Conservative and Nonconservative Torques in Optical Binding

We show that in the canonical case of two lossless spheres that are electromagnetically coupled there is interplay between conservative and nonconservative forces, which is controlled by the polarization of the bounding field. We demonstrate that this phenomenon leads to new mechanisms to induce torques on spherically symmetric, optically isotropic, and lossless objects. The electromagnetic interaction can be exploited to apply orbital torque about the mutual center of mass of the bounded spheres as well as spin around the individual axes. When the incident field is linearly polarized, the torques are mostly conservative and affect only transient behaviors while for circularly polarized fields, the torques are entirely nonconservative, resulting in steady rotations. Means to control the magnitudes of orbital and spin torques are presented and applications to nanorotator machines are discussed.

Figure 1

Optical binding in elliptically polarized light. Apart from the binding force ${\mathbf{F}}_R$, interacting particles experience tangential forces ${\mathbf{F}}_T$. Note the existence of differential forces $\Delta \mathbf{F}$ leading to individual spinning in addition to common orbiting of particles around the system’s center of mass.

Figure 2

Torques in an optically bound system of silica spheres of radius $a=0.1{\lambda }_m$ solid lines, $a=0.2{\lambda }_m$ dashed lines, $a=0.4{\lambda }_m$ dot-dashed lines: (a) orbital torque about the system’s center of mass and (b) spin torque of a sphere about its own axis. The spheres are in water and are excited with a field polarized linearly at an angle $\theta$ with respect to the optical binding vector (${\Gamma }_0={10}^4|{\mathbf{E}}_I{|}^2{a}^4/\lambda$).

Figure 3

Magnitude of orbital torque as a function of the radius of interacting spheres for the first (curve 1, blue) and second (curve 2, red) stationary orbits. The plus symbols indicate regions where the torque has opposite sign. The dashed lines indicate the analytical predictions based on Eq. (4) for Rayleigh particles. The calculations are for silica spheres in water excited with a plane wave of intensity $50\mathrm{mW}/\mu {\mathrm{m}}^2$ and wavelength in vacuum $\lambda =532\mathrm{nm}$. The black line shows the magnitude of torque due to Brownian force at 290 K in the first stationary orbit. The inset depicts the symmetric potential energy landscape and the trajectory of a bound particle due to nonconservative orbital torques.

Figure 4

Magnitude of spin torque ${\Gamma }_s$ as a function of the radius of interacting spheres for the first (curve 1, blue) and second (curve 2, red) stationary orbits. The plus symbols indicate regions where the torque has opposite sign. The calculations are for the same conditions as in Fig. 3. The black line shows the magnitude of absorption-induced spin torque of one silica sphere with refractive index $n_i=1.59+{10}^{-7}i$.