The integral equation method for electromagnetic scattering problem at oblique incidence

We consider the scattering of electromagnetic waves scattered by an infinitely long impedance cylinder at oblique incidence, which is modeled as a system of a pair of the two-dimensional Helmholtz equations with coupled oblique boundary conditions. The solvability of such a scattering problem is proven by using the boundary integral equation method. By expressing the scattered fields in the form of single-layer potentials, our oblique scattering problem is transformed to a system of two integral equations. It is not a usual Fredholm system of the second kind as that in the case of normal incidence, since the system involves the tangential derivatives of the single-layer potential. By relating it to the Cauchy integral operator, we show that this system of operators is of Fredholm type with index 0. Therefore, the solvability of the integral system follows from the uniqueness of its solutions due to the Fredholm theory. A numerical scheme for solving the integral equations is also presented with some numerics. The numerical results illustrate the validity and efficiency of the proposed method.