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

Energy flow and momentum under paraxial regime