Brownian motion in a nonhomogeneous force field and photonic force microscope

The photonic force microscope (PFM) is an opto-mechanical technique that uses an optically trapped probe to measure forces in the range of pico to femto Newton. For a correct use of the PFM, the force field has to be homogeneous on the scale of the Brownian motion of the trapped probe. This condition implicates that the force field must be conservative, excluding the possibility of a rotational component. However, there are cases where these assumptions are not fulfilled. Here, we show how to expand the PFM technique in order to deal with these cases. We introduce the theory of this enhanced PFM and we propose a concrete analysis workflow to reconstruct the force field from the experimental time series of the probe position. Furthermore, we experimentally verify some particularly important cases, namely, the case of a conservative and of a rotational force field.

Figure 1

(Color online) Examples of force fields that cannot be correctly probed with a classical PFM because they vary on the scale of the Brownian motion of the trapped probe, i.e., in a range of tens to hundreds of nanometers (a possible range is indicated by the red bars): (a) forces produced by a surface plasmon polariton in the presence of a patterned surface on a $50\text{−}\mathrm{nm}$-radius dielectric particle (reproduced from [32]); (b) trapping potential for a $10\text{−}\mathrm{nm}$-diam dielectric particle near a $10\text{−}\mathrm{nm}$-wide gold tip in water illuminated by a $810\mathrm{nm}$ monochromatic light beam (reproduced from [34]); and (c) force field acting on a $500\text{−}\mathrm{nm}$-radius dielectric particle in the focal plane of a highly focused Laguerre-Gaussian beam (reproduced from [35]).

Figure 2

Stability diagram. Assuming $\varphi >0$, the stability of the system is shown as a function of the parameters $\Omega ∕\varphi$ and $\mathrm{\Delta }\varphi ∕\varphi$. The white region satisfies the stability conditions (9). The dashed lines represent the $\mathrm{\Delta }\varphi =\mid \Omega \mid$ and $\mathrm{\Delta }\varphi =\varphi$ curves. The dots represent the parameters that are further investigated in Figs. 3, 4, 5.

Figure 3

(Color online) Brownian motion in a force field [Eq. (7)]. The arrows show the force field vectors for various values of the parameters $\mathrm{\Delta }\varphi ∕\varphi$ and $\Omega ∕\varphi$. The shadowed areas show the probability distribution function (PDF) of the probe position in the corresponding force field.

Figure 4

(Color online) Autocorrelation and cross-correlation functions [Eqs. (13, 14, 15, 16)] for various values of the parameters $\mathrm{\Delta }\varphi ∕\varphi$ and $\psi ∕\varphi$: $r_{xx}$ (black continuous line), $r_{yy}$ (black dotted line), $r_{xy}$ (gray—green in the online version–continuous line), and $r_{yx}$ (gray—green in the online version—dotted line).

Figure 5

(Color online) Functions (25, 26) independent from choice of the reference system for various values of the parameters $\mathrm{\Delta }\varphi ∕\varphi$ and $\Omega ∕\varphi$: $S_{ACF}\left(\mathrm{\Delta }t\right)$ (black line) and $D_{CCF}\left(\mathrm{\Delta }t\right)$ (gray—green in the online version—line).

Figure 6

Functions that depend on the orientation of the coordinate system [Eqs. (27, 28)] for various values of the angle with respect to the coordinate system in Sec. 2B1. The markers highlight the values for $\mathrm{\Delta }t=0$.

Figure 7

(Color online) Data analysis of numerically simulated time series. [(a) and (b)] Probability density function for a Brownian particle under the influence of the force-field (simulated data $30\mathrm{s}$ at $16\mathrm{kHz}$); in (a) the force field is purely conservative, while in (b) it has a rotational component. [(c) and (d)] Invariant function, $S_{ACF}\left(\mathrm{\Delta }t\right)$ (black line) and $D_{CCF}\left(\mathrm{\Delta }t\right)$ (gray—red in the online version—line) (calculated from the simulated data windowed in $1\mathrm{s}$ intervals and averaged). [(e) and (f)] Force fields reconstructed from the simulated data.

Figure 8

(Color online) Experimental setup: 1, $100\times 1.3\mathrm{NA}$ objective; 2, $50\times$ objective.

Figure 9

(Color online) Conservative force field. (a) Experimental configuration with two spinning spheres (the horizontal $x$ and vertical $y$ axes are centered on the probe particle) and (b) hydrodynamic component of the force field (from hydrodynamic theory). The black bar represents a length of $1\mathrm{\mu }\mathrm{m}$. (c) Experimental invariant functions $S_{ACF}\left(\mathrm{\Delta }t\right)$ (black line) and $D_{CCF}\left(\mathrm{\Delta }t\right)$ (gray—red in the online version—line), and respective fitting to the theoretical shape (dotted lines). (d) Experimental probability density function and reconstructed total force field; inset: reconstructed hydrodynamic force field. The reconstruction is accurate only over the region visited by the Brownian particle. Ten datasets obtained during $15\mathrm{s}$ with sampling rate $2000\mathrm{Hz}$ were acquired and analyzed.

Figure 10

(Color online) Rotational force field. (a) Experimental configuration with four spinning spheres (the horizontal $x$ and vertical $y$ axes are centered on the probe particle) and (b) hydrodynamic component of the force field (from hydrodynamic theory). The black bar represents a length of $3\mathrm{\mu }\mathrm{m}$. (c) Experimental invariant functions $S_{ACF}\left(\mathrm{\Delta }t\right)$ (black line) and $D_{CCF}\left(\mathrm{\Delta }t\right)$ (gray—red in the online version—line), and respective fitting to the theoretical shape (dotted lines). (d) Experimental probability density function and reconstructed total force field; inset: reconstructed hydrodynamic force field. The reconstruction is accurate only over the region visited by the Brownian particle. Ten datasets obtained during $15\mathrm{s}$ with sampling rate $2000\mathrm{Hz}$ were acquired and analyzed.