t - Computational Optics Group

Lecture Notes (1 of 4)
“The Finite Element Time Domain Method
(FETD)
for Computational Electrodynamics”
Benedikt Oswald
benedikt.oswald@psi.ch
Phone +41 (0)56 310 32 12
WBGB 131
Paul Scherrer Institut (www.psi.ch)
Large Research Facilities Department
Revisions -2010 March 29, ETH

Passion is required for any great work,
and for the revolution passion and
audacity are required in big doses
Ernesto 'Che' Guevara, Letter to his parents.

Acknowledgements
It is my pleasure to acknowledge various contributions to this lecture:
in alphabetic order
Adelmann, Andreas – PSI group leader
Burri, René – world renowned Swiss (Magnum Photos) photographer
Fomins, Aleksejs – summer student contributing to hades3d
Garvey, Terrence – head of ABK
Hafner, Christian – for inviting me
Leidenberger, Patrick – collaborator and esteemed colleague

Literature References
Jin, Jianming; “The Finite Element Method in Electromagnetics”; 2 nd edition, John Wiley & Sons; 2002. ISBN 0-
471-43818-9. Note: reference work
Volakis, John L., and Chatterjee, Arindam, and Kempel, Leo C.; “Finite Element Method for Electromagnetics –
Antennas, Microwave Circuits, and Scattering Applications”; Wiley Interscience & IEEE Press; 1998. ISBN 0-7803-
3425-6. Note: reference work
Novotny, Lukas and Hecht, Bert; “Principles of Nano-Optics”; Cambridge University Press; 2006. ISBN 13-978-
0-521-83224-3. Note: motivates applications
Taflove, Allen and Hagness, Susan, C.; “Computational Electrodynamics – The Finite Difference Time Domain
Method”, 3 rd edition; Artech House; 2005; ISBN 1-58053-832-0. Note: general inspiration for time domain methods
Zienkiewicz, O. C. and Taylor, R. L.; “The Finite Element Method – Volume 1: The Basics”; 5 th edition;
Butterworth-Heinemann; 2000; ISBN 0 7506 5049 4. Note: fundamentals on finite element methods by its
pioneers in engineering sciences.
And my sincerest apology for all forgotten references, it was not by intention!

Overview
Introduction, Motivation and Governing Equation
Finite Element Method as a Volume Approach
Practical Realization of the Algorithms
Case Studies and Modeling Accuracy

Introduction, Motivation and Governing Equations (1)
Maxwell's equations Constitutive relations
∇× H =j ∂ D
∂ t
∂ B
∇×E=−
∂t
∇⋅D= el
∇⋅B=0
D=⊗E
D=⋅E
j=⋅E
B= 0 ⋅H
Dispersive, scalar
Non-dispersive, scalar
Ohm's law
James Clerk Maxwell
Source wikivisual

Introduction, Motivation and Governing Equations (2)
E lectric field vector wave equation
aka. curl-curl equation:
general dielectric dispersive formulation
∇× 1
∇×E ∂
∂ t
E
∂2
∂
∗E =− 2
∂t ∂ t J 0
S ubject to boundary conditions and constitutive relations
E=E x , t x ∈ℝ with and 3

Introduction, Motivation and Governing Equations (3)
Why solve Maxwell's equations in the time domain ?
Model transient processes
Model non-linear processes: Raman scattering in near-field optics
Model processes over large bandwidths in one single run
Fields of applications ?
Nano-optics, scanning near-field optical microscope probes, optical antennas
Devices based on plasmonic resonances
Laser assisted field emission for advanced photo-cathodes in accelerators (SwissFEL)
Also, traditional microwave electromagnetics

Introduction, Motivation and Governing Equations (4)
Dispersive Dielectric Material – Debye Model
used for:
alcohols
Water
water in soils
∞
= s − ∞
t= ∞ t
= ∞
1 j
relative permittivity at infnite frequency (often 1.0)
change in relative permittivity due to pole
pole relaxation time
e− t
⋅U t
Source: wikipedia
Peter Joseph
William
Debye;Dutch
physicist and
physical chemist;
1884-1966.
Debye model in time domain
U t = unit step function

Introduction, Motivation and Governing Equations (5)
Dispersive Dielectric Material – Drude Model Source: wikipedia
used for:
Plasmas
Metals in the visible region
p
2
p = − ∞
2 − j
∞ relative permittivity at infnite frequency (often 1.0)
t= ∞ t 2
p
p
pole angular frequency
angular relaxation frequency
Paul Karl Ludwig
Drude; German
Physicist; 1863-
1906.
1−e − p t ⋅U t Drude model in time domain

Introduction, Motivation and Governing Equations (6)
Dispersive Dielectric Material – Lorentz Model
used for:
Solid in the visible
Metals in the visible
∞
= s − ∞
p
p
= ∞
pole angular frequency
t= ∞ t 2
p
2
p 2
2 j p p − 2
relative permittivity at infnite frequency (often 1.0)
change in relative permittivity due to pole
damping coeffcient associated with pole
p
2 − p
Source: wikipedia
Hendrik Antoon
Lorentz; Dutch
Physicist; 1853-
1928.
2 ⋅e− p t 2 2
⋅sin p−
pt
⋅U t
Lorentz model in time domain

