Pseudo-local effect medium theory
Authors: Song Tong-Tong , Luo Jie , Lai Yun ,   Collaborative Innovation Center of Advanced Microstructures, State Key Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China,   School of Physical Science and Technology, Soochow University, Suzhou 215006, China
Published: December 2020

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 <inline-formula><tex-math id="M7">\begin{document}${\overleftrightarrow \varepsilon ^{\rm{p}}}\left( \omega \right)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M7.png"/></alternatives></inline-formula> and permeability <inline-formula><tex-math id="M8">\begin{document}${\overleftrightarrow \mu ^{\rm{p}}}\left( \omega \right)$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M8.png"/></alternatives></inline-formula>, together with an additional wave vector <inline-formula><tex-math id="M9">\begin{document}${{{k}}_a}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M9.png"/></alternatives></inline-formula>. We find that the pseudo-local medium exhibits a unique blend of local and nonlocal characteristics. On the surface normal to <inline-formula><tex-math id="M10">\begin{document}${{{k}}_a}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M10.png"/></alternatives></inline-formula>, the pseudo-local medium is optically equivalent to its local medium counterpart. While on the surface parallel to <inline-formula><tex-math id="M11">\begin{document}${{{k}}_a}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M11.png"/></alternatives></inline-formula>, the abnormal wave phenomena induced by inherent nonlocality, such as negative refraction and total reflection, may occur. Furthermore, it is found that a <inline-formula><tex-math id="M12">\begin{document}$\text{π}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200196_M12.png"/></alternatives></inline-formula> 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.