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

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

Jan Pomplun Zuse Institut Berlin 3 3rd Workshop on Numerical Methods for Optical Nano Structures, Maxwell‘s Equations (1861) James Clerk Maxwell ( ) 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:

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

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

Jan Pomplun Zuse Institut Berlin 6 3rd Workshop on Numerical Methods for Optical Nano Structures, 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:

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

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

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

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

Jan Pomplun Zuse Institut Berlin 11 3rd Workshop on Numerical Methods for Optical Nano Structures, 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

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

Jan Pomplun Zuse Institut Berlin 13 3rd Workshop on Numerical Methods for Optical Nano Structures, 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)

Jan Pomplun Zuse Institut Berlin 14 3rd Workshop on Numerical Methods for Optical Nano Structures, 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

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

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

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

Jan Pomplun Zuse Institut Berlin 18 3rd Workshop on Numerical Methods for Optical Nano Structures, 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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Jan Pomplun Zuse Institut Berlin 33 3rd Workshop on Numerical Methods for Optical Nano Structures, 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

Jan Pomplun Zuse Institut Berlin 34 3rd Workshop on Numerical Methods for Optical Nano Structures, 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 , 2005.] FDTD FEM Waveguide Method

Jan Pomplun Zuse Institut Berlin 35 3rd Workshop on Numerical Methods for Optical Nano Structures, 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

Jan Pomplun Zuse Institut Berlin 36 3rd Workshop on Numerical Methods for Optical Nano Structures, 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

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

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

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

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

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

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

Jan Pomplun Zuse Institut Berlin 43 3rd Workshop on Numerical Methods for Optical Nano Structures, 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

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

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