In this work, a matrix method is applied to study the propagation of electromagnetic waves inside a non-slanted reflection grating. The elements of the matrix which characterizes the periodic medium are obtained in terms of Mathieu functions and their derivatives, and the expressions of the efficiencies of reflected and transmitted orders are calculated in terms of the elements of the matrix. In addition the band structure of a general reflection grating is studied with the layer matrix of one single period. The results obtained by this matrix method are firstly compared to the results obtained by Kogelnik's expressions in index-matched media showing good agreement. The comparison is also made for a reflection grating embedded in two media with different refractive indexes, showing good agreement with an FDTD method, but slight differences with respect to Kogelnik's Coupled Wave Theory.
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