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Finite Element Modelling of Photonic Crystals Ben Hiett J Generowicz, M Molinari, D Beckett, KS Thomas, GJ Parker and SJ Cox High Performance Computing
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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.
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High Performance Computing The Physics… F Solve Maxwell Equations with periodic boundary conditions F Numerical method based on Finite Element Method
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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
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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)
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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
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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
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High Performance Computing Elemental Equations Domain Discretisation Selection of Interpolation Function Derivation of the Elemental Equations Matrix Assembly Solution of the Eigensystem Visualisation
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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
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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
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High Performance Computing Visualisation Domain Discretisation Selection of Interpolation Function Derivation of the Elemental Equations Matrix Assembly Solution of the Eigensystem Visualisation
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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
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High Performance Computing Triangular Lattice Spectra Filling fraction = 80% Rod dielectric constant = 12.25 Unit Cell for calculation Periodic repeat of unit cell
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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
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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
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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…
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High Performance Computing Quasicrystal Spectra Filling fraction = 28% Rod dielectric constant = 12.25
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High Performance Computing Quasicrystal Spectra (2) Filling fraction = 28% Rod dielectric constant = 4.0804
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High Performance Computing Quasicrystal with Defect Filling fraction = 28% Rod dielectric constant = 12.25
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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
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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
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