we get equations Substituting into Eq. (52) the expansion (50) and analogous expansions for all other com- ponents of E and H , we arrive at where (Θ)$_{n}$$_{−}$$_{m}$ is the Toeplitz matrix. The following equations follow from the remaining four Maxwell equations: One can substitute Eq. (53) into Eq. (54) and obtain a system of the first order differential equations for the Fourier components of the electromagnetic field E n $_{y}$, E n $_{x}$, H n $_{y}$, H n x in the region 0 ≤ z ≤ h . Now we have to determine the Rayleigh coefficients R ( e ) $_{np}$, R ( h ) np for the specific periodic geometry profile. One can determine these coefficients by matching the solution of equations inside the corrugation region 0 ≤ z ≤ h with Rayleigh expansions (44) at z = h and expansions (48) at z = 0. This can be done by imposing the continuity conditions on each Fourier component of the fields E$_{y}$, E$_{x}$, H$_{y}$, H$_{x}$ at z = 0 and z = h . 29 dE n y dz = ik$_{y}$E n z − iωH n $_{x}$, dE n x dz = iα$_{n}$E n z + iωH n $_{y}$, (54) dH n y dz = ik$_{y}$H n z + ∑ m ( iωε )$_{n}$$_{−}$$_{m}$ E m x , dH n x dz = iα$_{n}$H n z − m ( iωε )$_{n}$$_{−}$$_{m}$ E m y . E n z = ∑ m ( 1 iωε ) n − $_{m}$( ik$_{y}$H m x − iα$_{m}$H m y ) , H n z = − 1 iω ( ik$_{y}$E n x − iα$_{n}$E n $_{y}$) , (53) E$_{z}$ = 1 iωε ( ik$_{y}$H$_{x}$ − ∂H$_{y}$ ∂x ) , (52) H$_{z}$ = − 1 iω ik$_{y}$E$_{x}$ − ∂E$_{y}$ ∂x . ik$_{y}$E$_{x}$ − ∂E$_{y}$ ∂x = − iωH$_{z}$, (51) ik$_{y}$H$_{x}$ − ∂H$_{y}$ ∂x = iωεE$_{z}$