Direct Measurement of the Nonconservative Force Field Generated by Optical Tweezers

The force field of optical tweezers is commonly assumed to be conservative, neglecting the complex action of the scattering force. Using a novel method that extracts local forces from trajectories of an optically trapped particle, we measure the three-dimensional force field experienced by a Rayleigh particle with 10 nm spatial resolution and femtonewton precision in force. We find that the force field is nonconservative with the nonconservative component increasing radially away from the optical axis, in agreement with the Gaussian beam model of the optical trap.

Figure 1

The local force acting on a particle in an optical trap can be determined from the time series of its Brownian motion. (a) A single beam optical trap formed by a focused Gaussian beam propagating in the positive $z$ direction (not drawn to scale). (b) Time series of the particle position along the $x$ axis. The two horizontal lines indicate the $x$ boundaries of the volume element shown in (a). Note: Only the blue and green sections actually cross the selected volume element. (c) Multiple paths crossing the same volume element (left) are analyzed to obtain the local force from the average drift component of the diffusing particle (right).

Figure 2

Experimental force field calculated from the local drift of a 200 nm particle. Left: force field in the $x\mathrm{\text{−}}y$ plane located at $z=0$. Right: force field in the $x\mathrm{\text{−}}z$ plane located at $y=0$. Laser power $P=26\mathrm{mW}$, $y$ polarized. $\Delta t=17\mu \mathrm{s}$. Errors for each dimension are within $\pm 1–\pm 8\mathrm{fN}$, going from the center to the outer region of the trap, with $\pm 8\mathrm{fN}$ corresponding to an error of $\pm 2.4\%$ assuming a 340 fN force.

Figure 3

Curl of the experimental, simulated, and theoretical force fields for a Rayleigh particle in a single beam gradient trap. (a) Curl from experiment averaged over a $\pm 90\mathrm{nm}$ range along the $z$ axis. The red circle indicates the approximate minimum. (b) Curl of the force field calculated from Brownian dynamics simulation with a 200 nm particle and $n=1.57$ at a laser power of 25 mW. Inset: curl of the force field calculated from Gaussian beam model.