Three-dimensional force-field microscopy with optically levitated microspheres

We report on the use of $4.7\text{−}\mu \mathrm{m}$-diameter, optically levitated, charged microspheres to image the three-dimensional force field produced by charge distributions on an Au-coated, microfabricated Si beam in vacuum. An upward-propagating, single-beam optical trap, combined with an interferometric imaging technique, provides optimal access to the microspheres for microscopy. In this demonstration, the Au-coated surface of the Si beam can be brought as close as $\sim 10\mu \mathrm{m}$ from the center of the microsphere while forces are simultaneously measured along all three orthogonal axes, fully mapping the vector force field over a total volume of $\sim {10}^6\mu {\mathrm{m}}^3$. We report a force sensitivity of $(2.5\pm 1.0)\times {10}^{-17}\mathrm{N}/\sqrt{\mathrm{Hz}}$, in each of the three degrees of freedom, with a linear response to up to $\sim {10}^{-13}\mathrm{N}$. While we discuss the case of mapping static electric fields using charged microspheres, it is expected that the technique can be extended to other force fields, using microspheres with different properties.

Figure 1

Schematic view of the free-space optical system. The output of the optical fiber carrying the trapping beam is first collimated, then deflected by a high-bandwidth ($f_{-3\mathrm{dB}}\sim 1\mathrm{kHz}$) piezo-mounted mirror in the Fourier plane of the trap, which produces translations at the trap position. Two aspheric lenses inside the vacuum chamber focus the trapping beam and recollimate the light transmitted through the MS. The transmitted beam is then interfered, on a quadrant photodiode (QPD), with a local oscillator (LO) beam, whose frequency is shifted by 500 kHz with respect to the trapping beam. The frequency shift is made prior to free space launch by two fiber-coupled acousto-optic modulators (not pictured), one for the trapping beam and one for both LO beams. Light backscattered by the MS is extracted, combined with another LO beam, and used to interferometrically measure the axial position of the MS. All components are only shown schematically and are not drawn to scale. PD, photodiode; PBS, polarizing beam splitter; and $\lambda /2$, half-wave plate.

Figure 2

Trap region: A drawing of the trap region is shown in the left panel, illustrating the 2-mm-diameter holes in the Au-coated pyramidal shielding electrodes through which the ACSB and the trapping beam are brought in and extracted. The trapping light is represented by the red conical feature. The panel to the right illustrates a detail of the end of the ACSB and trapped MS, along with the coordinate frame used in the data analysis. A separate arrow shows the direction of Earth's gravity.

Figure 3

Linearity of the force sensor in the $x$ direction, with residuals from perfectly linear behavior. Data are collected for a 41-Hz drive signal applied with a shielding electrode. The other two DOFs have comparable linearity.

Figure 4

Force noise micrographs with a charged MS, represented with respect to a stationary ACSB at various relative positions. Practically, changes in the relative position are produced by displacing the ACSB, yet for ease of representation, the figure shows the ACSB as stationary. The force vectors represent the MS response at a frequency far from a single-frequency AC voltage drive applied to the ACSB during these measurements. For the data in the top (bottom) panel, the horizontal component of the force vector represents $F_x$, the force in the $x$ direction ($F_y$, the force in the $y$ direction), while the $z$ component of the vector is always the force in the $z$ direction. In both cases, force vectors measured at five $y$ positions (with a different color indicating each $y$ position shown in the legend) in a regularly spaced grid ranging from $-40$ to $40\mu \mathrm{m}$ are plotted at each $(x,z)$ location.

Figure 5

From top to bottom: histograms of the measured force noise in the $x$, $y$, and $z$ directions at different grid locations. Also shown are the fits to normal distributions. The low noise and high linearity of the apparatus enable measurements with a dynamic range of over four orders of magnitude.

Figure 6

Vector field plots of $(F_x,F_z)$, top, and $(F_y,F_z)$, bottom, in an $xz$ plane of relative positions. As before, changes in the relative position are produced by displacing the ACSB, yet for ease of representation, the figure shows the ACSB as stationary. The black arrows represent the measured force, while the red arrows represent the best fit from the FEA of the $\mathbf{E}$ field produced by an ACSB with the same overall bias voltage as in the experiment. A few grid points are missing, due to data corruption introduced by the data acquisition instrumentation.

Figure 7

The rms of the $x$ and $y$ components of the measured force on the MS in the plane of the ACSB, as a function of separation from the ACSB. The data are compared to a model described in the text. The bands represents the standard deviation of the model, using different random implementations of the patch potentials. The fit of $F_{x,\mathrm{rms}}$ ($F_{y,\mathrm{rms}}$) implies voltage patches of size $\sim 0.8\mu \mathrm{m}$ ($\sim 0.7\mu \mathrm{m}$), assuming $V_{\mathrm{patch}}=100\mathrm{mV}$ [38, 39].