Thorsten Liebig, Andreas Rennings, and Daniel Erni

Transcription

1 8th Workshop on Numerical Methods for Optical Nanostructures, ETH Zürich, July 2-4, Zürich Switzerland, 2012 OpenEMS A Free and Open Source Cartesian and Cylindrical EC-FDTD Simulation Platform Supporting Multi-Pole Drude/Lorentz Dispersive Material Models for Plasmonic Nanostructures Thorsten Liebig, Andreas Rennings, and Daniel Erni General and Theoretical Electrical Engineering (ATE), Faculty of Engineering, University of Duisburg-Essen, and CENIDE Center for Nanointegration Duisburg-Essen, D Duisburg, Germany thorsten.liebig@uni-due.de Web: Abstract OpenEMS [1] is a full vectorial three-dimensional equivalent-circuit (EC) finite-difference time-domain (FDTD) simulation platform with complete support for either Cartesian or cylindrical coordinates. Because of its memory efficiency and speed, the FDTD method is known to be particularly well suited for large three-dimensional problems, where the latter can either be a large complex system [2] or a sophisticated optical nano-structure. It s worth noting that the EC-FDTD formulation has clear benefits such as a reduced numerical effort inside the iteration loop due to a reduced number of multiplications (which is intrinsic to the representation of fields with corresponding state variables such as voltages and currents) and the intuitive incorporation of (highly) dispersive materials. For example materials that are represented by multi-polar Drude/Lorentz type models [3] can be easily implemented along corresponding extensions of the associated EC (in the form of small tailored filter sections), which mimicks the material dispersion underlying e.g. plasmonic structures, wide band conducting sheet models or even more challenging material properties like biological tissue [4]. An additional virtue of the EC- FDTD formulation is its affinity to an energy-based stability criterion that tends to relax the usually applied Courant-Friedrich-Levy (CFL) stability criterion. To our knowledge there are only very few free and open source FDTD solver that support adaptable dispersive material models and there is virtually no available code that supports this type of material feature using fully graded meshing in Cartesian and cylindrical coordinates. OpenEMS can be deployed on any modern personal computer running Linux or Windows and with the support of MPI, openems is even ready to run on a Linux cluster (or a supercomputer). At the workshop we will present an insight into the EC-FDTD formulation for the Cartesian and cylindrical mesh and discuss our concept of «engine extensions» as a very systematic and simple approach to implement even complex material models without the need of modifications to the generic (and speed optimized) FDTD core iteration engine. Hence OpenEMS is open to the community to contribute many additional and exciting new engine extensions. Furthermore we will elucidate our user-friendly Matlab/Octave scripting interface to setup, control and evaluate the EC- FDTD simulation, as well as a simple graphical user interface providing a 2D/3D structural viewer. We will conclude with a plasmonic nanodevice example that exploits our cylindrical meshing feature. [1] T. Liebig, openems Website: [2] T. Liebig, D. Erni, A. Rennings, N. H. L. Koster, J. Fröhlich, ESMRMB 2011, Oct. 6-8, Leipzig, pp. 22, [3] A. Rennings, Elektromagnetische Zeitbereichssimulationen innovativer Antennen auf Basis von Metamaterialien. Dissertation, University of Duisburg-Essen, [4] S. Huclova, D. Erni, J. Fröhlich, Journal of Physics D: Applied Physics, 43(36), , 2010.

2 OpenEMS - A Free and Open Source Cartesian & Cylindrical EC-FDTD Simulation Platform Supporting Multi-Polar Drude/Lorentz Dispersive Material Models for Plasmonic Nanstructures Thorsten Liebig*, Andre Rennings, and Daniel Erni General and Theoretical Electrical Engineering (ATE), Faculty of Engineering, University of Duisburg-Essen, and CENIDE Center for Nanointegration Duisburg-Essen D-47048, Duisburg, Germany Outline Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 2 / 23

