14 September 2007 Description of caustic structures in non-linear media: envelope of characteristic trajectories for the non-linear Schrödinger equation
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Abstract
We describe the mode solutions for the Helmholtz Equation using the operator formalism. The study is extended to the structural solution for the focused non-linear Schrödinger equation (NLSE). With this treatment, we obtain for the NLSE a reduced partial differential equation, whose characteristic solution has an eikonal structure which allows us a geometrical analysis. Focusing region in non-linear media is described by means of an envelope region of eikonal trajectories establishing similar behaviors with caustic structures. In particular, if the boundary condition consists of a slit shape curve, the focusing profile corresponds with the evolute of the curve. In general, the profile satisfies a non-linear partial differential equation whose structure remains non-variable under changes of variables which may represents scaling or rotations. This feature permits us to extend the analysis to other kind of focusing regions, such as focusing vortex.
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J. C. Juarez-Morales, G. Martinez-Niconoff, J. Munoz-Lopez, "Description of caustic structures in non-linear media: envelope of characteristic trajectories for the non-linear Schrödinger equation", Proc. SPIE 6669, Seventh International Conference on Solid State Lighting, 66691F (14 September 2007); doi: 10.1117/12.732202; https://doi.org/10.1117/12.732202
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