poynting vector. As examples, the numerical results for the interference field composed of two or three circular polarized plane waves show that the spatial distribution pattern of the lateral OF is always determined by Poynting vectors of the illuminating field itself, while the particle size and material affect solely the magnitude but not the spatial profile of the lateral OF. In addition, the magnitude of the lateral OF is comparable to other components of the OF. The lateral OF is traced to the scattering force due to the interference between multipoles excited on the particle. These findings may enrich the concept of the lateral OF and also add an additional implement to the optical manipulation toolbox.

Figure 1

Schematic illustration of a lateral OF $F_z$ on a spherical particle immersed in an interference field composed of $n$ plane waves with all the wave vectors lying in the $xoy$ plane. All the plane waves have the same incident wavelength, amplitude, and polarization vector $(p,q)$ throughout the paper. The parameter $k_n$ denotes the wave vector of the $n\text{th}$ plane wave.

Figure 2

Lateral OF $F_z$ acting on different spherical particles: (a) polystyrene particle with $\varepsilon =2.53$ and radius $r=0.5\mu \mathrm{m}$, (b) Au particle with $\varepsilon =-50.89+3.54i$ and radius $r=0.7\mu \mathrm{m}$, and (c) polystyrene-Au core-shell particle with radii of core and shell $r=0.5$ and $1.2\mu \mathrm{m}$, respectively. The illuminated interference field is composed of three plane waves with the same parameters except different incident angles of $0^{\circ }, {120}^{\circ }$, and ${240}^{\circ }$. (d) Spatial distribution of the $z$-component Poynting vector for the interference field. The three plane waves have the same amplitude, the same left-circular polarization with $(p,q)=(1,i)$, and the same incident wavelength $\lambda =1.064\mu \mathrm{m}$

Figure 3

(a) Lateral OF on a polystyrene particle of radius $r=0.9\mu \mathrm{m}$ illuminated by two plane waves with the same left-circular polarization $(p,q)=(1,i)$ and different incident angles $0^{\circ }$ and ${50}^{\circ }$. (c) Lateral OF on an Au particle of radius $r=0.6\mu \mathrm{m}$ illuminated by two plane waves with the same right-circular polarization $(p,q)=(1,-i)$ but different incident angles $0^{\circ }$ and ${70}^{\circ }$. (b) and (d) Spatial patterns of the $z$-component Poynting vector for the optical fields in (a) and (c), respectively. The two plane waves have the same incident wavelength $\lambda =1.064\mu \mathrm{m}$.

Figure 4

Spatial profiles of (a) the $x$ component $F_x$ and (b) the $y$ component $F_y$ of the OF for the polystyrene particle and (c) $F_x$ and (d) $F_y$ for the Au particle. The other parameters are the same as in Fig. 2.

Figure 5

Lateral OF $F_z$ (blue dashed line), the $z$ components of the interception force $F_{\text{int}}^z$ (black dotted line), and the scattering force $F_{\text{sca}}^z$ (red solid line) versus the positions of the particle. Panels (a)–(d) correspond to the cases in Figs. 2, 3, 2, and 3, respectively. The particle is positioned at $(x,0.45,0)\mu \mathrm{m}$. The lateral force $F_z^{\text{FWS}}$ (green circles) based on the FWS is also plotted for comparison.

Figure 6

Phase diagrams of the lateral OF in $\varepsilon \text{−}\mu$ space for (a) lossless particles and (b) loss particles with $\mathrm{Im}\left(\varepsilon \right)=\mathrm{Im}\left(\mu \right)=0.1i$, where the particle has a radius of $r=0.5\mu \mathrm{m}$ and is illuminated by three circular polarized plane waves with different incident angles $0^{\circ }, {120}^{\circ }$, and ${240}^{\circ }$. The spherical plots of the $z$ component scattered irradiance by the particle with (c) $(\varepsilon ,\mu )=(2.53,2)$ and (d) $(\varepsilon ,\mu )=(3,3)$. The up-down asymmetry in the scattered irradiance in (c) and (d) along the $z$ axis represents the lateral OF.