2. Application of Finite Element Methods to Photonic Crystal Modelling B.P. Hiett D. Beckett, S.J. Cox, J. Generowicz, M. Molinari, K.S. Thomas High Performance Computing

3. High Performance Computing Photonic Crystals F Photonic Crystals: the presence of ‘photonic band gaps’. F Prohibited range of photon energies. F Huge potential in a range of applications. v Highly efficient narrow-band lasers, v integrated optical circuits, v high-speed optical communication networks. F Hence a need for accurate and efficient modelling.

4. High Performance Computing The Physics… F Solve Maxwell Equations with periodic boundary conditions F Restrict to linear, lossless dielectric materials F Separate time dependence by expanding fields into a set of harmonic modes F Numerical method based on Finite Element Method

5. High Performance Computing 2D Scalar Spectral Problem TE TM x y z For  (r) periodic in 2 dimensions and constant in the 3 rd Considering waves travelling only along the plane of periodicity

6. High Performance Computing Domain Discretisation Bridge waveguide structure courtesy of Martin Charlton, Southampton Microelectronics Research Group. UNIT-CELL PERIODICALLY TILED UNIT-CELLS REAL-THING (photo) pitch=300nm

7. High Performance Computing FEM with Periodic Boundaries Express the wave as a Bloch state: Approximate the solution using Galerkin's method: Metric Dirichlet Vector of B matrices

8. High Performance Computing F Computationally Expensive (~95%) F Needs to be efficient / optimised F Only compute eigenvalues of interest (lowest) F Exploit similarity of adjacent solutions F Search a larger sub-space to improve convergence Subspace Iterative Eigenvalue Solver

9. High Performance Computing Algorithm Performance F Near linear scaling against O(n²) for traditional plane wave expansion methods. F Avoid linear (1 st order) interpolation. IDEAL

10. High Performance Computing Results – Triangular Lattice F Triangular lattice of air-rods in a gallium arsenide substrate (  =11.4) F Exhibits a complete (overlapping TE & TM mode) band gap Results from Photonic Crystals, Joannopoulos et al. Results using the FEM Rhombic Unit Cell

11. High Performance Computing 12-fold Symmetric Quasicrystals F Based on tiling of dodecagons composed of squares and equilateral triangles F Possesses 12 fold rotational symmetry F Leads to a highly homogeneous band gap

12. High Performance Computing Quasicrystal Results FEM Results (Density of States) FDTD Results Experimental Results

13. High Performance Computing Future Work F 3D Vector FEM Implementation v Explore fully 3D periodic structures v Replace Lagrange scalar elements with vector edge elements v Deal with spurious modes F PBG Optimisation Software v Automated Mesh Generation v Suitable objective function

14. High Performance Computing Conclusion F 2D analysis of Photonic Crystals with FEM v Highly efficient in comparison to traditional Numerical Methods v Utilise Bloch functions to model crystals of infinite extent v Highly optimised eigenvalue solver v Excellent agreement with experimental and theoretical data F Provides an efficient & accurate photonic crystal modelling tool.