To the best of our knowledge, at the present time there is no answer to the fundamental question stated in the title that provides a complete and satisfactory physical description of the structured nature of Hermite-Gauss beams. The purpose of this manuscript is to provide proper answers supported by a rigorous mathematical-physics framework that is physically consistent with the observed propagation of these beams under different circumstances. In the process we identify that the paraxial approximation introduces spurious effects in the solutions that are unphysical. By removing them and using the property of self-healing, that is characteristic to structured beams, we demonstrate that Hermite-Gaussian beams are constituted by the superposition of four traveling waves.
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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