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Published December 15, 2021 | Submitted
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Snell's Law for Swimmers


Snell's law, which encompasses both refraction and total internal reflection, provides the foundation for ray optics and all lens-based instruments, from microscopes to telescopes. Refraction results when light crosses the interface between media of different refractive index, the dimensionless number that captures how much a medium retards the propagation of light. In this work, we show that the motion of self-propelled particles moving across a drag discontinuity is governed by an analogous Snell's law, allowing for swimmer ray optics. We derive a variant of Snell's law for swimmers moving across media of different viscosities. Just as the ratio of refractive indexes sets the path of a light ray, the ratio of viscosities is shown to determine the trajectories of swimmers. We find that the magnitude of refraction depends on the swimmer's shape, specifically the aspect ratio, as analogous to the wavelength of light. This enables the demixing of a polymorphic, many-shaped, beam of swimmers into distinct monomorphic, single-shaped, beams through a viscosity prism. In turn, beams of monomorphic swimmers can be focused by spherical and gradient viscosity lenses. Completing the analogy, we show that the shape-dependence of the total internal reflection critical angle can be used to create swimmer traps. Such analogies to ray optics suggest a universe of new devices for sorting, concentrating, and analyzing microscopic swimmers is possible.

Additional Information

T.D.R. and P.W.K.R. were funded by the U.S. Department of Energy Award Number Department of Energy award DE-SC0020993. P.W.K.R. was additionally funded by the Office of Naval Research award N00014-18-1-2649. D.O. was partially supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Award Number DE-SC-0010595 and partially through the Moore foundation. J.F.B. was funded by the National Science Foundation Grant 1803662. Author contributions. T.D.R. and P.W.K.R. developed the core concept of the work. J.F.B. and T.D.R. worked on the analytical theory. D.O. built the simulation framework. T.D.R. ran and analyzed the simulations. All authors discussed results and wrote the manuscript.

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August 20, 2023
October 23, 2023