Orbital rotation of spheroidal Mie particles driven by counter-propagating circularly-polarized beams

TL;DR

This paper addresses how a spheroidal submicron particle behaves when trapped by two counter-propagating circularly polarized Gaussian beams. It develops a theoretical framework that couples orbital motion around the beam axis to spin-induced rotation about the particle’s equator, deriving analytic relations that connect orbit radius and angular velocities to optical forces and torques, including two dynamic regimes: rapid proper rotation and phase synchronization. Numerical simulations, including Brownian noise, confirm the existence of these regimes and reveal how particle size, beam power, and temperature affect transitions, with rapid rotation persisting at room temperature and synchronization occurring only at cryogenic temperatures. The findings have potential applications in nanoscale gyroscopes and shape-selective sorting, and point to future work on stability to longitudinal perturbations and richer 3D attractor dynamics at higher intensity.

Abstract

We theoretically consider orbital rotation of a spheroidal submicron particle in the field of two counter-propagating circularly polarized Gaussian beams. We derived equations connecting the parameters of the circular orbits centered on the beams axis to the optical force and torque. The equations show that, besides orbital rotation, the spheroidal particle simultaneously rotates around its equatorial axis. We found that two distinct dynamic regimes are possible. The orbital motion can be accompanied by a rapid proper rotation with angular velocity an order of magnitude larger than the angular velocity of the orbital rotation. Alternatively, the orbital and proper rotations can be synchronized. The direction of orbital rotation can either coincide with or be opposite to the direction of rotation of the electric vector. The findings are confirmed by direct numerical simulations. The results can be of use in development of nano-scale gyroscopes as well in shape-selective sorting of submicron particles.

Figures (4)

• Figure 1: Spheroidal particle trapped by two counter-propagating coherent circularly polarized Gaussian beams. The sketch of the system is demonstrated on top. The coordinates in the $x0y$-plane are shown in the south-west corner, together with the definition of the rotating reference frame. In the south-east corner we illustrate the definition of the Euler angles.
• Figure 2: Optical force and torque on a spheroidal optical particle, $a_z/a_x=0.7$, subject to to the polarized counter-propagating Gaussian beams shown in Fig. \ref{['fig1']} as function of the first Euler angle $\alpha$ and azimuthal angle $\phi$; $z=0,~\beta=\pi/2$. The particle is located at the distance $r=1~{\mu\rm{m}}$ from the axis of the beams. The incident power of either beam is $1~\rm{mW}$. (a) $a_x=250~\mathrm{nm}$, and (b) $a_x=450~\mathrm{nm}$. The star on subplot (b) corresponds to $\alpha$ and $\phi$ in the solution Eq. \ref{['data3']}.
• Figure 3: Rotation in a circular orbit. The first column shows the first Euler angle $\alpha$ against time, the second column -- the $x$-coordinate against time, the third column -- the trajectory in $x{0}y$ plane. The red circle in the right row shows the initial condition. The initial kinetic energy is zero. (a) The evolution of variables $\alpha, x, y$ in course of time for a smaller particle $a_x=0.250~\mu{\rm m}$ with a rapid proper rotation. The beam power $P=10~{\rm mW}$. (b) The phase synchronized trajectory for a larger particle $a_x=0.450~\mu{\rm m}$. The beam power $P=0.45~{\rm mW}$. (c) Rapid rotation for $a_x=0.400~\mu{\rm m}$ with incident beam power $P=0.67~{\rm mW}$.
• Figure 4: Effect of Brownian force. (a) Rapid proper rotation trajectory from Fig. \ref{['fig3']} (a) at $T=300~{\rm K}$. (b) Phase-synchronized trajectory from Fig. \ref{['fig3']} (b) at $T=6~{\rm K}$. The mid-column shows the time dependence of the second Euler angle $\beta$.