Control of diffusion of nanoparticles in an optical vortex lattice

A two-dimensional periodic optical force field, which combines conservative dipolar forces with vortices from radiation pressure, is proposed in order to influence the diffusion properties of optically susceptible nanoparticles. The different deterministic flow patterns are identified. In the low-noise limit, the diffusion coefficient is computed from a mean first passage time and the most probable escape paths are identified for those flow patterns which possess a stable stationary point. Numerical simulations of the associated Langevin equations show remarkable agreement with the analytically deduced expressions. Modifications of the force field are proposed so that a wider range of phenomena could be tested.

Figure 1

An example of flow of type E. Symmetry transformations (see Sec. 3a) applied to the marked stationary points lead to all the stationary points.

Figure 2

Examples of flows types A–D (see Table 2). The deterministic flow lines from $\dot{\mathbf{r}}=\mathbf{f}\left(\mathbf{r}\right)$ [see Eq. (9)] are shown in gray, the stationary points are black dots, labeled by their character (see Table 1). Symmetry transformations (see Sec. 3a) applied to the marked stationary points lead to all the stationary points. The separatrices, delimiting the regions of attraction from the stable nodes, are shown in blue. The red arrowed lines denote the MPEP candidates.

Figure 3

Contours of $W\left(x,y\right)$ in a cell slightly larger than ${\left[0,\pi \right]}^2$ to accommodate the separatrices (in blue lines). The parameters are taken from type A in Fig. 2. This shows (and will be justified below) that the MPEP (horizontal red arrowed line) is along the lines $y=\pi /2$, where $W\left(x,y\right)$ is appreciably smaller than along the other MPEP (vertical red arrowed line) candidate lines $x=\pi /2$.

Figure 4

Contour lines for $K\left({\mathbf{r}}_{s4}\right)$ and for the prefactor in Eq. (25), where $\sqrt{\#}:=\sqrt{\left(1+c_{+}\right)\left(1+c_{-}\right)/\left(1-c_{+}{c}_{-}\right)}$. This last figure shows a remarkable regularity, but we do not have an argument in order to explain it.

Figure 5

The time-dependent diffusion coefficient (particle mean square displacement scaled with time), showing the two distinct dynamic regimes. At short times, nanoparticles are wandering around the stable node with a diffusion coefficient given by $\epsilon /2$, at large times, the (slow) diffusion is controlled by the transition rate for the particle to jump to a neighboring SN. Numerical results correspond to parameter of case A of Fig. 2.

Figure 6

Long time diffusion coefficient corresponding to case A of Fig. 2. Circles are results from numerical simulations of the Langevin equation. The solid line is the analytical result of Eq. (25) and the dashed line indicates the fast diffusion coefficient $\epsilon /2$. At large $\epsilon$, thermal diffusion overpowers the effect of optic forces.

Figure 7

The long time diffusion scaled with the thermal diffusion coefficient $(2D/\epsilon )$ for a set of parameters of the setup. Dashed lines correspond to the theoretical result of Eq. (25) and circles to simulations (same color code as lines). In (a) we present results against $(1+c_{-})$, showing an excellent agreement with theory. It is noted that, for the regime treated in this work $(1+c_{-})>0$, such scaling leads to $\beta$-independent trends. In (b) we plot the scaled diffusion against the lasers' phase shift $\varphi$, illustrating the huge jump in diffusivities this setup allows to control.