The k-space formulation of the n-dimensional scattering problem

The n-dimensional scattering problem is solved by means of a k-space formulation of the field equations, thereby replacing the conventional integral equation formulation by a set of two algebraic equations in two unknowns in two spaces (the constitutive equation being an algebraic equation in x-space). These equations are solved by an iterative method with the aid of the fast Fourier transform (FFT) algorithm connecting the two spaces, requiring very simple initial approximations. Since algebraic and FFT equations are used, the number of arithmetic multiple-add operations and storage allocations required for a numerical solution are reduced from the order of N sq (for solving the matrix equations resulting from the conventional integral equations) to the order of N(log base 2 of N) and N, respectively (where N is the number of data points required for the specification of the problem). The advantage gained in speed and storage is thus of the order of N/log base 2 of N and N, respectively. This method is thus considerably more efficient than the conventional matrix method, and permits exact numerical solutions for much larger problems. Arguments are presented toward the view that the field equations are more fundamental in k-space. The details and some numerical results of the application of this method to the three-dimensional electromagnetic scattering problems are presented as an example.