Nonlinear effects in optical trapping of titanium dioxide and diamond nanoparticles

Abstract

Optical trapping in biophysics typically uses micron-scale beads made of materials like polystyrene or glass to probe the target of interest. Using smaller beads made of higher-index materials could increase the time resolution of these measurements. We characterized the trapping of nanoscale beads made of diamond and titanium dioxide (TiO₂) in a single-beam gradient trap. Calculating theoretical expectations for the trapping stiffness of these beads, we found good agreement with measured values. Trap stiffness was significantly higher for TiO₂ beads, owing to notable enhancement from nonlinear optical effects, not previously observed for continuous-wave trapping. Trap stiffness was over 6-fold higher for TiO₂ beads than polystyrene beads of similar size at 70 mW laser power. These results suggest that diamond and TiO₂ nanobeads can be used to improve time resolution in optical tweezers measurements.

Figure 1

Nonlinear effects in trap optical trapping. (A) Schematic representation of optical tweezers (not to scale). A particle (bead) is trapped in a highly focused laser beam at the trapping plane (TP). The equilibrium position of the particle in TP (z_eq) is slightly displaced from the focal plane (FP) of the laser beam in the direction of laser propagation owing to the scattering force. Insets: forces experienced by the particle: the particle experiences only the gradient force along radial direction (bottom inset, yellow), but the sum (left inset, red) of the gradient (left inset, cyan) and the scattering (left inset, blue) forces along the axial direction. (B) Schematic representation of the beam directions and the resulting change in the momentum (inset) for particle with linear (red arrows) or nonlinear (blue arrows) interactions; a ray diagram is used for illustrative purposes only, as the beads are much smaller than the trap wavelength. (C) Change in refractive index for TiO₂ (red) and ND (black) particles with incident laser power, calculated from Eq. (1). To see this figure in color, go online.

Figure 2

Calculated forces experienced by a particle in the trapping plane in an optical trap at a laser power of 100 mW. (A and B) Forces along the radial direction for (A) TiO₂ and (B) ND particles. (C and D) Forces along the axial direction for (C) TiO₂ and (D) ND particles. Dashed lines represent force expected in the absence of nonlinearity (showing that nonlinearity increased the force on TiO₂ particles). Insets: expanded view near the focal and the trapping planes. Because of the scattering force, the equilibrium position of the particle is displaced by a distance z_eq, that depends on the material, particle size, and laser power. To see this figure in color, go online.

Figure 3

Size-dependence of trapping. (A and B) Calculated trap stiffness as a function of particle size and average laser power. Solid lines: stiffness in radial direction; dashed lines: stiffness in axial direction. (C and D) Equilibrium displacement of trapping plane with respect to the focal plane along axial direction. To see this figure in color, go online.

Figure 4

Stiffness and temporal resolution of trapped TiO₂ and ND particles. (A) Radial stiffness as a function of laser power. Dashed lines: theoretical estimates of stiffness with (gray) and without (red) nonlinear effects. (B) Time response of trapped particles estimated based on stiffness (black), decay time of the bead position autocorrelation (red), and response to abrupt jump in trap position (brown). (C) Estimated time resolution for single-molecule force spectroscopy measurements at 15 pN. Error bars represent standard error of the mean. To see this figure in color, go online.