Zuse Institute Berlin DFG Research Center MATHEON Finite Element Methods for Maxwell‘s Equations...

Zuse Institute Berlin

DFG Research Center MATHEON

Finite Element Methods for

Maxwell‘s Equations

Jan Pomplun, Frank SchmidtComputational Nano-Optics Group

Zuse Institute Berlin

3rd Workshop on Numerical Methods forOptical Nano Structures, Zürich 2007

3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007

Jan PomplunZuse Institut Berlin

2

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



3

Maxwell‘s Equations (1861)

James Clerk Maxwell (1831-1879)

0

B

D

jDH

BE

t

t

ED

HB

in many applications the fields are in steady state:

electric field Emagnetic field Hel. displacement field Dmagn. induction Banisotropic permittivity tensor anisotropic permeability tensor

xEetxE ti ,

time-harmonic Maxwell‘s Eq:

0

021

E

EE



4

Problem types

Time-harmonic

Maxwell‘s equations

Scattering

problems

Resonance

problems

Waveguide

problems



5

Setup for Scattering Problem

incE

scatE

scatinc EEE

scattered field(strictly outgoing)

total field

incomming field

scatterer



6

Scattering Problem

reference configuration (e.g. free space)

incEscatE

scatinc EEE

(strictly outgoing)

solution to Maxwell‘s Eq. (e.g. plane wave)

dirichlet data on boundary

ext

intint

intE

computational domaincomplex geometries (scatterer)

incomming field:



7

extscat2

scat1

intint2

int1

in ,0

in ,0

EE

EE

on ,)(

on ,)(

incscat1

int1

incscatint

nEEnE

nEEnE

Scattering: Coupled Interior/Exterior PDE

Coupling condition

radiating outward is scatE

Interior and scattered field

Radiation condition (e.g. Silver Müller)

scat scat



8

Resonance Mode Problem

Eigenvalue problem for

EE 21

2,E

Bloch periodic boundary condition for photonic crystal band gap computations.

Radiation condition for isolated resonators



9

Propagating Mode Problem

Structure is invariant in z-direction:

x

yz

021

EE

z

x

x

z

x

x

ikik

Propagating Mode: zik zeyxzyx ),(),,( EE

Eigenvalue problem

for zk,E

Image: B. Mangan, Crystal Fibre



10

Weak formulation of Maxwell‘s Equations

021 EE

1.) multiplication with vectorial test function :

V,021 ΦEΦEΦ

int,curl HVΦ

2.) integration over interior domain : int

0321

int

rdEΦEΦ

3.) partial integration:

V,rdrd 2321

intint

ΦFΦEΦEΦ

boundary values



11

Weak formulation of Maxwell‘s Equations

V,rdrd 2321

intint

ΦFΦEΦEΦ

rd)f(

rd),(

2

321

int

Fww

vwvwvw

a

define following bilinear and linear form:

weak formulation of Maxwell‘s equations:

Find such that

Va wwvw ,)f(),(

intcurl, HVvFind such that

hVa wwvw ,)f(),(

VVh vdiscretization

finite element space hV hh NVdim



12

Assembling of FEM System

Find such that

},,,{,)f(),( 21 hNa wwvw

VVh vhh NV )dim(},,,{ 21 hN basis:

i

iihv

ansatz for FEM solution:

},,2,1{,)f(),( hji

iij Njha

jfi

ijihAyields FEM system:

)(ff

),(

j j

ijji aA

with:

sparse matrix



13

Finite Element Construction Principles

Q

QQQ NVV )dim(,

},,,{ 21 QN

(e.g. triangle)Finite element consists of:• geometric domain

• local element space

• basis of local element space

hVConstruction of with finite elements:locally defined vectorial functions of arbitrary order that are related to small geometric patches (finite elements)

hQ VV

Q



14

Construction of Finite Elements for Maxwell‘s Eq.

E.g. eigenvalue problem: EE 21

Fields with lie in the kernel of the curl operator-> belong to eigenvalue

E0

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 begradients of the constructed discrete scalar functions



15

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

curl,H

2L

1H

On the discrete level we also want:

)(

),(

\)(

2

1

LS

curlHV

RHW

h

h

h



16

Construction of Vectorial Finite Elements (2D: (x,y))

RbabyaxWRH h ,,\1

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

hh VRbab

aW

,, constant functions

curl,HVh

First idea to extend :

Rbaybxbb

yaxaaii ,,

210

210

Q

hV



17

Vectorial Finite Elements (2D)

hVBasis of : 321 ,,

Rcba

x

yc

b

aVH h ,,,curl,

02

1

yb

xaBut: -> lies in the kernel of the curl operator,but hW



18

FEM solution of Maxwell‘s equtions

Maxwell‘s equations(continuous model)

Weak formulation

Discretization by FEM(discrete model)

Discrete solution

A posterior errorestimation

Error<TOL?

Refine mesh

(subdivide patches Q)

solutionno

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)



19

FEM-Refinement 1 Hexagonal photonic

crystal

0 refinements252 triangles

Uniform Refinement



20

FEM-Refinement 2


Hexagonal photonic crystal



21

FEM-Refinement 3





22

FEM-Refinement 4





23


FEM-Refinement 5


t (CPU) ~ 10s

(Laptop)



24

Plasmon waveguide (silver strip): Adaptive Refinement



25

Solution (intensity)



26

Adaptiv refined mesh



27

Zoom



28

Zoom with mesh



29

Zoom 2



30

Zoom 2 with mesh



31

10 2 10 3 10 4 10 510 -7

10 -6

10 -5

10 -4

10 -3

10 -2

10 -1

N um ber of U nknow ns

3 sec

10 2 10 3 10 4 10 5

90 sec

300 sec

1800 sec

perturbed P hCunperturbed P hC

quadratic FEs linear FEs

Benchmark: 2D Bloch Modes

Benchmark:convergence of Bloch modesof a 2D photonic crystal

JCMmode is 600* faster than a

plane-wave expansion (MPB by MIT)



32

Substrate

Cr line

Air

Triangular Mesh

Plane wave = 193nm

Benchmark problem: DUV phase mask



33

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

incidence field ofvector k



34

Convergence: TE-Polarization (0-th diffraction order)

resolutionhighest t with coefficienFourier 0~

tcoefficienFourier 0

~

~

:error

0,

0,

0,

0,0,

thy

thy

y

yy

A

A

A

AAA

•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 368-379, 2005.]

FDTD

FEM

Waveguide Method



35

Laser Guide Stars

ESO‘s very large telescopeParanal, Chile

January 2006:laser beam of several Watts createdfirst artificial reference star (laser guide star)

powerful laser589nm

laser guide star (~90km):luminating sodium layer

Hollow core photonic crystal fiberfor 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



36

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



37

FEM Investigation of HCPCFs

fundamental second fourth

Eigenmodes of 19-cell HCPCF:



39

Convergence of FEM Method (uniform refinement)

rela

tive e

rror

of

real p

art

of

eig

envalu

e

p: polynomial degree of ansatz functions

effn

dof



40

Convergence of FEM Method

rela

tive e

rror

of

real p

art

of

eig

envalu

e

Comparison: adaptive and uniform refinement

effn

dof



42

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!



44

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



45

Vielen Dank

Thank you!

Zuse Institute Berlin DFG Research Center MATHEON Finite Element Methods for Maxwell‘s Equations...

Documents

Transcript of Zuse Institute Berlin DFG Research Center MATHEON Finite Element Methods for Maxwell‘s Equations...