ON THE LACK OF EXTERNAL RESPONSE OF A NONLINEAR MEDIUM IN THE SECOND-HARMONIC GENERATION PROCESS

This paper concerns the scattering problem for a nonlinear medium of compact support, D, with second-harmonic generation. Such a medium, when probed with monochromatic light beams at frequency ω, generates additional waves at frequency 2ω. The response of the medium is governed by a system of two coupled semilinear PDEs for the electric fields at frequencies ω and 2ω. We investigate whether there are situations in which the generated 2ω wave is localized inside D, that is, whether the nonlinear interaction of the medium with the probing wave is invisible to an outside observer. This leads to the analysis of a semilinear elliptic system formulated in D with nonstandard boundary conditions. The analysis presented here sets up a mathematical framework needed to investigate a multitude of questions related to nonlinear scattering with second-harmonic generation.