Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

A group-theoretical approach to the paraxial propagation of Hermite–Gaussian modes based on the factorization method is presented. It is shown that the su(1,1) and the su(2) algebras generate the spectrum of propagation constants at any fixed transversal plane. The complete set of HG modes is decomposed into hierarchies that are used to establish the representation spaces of SU(1,1) and SU(2). The corresponding families of generalized coherent states are constructed and the variances of the quadratures and canonical variables are determined.