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

A CSI algorithm for reconstructing grating profiles is disclosed. Solving a volume integral equation for current density, J, employs the implicit construction of vector field, FS related to the electric field, ES, and current density, J, by selection of continuous components of E and J, F being continuous at one or more material boundaries, so as to determine an approximate solution of J. F is represented by at least one finite Fourier series with respect to at least one direction, x, y, and the step of numerically solving the volume integral equation comprises determining a component of J, by convolution of F, with a convolution operator, M comprising material and geometric structure properties in both directions. J may be represented by at least one finite Fourier series with respect to both directions. The continuous components can be extracted using convolution operators, PT and PN, acting on E and J.