Figure 1

(a) Polarization vector $\mathbf{P}$ induced in an anisotropic particle by an external electric field $\mathbf{E}$. Torque is generated when $\mathbf{P}$ is not aligned with the field. (b) The index ellipsoids of quartz and calcite are represented, where the region of maximum electric susceptibility is marked in red. Torque exerted on the particle tends to align the red region with the $\mathbf{E}$ vector. For quartz particles the applied torque tends to align the extraordinary axis (red areas) with the $\mathbf{E}$ vector, constraining two of three Euler angles. For calcite particles, the extraordinary axis is repelled by the $\mathbf{E}$ vector and the plane defined by the ordinary axes (red band) is attracted to the $\mathbf{E}$ field, constraining one of three Euler angles. In this case the extraordinary axis is either perpendicular [ellipsoid marked (1)] or parallel to the propagation vector $\mathbf{k}$ [ellipsoid marked (2)]. In the latter case, the birefringence of the particle is extinguished, allowing the particle to spin freely around the optical axis (along $\mathbf{k}$). (c) The alignment of the particle with the trap polarization is illustrated. (d) The torque detector measures the difference in intensities of the right circular and left circular components of the transmitted beam. (e) Schematic representation of the apparatus. The polarization rotator is described in the text. The input detector determines the polarization angle of the input trapping field. The torque detector and quadrant detector determine the torque and force acting on the particle.Reuse & Permissions

Figure 2

Video images recorded every 250 ms show a $\sim 1\mu \mathrm{m}$ diameter quartz particle being rotated counterclockwise in the angular trap. Quartz particles are obtained by centrifuging and collecting fractions from quartz powder. The particle shown is close to spherical but has an irregularity on one side (indicated by arrows) that allows its orientation to be recognized in video images.Reuse & Permissions

Figure 3

(a) The torque signal for a particle spinning in opposite directions at 11.5 rps, and nearly motionless at 0.5 rps. The dc offset is a measure of the drag on the spinning particle and the broadband fluctuations are primarily due to Brownian motion. (b) The power spectral density of the rotational fluctuations, and fit with Lorentzian function of the form $S(f)=A^2/(f^2+f_0^2)$ with $A^2=0.059{\mathrm{rad}}^2$ (Hz) and $f_0=75\mathrm{Hz}$. The inset shows the torque signal as a quasistationary particle is scanned by a rapidly rotating polarization vector (200 Hz). The amplitude $V_0=2.63\mathrm{V}$ implies a small-signal angle sensitivity of $0.19\mathrm{rad}/\mathrm{V}$. From these values we calculate stiffness $\kappa =3360\mathrm{pN}·\mathrm{nm}/\mathrm{rad}$, damping coefficient $\xi =7.1\mathrm{pN}·\mathrm{nm}·\mathrm{s}$, and torque sensitivity $638\mathrm{pN}·\mathrm{nm}/\mathrm{V}$. (c) The measured torque as a function of rotation rate, converted to $\mathrm{pN}·\mathrm{nm}$ using the calibration parameters defined in (b). From the slope of the linear regime we obtain $\xi =7.8\mathrm{pN}·\mathrm{nm}·\mathrm{s}$. (d) Measured torque and trap polarization angle as a function of time. Feedback to stabilize the torque at $100\mathrm{pN}·\mathrm{nm}$ ($0\mathrm{pN}·\mathrm{nm}$) is activated at 0.4 s (0.8 s).Reuse & Permissions