RANDRIAMIHAJA

A spectral element method is developed for determining the propagation constants of guided and leaky modes in planar multilayer waveguides. The originality of the technique lies in what we believe to be a novel combination of a hierarchical basis built with modified Legendre polynomials and a rational mapping that efficiently handles semi-infinite computational subdomains. To handle leaky waves, we extend the algebraic mapping to complex coordinates that act as perfectly matched layers (PMLs). The method thus naturally enforces both radiation conditions at infinity and boundary conditions at material interfaces, leading to improved convergence properties without additional constraints or numerical treatments. The efficiency and numerical precision of our method are demonstrated through comparisons with results from the literature for high-index-contrast dielectric and plasmonic waveguides, ARROW anisotropic structures, and quantum well waveguides.

In a waveguide, vertical (or transverse) modes are specific configurations of the electric and magnetic fields whose properties can be finely tuned using periodic structures. In this study, we investigate the excitation and propagation of vertical modes on a sinusoidal grating, periodically repeated along another spatial direction, referred to as a snake grating. This type of structure enables a significant enhancement of the coupling between incident light waves and vertical modes. Using the method of moments, in which a tensor product of a Fourier basis and Legendre polynomials serves as the expansion and projection functions, we demonstrate a strong sensitivity of vertical modes to the geometric parameters of the grating. The expected result is the excitation of vertical modes for both transverse magnetic (TM) and transverse electric (TE) polarizations on the snake grating, thus allowing more flexible light manipulation and the development of more efficient optical devices.

The problem of diffraction by snake gratings is presented and formulated as an eigenvalue eigenvector problem. A numerical solution is obtained thanks to the method of moments where a tensor product of pseudo-periodic functions and Legendre polynomials is used as expansion and test functions. The method is validated by comparison with the usual Fourier modal method (FMM) as applied to crossed gratings. Our method is shown to be more efficient than the FMM in the case of metallic gratings.

The problem of diffraction by slanted lamellar dielectric and metallic gratings in classical mounting is formulated as an eigenvalue eigenvector problem. The numerical solution is obtained by using the moment method with Legendre polynomials as expansion and test functions, which allows us to enforce in an exact manner the boundary conditions which determine the eigensolutions. Our method is successfully validated by comparison with other methods including in the case of highly slanted gratings.

An efficient numerical modal method for modeling a lamellar grating in conical mounting is presented. Within each region of the grating, the electromagnetic field is expanded onto Legendre polynomials, which allows us to enforce in an exact manner the boundary conditions that determine the eigensolutions. Our code is successfully validated by comparison with results obtained with the analytical modal method.