# Vectorial microlasers with designable topological charges based on M\"{o}bius-like correspondence in quasi-BICs

## TL;DR

This paper introduces a novel method to design compact vectorial microlasers with customizable topological charges using M"{o}bius-like correspondence in quasi-BICs, overcoming previous symmetry-based limitations.

## Contribution

The authors propose and experimentally demonstrate a compound cavity design based on M"{o}bius-like correspondence in quasi-BICs, enabling tunable topological charges in microlasers.

## Key findings

- Realization of vectorial lasing with topological charges from -5 to +5.
- Establishment of a one-to-one correspondence between cavity design and topological charge.
- Experimental validation of the M"{o}bius-like correspondence in photonic crystal slabs.

## Abstract

The ability to control topological properties of laser emission represents a fundamental advancement in photonic technology. Achieving topological laser 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 lasers. 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 their designability of lasing topological charges. Here, we propose and experimentally demonstrate a compound cavity design method based on the M\"{o}bius-like correspondence in quasi-BICs (q-BICs), by which compact vectorial microlasers 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 M\"{o}bius-like correspondence. By splicing PhC slab sectors utilizing this M\"{o}bius-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 $+5$ were experimentally realized. Our work establishes a novel BIC-based platform that enables designable topological lasing, providing a promising route toward compact topological sources.