Photonic Bandgap Calculations Using Finite Element Methods

Photonic crystals have been extensively studied due to their unusual ability to manipulate the flow of light. The most important property of a photonic crys-tal is its bandgap structure. Numerical methods are essential in the design and optimization of photonic crystal devices. Advances in various branches of pho-tonics technologies have established the need for the development of numerical and approximate methods for the analysis of the photonic bandgap structures that are not amenable to exact analytical studies. The focus of this thesis is on the finite element analysis and calculations for the photonic bandgap struc-ture of a photonic crystal in which the index of refraction varies alternatively between high-index regions and low-index regions.In this thesis, we develop the finite element methods for one-dimensional photonic crystals firstly. The electric and magnetic fields in photonic crystal are given by the Maxwell Equations. For one-dimensional case, we study the bandgap structure built with an elementary cell consisting of two sub-layers or four sub-layers by using a finite clement approach to solve the differential systems with the continuous jump conditions. The transmission properties of a photonic crystal structure by calculating the transmission coefficient repeatedly for a partition at frequency domain. By using this algorithm, we study the re-lationship between the gap and the number of periods, the lattice constant and the material dispersion. Then the optimal design of one-dimensional photonic crystals can be obtained by several numerical simulations. Our model and nu-merical method are easy to be extended to study the numerical design for the photonic bandgap in two-dimensional PhCs.Secondly, we study the photonic bandgap for two-dimensional photonic crystals composed of a square lattice of circular cylinders by using finite ele- ment menthods in this thesis. The problem is formulated as a generalized eigen-value problem, and we introduce a new mixed variational eigenvalue problem to describe the bandgap problem in photonic crystals. By modifying the linear form associated with the eigenvalue problem, we show the convergence analy-sis for the finite element approximations. The dispersion curves with the Bloch wave vector (α,β), given on the boundary of the irreducible Brillouin zone is calculated by a linear finite element approximation. We also show the gap maps reveals the dependence of the bandgaps on the ratio between the radius of the cylinders and the lattice constant for two-dimensional photonic crystals com-posed of a square lattice of circular cylinders. Numerical results are reported to illustrate the performance of the finite element method, and they show we can control the band structure by choosing the geometrical parameters of the elementary cell.