Introduction, Motivation and Governing Equations (7)
Dispersive Dielectric Material – Drude-Lorentz Model – especially for metals in the visible
= ∞ −
2
p 2 − j
Vial et al., Physical Review B, Vol 71, pp. 085416, 2006:
Gold dielectric data modeled by Drude & Drude-Lorentz
2
p
2
2 j − p
p 2
source ditto

Introduction, Motivation and Governing Equations (8)
Why finite elements ? Rationale ? Concept ?
In order to numerically solve
ordinary differential equations (ODE)
partial differential questions (PDE)
Integral equations (IE)
defined on multi-dimensional, geometrically complicated domains, we
subdivide the complicated domain into elements of simple shape
approximate the solution via functions of know shape but unknown coefficients, aka. Degrees of freedom
formulate systems of equations to calculate the DoF

Introduction, Motivation and Governing Equation (9)
Appeal of the finite element method – level of detail (LoD)
The figure represents an S EM comparison of
the emitter shape from a double-oxidation
mould with the emitter shape from a single
oxidation mould. Fig a) is a 45 degree image of
the emitter from the double oxidation mould,
showing the rounded joins between pyramid
faces. Fig. b) shows the blunted tip of this
emitter from above. Fig c) and d) are the
equivalent views for the emitter from the single
oxidation mould, which has a very sharp tip
(not more than 5 nm - this is about the limit of
the S E M resolution for looking at tips) and it
also has undesirable, sharp ridges where the
pyramid faces meet.
From: Fabrication of all-metal field
emitter arrays with controlled apex sizes
by molding, E. Kirk, S. Tsujino, T. Vogel,
K. Jefimovs, J. Gobrecht and A. Wrulich.
Journal of Vacuum Science and
Technology B, 27(4), 2009.
c)
a)
200 nm
20 nm
1μm 20 nm
b)

Introduction, Motivation and Governing Equation (10)
mesh – element types

Introduction, Motivation and Governing Equations (11)
Ultra-short History of the Finite Element Method in Electromagnetics
...admittedly somewhat arbitrary...
Zoltan Cendes Boris Galerkin
1870 - Rayleigh variational methods
1909 - Ritz ditto
1915 - Galerkin weighted residuals
1943 - Courant piecewise continuous trial functions
1947 - Prager-Synge ditto
1964 - Zienkiewicz ditto
Source CNN
1967 - Zienkiewicz et al. solution of three-dimensional field problems
1970 - Silvester et al. non-linear magnetic field analysis
1980ies - many authors electrodynamics problems, problems with spurious modes
1990ies - Cendes, Webb, edge element to avoid spurious modes
Lee, Nédelec et al. finite element time domain methods
Source wikipedia
Source times
Olek Zienkiewicz
2000nds many authors PML, ABC, boundary integrals, dispersive media in time domain

el anejo dieléctrico (1)
0
0
derivation of the Drude operator L
we need to evaluate: 0
2
p p
2
p p
∂ 2
∂ t 2 u t⊗E = 0
2
p p
2
p p
∂ 2
∂ t 2 −e− p t u t⊗E =− 0
∂ 2
∂ t 2 1−e− p t u t ⊗E
∂
∂ t t ⊗E = 0
− 0
we use: e − p t t⊗E=∫ −∞
thus
0
2
p p
∞
2
p
p
2
p
p
∂
∂ t
2
p
p
∂ E
∂ t
[ ∂
∂ t e− p t ut ⊗E ]
1 st part
2 nd part
∂
∂t [− p e− p t u te − p t t ⊗E]
e − pt ' t ' E t−t ' dt '=E t
∂ 2
∂ t 2 −e− p t 2 ∂
u t⊗E =0 p
∂ t e− p t ut ⊗E −0 2
p
p
∂ E
∂ t

el anejo dieléctrico (2)
we combine the 1 st and 2 nd part
0
0
2
p p
2
p p
∂ 2
∂ t 2 1−e− p t 2 ∂
u t ⊗E= 0 p
∂t e− p t u t⊗E
∂ 2
∂ t 2 1−e− p t 2
u t ⊗E= 0 p[−
p e − p t u te −p t t]⊗E
using the identity from the preceeding slide again, eventually we obtain
0
2
p p
thus
∂ 2
∂ t 2 1−e− p t 2
u t ⊗E= 0 p[
E− p e − p t u t⊗E ]
L Drude = 0
2
p
p
∂ 2
∂t 2 1−e− p t 2
u t⊗E =0 p[
E− p e − p t u t ⊗ E ]

End of 1 st Part
The revolution is not an apple that falls when it is ripe. You have to make it fall (Che)