DFG Research Center Matheon mathematics for key technologies and Zuse Institute Berlin Finite Element Methods for Optical Device Design Frank Schmidt Sven Burger, Roland Klose, Achim Schädle, Lin Zschiedrich http://www.zib.de/nano-optics
Computational Nano-Optics: Group Members Imbo Sim Nadja Roizengaft Christian Marwitz Sven Burger Roland Klose Benjamin Klettner Achim Schädle Lin Zschiedrich Frank Schmidt FEM for Optical Device Design 2 Frank Schmidt
Outline 1. Finite Element Method Variational Formulation Vectorial Finite Elements (Nedelec s Edge Elements) Adaptivity 2. Scattering Problems Metamaterials Photomasks 3. Eigenvalue/Resonance Problems Photonic Crystals Resonators FEM for Optical Device Design 3 Frank Schmidt
Time Harmonic Maxwell s Equations H( x, t) = e iωt H( x), E( x, t) = e iωt E( x) yield: 1 ε H = ω2 µh, µh = 0 1 µ E = ω2 ɛe ɛe = 0 FEM for Optical Device Design 4 Frank Schmidt
Variational Formulation Simple example: 1 ε r ( H) µ 0 ε 0 ω 2 H = F n H = 0 on Γ = Ω Define quantities: a(u, v) = (u, v) = d(u, q) = Ω Ω Ω 1 ( u) ( v) dx µ 0 ε 0 ω 2 ε r u v d x u q d x. Ω u v d x FEM for Optical Device Design 5 Frank Schmidt
Variational Formulation Simple example: 1 ε r ( H) µ 0 ε 0 ω 2 H = F n H = 0 on Γ = Ω Variational form: Find u V 0 such that a(u, v) = (F, v) for all v H(curl, Ω) H(curl, Ω) = { v L 2 (Ω) curl v L 2 (Ω) } V 0 = { w H(curl, Ω) d(w, q) = 0 q H 1}. FEM for Optical Device Design 6 Frank Schmidt
Galerkin Method ψ 1,..., ψ N H (curl) ansatz functions. Approximate solution: u = i N c i ψ i Galerkin method: Project variational problem onto subspace V spanned by the ansatz functions ψ 1,..., ψ N : a (u, ψ j ) = (F, ψ j ) ψ j V A i,j = a (ψ i, ψ j ) b j = (F, ψ j ) Algebraic problem: = Au = b, u = [c 1,..., c N ] FEM for Optical Device Design 7 Frank Schmidt
Triangulation Decomposition of the computational domain (triangles, tetrahedra, etc.) H and E are only tangentially continous on material boundaries. Normal component jumps across material interfaces. FEM for Optical Device Design 8 Frank Schmidt
Edge Elements Edge elements (Nedelec 1980) Elements of order 0, 1,2 with respect to the tangential component of the diagonal edge FEM for Optical Device Design 9 Frank Schmidt
Curl-Free and Div-Free Fields curl-free field: curl E = 0 E = φ div-free field: div E = 0 E = W FEM for Optical Device Design 10 Frank Schmidt
Field on Elementary Ring Resonator Global field composed from fields on local patches: FEM for Optical Device Design 11 Frank Schmidt
Numerical Analysis and Modelling FEM for Optical Device Design 12 Frank Schmidt
Scattering Problem Scattering of a plane wave on a bounded obstacle Task: Compute H on Ω from 1 ε H ω2 µh = 0, µh = 0 + incident field and its normal derivative on Ω + inner boundary conditions + (quasi periodic boundary conditions) + transparent boundary conditions FEM for Optical Device Design 13 Frank Schmidt
Beam Passing a Plane Lens Micro lens embedded in free space: intensity Micro lens embedded in free space: phase FEM for Optical Device Design 14 Frank Schmidt
Basic idea: PML and Pole Condition PML: Consider complex continuation on exterior rays δη a~ a γ 0 γ 1 Curved ray Straight rays Pole condition: Consider Laplace transform on exterior rays FEM for Optical Device Design 15 Frank Schmidt
Scattering Problems: Metamaterials Periodic array: Field on elementary cell: SEM of miniaturized gold split-ring resonators with dimension smaller than optical wavelength (courtesy of CFN Karlsruhe). FEM for Optical Device Design 16 Frank Schmidt
