Based on the expressions of electric field distributions of non-paraxial Gaussian beams in the oblate ellipsoid coordinate system, an expression of far-field electric field amplitude distribution of the non-paraxial LG10 mode Gaussian beam is given. The characteristics of far-field electric field amplitude distributions of the Gaussian beams with non-paraxial LG00 mode and non-paraxial LG10 mode are analyzed. The matching effciencies of the far-field electric field amplitude distributions and the relative calculation errors of the second-order moment parameters of the electric field intensity distributions of the non-paraxial and paraxial Gaussian beams are calculated. It is shown that the far-field electric field amplitude distributions of the non-paraxial LG00 mode and LG10 mode Gaussian beams tend to coincide with the far-field electric field amplitude distributions of the paraxial LG00 mode and LG10 mode Gaussian beams when the half of focal length of the oblate ellipsoid coordinate system is much larger than the wavelength.