Scaling symmetry and conserved charge for shape-invariant optical fields
Omar El Gawhary and Sergio Severini
Accepted
In this work we present an extensive study of the scaling symmetry typical of a paraxial wave theory. In particular, by means of a Lagrangian approach we derive the conservation law and the corresponding generalized charge associated to the scale invariance symmetry. On a general ground, such a conserved charge, qs say, can take any value that remains constant during propagation. However, it is explicitly proven that for the whole class of physically realizable shape invariant fields, i.e. fields whose intensity distribution maintains its shape on propagation, qs must necessarily vanish. Finally, an interesting relation between such charge qs and the effective radius of a beam, as introduced by Siegman some years ago, is derived.