A boundary Variation method for the analysis of both infinite periodic and finite aperiodic waveguide grating couplers in two dimensions is introduced. Based on a previously introduced boundary variation method for the analysis of metallic and transmission gratings [J. Opt. Soc. Am. A 10, 2307, 2551 (1993)], a numerical algorithm suitable for waveguide grating couplers is derived. Examples of the analysis of purely periodic grating couplers are given that illustrate the convergence of the scheme. An analysis of the use of the proposed method for focusing waveguide grating couplers is given, and a comparison with a highly accurate spectral collocation method yields excellent agreement and illustrates the attractiveness of the proposed boundary variation method in terms of speed and achievable accuracy. (C) 2000 Optical Society of America [S0740-3232(00)01709-9] OCIS codes: 050.0050, 050.1960, 000.4430.