A note on the Airy beams in the light of the symmetry algebra based approach

The Airy function based solutions of the paraxial wave equation in planar geometry are framed within the unifying context of a well-known Lie algebra formalism, which is shown to account for both infinite and finite energy solutions. In fact, the finite energy solutions can be obtained by complexification of the relevant propagation parameter. Also, they are alternatively deduced by taking advantage of the added degree of freedom they display, inherent in the symmetry algebra based approach, i.e. the eigenvalue of the operator (in the symmetry algebra), from which the solutions can be understood to originate.