Separable Two-Dimensional Landscapes

Figure 4: Schematic representation of a landscape sinusoidally modulated in both the $\hat x$ and $\hat y$ directions according to Eq. (48).

The one-dimensional potential of Eq. (25) is one of the few landscapes that allows for exact solutions of the equations of motion. Two-dimensional landscapes can be solved analytically only if the potential can be written as a sum of modulations in the $\hat x$ and $\hat y$ directions. In particular, we can consider separate sinusoidal modulation with the same period in the two directions:

$$I(\ensuremath{{\mathbf{r}}}\xspace ) = I_0 \, \left[ \sin (k_0 y) + \sin (k_0 x) \right],$$ (47)

shown schematically in Fig. 4. This landscape is interesting mainly because it leads to decoupled equations of motion:

$$\frac{dx}{dt} = \xi^{-1} V_0 k_0 \, \cos (k_0 x) + v_0 \, \cos \theta + \gamma(t)$$ (48)

$$\frac{dy}{dt} = \xi^{-1} V_0 k_0 \, \cos (k_0 y) + v_0 \, \sin \theta + \gamma(t),$$ (49)

where $\gamma(t) = \Gamma/\xi$. Nevertheless, such a landscape could be implemented experimentally using optical forces. For example, mutually incoherent pairs of laser beams intersecting at right angles would lead to a potential of the form given in Eq. (48), as would pairs with orthogonal polarization.

Since the motions in the $\hat x$ and $\hat y$ directions are independent in this case, the same exponential sensitivity to particle size, Eq. (32) is obtained, in the absence of thermal forces. As well, the same integration can be used to determine the average deflection angle for free-flowing particles, analogous to Eq. (38):

$$\tan \psi = \frac{\sqrt{\sin^2\theta - \eta^2}}{\sqrt{\cos^2\theta - \eta^2}}.$$ (50)

Similarly, for finite temperatures, the mean deflection angle is given by

$$\tan \psi = \tan \theta \; \frac{1 + \frac{2 \sin\theta}{\eta} \,... ...ta} \, \mathrm{Im}\left[S_1\left(\tau, \frac{\cos \theta}{\eta}\right)\right]}.$$ (51)

Figure 5: (a) Deflection as a function of orientation for a separable two-dimensionally modulated landscape at $\eta (a) = 0.1$, 0.2, and 0.3. The diagonal dashed line indicates the result with no landscape. (b) Size dependence of the deflection angle for $\eta _0 = 0.4$ and $\tan \theta = 0.441$.

The zero temperature deflection angle is plotted in Fig. 5 as a function of the angle $\theta$ of the driving force, for a fixed value of the normalized potential $\eta$. Also plotted is $\psi$ as a function of particle size $a$ for a fixed $\theta$. The results can be seen to be similar to those obtained with one-dimensional fringes. In other words, no qualitative difference is obtained in this case by modulating in two directions rather than just one.

In order to see new effects of increased dimensionality, it is necessary to consider landscapes that cannot be separated into one-dimensional terms; i.e., landscapes where the motion in one dimension depends on the position in the other. Analytical solutions are not available for such landscapes. However, it is possible to develop limiting arguments that illustrate the novel features of transport in such landscapes, including the continued possibility for sorting that is exponentially sensitive to particle size.