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Finite Element Methods for Maxwell‘s Equations

Finite Element Methods for Maxwell‘s Equations. 3rd Workshop on Numerical Methods for Optical Nano Structures, Zürich 2007. Jan Pomplun, Frank Schmidt Computational Nano-Optics Group Zuse Institute Berlin. Outline. Problem formulations based on time-harmonic Maxwell‘s equations

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Finite Element Methods for Maxwell‘s Equations

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  1. Finite Element Methods for Maxwell‘s Equations 3rd Workshop on Numerical Methods for Optical Nano Structures, Zürich 2007 Jan Pomplun, Frank Schmidt Computational Nano-Optics Group Zuse Institute Berlin

  2. Outline • Problem formulations based on time-harmonic Maxwell‘s equations • Scattering problems • Resonance problems • Waveguide problems • Discrete problem • Weak formulation of Maxwell‘s Equations • Assembling og FEM system • Contruction principles of vectorial finite elements • Refinement strategies • Applications • PhC benchmark with MIT-package • BACUS benchmark with FDTD • Optimization of hollow core PhC fiber 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  3. Maxwell‘s Equations (1861) James Clerk Maxwell (1831-1879) electric field E magnetic field H el. displacement field D magn. induction B anisotropic permittivity tensor e anisotropic permeability tensor m in many applications the fields are in steady state: time-harmonic Maxwell‘s Eq: 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  4. Time-harmonic Maxwell‘s equations Scattering problems Resonance problems Waveguide problems Problem types 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  5. E scat Setup for Scattering Problem incomming field scattered field (strictly outgoing) scatterer total field 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  6. Scattering Problem incomming field: (strictly outgoing) E solution to Maxwell‘s Eq. (e.g. plane wave) scat G dirichlet data on boundary reference configuration (e.g. free space) computational domain complex geometries (scatterer) 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  7. Scattering: Coupled Interior/Exterior PDE Interior and scattered field Coupling condition Radiation condition (e.g. Silver Müller) scat scat 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  8. Resonance Mode Problem Eigenvalue problem for Radiation condition for isolated resonators Bloch periodic boundary condition for photonic crystal band gap computations. 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  9. Propagating Mode Problem Structure is invariant in z-direction: y z x Image: B. Mangan, Crystal Fibre Propagating Mode: Eigenvalue problem for 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  10. Weak formulation of Maxwell‘s Equations 1.) multiplication with vectorial test function : 2.) integration over interior domain : boundary values 3.) partial integration: 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  11. Weak formulation of Maxwell‘s Equations define following bilinear and linear form: finite element space weak formulation of Maxwell‘s equations: Find such that discretization Find such that 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  12. Assembling of FEM System Find such that basis: ansatz for FEM solution: yields FEM system: with: sparse matrix 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  13. Finite Element Construction Principles Construction ofwith finite elements: locally defined vectorial functions of arbitrary order that are related to small geometric patches (finite elements) • Finite element consists of: • geometric domain • local element space • basis of local element space (e.g. triangle) 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  14. Construction of Finite Elements for Maxwell‘s Eq. Finite elements should preserve mathematical structure of Maxwell‘s equations (i.e. properties of the differential operators)! E.g. eigenvalue problem: Fields with lie in the kernel of the curl operator -> belong to eigenvalue For the discretized Maxwell‘s equations: Fields which lie in the kernel of the discrete curl operator should be gradients of the constructed discrete scalar functions 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  15. De Rham Complex On simply connected domains the following sequence is exact: • The gradient has an empty kernel on set of non constant functions in • The range of the gradient lies in and • is exactly the kernel of the curl operator • The range of the curl operator is the whole On the discrete level we also want: 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  16. Construction of Vectorial Finite Elements (2D: (x,y)) Starting point: Finite element space for non constant functions (polynomials of lowest order) on triangle : Exact sequence: gradient of this function space has to lie in constant functions First idea to extend : 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  17. Vectorial Finite Elements (2D) But: -> lies in the kernel of the curl operator,but Basis of : 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  18. FEM solution of Maxwell‘s equtions • Following examples • computed with JCMsuite: • 2D, 3D, cylinder