Periodically-modulated unipolar and bipolar orders in nematic fluids towards miniaturized nonlinear vectorial optics

Complex polar topological configurations are the foundation for exploring exotic phases and nontrivial emergent phenomena in condensed-matter physics. While numerous topological configurations like skyrmions and merons are known to occur spontaneously in various materials, achieving the designability of magnetically- and electrically-polar textures in a controlled manner is a long-standing challenge in domain engineering. Here, we discover an unprecedented mesophase that exhibits unipolar and bipolar orders, where both apolar and polar orientational order parameters are modulated. The polar state enables defect-free polar patterning with designability and controllability on large scales. A case study demonstrates the facile generation of second-harmonic perfect vector beams through photopatterned topological polar liquid crystal superstructures. Remarkably, this single-layer, micrometer-thin-film device attains functionalities comparable to multiple conventional optical components, establishing a foundation for advanced nonlinear vectorial optics at the micrometer scale.