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Finite Element Methods for Maxwells Equations

3rd Workshop on Numerical Methods for Optical

Nano Structures, Zürich 2007

- Jan Pomplun, Frank Schmidt
- Computational Nano-Optics Group
- Zuse Institute Berlin

Outline

- Problem formulations based on time-harmonic

Maxwells equations - Scattering problems
- Resonance problems
- Waveguide problems
- Discrete problem
- Weak formulation of Maxwells 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

Maxwells Equations (1861)

James Clerk Maxwell (1831-1879)

electric field E magnetic field H el.

displacement field D magn. induction

B anisotropic permittivity tensor e anisotropic

permeability tensor m

in many applications the fields are in steady

state

time-harmonic Maxwells Eq

Problem types

Setup for Scattering Problem

incomming field

scattered field (strictly outgoing)

scatterer

total field

Scattering Problem

incomming field

(strictly outgoing)

E

solution to Maxwells Eq. (e.g. plane wave)

scat

G

dirichlet data on boundary

reference configuration (e.g. free space)

computational domain complex geometries

(scatterer)

Scattering Coupled Interior/Exterior PDE

Interior and scattered field

Coupling condition

Radiation condition (e.g. Silver Müller)

scat

scat

Resonance Mode Problem

Eigenvalue problem for

Radiation condition for isolated resonators

Bloch periodic boundary condition for photonic

crystal band gap computations.

Propagating Mode Problem

Structure is invariant in z-direction

y

z

x

Image B. Mangan, Crystal Fibre

Propagating Mode

Eigenvalue problem for

Weak formulation of Maxwells Equations

1.) multiplication with vectorial test function

2.) integration over interior domain

boundary values

3.) partial integration

Weak formulation of Maxwells Equations

define following bilinear and linear form

finite element space

weak formulation of Maxwells equations

Find such

that

discretization

Find such that

Assembling of FEM System

Find such that

basis

ansatz for FEM solution

yields FEM system

with

sparse matrix

Finite Element Construction Principles

Construction of with finite

elements locally defined vectorial functions of

arbitrary order that are related to small

geometric patches (finite elements)

- Finite element consists of
- geometric domain
- local element space
- basis of local element space

(e.g. triangle)

Construction of Finite Elements for Maxwells Eq.

Finite elements should preserve mathematical

structure of Maxwells equations (i.e.

properties of the differential operators)!

E.g. eigenvalue problem

Fields with lie in the kernel of

the curl operator -gt belong to eigenvalue

For the discretized Maxwells equations Fields

which lie in the kernel of the discrete curl

operator should be gradients of the constructed

discrete scalar functions

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

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

Vectorial Finite Elements (2D)

But

-gt lies in the kernel of the curl operator,but

Basis of

FEM solution of Maxwells equtions

- 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)

Maxwells equations (continuous model)

Scattering, resonance, waveguide

Weak formulation

Discretization by FEM (discrete model)

Finite element construction, assembling

Discrete solution

Refine mesh (subdivide patches Q)

A posterior error estimation

no

ErrorltTOL?

solution

FEM-Refinement 1

Uniform Refinement

Hexagonal photonic crystal

0 refinements 252 triangles

FEM-Refinement 2

Hexagonal photonic crystal

1 refinements 1008 triangles

FEM-Refinement 3

Hexagonal photonic crystal

2 refinements 4032 triangles

FEM-Refinement 4

Hexagonal photonic crystal

3 refinements 16128 triangles

FEM-Refinement 5

Hexagonal photonic crystal

t (CPU) 10s (Laptop)

4 refinements 64512 triangles

Plasmon waveguide (silver strip) Adaptive

Refinement

Solution (intensity)

Adaptiv refined mesh

Zoom

Zoom with mesh

Zoom 2

Zoom 2 with mesh

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)

Benchmark problem DUV phase mask

? 193nm

Plane wave

Substrate

Cr line

Air

Triangular Mesh

Benchmark Geometry

- extremely simple geometry
- simple treatment of incident field
- -gt well suited for benchmarking methods
- geometric advantages of FEM are not put into

effect

Convergence TE-Polarization (0-th diffraction

order)

FDTD

- All solvers show "internal" convergence
- Speeds of convergence differ significantly

Waveguide Method

FEM

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.

Laser Guide Stars

- Adaptive optics system
- corrects the atmospheres blurring effect

limiting the image quality - needs a relatively bright reference star
- observable area of sky is limited!

laser guide star (90km) luminating sodium layer

January 2006 laser beam of several Watts

created first artificial reference star (laser

guide star)

Hollow core photonic crystal fiber for guidance

of light from very intense pulsed laser

powerful laser

ESOs very large telescope Paranal, Chile

589nm

Hollow core photonic crystal fiber

hollow core

- guidance of light in hollow core
- photonic crystal structure
- prevents leakage of radiation
- to the exterior

exterior air

- high energy transport possible
- small radiation losses!
- Roberts et al., Opt. Express 13, 236 (2005)

transparent glass

Courtesy of B. Mangan, Crystal Fibre, DK

- Goal
- calculation of leaky propagation modes inside

hollow core - optimization of fiber design to minimize

radiation losses

FEM Investigation of HCPCFs

Eigenmodes of 19-cell HCPCF

second

fundamental

fourth

FEM Investigation of HCPCFs

eigenvalues effective refractive index

symmetry

unknowns 1st eigenvalue 2nd eigenvalue

861289

438297

218504

TE

TE transversal electric field 0 TM

transversal magnetic field 0

TM

Convergence of FEM Method (uniform refinement)

relative error of real part of eigenvalue

dof

p polynomial degree of ansatz functions

Convergence of FEM Method

Comparison adaptive and uniform refinement

relative error of real part of eigenvalue

dof

Convergence of FEM Method

adaptive refinement

relative error of imaginary part of eigenvalue

dof

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!

Optimization of HCPCF design number of cladding

rings

imaginary part of eigenvalue

w 50nm t 170nm r 300nm L 1550nm

number of cladding rings n

Conclusions

- Mathematical formulation of problem types for

time-harmonic Maxwells 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

Vielen Dank

Thank you!