1. Finite Element Modelling of Photonic Crystals Ben Hiett J Generowicz, M Molinari, D Beckett, KS Thomas, GJ Parker and SJ Cox High Performance Computing

2. High Performance Computing Photonic Crystals F Photonic Crystals: the presence of ‘photonic band gaps’ 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.

3. High Performance Computing The Physics… F Solve Maxwell Equations with periodic boundary conditions F Numerical method based on Finite Element Method

4. High Performance Computing Advantages of FEM F Simple and intuitive representation of the photonic crystal structure F Accurate expression of the sharp discontinuities in dielectric constant

5. High Performance Computing Advantages of FEM F Adaptive mesh refinement allows improvement of solution in specific areas of high relative error F The global matrices comprising the eigenvalue problem are SPARSE. Hence the method scales (almost) linearly in terms of computation and memory requirements Visualisation of a Global Matrix (non-zero elements highlighted)

6. High Performance Computing Domain Discretisation Selection of Interpolation Function Derivation of the Elemental Equations Matrix Assembly Solution of the Eigensystem Visualisation Unit cellPeriodically tiled unit cells

7. High Performance Computing Interpolation Function F Linear, quadratic, cubic… F Trade off between computation/memory cost and solution accuracy Domain Discretisation Selection of Interpolation Function Derivation of the Elemental Equations Matrix Assembly Solution of the Eigensystem Visualisation

8. High Performance Computing Elemental Equations Domain Discretisation Selection of Interpolation Function Derivation of the Elemental Equations Matrix Assembly Solution of the Eigensystem Visualisation

9. High Performance Computing Matrix Assembly Domain Discretisation Selection of Interpolation Function Derivation of the Elemental Equations Matrix Assembly Solution of the Eigensystem Visualisation 1 2 3 F Global matrix assembly via local to global node mapping of elemental matrices

10. High Performance Computing Solution of the Eigensystem F Computationally Expensive (~95%) F Needs to be efficient F Sub-space iterative technique F Only compute eigenvalues of interest (lowest) F Exploit similarity of adjacent solutions F Search a larger sub-space to improve convergence Domain Discretisation Selection of Interpolation Function Derivation of the Elemental Equations Matrix Assembly Solution of the Eigensystem Visualisation

11. High Performance Computing Visualisation Domain Discretisation Selection of Interpolation Function Derivation of the Elemental Equations Matrix Assembly Solution of the Eigensystem Visualisation

12. High Performance Computing Convergence F Does the method work in practice? v Converges to analytic solution for free space v Verified against other structures in the literature v Compared with experiment Mesh Granularity (no. of elements) Error

13. High Performance Computing Triangular Lattice Spectra Filling fraction = 80% Rod dielectric constant = 12.25 Unit Cell for calculation Periodic repeat of unit cell

14. High Performance Computing Computational Efficiency F Can get more accurate solution by v Using a finer mesh v Using a higher order interpolation function A compromise is necessary

15. High Performance Computing Which is Best ? First order is cheap but inaccurate Higher order gives better accuracy for same compute time Increased Accuracy Reduced Time

16. High Performance Computing Interesting geometries F Exhibit large photonic band gaps F Substrate material can have low dielectric constant F Practical to manufacture F 12-fold symmetric quasicrystals…

17. High Performance Computing Quasicrystal Spectra Filling fraction = 28% Rod dielectric constant = 12.25

18. High Performance Computing Quasicrystal Spectra (2) Filling fraction = 28% Rod dielectric constant = 4.0804

19. High Performance Computing Quasicrystal with Defect Filling fraction = 28% Rod dielectric constant = 12.25

20. High Performance Computing Conclusion F Finite Elements have advantages v simple and intuitive crystal representation v Dielectric discontinuities are modelled accurately v Resultant eigenvalue problem is SPARSE F Can use FEM modelling to tune photonic crystal properties F Also have fully 3D extension

21. High Performance Computing Further Information F E-mail: B.P.Hiett@soton.ac.ukB.P.Hiett@soton.ac.uk F Web:www.photonics.n3.net High Performance Computing Centre