symm. solver for • scattering, resonance and • propagation mode problems • Vectorial Finite Elements up • to order 9 • Adaptive grid refinement • Self adaptive PML • (inhomogeneous exterior domians) Maxwell‘s equations (continuous model) Scattering, resonance, waveguide Weak formulation Discretization by FEM (discrete model) Finite element construction, assembling Discrete solution Refine mesh (subdivide patches Q) A posterior error estimation no Error<TOL? solution 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  19. FEM-Refinement 1 Uniform Refinement Hexagonal photonic crystal 0 refinements 252 triangles 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  20. FEM-Refinement 2 Hexagonal photonic crystal 1 refinements 1008 triangles 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  21. FEM-Refinement 3 Hexagonal photonic crystal 2 refinements 4032 triangles 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  22. FEM-Refinement4 Hexagonal photonic crystal 3 refinements 16128 triangles 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  23. FEM-Refinement 5 Hexagonal photonic crystal t (CPU) ~ 10s (Laptop) 4 refinements 64512 triangles 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  24. Plasmon waveguide (silver strip): Adaptive Refinement 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  25. Solution (intensity) 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  26. Adaptiv refined mesh 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  27. Zoom 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  28. Zoom with mesh 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  29. Zoom 2 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  30. Zoom 2 with mesh 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  31. Benchmark: 2D Bloch Modes Benchmark: convergence of Bloch modes of a 2D photonic crystal JCMmode is 600* faster than a plane-wave expansion (MPB by MIT) 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  32. Benchmark problem: DUV phase mask  = 193nm Plane wave Substrate Cr line Air Triangular Mesh 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  33. air substrate Benchmark Geometry • extremely simple geometry • simple treatment of incident field • -> well suited for benchmarking methods • geometric advantages of FEM are not put into effect 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  34. Convergence: TE-Polarization (0-th diffraction order) FDTD • All solvers show "internal" convergence • Speeds of convergence differ significantly Waveguide Method FEM [S. Burger, R. Köhle, L. Zschiedrich, W. Gao, F. Schmidt, R. März, and C. Nölscher. Benchmark of FEM, Waveguide and FDTD Algorithms for Rigorous Mask Simulation. In Photomask Technology, Proc. SPIE 5992, pages 368-379, 2005.] 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  35. Na Laser Guide Stars • Adaptive optics system: • corrects the atmosphere‘s blurring effect limiting the image quality • needs a relatively bright reference star • observable area of sky is limited! laser guide star (~90km): luminating sodium layer January 2006: laser beam of several Watts created first artificial reference star (laser guide star) Hollow core photonic crystal fiber for guidance of light from very intense pulsed laser powerful laser ESO‘s very large telescope Paranal, Chile 589nm 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  36. Hollow core photonic crystal fiber hollow core • guidance of light in hollow core • photonic crystal structure • prevents leakage of radiation • to the exterior exterior: air • high energy transport possible • small radiation losses! • [Roberts et al., Opt. Express 13, 236 (2005)] transparent glass Courtesy of B. Mangan, Crystal Fibre, DK • Goal: • calculation ofleaky propagation modesinside hollow core • optimization of fiber design tominimize radiation losses 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  37. FEM Investigation of HCPCFs Eigenmodes of 19-cell HCPCF: second fundamental fourth 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  38. Convergence of FEM Method (uniform refinement) relative error of real part of eigenvalue dof p: polynomial degree of ansatz functions 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  39. Convergence of FEM Method Comparison: adaptive and uniform refinement relative error of real part of eigenvalue dof 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  40. Optimization of HCPCF design • geometrical parameters of HCPCF: • core surround thickness t • strut thickness w • cladding meniscus radius r • pitch L • number of cladding rings n Flexibility of triangulations allow computation of almost arbitrary geometries! 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  41. Conclusions • Mathematical formulation of problem types for time-harmonic Maxwell‘s Eq. • Discretization with Finite Element Method • Construction of appropriate vectorial Finite Elements • Benchmarks with FDTD and PWE method showed • much faster convergence of FEM method • Application: Optimization of PhC-fiber design 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

  42. Vielen Dank Thank you! 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007

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