The Taylor Interpolation through FFT Algorithm for Electromagnetic Wave   Propagation and Scattering
  The Taylor Interpolation through FFT (TI-FFT) algorithm for the computation of the electromagnetic wave propagation in the quasi-planar geometry within the half-space is proposed in this article. There are two types of TI-FFT algorithm, i.e., the spatial TI-FFT and the spectral TI-FFT. The former works in the spatial domain and the latter works in the spectral domain. It has been shown that the optimized computational complexity is the same for both types of TI-FFT algorithm, which is N_r^{opt} N_o^{opt} O (N log_2 N) for an N = N_x \times N_y computational grid, where N_r^{opt} is the optimized number of slicing reference planes and N_o^{opt} is the optimized order of Taylor series. Detailed analysis shows that N_o^{opt} is closely related to the algorithm's computational accuracy \gamma_{TI}, which is given as N_o^{opt} ~ - ln(\gamma_{TI}) and the optimized spatial slicing spacing between two adjacent spatial reference planes \delta_z^{opt} only depends on the characteristic wavelength \lambda_c of the electromagnetic wave, which is given as \delta_z^{opt} ~ 1/17 \lambda_c. The planar TI-FFT algorithm allows a large sampling spacing required by the sampling theorem. What's more, the algorithm is free of singularities and it works particularly well for the narrow-band beam and the quasi-planar geometry. 
Shaolin Liao