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
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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Problem types
Time-harmonic
Maxwell‘s equations
Scattering
problems
Resonance
problems
Waveguide
problems
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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Setup for Scattering Problem
incE
scatE
scatinc EEE
scattered field(strictly outgoing)
total field
incomming field
scatterer
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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:
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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)
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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FEM-Refinement 1 Hexagonal photonic
crystal
0 refinements252 triangles
Uniform Refinement
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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FEM-Refinement 2
1 refinements1008 triangles
Hexagonal photonic crystal
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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FEM-Refinement 3
2 refinements4032 triangles
Hexagonal photonic crystal
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
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FEM-Refinement 4
3 refinements16128 triangles
Hexagonal photonic crystal
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
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Hexagonal photonic crystal
FEM-Refinement 5
4 refinements64512 triangles
t (CPU) ~ 10s
(Laptop)
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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Plasmon waveguide (silver strip): Adaptive Refinement
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
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Solution (intensity)
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
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Adaptiv refined mesh
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
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Zoom
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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Zoom with mesh
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
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Zoom 2
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
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Zoom 2 with mesh
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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)
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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Substrate
Cr line
Air
Triangular Mesh
Plane wave = 193nm
Benchmark problem: DUV phase mask
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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FEM Investigation of HCPCFs
fundamental second fourth
Eigenmodes of 19-cell HCPCF:
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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Convergence of FEM Method
rela
tive e
rror
of
real p
art
of
eig
envalu
e
Comparison: adaptive and uniform refinement
effn
dof
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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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!
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
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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
3rd Workshop on Numerical Methods forOptical Nano Structures, 10.07.2007
Jan PomplunZuse Institut Berlin
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Vielen Dank
Thank you!