Homogenized Equivalent Substrate Applied to Simplifying Boundary Condition of Metasurface on the Substrate Interface

In this letter, we convert the boundary condition of the metasurface on the substrate interface into that in a homogenized substrate with almost arbitrary permittivity. The key is that when the metasurface is placed on the substrate interface, the substrate interface with a small thickness of $h_{1}$ + $h_{2}$ acts as the 'background material' of the metasurface. Following this thought, the substrate interface with $h_{1}$ + $h_{2}$ could be physically equivalent to a kind of homogenized substrate layer (with $h_{1}$ + $h_{2}$ ) according to the definition of permittivity. Thereby, when the wave impinges onto the metasurface in the equivalent substrate (with $h_{1}$ + $h_{2}$ ), the external electromagnetic response will be the same as that of the metasurface on the substrate interface (with $h_{1}+ h_{2}$ ). Finally, the transmission matrix of the metasurface on the substrate interface can be related to that in the homogenized substrate. An example is given to verify the conversion between two boundary conditions. The method can greatly speed up the theoretical analysis of the metasurface on the substrate interface because only the metasurface in a homogenized substrate needs to be solved.