Planar near-field measurements are a common method to characterize antenna performance at millimeter and submillimeter wavelengths. Far-field beam patterns are then calculated from those measurements. However, different approximations have been traditionally used for this calculation without careful analysis. In this paper, we derive a Near-field-to-Far-field Transformation (NFT) based on diffraction theory and the full components of electrical fields based on the boundary condition in the far-field region. We analyze the systematic errors introduced by Fourier Transformation (FT) and the probe antenna by comparison with simulations. We further apply our data analysis procedures in the near-field measurement of a corrugated horn. Measurement and simulation results suggest that when all the procedures are conducted accurately, we are able to reconstruct far-field beam patterns from planar near-field measurements with high accuracy.