The study of structured laser beams has been one of the most active fields of research for decades, particularly in exploring laser beams with orbital angular momentum. The direct generation of structured beams from laser resonators is deeply associated with the formation of transverse modes. The wave representations of transverse modes of spherical cavities are usually categorized into Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) modes for a long time. Enormous experimental results have revealed that the generalized representation for the transverse modes is the Hermite–LG (HLG) modes. We make a detailed overview for the theoretical description of the HLG modes from the representation of the spectral unitary group of order 2 in the Jordan–Schwinger map. Furthermore, we overview how to derive the integral formula for the elliptical modes based on the Gaussian wave-packet state and the inverse Fourier transform. The relationship between the HLG modes and elliptical modes is linked by the quantum Fourier transform. The most striking result is that the HLG modes can be exactly derived as the superposition of the elliptical modes without involving Hermite and Laguerre polynomials. Finally, we discuss the application of the HLG modes in characterizing the propagation evolution of the vortex structures of HG beams transformed by an astigmatic mode converter. This overview certainly provides not only a novel formula for transverse modes, but also a pedagogical insight into quantum physics.
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