Photopatterned liquid crystal elastomer templates living tissue with predesigned alignment patterns and topological defects.

Active systems composed of self-propelled units show fascinating transitions from Brownian-like dynamics to collective coherent motion. Swirling of swimming bacteria is a spectacular example. This study demonstrates that a nematic liquid crystal environment patterned as a spiral vortex controls individual-to-collective transition in bacterial swirls and defines whether they expand or shrink. In dilute dispersions, the bacteria swim along open spiral trajectories, following the preimposed molecular orientation. The trajectories are nonpolar. As their concentration exceeds some threshold, the bacteria condense into unipolar circular swirls resembling stable limit cycles. This collective circular motion is controlled by the spiral angle that defines the splay-to-bend ratio of the background director. Vortices with dominating splay shrink the swirls toward the center, while vortices with dominating bend expand them to the periphery. Spiraling vortices of ${45}^{\circ }$ with splay-bend parity produce the most stable swirls. All the dynamic scenarios are explained by hydrodynamic interactions of bacteria mediated by the patterned passive nematic environment and by the coupling between the concentration and orientation. The acquired knowledge of how to control individual and collective motion of microswimmers by a nematic environment can help in the development of microscopic mechanical systems.

In this paper, we analyze a few aspects of Pancharatnam–Berry (PB) phase optical lenses. First, we provide theoretical formulas on how the optical efficiency, focal length, and point spread function depend on wavelength and validate them with numerical calculations based on Fresnel diffraction theory. Second, we perform numerical studies on how optical efficiency is affected by discretization of the PB phase. We find that phase discretization significantly lowers the optical efficiency for low $f$-number PB lenses.