Modeling diffraction gratings by modal methods and Chebyshev collocation Dirichlet-to-Neumann map method

Abstract

Diffraction gratings are periodic structures with many practical applications, such as monochromators, spectrometers, lasers, wavelength division multiplexing devices, optical pulse compressing devices and other optical instruments. Numerical methods are essential in the design, analysis and optimization of grating structures. In principle, when the problem is formulated on one period of the structure, it can be solved by standard numerical methods, such as the finite element method (FEM). However, these general methods give rise to large, complex, indefinite linear systems that are relatively expensive to solve. Less general methods that take advantage of available geometric features are often more efficient. Existing methods for diffraction gratings include the analytic modal method, the Fourier modal method (FMM), the finite difference modal method, the differential method and the integral equation method, etc. All modal methods require that the structure consists of uniform layers, so that the wave field can be expanded in eigenmodes in each layer. Computing the eigenmodes in each layer is usually the most expensive part of the method. In this thesis, we first review the analytic modal method and the Fourier modal method. FMM calculates the eigenmodes based on Fourier series expansions. Since it is relatively easy to implement, FMM is extremely popular. Next, we derive a fourth order finite difference modal method. All modal methods need to solve the eigenvalue problems. Since the eigenvalue problems are relatively expensive to solve, we develop a Dirichlet-to-Neumann (DtN) map method for diffraction gratings with uniform layers. Instead of computing the eigenmodes in each layer, we calculate an operator that maps the wave field to its normal derivative at the boundaries of the layer. In practice, this operator, the so-called DtN map, is approximated by a matrix, and it is efficiently calculated using a highly accurate Chebyshev collocation method and a fourth order finite difference method to discretize the uniform and periodic directions, respectively. The DtN formalism has been previously used to analyze periodic arrays of cylinders and piecewise uniform waveguides. For circular cylinders, the DtN maps are constructed from cylindrical harmonics. For uniform waveguide segments, the Chebyshev collocation method was used with a second order finite difference method in the transverse direction to approximate the DtN maps. In our work, the fourth order finite difference scheme is used to discretize the periodic direction. As illustrated in numerical examples, our new method is more accurate than FMM, when the same degrees of freedom are used in the discretization, and it is also more efficient than FMM, since the time consuming eigenvalue decomposition is avoided and the DtN map can be calculated efficiently. Finally, the Chebyshev collocation DtN map method is extended to diffraction gratings in conical mounting.