Analysis and Optimization of Transmission Resonances Through Periodic Plasmonic Structures

This dissertation is focused on electromagnetic wave propagation through periodic structures. The work consists of two main parts. In the first part we deal with electromagnetic radiation through periodic aperture arrays in an "infinitesimally thin perfect electric conductor." Electromagnetic radiation through a bounded medium with two-dimensional geometry and finite width is considered as the second model. First of all, we formulate a mathematical model for electromagnetic transmission through periodic hole arrays in a thin perfect electric conductor, where we can obtain an explicit linear operator equation for the tangential components of the electric and magnetic fields inside the apertures. The linear operator is regularized to ensure a stable numerical solution. We then analyze the solvability of the linear operator equation with a truncated, regularized and mollified operator. The extension of our model to the case of an arbitrary incoming wave by adding a Bloch condition is included. We establish the conservation of energy, stating that the total energy of the reflected and transmitted waves is equal to the energy of the incident wave. We then introduce some numerical experiments and show numerically that energy dissipation occurs with the regularization. As the second model, we consider electromagnetic radiation through a bounded medium. It is assumed that the material parameters are constant in one direction, so that the problem can be formulated in two-dimensional geometries for appropriate polarizations. We investigate time-harmonic electromagnetic wave propagation through nonmagnetic heterogeneous media for which the complex dielectric coefficients have opposite signs in the real part. It is known that the imaginary part accounts for energy absorption. We formulate an equivalent variational problem over a bounded region with transparent boundary conditions and then show that the problem has a unique solution over all range of incidence angles, provided energy dissipation is present. The problem of designing a structure for which a desired transmission energy spectrum is attained, is considered. The optimal design problem is stated as a minimization problem. We then show that the minimization problem with an appropriate admissible set has at least one solution. We present the numerical experiment that indicates surface plasmons occur at the interface of some structure and then show the numerical experiment of the transmission energy spectrum for some structure for a various value of the dielectric coefficient.