Polarization-induced torque in optical traps

In the field of optical trapping and micromanipulation it is well known that linearly polarized Gaussian beams, which possess no inherent angular momentum, can exert an orienting torque on optically or geometrically anisotropic particles. Conservation of angular momentum requires that the application of such a torque be compensated for by an equivalent, and opposite, angular momentum flux in the beam. In the following paper we analyze this effect in terms of both the scattered field, and the mechanical torque experienced by the particle. It is demonstrated that, in general, the scattered field has a complicated form, carrying both spin and orbital angular momentum. However, we show that the variation of torque with rotation angle is identically equal to $A+B\sin (2\alpha +\beta )$ for arbitrarily shaped particles, where $A$, $B$, and $\beta$ are constants and $\alpha$ is the angular displacement of the major axis of the particle from the polarization direction. The scattered field, and the mechanical torque, are seen to reduce to qualitatively distinct forms that depend on the symmetry group of the scattering particle. Our findings are verified and illustrated by a series of numerical calculations of the forces and torques experienced by arbitrarily shaped particles trapped in linearly polarized Gaussian beams.

Figure 1

Longitudinal section showing the intensity along the axis of a focused beam in a FDTD simulation, generated via (a) the paraxial approximation, and (b) a multipole expansion. The position of the source is shown. The gray regions indicate the positions of the absorbing boundary layers.

Figure 2

Illustration of particles with (a) $C_1$, (b) $C_4$, and (c) $D_4$ symmetry about the vertical axis. The dimensions of the particles are indicated: the drawings are to scale.

Figure 3

$z$ component of the electric field near to a $C_1$ particle, taken from a FDTD simulation of a particle at the focus of an optical trap. The arrow shows the reference axis of the particle, and the spot marks the center of rotation used in subsequent calculations.

Figure 4

Normalized torque function for the $C_1$ particle (open circles) shown in Fig. 2a, as it is rotated in a plane polarized optical trap. A $\sin \left(2\alpha \right)$ fit is also shown (solid line).

Figure 5

Normalized torque for a $C_2$ particle rotating about the beam axis (open circles). The solid line is a $\sin \left(2\alpha \right)$ fit.

Figure 6

Normalized torque for a $D_2$ dumbbell-shaped particle, centered in the beam (solid circles) and displaced so that one end is in the center (open circles). The solid lines are fits to $\sin \left(2\alpha \right)$.

Figure 7

Normalized torque for a $C_4$ particle acquired from simulations with lattice spacings of $7\mathrm{nm}$ (open circles) and $14\mathrm{nm}$ (solid circles).

Figure 8

(a) Normalized torque for a $D_3$ particle acquired from FDTD simulations with lattice spacings of $7\mathrm{nm}$ (open circles), $14\mathrm{nm}$ (solid circles), and $3.5\mathrm{nm}$ (diamonds). The solid line is a fit to $\sin \left(3\alpha \right)$. (b) Normalized torque for a $D_4$ particle with lattice spacing $7\mathrm{nm}$. The solid line is a fit to $\sin \left(4\alpha \right)$.