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Finite Element Methods for Maxwell

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Title: Finite Element Methods for Maxwell


1
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

2
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

3
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
4
Problem types
5
Setup for Scattering Problem
incomming field
scattered field (strictly outgoing)
scatterer
total field
6
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)
7
Scattering Coupled Interior/Exterior PDE
Interior and scattered field
Coupling condition
Radiation condition (e.g. Silver Müller)
scat
scat
8
Resonance Mode Problem
Eigenvalue problem for
Radiation condition for isolated resonators
Bloch periodic boundary condition for photonic
crystal band gap computations.
9
Propagating Mode Problem
Structure is invariant in z-direction
y
z
x
Image B. Mangan, Crystal Fibre
Propagating Mode
Eigenvalue problem for
10
Weak formulation of Maxwells Equations
1.) multiplication with vectorial test function

2.) integration over interior domain
boundary values
3.) partial integration
11
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
12
Assembling of FEM System
Find such that
basis
ansatz for FEM solution
yields FEM system
with
sparse matrix
13
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)
14
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
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

On the discrete level we also want
16
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
17
Vectorial Finite Elements (2D)
But
-gt lies in the kernel of the curl operator,but
Basis of
18
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
19
FEM-Refinement 1
Uniform Refinement
Hexagonal photonic crystal
0 refinements 252 triangles
20
FEM-Refinement 2
Hexagonal photonic crystal
1 refinements 1008 triangles
21
FEM-Refinement 3
Hexagonal photonic crystal
2 refinements 4032 triangles
22
FEM-Refinement 4
Hexagonal photonic crystal
3 refinements 16128 triangles
23
FEM-Refinement 5
Hexagonal photonic crystal
t (CPU) 10s (Laptop)
4 refinements 64512 triangles
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
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)
32
Benchmark problem DUV phase mask
? 193nm
Plane wave
Substrate
Cr line
Air
Triangular Mesh
33
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

34
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.
35
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
36
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

37
FEM Investigation of HCPCFs
Eigenmodes of 19-cell HCPCF
second
fundamental
fourth
38
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
39
Convergence of FEM Method (uniform refinement)
relative error of real part of eigenvalue
dof
p polynomial degree of ansatz functions
40
Convergence of FEM Method
Comparison adaptive and uniform refinement
relative error of real part of eigenvalue
dof
41
Convergence of FEM Method
adaptive refinement
relative error of imaginary part of eigenvalue
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!
43
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
44
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

45
Vielen Dank
Thank you!
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