Originating an integral formula and using the quantum Fourier transform to decompose the Hermite-Laguerre-Gaussian modes into elliptical orbital modes

We exploit the SU(2) representation of the Hermite-Laguerre-Gaussian (HLG) mode to manifest the successive transformation between Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes. We theoretically confirmed that the time-dependent coherent state for the HLG modes can be simplified as a closed form of Gaussian wave packet. We further employ the explicit closed form to originate an integral of the Gaussian wave-packet state over the elliptical orbit to represent the elliptical orbital mode with fractional orbital angular momentum. On the other hand, we also derive the elliptical orbital mode as the superposition of the degenerate HLG modes. By using the derived formulae and the quantum Fourier transform, the HLG mode is inversely expressed as the superposition of the elliptical orbital modes. The derived representation unambiguously reveals the connection between HLG modes and bundles of elliptical orbits.