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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Proceedings Volume 6669, Seventh International Conference on Solid State Lighting; 66691F (2007); doi: 10.1117/12.732202
Event: Optical Engineering + Applications, 2007, San Diego, California, United States
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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KEYWORDS
Partial differential equations

Nonlinear optics

Refractive index

Wave propagation

Prototyping

Coherence (optics)

Diffraction

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