3 Outline Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 3 / 23 EC-FDTD Yee Scheme in Cartesian Coordinates Difference to the conventional FDTD scheme: Edge voltages, e.g.: v x =Δ x E x instead of the electric field E x Edge currents, e.g.: i y = Δ y H y instead of the magnetic field H y Ex(nz + 1) Ay Ex(nz + 1) Hx Ñ Ez(nx + 1) Hx Ñ Ez(nx + 1) Ez Hy Ez Hy N Ey Hz N Ey Hz Hz(ny 1) Ex Ãx Hz(ny 1) Ex Hy(nz 1) Hy(nz 1) This yields the update equations for the equivalent circuit (EC) FDTD: v x = i y = 2Cx ΔtGx 2C x +Δ tg x v x + 2Δ t (i y i y(n z 1) i z + i z(n y 1)) 2C x +Δ tg x 2Ly ΔtRy 2Δ t i y (v z v z(n x + 1) v x + v x(n z + 1)) 2L y +Δ tr y 2L x +Δ tr y Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 4 / 23

4 Advantages of the EC-FDTD Scheme The EC-FDTD and conventional FDTD scheme are numerically equivalent with all the corresponding (dis)advantages of the FDTD method Only two coefficients for each update equation instead of three 33% reduced coefficient memory usage 33% less multiplications necessary inside the FDTD engine Intuitive energy-based stability criteria [1] Using capacitors and inductors delivers a deeper inside into material interpolation/averaging Easy integration of advanced dispersive material models as an equivalent circuit filter (see below) All known advanced FDTD features (e.g. PML boundary conditions) canbeusedintheec-fdtdscheme Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 5 / 23 EC-FDTD Yee Scheme in Cylindrical Coordinates Er(nz + 1) Er(nz + 1) A α Ez Hr Hα Ez(nr + 1) Ez Hr Hα Ez(nr + 1) Eα Eα Hz Hz Er Er Hz(nr 1) Ãr Hz(nr 1) Hα(nz 1) Hα(nz 1) Update equations: v r = i α = 2Cr ΔtGr 2C r +Δ tg r v n t 1 r + 2Δt 2C rδ tg r (i α i α(n z 1) i z + i z(n α 1)) 2Lα ΔtRα 2L α +Δ tr α i α 2Δ t (v z v z(n r + 1) v r + v r(n z + 1)) 2L r +Δ tr α Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 6 / 23

5 Differences in Cylindrical and Cartesian EC-FDTD Identical update equations (layout) Identical coefficient format Identical calculations of the equivalent circuit parameters C, G, L, R Identical support for e.g. PML, dispersive materials, etc. Sole difference in calculation of the edge length and surface areas Cartesian Mesh: Cylindrical Mesh: C x,y,z = ε Ãx,y,z Δ x,y,z G x,y,z = κ Ãx,y,z Δ x,y,z L x,y,z = μ Ax,y,z Δ x,y,z R x,y,z = σ Ax,y,z Δ x,y,z C r,z = ε Ãr,z Δ r,z and C α = ε Ãα rδ α G r,z = κ Ãr,z Δ r,z and G α = κ Ãα rδ α L r,z = μ Ar,z Δ r,z and L α = μ Aα r Δ α R r,z = σ Ar,z Δ r,z and R α = σ Aα r Δ α Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 7 / 23 Cylindrical EC-FDTD: Types of Mesh In cylindrical coordinates, different stages of mesh complexity are possible: primary mesh dual mesh primary mesh dual mesh primary mesh dual mesh r α α r α (a) partial cylindrical mesh (b) closed-α mesh (c) full cylindrical mesh Some special treatments are necessary: Closed-α mesh: A special boundary condition in ±α direction is needed Full cylindrical mesh: A special EC-FDTD operator at r = 0 necessary Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 8 / 23

6 Drawback of a full cylindrical mesh The full cylindrical mesh has very tiny cells around the singularity: CFL yields a very small time step and very long simulation time α α Solution: Using sub-grids around r = 0 with a reduced α-resolution! The result is a much larger time step for all cascaded grids Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 9 / 23 Using the EC-FDTD scheme it is quite intuitive to incorporate dispersive material models, e.g. the lossy multi-polar Drude model The additional update equations have to be calculated using an auxiliary differential equation (e.g. as an engine extension) The Drude/Lorentz model is well suited to efficiently model metals at optical frequencies Rigorous energy-based stability proof possible [1] For more detailed information about dispersive material models in the EC-FDTD formulation have a look at: [1] Rennings A., Elektromagnetische Zeitbereichssimulationen innovativer Antennen auf Basis von Metamaterialien., PhD Thesis, University of Duisburg-Essen, Sept Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 10 / 23

