Modeling Diffraction Gratings By Modal Methods And Chebyshev Collocation Dirichlet-to-Neumann Map Method

Diffraction gratings are periodic structures with many practical applications, suchas monochromators, spectrometers, lasers, wavelength division multiplexing devices,optical pulse compressing devices and other optical instruments. Numerical methodsare 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 bystandard numerical methods, such as the finite element method (FEM). However, thesegeneral methods give rise to large, complex, indefinite linear systems that are relativelyexpensive to solve. Less general methods that take advantage of available geometricfeatures are often more efficient. Existing methods for diffraction gratings includethe analytic modal method, the Fourier modal method (FMM), the finite differencemodal method, the differential method and the integral equation method, etc. All modalmethods require that the structure consists of uniform layers, so that the wave field canbe expanded in eigenmodes in each layer. Computing the eigenmodes in each layer isusually the most expensive part of the method.In this thesis, we first derive and implement the analytic modal method and theFourier modal method. FMM calculates the eigenmodes based on Fourier series ex-pansions. 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 tosolve the eigenvalue problems. Since the eigenvalue problems are relatively expen-sive to solve, we develop a Dirichlet-to-Neumann (DtN) map method for diffractiongratings with uniform layers. Instead of computing the eigenmodes in each layer, wecalculate an operator that maps the wave field to its normal derivative at the boundariesof the layer. In practice, this operator, the so-called DtN map, is approximated by amatrix, and it is efficiently calculated using a highly accurate Chebyshev collocationmethod and a fourth order finite difference method to discretize the uniform and peri-odic directions, respectively. The DtN formalism has been previously used to analyzeperiodic arrays of cylinders and piecewise uniform waveguides. For circular cylinders, the DtN maps are constructed from cylindrical harmonics. For uniform waveguidesegments, the Chebyshev collocation method was used with a second order finite dif-ference 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. Asillustrated in numerical examples, our new method is more accurate than FMM, whenthe same degrees of freedom are used in the discretization, and it is also more efficientthan FMM, since the time consuming eigenvalue decomposition is avoided and theDtN map can be calculated efficiently. Finally, the Chebyshev collocation DtN mapmethod is extended to diffraction gratings in conical mounting.