Description of Caustic Structures in Non-Linear Media: Envelope of Characteristic Trajectories for the Non-Linear Schroedinger Equation

摘要

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 Schrodinger 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.