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

2. Jan Pomplun Zuse Institut Berlin 2 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 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

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

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

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

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

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

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

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

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

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

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

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

14. Jan Pomplun Zuse Institut Berlin 14 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Construction of Finite Elements for Maxwell‘s Eq. E.g. eigenvalue problem: Fields with lie in the kernel of the curl operator -> belong to eigenvalue Finite elements should preserve mathematical structure of Maxwell‘s equations (i.e. properties of the differential operators)! 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

15. Jan Pomplun Zuse Institut Berlin 15 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 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:

16. Jan Pomplun Zuse Institut Berlin 16 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 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 :

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

18. Jan Pomplun Zuse Institut Berlin 18 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 FEM solution of Maxwell‘s equtions Maxwell‘s equations (continuous model) Weak formulation Discretization by FEM (discrete model) Discrete solution A posterior error estimation Error<TOL? Refine mesh (subdivide patches Q) solution no Scattering, resonance, waveguide Finite element construction, assembling 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)

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

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

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

22. Jan Pomplun Zuse Institut Berlin 22 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 FEM-Refinement 4 3refinements 16128triangles Hexagonal photonic crystal

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

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

25. Jan Pomplun Zuse Institut Berlin 25 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Solution (intensity)

26. Jan Pomplun Zuse Institut Berlin 26 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Adaptiv refined mesh

27. Jan Pomplun Zuse Institut Berlin 27 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Zoom

28. Jan Pomplun Zuse Institut Berlin 28 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Zoom with mesh

29. Jan Pomplun Zuse Institut Berlin 29 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Zoom 2

30. Jan Pomplun Zuse Institut Berlin 30 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Zoom 2 with mesh

31. Jan Pomplun Zuse Institut Berlin 31 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 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)

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

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

34. Jan Pomplun Zuse Institut Berlin 34 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Convergence: TE-Polarization (0-th diffraction order) All solvers show "internal" convergence Speeds of convergence differ significantly [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.] FDTD FEM Waveguide Method

35. Jan Pomplun Zuse Institut Berlin 35 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Laser Guide Stars ESO‘s very large telescope Paranal, Chile January 2006: laser beam of several Watts created first artificial reference star (laser guide star) powerful laser 589nm laser guide star (~90km): luminating sodium layer Hollow core photonic crystal fiber for guidance of light from very intense pulsed laser 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! Na

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

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

38. Jan Pomplun Zuse Institut Berlin 38 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 FEM Investigation of HCPCFs unknowns1st eigenvalue2nd eigenvalue 861289 438297 218504 symmetry TE TM TE: transversal electric field = 0 TM: transversal magnetic field = 0 eigenvalues: effective refractive index:

39. Jan Pomplun Zuse Institut Berlin 39 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Convergence of FEM Method (uniform refinement) relative error of real part of eigenvalue p: polynomial degree of ansatz functions dof

40. Jan Pomplun Zuse Institut Berlin 40 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Convergence of FEM Method relative error of real part of eigenvalue Comparison: adaptive and uniform refinement dof

41. Jan Pomplun Zuse Institut Berlin 41 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Convergence of FEM Method relative error of imaginary part of eigenvalue adaptive refinement dof

42. Jan Pomplun Zuse Institut Berlin 42 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 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!

43. Jan Pomplun Zuse Institut Berlin 43 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Optimization of HCPCF design: number of cladding rings number of cladding rings n imaginary part of eigenvalue w = 50nm t =170nm r =300nm L =1550nm

44. Jan Pomplun Zuse Institut Berlin 44 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 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

45. Jan Pomplun Zuse Institut Berlin 45 3rd Workshop on Numerical Methods for Optical Nano Structures, 10.07.2007 Vielen Dank Thank you!