Multicoated gratings: a differential formalism applicable in the entire optical region

Abstract

We present a new formalism for the diffraction of an electromagnetic plane wave by a multicoated grating. Its basic feature lies in the use of a coordinate system that maps all the interfaces onto parallel planes. Using Maxwell's equations in this new system leads to a linear system of differential equations with constant coefficients whose solution is obtained through the calculation of the eigenvalues and eigenvectors of a matrix in each medium. Through classical criteria, our numerical results have been found generally to be accurate to within 1%. The serious numerical difficulties encountered by the previous differential formalism for highly conducting metallic gratings completely disappear, whatever the optical region. Furthermore, our computer code provides accurate results for metallic gratings covered by many modulated dielectric coatings or for highly modulated gratings. We give two kinds of applications. The first concerns the use of dielectric coatings on a modulated metallic substrate to minimize the absorption of energy. Conversely, the second describes the use of highly modulated metallic gratings to increase this absorption.

Propagative and Evanescent Waves Diffracted by Periodic Surfaces: Perturbation Method

“…This situation only occurs twice with matrix elements of M (1) , ψ −1 |hM (1) |ψ 0 and ψ 0 |hM (1) |ψ −1 , and twice with matrix elements of M (2) , ψ −1 |h 2 M (2) |ψ 0 and ψ 0 |h 2 M (2) |ψ −1 , which are small.…”

“…where [a (1) ] and [a (2) ] are the matrix of which elements given in (98) are respectively the coefficients a (1) mi and a …”

“…Then the result given by Equation (98) suggests that a suitable definition of "smallness" for the perturbations hM (1) and h 2 M (2) is that the relevant matrix elements (98) and (99) we can see that the perturbed wave vector |ψ m can be viewed as a linear combination of unperturbed eigenvector |ψ (0) i ; the coefficients a (1) mi and a (2) mi tell us "how much" of the ith unperturbed vectors contribute to the mth perturbed vector. We say that the unperturbed vector |ψ Since a (1) mi and a (2) mi are inversely proportional to eigenvalues separation between the mth and ith unperturbed diffracted directions we expect directions that are closest together in diffracted directions to mix most strongly with |ψ (0) m . This fact provides a clue on how to avoid calculating the set of terms in Equation (98) depending on the degree of accuracy desired, we need only evaluate terms for the diffracted directions closest in mth direction.…”

“…Moreover, the Equation (97) indicates that the mth perturbed eigenvalue r m associated to the mth diffracted direction is given by the eigenvalue r (0) m of the corresponding unperturbed problem plus the expectation values of the first order perturbation hM (1) and second order perturbation h 2 M (2) . These terms can be of either sign and thus can raise or lower the value r m from the unperturbed value r (0) m .…”

“…The main feature of this method is the use of a translation coordinate system that maps the profile to a plane and enables to express the problem as an eigenvalue problem which can be easily numerically solved by standard algorithms [1][2][3][4]. The eigenvalues correspond to diffracted directions and the eigenfunctions correspond to fields of diffracted waves in these directions.…”