7 Outline openems - Engine Extensions Concept Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 11 / 23 openems - Engine Extensions Concept Full 3D EC-FDTD in Cartesian and cylindrical coordinates Multi-polar Drude type material model Powerful and easy to use Matlab/Octave scripting interface: Coordinate dependent material and excitation definition Many geometrical primitives: e.g. cubes, cylinder, spheres, wires, curves... Apply arbitrary affine transformations on all geometrical primitives Simple graphical user interface to review the defined structures Access and process raw or interpolated field dumps in TD or FD Fast multi-threading, near-to-far-field transformation Multi-platform: Linux and Windows fully supported Relies on open source libraries such as boost, hdf5, vtk, fparser... FDTD engine utilizing SSE and multi-threading for higher speed Support for remote and cluster simulation using MPI Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 12 / 23

8 openems - Engine Extensions Concept openems - Engine Extensions Concept EC-FDTD Engine Extension read read write read Generic EC-FDTD Engine voltage updates openems uses a generic and highly optimized FDTD engine Additional FDTD features are encapsulated in engine extensions Extension designer doesn t need to worry about engine internals read write current updates Currently available extensions: PML, Drude material, conducting sheet, Excitation, MPI, cylindrical sub-grids, etc. Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 13 / 23 Outline Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 14 / 23

9 Microwave Patch Antennas Tutorial: Wireless LAN Patch Antenna Example: (a) Conventional / Cartesian Patch Antenna (b) Conformal / Cylindrical Patch Antenna Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 15 / 23 Microwave Helical Antennas Tutorial: Wireless LAN Helical Antenna Example: (a) Helical Antenna using a Cartesian or Cylindrical grid (b) Helical Antenna Array using a Cartesian grid Many more microwave tutorials and examples are included in the openems package... Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 16 / 23

10 Traveling Wave - Magnetic Resonance Imaging I Metabore as a flexible excitation scheme: Simulation Info: CRLH ring antennas with very tiny features 100μm Large overall volume: = 64cm, L = 3m Virtual Family human phantom included 30 million cells or 180 million unknowns Simulation time: 15h (Core-i7) openems is the only known tool to handle this task efficiently Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 17 / 23 Traveling Wave - Magnetic Resonance Imaging II Goal: Narrow window of exposure in a desired body region (e.g. abdomen) Process: 1 Perform one simulation for each ring and each polarization 2 Read all field dumps to setup a field database 3 Optimize the excitation distribution (amplitude and phase vector) for the desired goal function Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 18 / 23

11 Plasmonic (not very nano) Device I An openems show case in the area of plasmonic nano devices: Silver disk (Drude material model) with a concentric Bragg surface grating AG disk : nm grating period: 585 nm grating width: 60 nm grating height: 20 nm Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 19 / 23 Plasmonic (not very nano) Device II Some simulation figures: Wavelength range of excitation (Gaussian pulse): 400 nm nm Simulation domain: 22 μm diameter and height of 7.2 μm 43 million cells (MC) or 258 million unknowns 4 cascaded sub-grids for a reasonable time step Simulation time 17h (Core-i7 CPU) Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 20 / 23

12 Outline Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 21 / 23 The EC-FDTD scheme for Cartesian and cylindrical grids has been presented Cascaded sub-grids are not optional but mandatory for en efficient simulation of full cylindrical grids Using engine extensions allows for an easy and clean integration of (new) advanced and complex FDTD features openems is well suited for a wide range of applications, from RF to plasmonic nano devices using Cartesian or cylindrical grids openems EC-FDTD engine already showing a good speed performance (up to 150 MC/s on a single Core-i7, some GC/s on a cluster) More interesting tutorial examples for plasmonic devices would be desirable Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 22 / 23

13 Further Reading For further information: openems Website: openems Forum: openems Development: openems is a free and open source software Feel free to download, evaluate and contribute Thank you for your attention! Thorsten Liebig thorsten.liebig@uni-due.de openemsa - Free and Open Source EC-FDTD Simulation Platform 23 / 23