II. THE RAYLEIGH METHOD AND ITS IMPROVEMENT. Generalized radiation condition In the canonical problem of diffraction gratings, a periodic surface, S, denoted y=f(x)=f(x+p), separates space into two regions. It is assumed that the electric vector for the polarization s is parallel to the z axis and that the z component of the E field is denoted by u, with the subscripts i, s or t indicating incident, scattered or transmitted fields.Fig. 1: A periodic surface. Omitting the time factor exp(it), the incident field ui(x, y)= p1/2exp(i xiy) (1) where  =k0sini, =k0cos i, i the incident angle measured counterclockwise from the positive y axis as shown in Fig. 1, k0=2/=/c,  and c the wavelength and speed of the radiation that propagates in the void. To formulate this problem, in reference [3] M. Cadilhac considers a function u in the half-plane y > max f(x) with the three hypotheses: a) u is limited for large yb ) u is a solution of the Helmholtz equation  2u + k2u = 0 with real .c) Once y is fixed, u can be analyzed according to Fourier as a function of x, in a certain sense (u is for example a square that can be integrated into x; more generally, it is a tempered distribution). To solve the boundary value problem is to determine the function u that satisfies the Helmholtz equation in the domain : y>f(x) instead of the half-plane y>max f(x), and pre-assigned values ​​on the boundary : y =f(x) (Dirichlet problem). We must therefore invoke RH which, based on the postulate that one and the same representation describes the field in the 'external' area y>maxf(x) as well as in the 'internal' area f(x)≤y≤maxf( x)of the surface periodic. The so-called Rayleigh expansion is widely used in periodic surface problems and has the form of ur(x,y)... ... center of the paper ......essential functions in (8) in the Fourier series exp (i f(x))=l Vlexp(i2lx/p) (9a) and exp(imf(x))=l Mlmexp(i2 lx/p) (9b) with Vl= p10p exp(i f(x))exp(i2lx/p)dx (9c) and Mlm= p10p exp(i mf(x))exp( i2lx/p)dx (9d) substituting (9a) and (9b) in (8) then multiplying by n*, taking integrals over the period p on both sides, due to the orthogonality of m, we obtain that VnmMmn, mRm=0 (10)The matrix-vector equation (10) can be used to determine the unknown vector Rm, and us(x, y) in turn, can be calculated. Similarly, based on (7) we can formulate ERMVnm N mn, mJm=0 (11) where Nlmp10p(Zm/2)exp(im (f (x)+h))exp(i2lx/p)dx=(Zm/2)exp(imh)Mlm (12)(11) is the matrix-vector equation to determine the unknown vector Jm and uR(x, y) can also be calculated.