Finite Element Method
general purpose tool for finding approximate solution
to partial differential equations as well as integral
equations
12
solution strategy: reduction to a set of linear equations for
steady state problems or to a set of ordinary differential
equation for time harmonic problems
Alexander Hrennikoff (1941) and Richard Courant (1942)
problems on elasticity and structural analysis in
aeronautics
essential: discretization of a continuous domain into
discrete sub-domain, called elements
Note: most of the following material were taken from a tutorial
(A Tutorial on the Finite Element Method) by Arashi Mafi
Finite Element Method
detail here the basic idea for a 1D and 2D scalar FEM
13
commercial programs are available that employ such
computational technique (Comsol, JCM wave)
strength is the ability to retain various physical
models in the simulation simultaneous
takes advantage of a large computational back bone
to solve linear equations
An Analysis of the Finite Element Method
William Gilbert Strang und George J. Fix
1D problem: basic set up
14
domain is partitioned into segments (elements)
xi position of the boundaries Ui value of the function at the node
solution to the PDE is the linear interpolation among the nodes
Global vs. local labeling
15
two nodes of the e’s element identified as andx1e x2
elocal index
(at discussing a single node)
e’s element identified in the domain [xe!1, xe]global index
(at discussing the final assemblage)
Example for a 1D FEM
16
R(x) = !2x"(x) + #2"(x) = 0
Neumann boundaries: !x"(x) |x=!!2 , !
2= 0
in the interval:
!!!
2,!
2
"
analytical solution: !(x) = A sin("x)! = n (integer)
weak formulation:
!!!
2,!
2
"is partitioned into
small elements
introducing a number
of weight functions Qie(x)
in a strict sense, is everywhere zeroR(x)
Weight functions
17
! x2e
x1e
Qie(x)R(x)dx = 0
R(x) = !2x"(x) + #2"(x) = 0
introduction and integration by parts gives
! x2e
x1e
"!!xQi
e(x)!x"(x) + #2Qie(x)"(x)
#dx = 0
Discretization
18
partitioning of the domain into elements equal size
!!!
2,!
2
"N
xe =!e
N! !
2
linear approximation of the function on each element!!e(x) = pex + qe
(valid only on the e’the element, zero everywhere else)
!(x) =N!
e=1
!e(x)
Discretization
19
has to satisfy and !e(x) !e(x1e) = U1
e !e(x2e) = U2
e
pe =U2
e ! U1e
x2e ! x1
e
qe =U1
e x2e ! U2
e x1e
x2e ! x1
e
can be rearranged to !e(x) !e(x) =2!
i=1
U ieW
ie(x)
W 1e (x) =
x2e ! x
x2e ! x1
e
W 2e (x) =
x! x1e
x2e ! x1
e
W ie(x
je) = !ij
Lagrange linear polynomials:
Discretization
20
Solution can be written as
!(x) =N!
e=1
2!
i=1
U ieW
ie(x)
One has to solve for the unknown U ie
By choosing the weighting function to be Qie(x) = W i
e(x)N!
m=1
2!
j=1
U je
" x
xe!1
#!!xW i
e(x)!xW jm(x) + "2W i
e(x)W jm(x)
$dx = 0
e
Discretization
21
2!
j=1
U je
" x
xe!1
#!xW i
e(x)!xW je (x)
$dx = "2
2!
j=1
U je
" x
xe!1
#W i
e(x)W je (x)
$dx
Aije =
! x
xe!1
!xW ie(x)!xW j
e (x)dx
Ae =N
2
!1 !1!1 1
"
Bije =
! x
xe!1
W ie(x)W j
e (x)dx
Be =1
3N
!2 11 2
"
Aije U j
e = !2Bije U j
eonly related to the e’th elements
e e
ee
Assemblage
22
Solution is redundant as U2e!1 = U1
e
Assembly of elements removes ambiguity
Example at the case of N = 3three elements with three matrix equations
Aije U j
e = !2Bije U j
e e = 1, 2, 3
switching from a local to a global scheme
U0 = U11 , U1 = U1
2 = U21 , U2 = U2
2 = U31 , U3 = U3
2
Assemblage
23
!A11
1 A121
A211 A22
1
" !U0
U1
"= !2
!B11
1 B121
B211 B22
1
" !U0
U1
"
!A11
2 A122
A212 A22
2
" !U1
U2
"= !2
!B11
2 B122
B212 B22
2
" !U1
U2
"
!A11
3 A123
A213 A22
3
" !U2
U3
"= !2
!B11
3 B123
B213 B22
3
" !U2
U3
"
six equations with only four unknowns
adding up equations containing the same unknowns
eigenvalue is no unknown as it is determined automatically
Assemblage
24
A211 U0 + A22
1 U1 = !1(B211 U0 + B22
1 U1)
A112 U1 + A12
2 U2 = !1(B112 U1 + B12
2 U2)+
A211 U0 + (A22
1 + A112 )U1 + A12
2 U2 = !1(B211 U0 + (B11
2 + B221 )U1 + B12
2 U2)=
AU = !2BU
Assemblage
25
A =
!
