Correction of order three for the expansion of two dimensional electromagnetic fields perturbed by the presence of inhomogeneities of small diameter

The derivation of the correction of order 3 for the expansion of 2 dimensional electromagnetic fields perturbed by the presence of dielectric inhomogeneities of small diameter was completed in [3]. However previous numerical work such as that in [6] and in [14] do not corroborate the existence of these correcting terms. The inhomogeneities used in all those numerical simulations were collections of ellipses. In this paper we propose to elucidate this discrepancy. We prove that the correction of order 3 is zero for any inhomogeneity that has a center of symmetry. We present numerical experiments for asymmetric inhomogeneities. They illustrate the importance of the correction of order 3. Finally we prove that numerical schemes based on the usual quadrature for solving mixed linear integral equations on a smooth contour with smooth integration kernels and kernels involving logarithmic singularities preserve at the discrete level the fact that correcting terms of order 3 are zero for inhomogeneities that are symmetric about their center.