Effective medium theory of checkboard structures in the long-wavelength limit

Effective medium theory is a powerful tool to solve various problems for achieving multifarious functionalities and applications. In this article, we present a concise empirical formula about effective permittivity of checkboard structures for different directions. To verify our empirical formula, we perform simulations of checkboard periodic structures in squares, rectangles, and sectors in two dimensions. Our results show that the formula is valid in a large range of parameters. This work provides a new way to understand and design composite materials, which might lead to further optical applications in transformation optics.