""#
A111 A12
1 0 0A21
1 A221 + A11
2 A122 0
0 A212 A22
2 + A113 A12
3
0 0 A213 A22
3
$
%%&
B =
!
""#
B111 B12
1 0 0B21
1 B221 + B11
2 B122 0
0 B212 B22
2 + B113 B12
3
0 0 B213 B22
3
$
%%&
U = (U0, U1, U2, U3)
Assemblage
26
Solution for elements is straight forwardN
A =3N
!
"""""#
2 1 0 0 01 4 1 0 0
0 1. . . 1 0
0 0 1 4 10 0 0 1 2
$
%%%%%&
B =N
2
!
"""""#
!1 1 0 0 01 !2 1 0 0
0 1. . . 1 0
0 0 1 !2 10 0 0 1 !1
$
%%%%%&
Results
27
!1.57 !0.57 0.43 1.43!1
!0.5
0
0.5
1
0 1 2 3 4 5 6 7 8 9 10
Eigenvalue number
0
1
2
3
4
5
6
7
8
9
10
11
Eig
en
va
lue
two solutions for
(Mathematica)
N = 10
Results
28
!1.57 !0.57 0.43 1.43!1
!0.5
0
0.5
1
0 1 2 3 4 5 6 7 8 9 10
Eigenvalue number
0
1
2
3
4
5
6
7
8
9
10
11E
ige
nva
lue
analytica
l solutio
n
What will happen in this lecture
discussing the meaning of a Green’s function
discretizing the volume integral
another general purpose method based on volume integral
1
Green’s function in 1D, 2D and 3D
some examples for possible applications
numerical peculiarities
Scattering problem solved by the Greens function6 Scattering calculations with the Green’s tensor technique
!1
!2
!3
!4
!5
y
z
!(r)
!(r)
E0
E
Figure 2.1. Typical geometry under study. Several scatterers with permittivity!(r) are embedded in a stratified background composed of L layers withrespective permittivity !l, l = 1, . . . , L. Note that the first and last layersare semi–infinite media.
the Green’s tensor. We start with the general 3D case and then partic-ularize the formalism for 2D geometries. The detailed derivation of theGreen’s tensors associated with a stratified medium (3D and 2D) will bepresented in chapter 3.
2.1 Electric field integral equation
When a scattering system is illuminated with an incident electric fieldE0(r) propagating in the background, the total field E(r) is a solutionof the vectorial wave equation [37]:
!"!" E(r) # k20!(r)E(r) = 0 , (2.1)
where k20 = "2!0µ0 is the vacuum wave number. The incident field E0(r)
must fulfill the vectorial wave equation for the bare stratified background:
!"!" E0(r) # k20!!E
0(r) = 0 , r $ layer # . (2.2)
Introducing the dielectric contrast
!!(r) = !(r) # !! , r $ layer # , (2.3)
Thesis of M. Paulus @ ETHZ
Scattering problem solved by the Greens function6 Scattering calculations with the Green’s tensor technique
!1
!2
!3
!4
!5
y
z
!(r)
!(r)
E0
E
Figure 2.1. Typical geometry under study. Several scatterers with permittivity!(r) are embedded in a stratified background composed of L layers withrespective permittivity !l, l = 1, . . . , L. Note that the first and last layersare semi–infinite media.
the Green’s tensor. We start with the general 3D case and then partic-ularize the formalism for 2D geometries. The detailed derivation of theGreen’s tensors associated with a stratified medium (3D and 2D) will bepresented in chapter 3.
