Subfemtonewton force fields measured with ergodic Brownian ensembles

We demonstrate that radiation pressure force fields can be measured and reconstructed with a resolution of 0.3 fN (at a $99.7\%$ confidence level) using an ergodic ensemble of overdamped colloidal particles. The outstanding force resolution level is provided by the large size of the statistical ensemble built by recording all displacements from all diffusing particles, regardless of trajectory and time. This is only possible because the noise driving the particles is thermal, white, and stationary, so that the colloidal system is ergodic, as we carefully verify. Using an ergodic colloidal dispersion for performing ultrasensitive measurements of external forces is not limited to nonconservative optical force fields. Our experiments therefore give way to interesting opportunities in the context of weak force measurements in fluids.

Figure 1

Schematics of the experimental setup. (a) Using a half-wave plate (HWP), a polarizing beam-splitter (PBS), and two mirrors (M1, M2), a linearly polarized single-mode ${\mathrm{TEM}}_{00}$ horizontal laser beam (633 nm, 200 mW) can either be directed toward the sample along the $y$ axis (single-beam mode) or be split into two noninterfering (crossed polarizations) counterpropagating beams of identical intensity (using the HWP for the fine intensity balance). A microscope objective (NA = 0.25, $20\times$) collects the fluorescence of dye-doped melamine spheres (diameter $d=940\pm 50$ nm, from microParticles GmbH) diffusing in water inside the cell. The particles are imaged on a CCD camera at a frame rate $f=120$ Hz and tracked using an algorithm adapted from [20]. A filter F eliminates any stray light coming from the laser. (b) The cell consists of a quartz cuvette of dimensions—$10\left(x\right)\times 2\left(y\right)\times 35\left(z\right){\mathrm{mm}}^3$—chosen such that the imaging region-of-interest is located far away from any wall, allowing us to neglect safely any perturbation of the wall on the diffusion dynamics. The profile of the laser sent through the cuvette is set using 400-mm focusing lenses (L1,L2) for large waist $w_0=65\mu \mathrm{m}$ and Rayleigh range $z_r=18$ mm, so that the illumination is a plane wave with a Gaussian distribution of intensity in the transverse $z$ direction. (c) Region-of-interest—$350\left(y\right)\times 250\left(z\right)\mu {\mathrm{m}}^2$—imaged (using lens L3) from the center of the illumination zone. The right-hand side arrow corresponds to the laser propagation direction. In this single-beam mode using a fairly high-intensity, $0.37\mathrm{kW}{\mathrm{cm}}^{-2}$, the recorded trajectories clearly reveal the shift induced along $y$ by radiation pressure. In such conditions of high intensity, the diffusion along $z$ results from the combination of sedimentation and laser-induced convection—see main text and Appendix pp1. The crosses indicate the initial positions taken by the particles: Due to strong convective drag, the particles are diffusing upwards.

Figure 2

(a) Allan standard deviations calculated for 15 experiments performed in the dual-beam mode, using different laser intensities (from $0.006\mathrm{kW}/{\mathrm{cm}}^2$ to $0.221\mathrm{kW}/{\mathrm{cm}}^2$) as function of time lag $\mathrm{\Delta }$. The close agreement with the Allan standard deviation theoretically expected for a thermal white noise is clear with the red continuous line that corresponds to $\sigma \left(\mathrm{\Delta }\right)=\sqrt{2k_{\mathrm{B}}T/\left(\gamma \mathrm{\Delta }\right)}$—see Appendix pp2 for a demonstration. Note that the increased fluctuations in the Allan standard deviations at large time lags only comes from a reduction in statistics. (b) The covariance and correlation of covariance evaluated from all the available displacements are plotted as a function of the time lag $\mathrm{\Delta }$ to support the conclusion of the Allan variance. They both show the uncorrelated nature of the noise over a large temporal window of time lags. (c) Experimental time ensemble average MSD measured along the $y$ axis as a function of time lag $\mathrm{\Delta }$ for the different laser intensities given in panel (d). The superimposed dashed line shows the $2D\mathrm{\Delta }$ MSD theoretically expected for a thermal white noise. (d) Diffusion coefficient $D_y$ (blue circles) extracted, from the different $y$-axis MSD shown in panel (c), by fitting the first 15 points of the MSD—following Ref. [23]. Associated error bars at a 99.7% confidence level on the MSD linear regression are displayed, taking into account tracking errors discussed in Appendix pp3. As seen on this panel, experimental values for $D_y$ are in a good agreement with the value $D=k_{\mathrm{B}}T/\gamma$ calculated at given temperature ($T=298.7\mathrm{K}$), viscosity ($\eta =0.88\times {10}^{-3}$ Kg ${\mathrm{m}}^{-1}$ ${\mathrm{s}}^{-1}$) and particle diameter ($d=940\text{nm}$). The gray zone corresponds to uncertainty in the diffusion coefficient coming from errors in temperature ($\delta T=\pm 1.5\mathrm{K}$), viscosity ($\delta \eta =\pm 0.018\times {10}^{-3}$ Kg ${\mathrm{m}}^{-1}$ ${\mathrm{s}}^{-1}$) and particle diameter ($\delta d=\pm 50\mathrm{nm}$).

