Ray Optics for Gliders

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.