Physical Origin of the Effective Parameters at Boundaries in Finite Difference Schemes

Using the Observable form of Maxwell's equations, we reveal that effective parameters at materials boundaries emerge naturally as anisotropic transfer functions. The complexity of the boundary dictates the order of these functions. Employing the residue-pole expansion, we describe the relation between field components as a system of ODEs solvable through the auxiliary differential equation approach. Not only can be used in Finite difference methods, the approach can be also employed to model complex surface structures.