O. El Gawhary , S. Severini, "Scaling symmetry and conserved charge for shape-invariant optical fields", Phys. Rev. A, 87, 023811, 2013.
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 with the scale invariance symmetry. In general, such a conserved charge,qssay, 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, that is, fields whose intensity distribution maintains its shape on propagation,qsmust necessarily vanish. Finally, an interesting relation between such chargeqsand the effective radius of a beam, as introduced by Siegman some years ago, is derived.
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