This chapter presents studies periodic complementary metallic structures at the substrate interface. In the first theory, an approximate extension is conducted for the of the Babinet principle, where the dual substrate is adopted. In the second theory, Based on the above approximate Babinet principle, dual boundary conditions and impedance conditions is used to reconstruct the fields of complementary metasurface. Both theories yield reliable results for moderate incident angles but exhibit noticeable errors near grazing incidence. To enable theoretical analysis under large-angle conditions, the third theory introduces a virtual effective medium with near-zero thickness at the substrate interface. This approach provides a viable mathematical framework for analysis. Notably, theoretical results show strong consistency with simulation data even under large-angle incidence. Finally, based on the zero-thickness effective medium, this chapter investigates the generalized Babinet principle applicable to arbitrary complementary structures at the substrate interfaces.