Eigenstates of the full Maxwell equations for a two-constituent composite medium and their application to a calculation of the local electric field of a time dependent point electric dipole in a flat-slabs microstructure

Abstract

An exact calculation of the local electric field ${\bf E}({\bf r})$ is described for the case of a time dependent point electric dipole ${\bf p}e^{-i\omega t}$ in the top layer of an $\epsilon_2$, $\epsilon_1$, $\epsilon_2$ three parallel slabs composite structure, where the $\epsilon_1$ layer has a finite thickness $2d$ but the $\epsilon_2$ layers are infinitely thick. For this purpose we first calculate all the eigenstates of the full Maxwell equations for the case where $\mu=1$ everywhere in the system. The eigenvalues appear as special, non-physical values of $\epsilon_1$ when $\epsilon_2$ is given. These eigenstates are then used to develop an exact expansion for the physical values of ${\bf E}({\bf r})$ in the system characterized by physical values of $\epsilon_1(\omega)$ and $\epsilon_2(\omega)$. Results are compared with those of a previous calculation of the local field of a time dependent point charge in the quasi-static regime. Numerical results are shown for the local electric field in practically important configurations where attaining an optical image with sub-wavelength resolution has practical significance.