Pseudo-local effect medium theory

Effective medium theory is of great importance for using the artificial microstructure materials to extend the optical parameters. In this article, we develop a new kind of effective medium theory for artificial microstructures with nonlocal effects, like photonic crystals, which we name the pseudo-local effective medium theory. The optical properties of the pseudo-local effective medium are described by effective local permittivity \begin{document}${\overleftrightarrow \varepsilon ^{\rm{p}}}\left( \omega \right)$\end{document}and permeability \begin{document}${\overleftrightarrow \mu ^{\rm{p}}}\left( \omega \right)$\end{document}, together with an additional wave vector \begin{document}${{{k}}_a}$\end{document}. We find that the pseudo-local medium exhibits a unique blend of local and nonlocal characteristics. On the surface normal to \begin{document}${{{k}}_a}$\end{document}, the pseudo-local medium is optically equivalent to its local medium counterpart. While on the surface parallel to \begin{document}${{{k}}_a}$\end{document}, the abnormal wave phenomena induced by inherent nonlocality, such as negative refraction and total reflection, may occur. Furthermore, it is found that a \begin{document}$\text{π}$\end{document}phase shift is added to transmission wave through the pseudo-local medium composed of odd number of unit cells under all incident angles. Based on this unique feature, an all-angle phase grating is proposed. Our work opens a route towards the advanced optical devices based on the pseudo-local effective media.