Curvilinear coordinate generalized source method for gratings with sharp edges

Abstract

High-efficient direct numerical methods are currently in demand for optimization procedures in the fields of both conventional diffractive and metasurface optics. With a view of extending the scope of application of the previously proposed Generalized Source Method in the curvilinear coordinates, which has theoretical O (N log N ) asymptotic numerical complexity, a new method formulation is developed for gratings with sharp edges. It is shown that corrugation corners can be treated as effective medium interfaces within the rationale of the method. Moreover, the given formulation is demonstrated to allow for application of the same derivation as one used in classical electrodynamics to derive the interface conditions. This yields continuous combinations of the fields and metric tensor components, which can be directly Fourier factorized. Together with an efficient algorithm the new formulation is demonstrated to substantially increase the computation accuracy for given computer resources.

Čerenkov radiation in vacuum from a superluminal grating

“…Our previous work is based on the coordinate translation method originally proposed by Chandezon et al [30,31], where they consider static corrugated interfaces and match Maxwell's boundary conditions directly at the interfaces with the help of differential geometry. Using this method, it is straightforward to take the structure of the interfaces into consideration, and it has been utilised to calculate structured surfaces of various media, including anisotropic, plasmonic and dielectric materials [32][33][34][35][36] It is also worth noting that there is a series of studies, which confirm that the method works well for smooth shallow corrugations and propose possible ways to improve the method so that they can handle deep corrugation even with sharp edges [37][38][39][40][41][42][43]. In these works, local distortion of the coordinate systems is applied instead of the global translation of the coordinate in order to improve the convergence.…”