Quantum state for the two-variable Hermite polynomial–Gaussian laser modes of the electromagnetic field

In Ref. [7] by using the plane-wave representation of the fundamental Gaussian mode as a seed function, Enderlein and Pampaloni derived higher-order beam modes (Laguerre–Gaussian laser beams) of the electromagnetic field by acting with differential operators on this fundamental solution. In this paper we show that their approach can be improved by directly performing the integration with the aid of the two-variable Hermite polynomials, in so doing these modes can exhibit some new good characters. Their relationship to the eigenmodes of light propagation in the quadratic graded-index (GRIN) medium is pointed out. By introducing the two-mode entangled state representation the quantum states corresponding to these modes are also found which turns out to be the Hermite-polynomial excitation on the two-mode squeezed vacuum state.