Vectorial lasing with designable topological charges based on Möbius-like correspondence in quasi-BICs

The ability to control topological properties of laser emission represents a fundamental advancement in photonic technology. Achieving topological lasing in a single compact photonic structure is crucial for device integration and miniaturization, but faces significant challenges for designing both the high-quality (high-Q) mode and radiative topological configurations. Recently, bound states in the continuum (BICs), as extraordinary states possessing both ultrahigh Q factors and polarization topological charges, have been demonstrated as a promising platform for compact topological lasing. However, as the cornerstone of BIC lasing's non-trivial properties, topological charges of BICs are protected by real-space structural symmetries, which simultaneously impose fundamental limitations that hinder the designability of lasing topological charges. Here, we propose and experimentally demonstrate a compound cavity design method based on the Möbius-like correspondence in quasi-BICs (q-BICs), by which compact vectorial lasing with designable topological charges can be realized. We reveal the hidden connection between real-space symmetry breaking and eigen-polarizations of q-BICs from the triangular photonic crystal (PhC) slab, manifesting as a Möbius-like correspondence. By splicing PhC slab sectors utilizing this Möbius-like correspondence, we establish a one-to-one correspondence between compound cavities and their lasing topological charges. Vectorial lasing with designable topological charges from -5 to +5 was experimentally realized. Our work establishes a novel BIC-based platform that enables designable topological lasing, providing a promising route toward compact topological sources.