Torque Detection using Brownian Fluctuations

We report the statistical analysis of the movement of a submicron particle confined in a harmonic potential in the presence of a torque. The absolute value of the torque can be found from the auto- and cross-correlation functions of the particle’s coordinates. We experimentally prove this analysis by detecting the torque produced onto an optically trapped particle by an optical beam with orbital angular momentum.

Figure 1

(a) Harmonic potential well $U(x,y)=\frac{k}{2}(x^2+y^2)$ ($k$ is the restoring force constant or stiffness of the harmonic oscillator) with a Brownian particle inside and with an external torque acting on the particle. (b) Experimental setup: 1—trapping 785 nm laser beam, 2—532 nm beam, 3—holographic mask, 4—Dove prism, 5—$100\times 1.3NA$ objective, 6—collimating $40\times$ objective, 7—quadrant photodetector. (c) Position of the sphere in the chamber when only the 532 nm LG propagates, and (d), when both the 532 nm and the 785 nm beams propagate in the chamber. The arrows show the propagation direction of the trapping and LG beams.

Figure 2

Experimental unbiased auto- and cross-correlation functions in the presence of the torque induced by a LG beam with $l=+10$. The trap force constant $k$ is low enough ($0.9\mathrm{fN}/\mu \mathrm{m}$) not to significantly influence the rotational motion of the sphere. The continuous lines show the mean values obtained using one series of data acquisition (acquisition time 60 s, sampling rate $f_s=1\mathrm{kHz}$). In the insets: (a) time traces for the $x$ (black) and $y$ (gray) coordinates; (b) histogram of the $x$ coordinate; (c) vector force field acting on the particle in the $xy$ plane.

Figure 3

Experimental auto- and cross-correlation functions in presence of the torque induced by a LG beam with $l=+10$. The trap force constant $k$ is high enough ($16\mathrm{fN}/\mu \mathrm{m}$) to confine the sphere. The continuous lines show the mean values obtained using five series of data acquisition (acquisition time 60 s, sampling rate $f_s=1\mathrm{kHz}$). The dotted lines show the fitting to the theoretical shape (the fitting was made on the central part of the curve for $\Delta t=[-2\mathrm{s},2\mathrm{s}]$). In the insets: (a) time traces for the $x$ (black) and $y$ (gray) coordinates; (b) histogram of the $x$ coordinate and in black the fitting to a Gaussian distribution; (c) vector force field acting on the particle in the $xy$ plane.

Figure 4

The same as in Fig. 2 when the torque is produced by a LG with $l=-10$.