Photomask Simulation 3D geometry Triangulation FEM for Optical Device Design 17 Frank Schmidt
Scattering Problems: EUV Lithography Line mask with EUV grating Field distribution FEM for Optical Device Design 18 Frank Schmidt
EUV Domain Decomposition Talk by Lin Zschiedrich this morning! FEM for Optical Device Design 19 Frank Schmidt
Photomask simulation - geometry 2D 2D geometry Triangulation FEM for Optical Device Design 20 Frank Schmidt
Benchmark Photomask Simulation 2D - TE FEM for Optical Device Design 21 Frank Schmidt
Benchmark Photomask Simulation 2D - TM FEM for Optical Device Design 22 Frank Schmidt
Eigenvalue Problems: Bloch Modes Periodic materials: Bloch periodicity: a 2 a 1 2D Photonic crystal (courtesy of V. Lehmann, Infineon Technologies). a mn = m a 1 + n a 2, m, n Z ɛ ( x) = ɛ ( x + a mn ), µ ( x) = µ ( x + a mn ) H ( x + a mn ) = e i k a mn H ( x). FEM for Optical Device Design 23 Frank Schmidt
Eigenvalue Problems: Bloch Modes = Tasks: Lattice periodic field: h ( x) = e i k x H ( x) ( +i ) 1 k ε ( +i ) k h = ω 2 µh, ( +i ) k µh = 0 + quasi-periodic boundary conditions + (transparent boundary conditions). Given k, compute the Blochvector h and the positiv real number ω 2, or, Given ω 2, compute the Blochvector h and the complex wavevector k FEM for Optical Device Design 24 Frank Schmidt
Example: Scaffold (3D) (a) (b) First eigenvalues at X-Point Step N o DOF CPU time ω 1 ω 2 ω 3 ω 4 0 3450 00:00:22.95 2.73648e-01 2.73967e-01 4.27901e-01 4.28753e-01 1 27572 00:07:36.46 2.73074e-01 2.73113e-01 4.26632e-01 4.26746e-01 2 220520 01:04:18.70 2.72794e-01 2.72801e-01 4.25991e-01 4.26006e-01 3 1764048 08:48:01.65 2.72681e-01 2.72683e-01 4.25734e-01 4.25737e-01 FEM for Optical Device Design 25 Frank Schmidt
Example: Scaffold (3D) continued Talk by Roland Klose this afternoon! FEM for Optical Device Design 26 Frank Schmidt
Photonic crystal from MPI Halle with measured data (air) (SiO2) (Si) (SiO2) Unit cell with hole Optical fields Band diagram Problem: Off plane radiation transparent boundary conditions FEM for Optical Device Design 27 Frank Schmidt
Micro Cavity Micro resonator embedded in free space Resonating mode Problem: Radiation in all directions transparent boundary conditions FEM for Optical Device Design 28 Frank Schmidt
Micro Cavity Micro resonator embedded in free space Resonating mode Problem: Radiation in all directions transparent boundary conditions FEM for Optical Device Design 29 Frank Schmidt
Comparison between PWM (MIT) and FEM (ZIB) 2D 10 1 ZIB JCM (linear) 10 2 MIT MPB 10 3 Relative error 10 4 10 5 (2.5 sec) (5 min) (30 min) 10 6 ZIB JCM (quadr.) (1.5 min) 10 7 10 1 10 2 10 3 10 4 10 5 10 6 10 7 Number of unknowns FEM for Optical Device Design 30 Frank Schmidt
Comparison between PWM (MIT) and FEM (ZIB) 3D 10 1 3D Photonic Crystal Convergence MIT package (PWM) 10 2 rel. error band 1 band 2 10 3 ZIB (FEM, quadratic elem.) 10 4 10 2 10 3 10 4 10 5 10 6 N FEM for Optical Device Design 31 Frank Schmidt
Conclusions FEM for time-harmonic Maxwell s equations is a versatile method to solve problems from optics. It is both applicable to eigenvalue problems as well as scattering problems. Numerical methods with optimal complexity are available. Adaptivity helps to overcome complexity barriers. FEM for Optical Device Design 32 Frank Schmidt