Methods and apparatus for calculating electromagnetic scattering properties of a structure and for reconstruction of approximate structures

Algorithm for reconstructing grating profile in a metrology application is disclosed to numerically solve a volume integral equation for a current density. It employs implicit construction of a vector field that is related to electric field and a current density by a selection of continuous components of vector field being continuous at one or more material boundaries, so as to determine an approximate solution of a current density. The vector field is represented by a finite Fourier series with respect to at least one direction, x, y. Numerically solving volume integral equation comprises determining a component of a current density by convolution of the vector field with a convolution operator, which comprises material and geometric properties of structure in the x, y directions. The current density is represented by a finite Fourier series with respect to x, y directions. Continuous components are extracted using convolution operators acting on electric field and current density.