Methods and apparatus for modeling electromagnetic scattering properties of microscopic structures and methods and apparatus for reconstruction of microscopic structures

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 ID 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. The improved volume-integral method may be incorporated into a forward diffraction model in metrology tools for reconstructing an approximate structure of an object, for example to assess critical dimensions (CD) performance of a lithographic apparatus.