Linear Fringes

In part to motivate a discussion of periodic potential energy landscapes, we first consider how objects traverse a single trench or barrier arranged at an angle to the driving force. This kind of landscape may be realized, for example, by creating a linear optical trap with a cylindrical lens or a diffractive line generator. Because such an optical landscape can act as either a barrier or a trench, depending on the sign of $\alpha$, we will refer to both as fringes. In either case, a fringe aligned with the $\hat x$ axis inhibits transport in the transverse direction. We model the landscape as a Gaussian profile of intrinsic width $w$,

$$I(\ensuremath{{\mathbf{r}}}\xspace ) = I_0 \, \exp\left(-\frac{y^2}{2 w^2}\right).$$ (17)

Using the object's form factor as defined in Eq. (14), the associated potential is

$$V(\ensuremath{{\mathbf{r}}}\xspace ) = 2\pi\alpha I_0 \, \frac{a^3 w}{\sigma(a)} \, \exp\left(-\frac{y^2}{2 \sigma^2(a)}\right).$$ (18)

The fringe's apparent width to a particle of size $a$ is broadened to $\sigma(a) = \sqrt{a^2 + w^2}$.

In the limit that thermal forces may be ignored, the equations of motion reduce to the deterministic form

$$\frac{dx}{dt} = v_0 \, \cos \theta$$ (19)

$$\frac{dy}{dt} = \xi^{-1} \, F_y(y) + v_0 \, \sin \theta,$$ (20)

where the landscape-free drift speed is $v_0 = \xi^{-1} F_0$ and

$$F_y(y) = 2\pi \, \alpha I_0 \, \frac{a^3 w}{\sigma^3(a)} \, y \, \exp\left(- \frac{y^2}{2 \sigma^2(a)}\right).$$ (21)

The landscape's restoring force $F_y(y)$ reaches a maximum at a distance $y = \ensuremath{y_\text{max}}\xspace$ from the fringe's axis, with $\ensuremath{y_\text{max}}\xspace = \sigma(a)$ for our particular example. If $F_0 \sin \theta > F_y(\ensuremath{y_\text{max}}\xspace )$ then a particle can cross the barrier. Such particles may be said to escape the barrier. By contrast, particles for whom the barrier is insurmountable travel unimpeded along the $\hat x$ direction at speed $v_x = v_0 \cos \theta$. Such particles are said to be locked in to the landscape.

The marginal angle $\theta _m$ at which an object just barely remains locked in to the barrier determines which objects are deflected and which are not. The dependence of $\theta _m$ on particle size and other characteristics establishes the sensitivity of the sorting technique. Referring to Eqs. (20) and (21), the condition for locked-in transport,

$$\sin \theta \leq \sin \theta_m \equiv \frac{F_y(\ensuremath{y_\text{max}}\xspace )}{F_0}$$ (22)

$$= \frac{\lvert \alpha \rvert \, I_0}{F_0} \, \frac{2 \pi}{\sqrt{e}} \, \frac{a^3 w}{a^2 + w^2},$$ (23)

applies both to attractive trenches ( $\ensuremath{y_\text{max}}\xspace =+\sigma$) and repulsive barriers ( $\ensuremath{y_\text{max}}\xspace =-\sigma$). The general result, Eq. (22), applies even if $f(\ensuremath{{\mathbf{r}}}\xspace )$ is not separable because, in this case at least, $I(\ensuremath{{\mathbf{r}}}\xspace )$ is independent of $x$.

The particular result in Eq. (23) shows that the marginal lock-in angle depends only algebraically on size, and only linearly on other properties through $\alpha$. This is neither better nor worse that the performance offered by other established techniques such as gel electrophoresis or flow-field fractionation (8). One substantial benefit offered by selective transport across a fringe is its ability to process a continuous stream of objects rather than being restricted to discrete batches. The selected fraction, moreover, can be tuned continuously, for example by adjusting $I_0$, $F_0$, $w$, or $\theta$. Optical implementations also can be optimized by varying the wavelength of light, in which case resonances might be exploited as a complementary mechanism for size separation.

Although a single fringe's performance is somewhat lackluster, one might expect multiple fringes to fare better. The first step along this direction is to consider a pair of parallel Gaussian fringes. The effective potential is the sum of two single-fringe potentials:

$$\begin{multline} V(\ensuremath{{\mathbf{r}}}\xspace ) = 2\pi\alpha I_0 \, \f... ...exp\left(-\frac{\left(y-b/2\right)^2}{2 \sigma^2}\right) \right], \end{multline}$$

where $b$ is the fringe separation. If $b < \sigma$, the two fringes overlap enough that the landscape resembles a single, broadened fringe. Again, no more than algebraic selectivity should be expected. In the opposite limit, $b \gg \sigma$, the fringes are independent, and particles cross the double barrier with the same facility with which they cross one. Neither of these cases offers benefits over the single fringe.

For intermediate $b$, on the other hand, the landscape consists of two unequal barriers, the smaller of which lies between the two fringes. The smaller barrier's height depends strongly on $b/\sigma$, which, in turn, depends on the particle size $a$. This lower intermediate barrier does not affect the fringes' overall ability to separate objects, which is dominated by the larger barrier. It suggests the possibility, however, that transport across $N$ overlapping fringes could be highly sensitive to particle size. Those particles not able to jump the inter-fringe barriers will be locked in and swept aside while others will hop from one fringe to the next across the field. Highly selective sorting is thus possible if edge effects due to the first or last fringe (in the case of trenches or barriers, respectively) can be circumvented.