Rigorous coupled-wave analysis method based on vector theory analysis is derived and discussed by using Maxwell equations and electromagnetic boundary conditions. A general diffraction analysis numerical code based on the Rigorous Coupled-wave Analysis (RCWA) is written so as to design and analyze the diffraction properties of grating, and the diffraction efficiency and the convergence of the RCWA are investigated for the TE mode. The calculated diffraction efficiency is shown to converge to the correct value with an increasing number of space harmonics over a wide range of parameters, including very deep gratings (d/λ>10).