We analyze the diffraction field when changes in the curvature function of the boundary condition are implemented. The study is performed using differential geometry models with a curvature function displaying local behavior. Depending on the sign of curvature, we classify the diffraction field as elliptic, hyperbolic, or parabolic. In particular, it is shown that the optical field is organized around the parabolic regions, which correspond to focusing regions. The model is experimentally corroborated by applying a coordinate transformation to the transmittance of a zone plate. The reason to use this transmittance comes from the fact that its diffraction field displays multiple foci allowing identification, description, and control of bifurcations and morphogenesis effects, which are studied using the curvature function.
We analyze the resonant interaction between cumulus of nano-particles distributed on a two-dimensional array controlling the polarization states on the illumination, this allows controlling the dipole moment induced in a tunable-way obtaining an analytic expression for the refractive index. The resonant effects depend on the parameters that characterize the spatial distribution of the particle arrangement. We present two cases, firstly the interaction is described using a linear polarization on a linear particle array, and secondly it is obtained using circular polarization inducing resonant interaction between ring-particle kind structures. The refractive index associated to both configurations is implemented in the Fresnel equations for the study of the reflectivity and transmittance of optical fields. As a main result of the analysis is that we can to identify and control the parameters required for the synthesis of metamaterials. Computer simulations are presented.
The phase function of optical fields collapse on focusing regions generating a discontinuity in the amplitude function, this induces sources or sinks that corresponds with the topological charge. When the previous comments are transferred to the Plasmon optics context, the discontinuity of the electromagnetic field generates a real distribution of electric charge. This distribution has associated a geometry which can be obtained from the boundary condition of the field. A dynamical character can be implemented on the charge distribution using partial coherence processes in the illumination configuration for the synthesis of the Plasmon field, generating local current distributions modifying selectively the electromagnetic field properties. The model is performed using as a prototype the interaction between plasmon fields Pearcey and Airy kind. Both of them have associated a catastrophe function to the phase function, this mathematical representation allows us to identify and quantify the discontinuity of the electromagnetic field. The computational simulations show that the charge/current distributions present non-linear effects, which offers applications for tunable spectroscopy, plasmonic tweezers, etc.