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 $\mathbf{E}(\mathbf{r})$ is described for the case of a time dependent point electric dipole $\mathbf{p}{e}^{-i\omega t}$ 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 $\mathbf{E}(\mathbf{r})$ 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.