Subfemtonewton Force Spectroscopy at the Thermal Limit in Liquids

We demonstrate thermally limited force spectroscopy using a probe formed by a dielectric microsphere optically trapped in water near a dielectric surface. We achieve force resolution below 1 fN in 100 s, corresponding to a 2 Å rms displacement of the probe. Our measurement combines a calibrated evanescent wave particle tracking technique and a lock-in detection method. We demonstrate the accuracy of our method by measurement of the height-dependent force exerted on the probe by an evanescent wave, the results of which are in agreement with Mie theory calculations.

Figure 1

Setup. An optical trap is formed by a 660 nm laser beam focused by a high-NA water immersion objective (OBJ). The trap confines polystyrene microspheres in water near an AR coated glass surface. The bead-surface separation is controlled by vertical translation of the objective. A low-intensity evanescent wave used for detection is produced by total internal reflection of a 637 nm (probe) beam. Probe light scattered by the particle is collected by the objective and sent to a low-noise photodiode (PD1). A 660 nm notch filter and two 637 nm bandpass filters placed in front of PD1 eliminate spurious signals from scattered trap and probe beams. A second, high intensity, 785 nm (pump) evanescent wave exerts a downward gradient optical force on the particle. A chopper modulates the intensity of the pump beam, which is monitored by PD2. This signal and the detection signal are digitized and recorded by a computer for lock-in signal processing.

Figure 2

Power spectral density of the position fluctuations of an optically trapped and optically driven Brownian probe. At low frequencies $1/f$ noise dominates and the PSD deviates from theory. A lock-in peak centered at approximately 23 Hz is visible in the frequency domain though the driven motion is obscured by thermal fluctuations in the time domain, as illustrated by the inset. The lock-in amplitude is determined algorithmically using the pump reference signal.

Figure 3

Measured rms uncertainty in force as a function of measurement time for three beads of different radii. Each data point is obtained by calculating the standard deviation of a series of repeated independent measurements resampled from a 1000 s time series. The maximum number of independent measurements used is 25 (for the shortest time points) and minimum is 5 (for the longest time points). Comparisons are made with the predicted thermal noise limit calculated from measured damping coefficients (solid lines). Contour lines of constant damping strength ($\gamma$) are shown in the background in color. Dotted line indicates 1 fN level.

Figure 4

Measured optical force on a $2.90\mu \mathrm{m}$ and a $1.06\mu \mathrm{m}$ diameter polystyrene bead in the presence of an evanescent field. The angle of incidence of the pump beam used to generate the optical force was measured to be 62.1° $\pm 0.1°$ from normal for both measurements (above the critical angle of 61.04°). Solid lines represent analytical Mie theory prediction of the attractive optical force in the corresponding system. The inset displays the same data on semilog axes.