Energy flow and momentum under paraxial regime

The paraxial model of propagation is an approximation to the model described by the d'Alembert equation. It is widely used to describe beam propagation and near-field diffraction patterns. Therefore, its use in optics and acoustics engineering is rather general. On the other hand, energetic balance and momentum in the electromagnetic or acoustic frameworks are well-known and lay in their own physical context. When dealing with paraxial solutions, these analyses are not so clear since paraxial propagation is not supported by the electromagnetic or mechanic theory. The present document establishes the fundamental energy and momentum analysis for paraxial solutions based on the classical approach by studying a Lagrangian density associated to the paraxial equation. Solutions of the paraxial wave equation, such as plane waves, Green's function and Gaussian beams, are studied under this scheme.