  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.
