Tailoring azimuthal optical force on lossy chiral particles in Bessel beams

Based on the Mie scattering theory and Maxwell stress tensor method, we investigate the transverse optical force (TOF) acting on chiral particles illuminated by a zero-order Bessel beam. It is demonstrated that the particle chirality can induce an azimuthal optical force (AOF), resulting in orbital motion of particles around the optical beam axis. The AOF depends strongly on particle loss as well as the handedness of chirality, with its amplitude capable of changing by over an order of magnitude by particle's chiral loss. The other component of TOF, the radial optical force (ROF), is much less sensitive to the magnitude and handedness of the particle chirality as well as the loss when the chirality is small. Analytical result based on dipole approximation reveals that the AOF arises from the direct coupling of particle chirality to both the spin angular momentum (SAM) and optical vorticity (curl of Poynting vector), exhibiting a conversion of optical SAM of an incident beam to mechanical orbital angular momentum of an illuminated particle. Differently, the ROF originates from the transverse gradient force. In addition, particle chirality yields a negative contribution to the gradient force; thus the ROF can be attenuated and even reversed in direction when particle chirality is sufficiently large. These characteristics of TOF might find applications in chirality detection as well as sorting chiral particles of different handedness and separating them from conventional ones.

Figure 1

ROF $F_r$ (a) and the AOF $F_{\varphi }$ (b) vs the displacement $r$ from the beam axis on a chiral particle illuminated by a zero-order Bessel beam with TM polarization and cone angle $\alpha ={30}^{\circ }$. The green lines, blue lines, red lines, and black lines denote, respectively, the chiral particle with $(\varepsilon ,\kappa )=(2.5,0),(2.5,0.3),(2.5+0.4i,0.3+0.1i)$, and $(2.5+0.4i,-0.3+0.1i)$. The incident wavelength is $\lambda =1.064\mu \text{m}$. The particle has radius $r_s=0.1\mu \text{m}$ and relative magnetic permeability $\mu =1$.

Figure 2

Time-averaged Poynting vectors for a particle with $\kappa =0$ (a) and $\kappa =0.3$ (b) immersed in a zero-order Bessel beam with TM polarization propagating along $z$. The permittivity $\varepsilon =2.5$ is set for both cases and other parameters are the same as in Fig. 1. The particle chirality breaks the up-down symmetry and creates the unbalanced photon momentum above and below the particle, inducing an AOF that manifests itself as a $y$ component of optical force.

Figure 3

Contributions of decomposed ROF $F_{r1}$ and $F_{r2}$ (a) and decomposed AOF $F_{\varphi 1},F_{\varphi 2}$, and $F_{\varphi 3}$ (b) and (c), defined, respectively, in Eqs. (17, 18, 19, 20), vs particle displacement $r$ from the beam axis. In all the panels, the black lines, red lines, and blue lines denote the cases with $(\varepsilon ,\kappa )=(2.5,0.3),(2.5+0.4i,0.3+0.1i)$, and $(2.5+0.4i,-0.3+0.1i)$, respectively. Panel (b) shows a comparison between the cases for lossless particle and lossy particle with positive handedness $\text{Re}\left(\kappa \right)>0$, while panel (c) displays a comparison of lossy chiral particles with opposite handedness $\left[\text{Re}\right(\kappa )>0$ and $\text{Re}\left(\kappa \right)<0]$. The superscripts “$a$” and “$b$” denote the lossy cases with positive $\left[\text{Re}\right(\kappa )>0]$ and negative $\left[\text{Re}\right(\kappa )<0]$ handedness, respectively.

Figure 4

ROF $F_r$ (a) and AOF $F_{\varphi }$ (b) vs the displacement $r$ for the chiral particles with three different permittivity losses corresponding to $\varepsilon =2.5+0.35i,\varepsilon =2.5+0.4i$, and $\varepsilon =2.5+0.45i$. The chirality is $\kappa =-0.3+0.1i$, kept unchanged.

Figure 5

ROF $F_r$ (a) and AOF $F_{\varphi }$ (b) vs the displacement $r$ for the chiral particles with three different chirality-deriving losses, corresponding to $\kappa =-0.3+0.08i,\kappa =-0.3+0.1i$, and $\kappa =-0.3+0.12i$. The permittivity is $\varepsilon =2.5+0.4i$, kept unchanged.

Figure 6

ROF $F_r$ (a) and AOF $F_{\varphi }$ (b) vs the real part of the chirality $\text{Re}\left(\kappa \right)$ for a chiral particle immersed in a Bessel beam and with different constitutive parameters. Black: $\varepsilon =2.5$ and $\kappa =\text{Re}\left(\kappa \right)$; red: $\varepsilon =2.5+0.4i$ and $\kappa =\text{Re}\left(\kappa \right)$; blue: $\varepsilon =2.5+0.4i$ and $\kappa =\text{Re}\left(\kappa \right)+0.1i$. The particle is positioned at a distance $r=0.4\mu \text{m}$ from the beam axis. Other parameters are the same as in Fig. 1.

Figure 7

(a) Electric part $F_{r1}$ and the magnetic part $F_{r2}$ of the gradient force, and (b) the vorticity term $F_{\varphi 1}$, the direct coupling term $F_{\varphi 2}$, and the higher-order coupling term $F_{\varphi 3}$ as a function of $\text{Re}\left(\kappa \right)$ for a chiral particle with $\varepsilon =2.5+0.4i$ and $\kappa =\text{Re}\left(\kappa \right)+0.1i$. The particle is located at $r=0.4\mu \text{m}$ away from the beam axis of a Bessel beam. Other parameters are the same as in Fig. 1.