Figure 3

Ergodic parameters $\epsilon$ (blue circles) displayed, for all time lags, for all the 15 experiments gathered in Figs. 2 and 2. These results are all corrected for tracking errors that particularly impact the data at small time lags, as discussed in details in Appendix pp6. As clearly seen, $\epsilon \left(\mathrm{\Delta }\right)$ follows the evolution expected for an ergodic free Brownian motion. All ratio $\rho \left(\mathrm{\Delta }\right)=〈\overline{\delta y^2\left(\mathrm{\Delta }\right)}〉/〈y^2\left(\mathrm{\Delta }\right)〉$ (red squares) evaluated for the same experiments remain, as expected, close to 1 for all time lags $\mathrm{\Delta }$. We note that both quantities establish the necessary and sufficient conditions for ergodicity discussed in Ref. [24].

Figure 4

(a) Maximal radiation pressure force estimator $F_0$ evaluated as a function of different laser intensities in the single-beam configuration of the experiment sketched in Fig. 1. The estimators (blue points) show the expected linear dependence on the laser power and the continuous black line is a linear fitting performed as a reference. The error bars on the estimators are discussed in Appendix pp7. The Mie calculation (red line) done for the conditions of the experiment (plane wave approximation, spherical dielectric -melamine- particles) is also shown, with its uncertainty (light red surface) detailed in Appendix pp8. The blue surfaces centered around the $F_0=0$ value display the thermal limit at 1, 2, and 3 successive confidence levels. Each experiment corresponds to a given $N$ value ($N=418408,418555,360323,393992,332772,275906,554426$ for increasing laser intensities), hence a given thermal limit in the force detection. (b) With the estimator $F_0$, the entire radiation pressure force profile can be reconstructed. A $F_y\left(z\right)$ profile is shown as an example, corresponding to a $0.18\mathrm{kW}{\mathrm{cm}}^{-2}$ laser intensity (i.e., 15 mW total laser power). The blue surface describes the uncertainty associated with the force profile reconstruction—see main text.

Figure 5

Radiation pressure forces measured zone-by-zone (in successive layers of identical thicknesses $18.4\mu \mathrm{m}$ corresponding to the horizontal widths of all error bars) along the $y$ optical axis for a laser intensity of $0.31\mathrm{kW}/{\mathrm{cm}}^2$—corresponding to a total laser power of 25 mW. These measurements, together with their associated error bars—following the analysis presented in Appendix pp7— are compared to the reconstructed profile using our statistical ensemble approach (blue line) including uncertainties (blue surface) and Mie calculations (red line). The number statistics for each layer of data (from left to right: 8 633, 17 785, 41 573, 45 241, 51 718, 67 707, and 43 249 recorded displacements) impacts the resolution, as clearly seen from the variable vertical widths of the corresponding error bars.

Figure 6

(a) Diffusion coefficients $D_r$ measured along the $r=y$ (blue points) and $r=z$ (red points) axes for different laser intensities. The value of the diffusion coefficient expected from a fixed temperature (measured on average at $T=302.2$ K), viscosity ($\eta =0.8145\times {10}^{-3}$ Kg ${\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$), and particle diameter ($d=940$ nm) is also displayed (black continuous line) with a light-gray surface including errors in the determination of temperature ($\delta T=\pm 1$ K), viscosity ($\delta \eta =\pm 0.018\times {10}^{-3}$ Kg ${\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ due to $\delta T$), and particle size dispersion ($\delta d=\pm 50$ nm). The error bars on the experimental determinations of $D_y$ and $D_z$ are given at a 95% confidence level from the linear regression taken on the variances of the displacement ${\sigma }^2[y(\mathrm{\Delta }+t)-y\left(t\right)]$ along $y$, and ${\sigma }^2[z(\mathrm{\Delta }+t)-z\left(t\right)]$ along $z$, respectively. These variances are shown on (b), noting ${\sigma }_r^2\left(\mathrm{\Delta }\right)={\sigma }^2[r(\mathrm{\Delta }+t)-r\left(t\right)]$ with $r=y$ (blue triangles) and $r=z$ (red squares). These variances, plotted as function of time lag $\mathrm{\Delta }$, are calculated for seven experiments done under the different laser intensities shown on panel (a).

Figure 7

Integration surface for Eq. (E12) on which the two sectors $[t_2>t_1]$ and $[t_2<t_1]$ are distinguished. This defines the appropriate change of variables $(t_1,t_2)\leftrightarrow (t_1,t^{\prime })$, with the line $t_2=t_1+t^{\prime }$ crossing the $t_2=0$ axis at $-t^{\prime }$ and the $t_2=\mathcal{T}-\mathrm{\Delta }$ axis at $\mathcal{T}-\mathrm{\Delta }+t^{\prime }$.

Figure 8

Comparison between ergodic parameters $\epsilon$ before $[{\epsilon }_{\text{exp}}\left(\mathrm{\Delta }\right)$, red triangles] and after correction $[{\epsilon }_0\left(\mathrm{\Delta }\right)$, blue circles], revealing a clear difference at small time lags $\mathrm{\Delta }$ but almost none for larger $\mathrm{\Delta }$. The superimposed dashed line corresponds to the ergodic long-time limit of free Brownian motion calculated in Appendix pp5. These data correspond to the lowest intensity ($0.006\mathrm{kW}/{\mathrm{cm}}^2$) experiment shown in Fig. 2.