Abstract
In the description of physical systems it is common to discard singular solutions to second order differential equations due to their apparent lack of physical meaning. Nevertheless, it has been demonstrated, using a mathematical-physics approach, that singular solutions can be used in the description of optical beams. In this paper, we construct and study paraxial traveling-waves using the full set of solutions to the paraxial wave equation, and prove that they diverge at infinity. We ascribe that non-physical effect to the paraxial approximation of the Helmholtz equation. Despite this, we show that these traveling waves provide a mathematical-physics framework that unveils orbital angular momentum carrying Laguerre–Gauss beam as the superposition of these traveling waves, and permits a physical description of the self-healing process.
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