2.1 Electric field integral equation
When a scattering system is illuminated with an incident electric fieldE0(r) propagating in the background, the total field E(r) is a solutionof the vectorial wave equation [37]:
!"!" E(r) # k20!(r)E(r) = 0 , (2.1)
where k20 = "2!0µ0 is the vacuum wave number. The incident field E0(r)
must fulfill the vectorial wave equation for the bare stratified background:
!"!" E0(r) # k20!!E
0(r) = 0 , r $ layer # . (2.2)
Introducing the dielectric contrast
!!(r) = !(r) # !! , r $ layer # , (2.3)
Thesis of M. Paulus @ ETHZ
Formulation of the scattering problem
electric field is a solution to the vectorial wave equation
!"!"E(r)# k20!(r)E(r) = 0
! · !(r)E(r) = 0
!"!"E(r)# k20!B(r)E(r) = k2
0 [!(r)# !B(r)]E(r)
properties of the medium are decomposed into background and a contribution by the scatterer
!(r) = !B(r) + !!(r)
time harmonic oscillating field with a fixed frequency e!i!t
Formulation of the scattering problem
electric field is a solution to the vectorial wave equation
time harmonic oscillating field with a fixed frequency
!"!"E(r)# k20!(r)E(r) = 0
! · !(r)E(r) = 0
!"!"E(r)# k20!B(r)E(r) = k2
0!!(r)E(r)
properties of the medium are decomposed into background and a contribution by the scatterer
!(r) = !B(r) + !!(r)
e!i!t
Formulation of the scattering problem
electric field is a solution to the vectorial wave equation
time harmonic oscillating field with a fixed frequency
!"!"E(r)# k20!(r)E(r) = 0
! · !(r)E(r) = 0
!"!"E(r)# k20!B(r)E(r) = k2
0!!(r)E(r)
Lippmann-Schwinger equation
e!i!t
Basic idea of a Greens functions
solution to such a inhomogenous differential equation is given by the sum of the homogenous solution:
!"!"E0(r)# k20!B(r)E0(r) = 0incident
field
E(r) = E0(r) + ES(r)
partial
solution(scattered
field) Greens function of the
system
7
ES(r) = k20
!!!(r)G(r, r!) · E(r!)dr!´
Lippmann-Schwinger equation
in general challenging to solve because appears on both sidesE(r)
simplifications are possible, e.g. first order Born series
E(r) ! E0(r)
what is this Greens function and how it looks
for a simple isotropic media?
how to solve this equation can be done once we know the Greens function
8
E(r) = E0(r) + k20
!!!(r)G(r, r!) · E(r!)dr!´
Properties of the Greens functions
solution to a wave equation with a point source term
k20 =
!2
c2with: and 1 =
!
"1 0 00 1 00 0 1
#
$
!"!"G(r, r!)# k20!(r)G(r, r!) = 1"(r# r!)
point source is represented by three orthogonal dipoles
G(r, r!) =
!
"Gxx Gxy Gxz
Gyx Gyy Gyz
Gzx Gzy Gzz
#
$
! · !(r)G(r, r!) = "! · "(r" r!)1
9
Basic idea of a Greens functions
point like excitation of a field in space
10
Greens function describes the response of an
environment to this singular excitation
e.g. the field value in every point upon excitation atr r!
2D Greens function free space
Basic idea of a Greens functions
point like excitation of a field in space
11
Greens function describes the response of an
environment to this singular excitation
e.g. the field value in every point upon excitation atr r!
2D Greens function half space
(position dependent!)
Basic idea of a Greens functions
point like excitation of a field in space
12
Greens function describes the response of an
environment to this singular excitation
e.g. the field value in every point upon excitation atr r!
2D Greens function half space + cylinder(position dependent!)
