Finite Element Methods for Optical Device Design

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1 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

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 Field on Elementary Ring Resonator Global field composed from fields on local patches: FEM for Optical Device Design 11 Frank Schmidt

12 Numerical Analysis and Modelling FEM for Optical Device Design 12 Frank Schmidt

13 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

14 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

15 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

16 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

17 Photomask Simulation 3D geometry Triangulation FEM for Optical Device Design 17 Frank Schmidt

18 Scattering Problems: EUV Lithography Line mask with EUV grating Field distribution FEM for Optical Device Design 18 Frank Schmidt

19 EUV Domain Decomposition Talk by Lin Zschiedrich this morning! FEM for Optical Device Design 19 Frank Schmidt

20 Photomask simulation - geometry 2D 2D geometry Triangulation FEM for Optical Device Design 20 Frank Schmidt

21 Benchmark Photomask Simulation 2D - TE FEM for Optical Device Design 21 Frank Schmidt

22 Benchmark Photomask Simulation 2D - TM FEM for Optical Device Design 22 Frank Schmidt

23 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

24 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

25 Example: Scaffold (3D) (a) (b) First eigenvalues at X-Point Step N o DOF CPU time ω 1 ω 2 ω 3 ω :00: e e e e :07: e e e e :04: e e e e :48: e e e e-01 FEM for Optical Device Design 25 Frank Schmidt

26 Example: Scaffold (3D) continued Talk by Roland Klose this afternoon! FEM for Optical Device Design 26 Frank Schmidt

27 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

28 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

29 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

30 Comparison between PWM (MIT) and FEM (ZIB) 2D 10 1 ZIB JCM (linear) 10 2 MIT MPB 10 3 Relative error (2.5 sec) (5 min) (30 min) 10 6 ZIB JCM (quadr.) (1.5 min) Number of unknowns FEM for Optical Device Design 30 Frank Schmidt

31 Comparison between PWM (MIT) and FEM (ZIB) 3D D Photonic Crystal Convergence MIT package (PWM) 10 2 rel. error band 1 band ZIB (FEM, quadratic elem.) N FEM for Optical Device Design 31 Frank Schmidt

32 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