Download Adaptive finite element methods for optical device design

Finite Element Methods for
Maxwell‘s Equations
3rd Workshop on Numerical Methods for
Optical Nano Structures, Zürich 2007
Zuse Institute Berlin
Jan Pomplun, Frank Schmidt
Computational Nano-Optics Group
Zuse Institute Berlin
DFG Research Center
MATHEON
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 for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Maxwell‘s Equations (1861)
James Clerk Maxwell (1831-1879)


  E   t B
electric field E

 
  H  t D  j
magnetic field H

el. displacement field D   D  

magn. induction B
B  0


B  H


D  E
anisotropic permittivity tensor 
anisotropic permeability tensor 
 
 
 it
in many applications the fields are in steady state: E x , t   e E x 
time-harmonic Maxwell‘s Eq:
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
   1  E   2E  0
  E
 0
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Jan Pomplun
Zuse Institut Berlin
Problem types
Time-harmonic
Maxwell‘s equations
Scattering
problems
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
Resonance
problems
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Waveguide
problems
Jan Pomplun
Zuse Institut Berlin
Setup for Scattering Problem
incomming field Einc
Escat scattered field
(strictly outgoing)
scatterer
E  Einc  Escat total field
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Scattering Problem
incomming field: Einc
solution to Maxwell‘s Eq.
(e.g. plane wave)
Escat (strictly outgoing)
ext
 int
E int
dirichlet data on boundary
E  Einc  Escat
reference configuration (e.g. free space)
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
 int
G
computational domain
complex geometries (scatterer)
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Jan Pomplun
Zuse Institut Berlin
Scattering: Coupled Interior/Exterior PDE
Interior and scattered field
   1  Eint   2Eint  0, in int
   1  Escat   2Escat  0, in ext
Coupling condition
Eint  n
 (Escat  Einc )  n, on G
 1  Eint  n   1  (Escat  Einc )  n, on G
Radiation condition (e.g. Silver Müller)
Escat is outward radiating
scat
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
scat
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Jan Pomplun
Zuse Institut Berlin
Resonance Mode Problem
   1  E   2E
Eigenvalue problem
for E,  2
Radiation condition for isolated resonators
Bloch periodic boundary condition for photonic
crystal band gap computations.
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Propagating Mode Problem
Structure is invariant in z-direction:
y
z
x
Image: B. Mangan, Crystal Fibre
Propagating Mode:
E( x, y, z )  E( x, y)eikz z
 x 
 
  1  x 
2
  x      x   E   E  0
 ik 
 ik 
 z
 z
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
Eigenvalue problem
for
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E, k z
Jan Pomplun
Zuse Institut Berlin
Weak formulation of Maxwell‘s Equations
   1  E   2 E  0
1.) multiplication with vectorial test function Φ V


 H curl , int  :
Φ     1  E   2 Φ E  0 , Φ  V
2.) integration over interior domain
 Φ    

int:

  E   2 Φ E d 3 r  0
1
 int
boundary values
3.) partial integration:
   Φ  


  E   2 Φ E d 3r 
1
 int
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Φ

F
d
r , Φ  V

G   int
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Jan Pomplun
Zuse Institut Berlin
Weak formulation of Maxwell‘s Equations
   Φ  


  E   2 Φ E d 3r 
1
 int
G   int
define following bilinear and linear form:
a ( w, v ) 
   w  
2
Φ

F
d
r , Φ  V



  v   2 w v d 3r
1
int
f( w)   w Fd 2 r
G
weak formulation of Maxwell‘s equations:
Find v V
 H curl, int  such that
discretization
a(w, v)  f( w ) , w V
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
finite element space Vh
dim Vh  N h  
Find
v Vh  V
such that
a(w, v)  f( w) , w Vh
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Jan Pomplun
Zuse Institut Berlin
Assembling of FEM System
dim( Vh )  N h
basis: {1 ,  2 , ,  N h }
v Vh  V
Find
such that
a(w, v)  f( w ) , w {1 ,  2 ,,  N h }
ansatz for FEM solution:
v   hii
i
 a( , )h  f( ) ,
j
i
i
j
j {1,2,, N h }
i
yields FEM system:
A h
ji i
 fj
with:
i
Aji  a( j , i )
f j  f ( j )
sparse matrix
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Finite Element Construction Principles
Construction of Vh with
finite elements:
locally defined vectorial
functions of arbitrary
order that are related to
small geometric patches
(finite elements)
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
Finite element consists of:
Q (e.g. triangle)
• geometric domain
• local element space VQ , dim( VQ )  N Q
• basis of local element space {1 , 2 ,,  NQ }
Q  
 VQ  Vh
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Jan Pomplun
Zuse Institut Berlin
Construction of Finite Elements for Maxwell‘s Eq.
Finite elements should preserve mathematical structure of
Maxwell‘s equations (i.e. properties of the differential operators)!
E.g. eigenvalue problem:    1  E   2E
Fields with E   lie in the kernel of the curl operator
-> belong to eigenvalue   0
For the discretized Maxwell‘s equations:
Fields which lie in the kernel of the discrete curl operator should be
gradients of the constructed discrete scalar functions
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
De Rham Complex
On simply connected domains the following sequence is exact:
1
• The gradient has an empty kernel on set of non constant functions in H  
• The range of the gradient lies in H curl,  and
is exactly the kernel of the curl operator
2
• The range of the curl operator is the whole L 
On the discrete level we also want:
Wh  H 1 () \ R
Vh  H (curl , )
S h  L2 ()
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Construction of Vectorial Finite Elements (2D: (x,y))
Starting point: Finite element space for non constant functions
(polynomials of lowest order) on triangle Q :
H 1   \ R  Wh  ax  by, a, b  R
Exact sequence: gradient of this function space has to lie in Vh  H curl, 
 a 

