Laser transverse modes with SU(2) representation

The direct generation of a structured beam from a laser resonator is closely related to the formation of transverse modes. For a long time, the representation of the transverse mode of the spherical cavity is usually divided into Hermite-Gaussian (HG) and Lager-Gaussian (LG) modes. A large number of experimental results show that the generalized representation of the transverse mode is the Hermite-Laguerre-Gaussian (HLG) mode. We give a detailed overview of the theoretical description of the HLG mode from the representation of SU(2) in the Jordan-Schwinger diagram. We further show that the HLG modes can be exactly derived as the superposition of the elliptical modes without involving Hermite and Laguerre polynomials.