h i g h l i g h t s• Ladder and shift operators for the Hermite-Gaussian modes are constructed.• These operators are used to obtain the generators of the dynamical algebras.• The Barut-Girardello and Perelomov coherent states are determined.• The uncertainty relations are considered in connection to the beam quality factor. a r t i c l e i n f o 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.

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Ladder and shift operators are determined for the set of Hermite-Gaussian modes associated with an optical medium with quadratic refractive index profile. These operators allow to establish irreducible representations of the su(1, 1) and su(2) algebras. Glauber coherent states, as well as su(1, 1) and su(2) generalized coherent states, were constructed as solutions of differential equations admitting separation of variables. The dynamics of these coherent states along the optical axis is also evaluated.

We report a new set of Laguerre-Gaussian wave-packets that propagate with periodical self-focusing and finite beam width in weakly guiding inhomogeneous media. These wave-packets are solutions to the paraxial form of the wave equation for a medium with parabolic refractive index. The beam width is defined as a solution of the Ermakov equation associated to the harmonic oscillator, so its amplitude is modulated by the strength of the medium inhomogeneity. The conventional Laguerre-Gaussian modes, available for homogenous media, are recovered as a particular case.