Greens function of the homogenous (free) space
GH(r, r!) =!1 +
!!k2
B
"eikBR
4!R
R =| R |=| r! r! |
P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw–Hill, New York, 1953)
solution to the wave vector equation with a point source
k2B =
!2
c2"B
last term is called the free space scalar Greens function
G0(R) =eikBR
4!R13
Greens function of the homogenous (free) space
explicit expression is given by:
GH(r, r!) =!1 +
ikBR! 1k2
BR21 +
3! 3ikBR! k2BR2
k2BR4
RR"
G0(R)
! = x2 + z2 k2! = k2
x + k2z
G2d0 (R) =
!G0(R)eiky(y!y!)dy"
=i
4H0 (!k!) eikyy
2D Greens function: decomposing a line source into a
string of point sources
Green’s tensor technique for scattering in two-dimensional stratified media
Michael Paulus1,2 and Olivier J. F. Martin1,*1Electromagnetic Fields and Microwave Electronics Laboratory, Swiss Federal Institute of Technology, ETH-Zentrum ETZ,
CH-8092 Zurich, Switzerland2IBM Research, Zurich Research Laboratory, CH-8803 Ruschlikon, Switzerland
!Received 1 February 2001; published 29 May 2001"
We present an accurate and self-consistent technique for computing the electromagnetic field in scattering
structures formed by bodies embedded in a stratified background and extending infinitely in one direction
!two-dimensional geometry". With this fully vectorial approach based on the Green’s tensor associated with thebackground, only the embedded scatterers must be discretized, the entire stratified background being accounted
for by the Green’s tensor. We first derive the formulas for the computation of this dyadic and discuss in detail
its physical substance. The utilization of this technique for the solution of scattering problems in complex
structures is then illustrated with examples from photonic integrated circuits !waveguide grating couplers withvarying periodicity".
DOI: 10.1103/PhysRevE.63.066615 PACS number!s": 42.25.!p, 42.79.Gn, 42.82.Et, 02.60.Cb
I. INTRODUCTION
The accurate computation of light scattering from par-ticles in the presence of a stratified background is extremelyimportant for the understanding of realistic structures.Ridges on a multilayered waveguide #1$, opaque regions on acontact lithography mask #2$, polarization gratings on atransparent backplane #3$, and nanowires deposited on a sub-strate for surface-enhanced Raman scattering #4$ all have incommon that dielectric or metallic scatterers are distributedin a medium consisting of several layers with different per-mittivities.Recently, we presented a technique for computing the
propagation and scattering of light in three-dimensional !3D"structures formed by a stratified background with embeddedscatterers of finite extension in all three dimensions #5,6$.This approach is based on the Green’s tensor associated withthe stratified background. In this paper, we extend this tech-nique to two-dimensional !2D" geometries, i.e., systems witha translation symmetry in one direction.A typical 2D system that we want to study is shown in
Fig. 1. Several scatterers described by the permittivity %(r)are embedded in a stratified background and illuminated withan incident field E0. The stratified background is composedof L layers with relative permittivity % l , l"1, . . . ,L , and thescatterers extend infinitely along the y axis so that the mate-rial system is invariant in that direction. If also the excitationhas such a translation symmetry, we can restrict the study ofthe 3D system !Fig. 1" to a 2D cross section in the xz plane!Fig. 2". We then define the coordinate r! parallel to thisplane,
r"!r! ,ry""!rx ,rz ,ry", !1"
and the parallel wave vector k! ,
k"!k! ,ky""!kx ,kz ,ky". !2"
Let us emphasize that it is not necessary that also the inci-dent field E0 propagates in the xz plane !Fig. 1". The soleconstraint is that E0 has an exp(iky
0y) dependence on the sym-
metry direction y. For example, a plane wave
E0!r,t ""E0exp! ik0r!i&t ""E0exp! ik!0r!!i&t "exp! iky
0y "!3"
at oblique incidence on the structure fulfills this condition!Figs. 1 and 2".However, if E0 propagates in the xz plane (ky
0"0), it ispossible to decompose the total field into a transverse electric!TE" part with the electric field in the xz plane, and a trans-verse magnetic !TM" part with the electric field parallel to
*Correspondence author. Email address: [email protected]
FIG. 1. Schematic view of a 2D scattering system. Several scat-
terers with permittivity %(r) are embedded in a stratified back-ground formed by L layers with permittivity % l , l"1, . . . ,L . Thescatterers are infinitely extended in the y direction. However, the
propagation of the incident field is not restricted and its wave vector
k0"k!0#ky
0 can have components parallel and perpendicular to the
xz plane. Similarly, the electric field can be split into two contribu-
tions: the p-polarized part Ep0 lying within the plane of incidence
formed by k0 and the z axis, and the s-polarized part Es0 standing
perpendicularly to this plane. If k0"k!0 (ky
0"0), p polarization isreferred to as TE and s polarization as TM.
