Ray Optics for Gliders

Abstract

Control of self-propelled particles is central to the development of many microrobotic technologies, from dynamically reconfigurable materials to advanced lab-on-a-chip systems. However, there are few physical principles by which particle trajectories can be specified and can be used to generate a wide range of behaviors. Within the field of ray optics, a single principle for controlling the trajectory of light�Snell's law� yields an intuitive framework for engineering a broad range of devices, from microscopes to cameras and telescopes. Here we show that the motion of self-propelled particles gliding across a resistance discontinuity is governed by a variant of Snell's law, and develop a corresponding ray optics for gliders. Just as the ratio of refractive indexes sets the path of a light ray, the ratio of resistance coefficients is shown to determine the trajectories of gliders. The magnitude of refraction depends on the glider's shape, in particular its aspect ratio, which serves as an analogue to the wavelength of light. This enables the demixing of a polymorphic, many-shaped, beam of gliders into distinct monomorphic, single-shaped, beams through a friction prism. In turn, beams of monomorphic gliders can be focused by spherical and gradient friction lenses. Alternatively, the critical angle for total internal reflection can be used to create shape-selective glider traps. Overall our work suggests that furthering the analogy between light and microscopic gliders may be used for sorting, concentrating, and analyzing self-propelled particles.

Active particles crossing sharp viscosity gradients

“…Regardless of where the particle is reflected, the entire scattering process is symmetric (about the point when as shown in Fig. 3 b) and hence obeys the reflection law for all particles when , as previously shown for gliders 44 .…”

“…As a result, if active particles approach a viscosity interface at a sufficiently shallow angle they can be reflected if swimming from low to high viscosity; otherwise, they simply cross the interface undergoing a degree of reorientation set by the relative viscosity difference. This is similar to the refraction or reflection of the light due to a change in refractive index and the law we derive governing the reorientation of neutral swimmers similar to Snell's law of ray optics (as previously shown for gliders on a frictional substrate 44 ).…”

“…Our results are also quite similar to recent theoretical work modeling gliders moving across a substrate features a jump in frictional properties. In particular, the functional form of the reorientation law we find is identical to that found for gliders 44 . However that work, and other studies where the propulsive force is similarly fixed 39 , shows reorientation towards higher viscosities as one might expect due to the modulation of drag alone.…”

“…The reorientation is independent of the speed of the particle due to the linearity of the Stokes equations, in this case both the thrust generated by the particle and the drag felt by the particle would be proportional to the mode. This form of reorientation law, , was found for gliders moving across a substrate featuring a jump in frictional properties 44 . In that case where , are torque-rotation and torque-translation resistance coefficients of the particle respectively.…”

Optimal active particle navigation meets machine learning (a)

“…In piecewise constant environments, the optimal trajectory is also piecewise constant, yielding Snell's law for active particles which involves a generalized refractive index that can also be negative as for light in meta-materials [88,89]. (See also [90].) Other exact solutions for the Euler-Lagrange equation can be obtained by exploiting conservation laws (symmetries) showing that the shortest path is typically not the fastest in complex environments.…”