Lagrangian Manifolds in the Theory of Wave Beams and Solutions of the Helmholtz Equation

This paper describes an approach to constructing the asymptotics of Gaussian beams, based on the theory of the canonical Maslov operator and the study of the dynamics and singularities of the corresponding Lagrangian manifolds in the phase space. As an example, we construct global asymptotics of Laguerre – Gauss beams, which are solutions of the Helmholtz equation in the paraxial approximation. Depending on the type of the beam and the emerging singularity on the Lagrangian manifold, asymptotics are expressed in terms of the Airy function or the Bessel function. One of the advantages of the described approach is that we can abandon the paraxial approximation and construct global asymptotics in terms of special functions also for solutions of the original Helmholtz equation, which is illustrated by an example.