Wh   , a, b  R  Vh
 b 

constant functions
 a0  a1 x  a2 y 

, ai , bi  R 
First idea to extend Vh : 
 b0  b1 x  b2 y 

3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Vectorial Finite Elements (2D)
 a1 x 
  0 -> lies in the kernel of the curl operator,but  Wh
But:   
 b2 y 
 a   y 

H curl,   Vh     c , a, b, c  R
 b    x 

Basis 1 ,  2 , 3  of Vh :
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
FEM solution of Maxwell‘s 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)
Maxwell‘s equations
(continuous model)
Scattering, resonance, waveguide Weak formulation
Finite element construction,
assembling
Discretization by FEM
(discrete model)
Discrete solution
Refine mesh
A posterior error
estimation
(subdivide patches Q)
no
Error<TOL?
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
solution
18
Jan Pomplun
Zuse Institut Berlin
Uniform Refinement
FEM-Refinement 1
0
252
Hexagonal photonic crystal
refinements
triangles
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
FEM-Refinement 2
1
1008
Hexagonal photonic crystal
refinements
triangles
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
FEM-Refinement 3
2
4032
Hexagonal photonic crystal
refinements
triangles
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
FEM-Refinement 4
Hexagonal photonic crystal
3
refinements
16128 triangles
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Hexagonal photonic crystal
FEM-Refinement 5
t (CPU) ~ 10s (Laptop)
4
refinements
64512 triangles
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Plasmon waveguide (silver strip): Adaptive Refinement
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Solution (intensity)
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Adaptiv refined mesh
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Zoom
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Zoom with mesh
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Zoom 2
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Zoom 2 with mesh
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Benchmark: 2D Bloch Modes
-1
10
unperturbed PhC
10-2
1800 sec
10-3

Benchmark:
convergence of
Bloch modes
of a 2D photonic crystal
perturbed PhC
10-4
3 sec
300 sec
10-5
quadratic FEs
linear FEs
-6
10
90 sec
-7
10
102
103
104 105
102 103
Number of Unknowns
104
105
JCMmode is 600* faster than a
plane-wave expansion (MPB by MIT)
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Benchmark problem: DUV phase mask
Plane wave
 = 193nm
Substrate
Cr line
Air
Triangular Mesh
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Benchmark Geometry
•extremely simple
geometry
•simple treatment of
incident field
•-> well suited for
benchmarking
methods
•geometric advantages
of FEM are not put
into effect
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
air
substrate
k  vector of field incidence
33
Jan Pomplun
Zuse Institut Berlin
Convergence: TE-Polarization (0-th diffraction order)
FDTD
• All solvers show "internal"
convergence
• Speeds of convergence
differ significantly
Waveguide
Method
FEM
~
Ay ,0  Ay ,0
error : A 
~
Ay ,0
Ay ,0  0th Fourier coefficien t
~
Ay ,0  0th Fourier coefficien t with highest resolution
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
[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.]
34
Jan Pomplun
Zuse Institut Berlin
Laser Guide Stars
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!
laser guide star (~90km):
luminating sodium layer
Na
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
ESO‘s very large telescope
Paranal, Chile
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
589nm
35
Jan Pomplun
Zuse Institut Berlin
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!
transparent glass
[Roberts et al., Opt. Express 13, 236 (2005)]
Courtesy of B. Mangan, Crystal Fibre, DK
Goal:
• calculation of leaky propagation modes inside hollow core
• optimization of fiber design to minimize radiation losses
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
36
Jan Pomplun
Zuse Institut Berlin
FEM Investigation of HCPCFs
Eigenmodes of 19-cell HCPCF:
fundamental
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
second
37
fourth
Jan Pomplun
Zuse Institut Berlin
neff 
relative error of real part of eigenvalue
Convergence of FEM Method (uniform refinement)
dof
p: polynomial degree of ansatz functions
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Convergence of FEM Method
neff 
relative error of real part of eigenvalue
Comparison: adaptive and uniform refinement
dof
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
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
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
Flexibility of triangulations
allow computation of almost
arbitrary geometries!
42
Jan Pomplun
Zuse Institut Berlin
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 for
Optical Nano Structures, 10.07.2007
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Jan Pomplun
Zuse Institut Berlin
Vielen Dank
Thank you!
3rd Workshop on Numerical Methods for
Optical Nano Structures, 10.07.2007
45
Jan Pomplun
Zuse Institut Berlin