content_image/374161.jpg content_image/374160.jpg content_image/374159.jpg content_image/374158.jpg
equations 

we get 

<img src='content_image/374161.jpg'>

<img src='content_image/374160.jpg'>

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. 

<img src='content_image/374159.jpg'>

The following equations follow from the remaining four Maxwell equations: 

<img src='content_image/374158.jpg'>

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 . 

