Our aim is to solve the electromagnetic problem of the diffraction of a plane wave by a one dimensional lamellar grating. In that case the solution to Maxwell's equations can be split into two canonical cases : the so-called transverse magnetic (TM) and transverse electric (TE) polarizations. These cases can be treated separately, reducing the problem to a scalar one. In this paper we only consider the TE polarization case in which the only non null component of the electric field is parallel to the grating grooves.

Since the grating is invariant in one direction the Maxwell's equations reduce to an eigenvalue problem for which a numerical solution is obtained by using the method of moments. First the unknown function is expanded in a series of spline functions and then the operator deduced from the Maxwell's equations is projected onto a set of test functions after a suitable inner product has been defined. The choice of the basis and test functions and their properties have an essential impact for the rate of convergence. One of the reasons for choosing splines functions is that they were successfully used in the signal processing field.

We can take advantage of their analytical definition as piecewise polynomials and their compact support. Concerning the test functions, we compare three possible choices : Dirac, gate or spline functions. Thanks to their attractive properties, these functions allow calculating analytically the matrix coefficients deduced from the inner product. The computational effort is therefore drastically minimized.