Determinations of angular stiffness in rotational optical tweezers

TL;DR

The paper develops a framework to determine angular trap stiffness $\\chi$ in rotational optical tweezers, highlighting that rotational dynamics require independent calibration from translational models. It introduces five passive analyses—EQP, MSD, ACF, PSD, and MLE—and a numerical torque-profile approach based on the T-matrix to compute $\\chi$, demonstrating consistent estimates across varying powers and beam configurations. It then investigates factors unique to rotational trapping, including the influence of an ancillary Helium–Neon measurement beam, shape-induced birefringence in spheroidal vaterite probes, and hydrodynamic and inertial corrections, showing that rotation is less sensitive to these effects than translation. The results establish a practical framework for characterising angular stiffness in ROTs and inform nanoparticle rotation experiments and rotational microrheology.

Abstract

Rotational optical tweezers are used to probe the mechanical properties of unknown microsystems. Quantifying the angular trap stiffness is essential for interpreting the rotational dynamics of probe particles. While methods for trap stiffness calibration are well established for translational degrees of freedom, angular trapping has been largely overlooked and is often assumed to behave analogously to translational dynamics. However, rotational and translational motions are sensitive to distinct experimental parameters and offer separate insights. This work covers passive analysis techniques for calibrating the angular trap stiffness and examines the influence of several factors unique to rotational optical tweezers. We show that the parameters of an ancillary measurement beam can be tuned to minimise its influence on angular trapping dynamics, while offering unprecedented improvements for nanoparticle analysis. We also explore the combined effects of shape-induced and material birefringence in spheroidal vaterite probes, and present a framework for assessing hydrodynamic and inertial contributions. These results provide a foundation for characterising rotational optical tweezers independent from translational models.

Figures (7)

• Figure 1: (A) Simplified schematic of the rotational optical tweezers system. (B) Circular polarisation configuration used to calibrate for angular displacement. The plot shows the conversion from volts to radians, where the black traces are segments taken from the time trace of the rotating vaterite and the red line is the mean. (C) Linear polarisation configuration and the measurement of the probe's azimuthal trajectory are used for angular trap-stiffness calibration.
• Figure 2: (A) Histogram of detections. (B) log(counts) histogram with quadratic fit to demonstrate harmonic potential. Fitting 99% of centred data. Experimental results of (C) MSD, (D) ACF, (E) PSD for a 1.5µm radius vaterite trapped at varying powers fitted by the expected model. (F) The linear relationship between optical toque and angle where overlaid lines have a gradient calculated using the MLE method.
• Figure 3: (A) The force profile for a $1\lambda$ radius probes. (B) The torque profile for a $1\lambda$ radius vaterite showing rotation about each axis. (C) The torque profile about $z$ for three differently sized probes. (D) The calculated $\chi$ for a given radius using the torque profile method. (E) Heatmap of the angular trap stiffness relative to beam size and beam convergence angle for a vaterite probe. (F) A heatmap of the optical torque on a vaterite probe when illuminated in circularly polarised light. In both (E) and (F), only the conditions where a stable spatial trap occurs are plotted, with the probe positioned at its equilibrium position along the beam axis as calculated from the force profile.
• Figure 4: The influence of the measurement beam on the equilibrium orientation of the probe and its angular trap stiffness for several probe sizes. (A) The shift in equilibrium angle is attributed to varying the numerical aperture of the measurement beam, which has a relative power of 0.1 of the trapping beam. (B) The shift in equilibrium and (C) relative change in angular trap stiffens, corresponding to the relative power of the measurement beam (NA 1.2).
• Figure 5: A heat map showing the He-Ne measurement beam power relative to the trapping IR beam that destroys the measurement. The dashed line separates the regions into cases where either the trap or alignment cannot be formed.