Controlling the wavefront and manipulating the polarization of the electromagnetic wave using an ultrathin flat device are highly desirable in many emerging fields. To shape the wavefront between two decoupled orthogonal circular polarization states, that is, the right-hand circular polarization (RCP) and the left-hand circular polarization (LCP), most state-of-the-art metasurfaces (MSs) combine the propagation phase and Pancharatnam–Berry phase into meta-atoms. This article proposes a different strategy to fully decouple the LCP and RCP and control their wavefronts independently. By taking advantage of the conductive and dielectric multi-material-integrated additive manufacturing technique, the proposed transmissive MS has an ultrathin thickness (0.11 free-space wavelength) and controls the LCP and RCP wavefronts independently under linearly polarized incidence illumination. The proposed meta-atom consists of a receiving antenna on the top, a transmitting antenna at the bottom with a strip-line connecting them. The strip-line introduces the same phase shifts for both RCP and LCP waves, while the transmitting antenna with in-plane rotation leads to the opposite phase shifts for RCP and LCP waves. Therefore, the phase delays from the strip-line and the angular rotation of the transmitting antenna provide two degrees of freedom, enabling independent beam shaping of LCP and RCP waves. Two MSs with different functionalities are printed for proof-of-concept, and the performances are experimentally verified.

We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tut48], and sampling can be reduced to counting [JVV86; JS96]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC.For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csa75]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery.

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