Improved convergence in the volume-
integral method (VIM) of calculating electromagnetic scattering properties of a structure is achieved by numerically solving a volume integral equation for a
vector field, F, rather than the
electric field, E. The
vector field, F, may be related to the
electric field, E, by a change of basis, and may be continuous at material boundaries where the
electric field, E, has discontinuities. Convolutions of the
vector field, F, are performed using
convolution operators according the finite Laurent rule (that operate according to a finite discrete
convolution), which allow for efficient matrix-vector products via 1D and / or 2D FFTs (Fast Fourier Transforms). An invertible
convolution-and-change-of-basis operator, C, is configured to transform the vector field, F, to the electric field, E, by performing a change of basis according to material and geometric properties of the periodic structure. After solving the volume integral for the vector field, F, an additional post-
processing step may be used to obtain the electric field, E, from the vector field, F. The vector field, F, may be constructed from a combination of field components of the electric field, E, and the
electric flux density, D, by using a normal-vector field, n, to filter out continuous components.