Explicit solutions of the inhomogeneous paraxial wave equation in a linear and quadratic approximation are applied to wave fields with invariant features, such as oscillating laser beams in a parabolic waveguide and spiral light beams in varying media. A similar effect of superfocusing of particle beams in a thin monocrystal film, harmonic oscillations of cold trapped atoms, and motion in magnetic field are also mentioned.
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Mahalov, Alex; Suazo, Erwin; Suslov, Sergei K. (2013). Spiral laser beams in inhomogeneous media. Optica Publishing Group. Collection. https://doi.org/10.6084/m9.figshare.c.3754661.v1