In this paper, scattering problem of the grating coupler is analyzed by the mode-matching method in the sense of least squares for the gaussian light beam incidence. This coupler has a periodic groove structure of finite extent, which is formed on the surface of the core layer of the symmetric thin-film waveguide. In the present method, the approximate scattered fields of each region of the grating coupler are described by the superpositions of the plane waves with band-limited spectra, respectively. These approximate wave functions are determined by the minimization of the mean-square boundary residual. This method results in the simultaneous Fredholm type integral equations of the second kind for these spectra. The first and second order approximate solutions of the integral equations are derived analytically and the coupling efficiency and scattered fields are analyzed on the basis of those solutions. A qualitative and physical consideration for the scattering problem of the grating coupler is presented with the fundamental data derived from approximate solutions in this paper.