Exploring the structured laser beam from the modified ray transfer matrix

We utilize vector decomposition and coordinate projection to enable the calculation of the trajectory of a 3D laser beam within a laser cavity using a 2-by-2 ray transfer matrix. By comparing experiments with theory, we have synthesized the ray transfer matrix within the laser cavity. This matrix provides a clear representation of the origin and deformation of the structured laser beam, as well as the connection between initial conditions and orbital angular momentum.