PHYSICAL REVIEW E, VOLUME 63, 066615
1063-651X/2001/63!6"/066615!8"/$20.00 ©2001 The American Physical Society63 066615-1
Thesis of M. Paulus @ ETHZ
Greens function of the homogenous (free) space
explicit expression is given by:
GH(r, r!) =!1 +
ikBR! 1k2
BR21 +
3! 3ikBR! k2BR2
k2BR4
RR"
G0(R)
1D Greens function: decomposing a sheet source into a
string of point sources (in two directions)
G1d0 (R) =
!G0(R)eiky(y!y!)eikx(x!x!)dy"dx"
=i
2kzeikz|z!z!|ei(kxx+kyy)
15
Greens function in the Fourier space
explicit expression is given by:
GH(r, r!) =!1 +
ikBR! 1k2
BR21 +
3! 3ikBR! k2BR2
k2BR4
RR"
G0(R)
GH(r, r!) =1
8!3k2B
!!!dk
"1k2
B ! kkk2 ! k2
B
#eik·R
decomposing the point source into a set of plane waves
makes it more suitable to calculate the Greens
function for an arbitrary stratified media 16
Reducing the 3D integral into a 2D integral
GH(r, r!) =1
8!3k2B
!!!dk
"1k2
B ! kkk2 ! k2
B
+ zz#
eik·R
! zz8!3k2
B
!!!dkeik·R
using calculus of residues
GH(r, r!) =i
8!2k2B
!!dkxdky
"1k2
B ! kB
kBz
#eikB ·R
! zzk2
B
"(R)
plane wave decomposition of the Greens function
with a singularity at the origin17
Reducing the 3D integral into a 2D integral
GH(r, r!) =1
8!3k2B
!!!dk
"1k2
B ! kkk2 ! k2
B
+ zz#
eik·R
! zz8!3k2
B
!!!dkeik·R
using calculus of residues
GH(r, r!) =i
8!2k2B
!!dkxdky
"1k2
B ! kB
kBz
#eikB ·R
! zzk2
B
"(R)
kBz =!
k2B ! k2
x ! k2y
kB(kBz ) =!
kxx + kyy + kBz z for z > z!
kxx + kyy ! kBz z for z < z! 18
N = 15 N = 31
N = 301
2D Greens function for the homogenous
space
Building a Greens function from plane waves
19
(Principle) Greens function for a stratified media
decomposing a point source into plane waves
separating into TE (s) and TM (p) polarized waves
G(r, r!) = ! zzk2
B
!(R)i
8"2
!!dkxdky
"ei[kx(x"x!)+ky(y"y!)]
" [hs(kx, ky; z, z!) + hp(kx, ky; z, z!)]
propagating each wave through the stratified media using matrix transfer technique
expressions for the tensors and are cumbersome to write down but contain all the information about the media
hs hp
M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media”, Phys. Rev. E, Vol 62, 5797 (2000)
20
Example
2D Greens tensor more intuitive
note that all Greens function do diverge at the origin
21
| Gzy || Gzx || Gzz |
(example from before)
Solving the scattering problem
come back to Lippmann-Schwinger Equation
integration volume limited to the volume occupied by the scatterer
source dyadic has to be taken explicitly into account
equation has to be discretized and solved
22
!V ! 0E(r) = E0(r) + lim
!
V !!Vk20!!(r)G(r, r") · E(r")dr" ! L · !!(r)
!BE(r)
E(r) = E0(r) + k20
!!!(r)G(r, r!) · E(r!)dr!´
´
actually necessary to use a smaller mesh when the dielec-tric contrast is larger. To that extent, one can expectthat the convergence of this scheme will be similar to thatobserved for scattering calculations in a homogeneousbackground. We refer the reader to Ref. 37, where thispoint was discussed in detail.
Keeping in mind that the discrete dielectric contrast!" i ! " i " "# depends on the permittivity of the layer #where mesh i is located, we can write the discretized sys-tem of equations that correspond to Eq. (4):
Ei ! Ei0 # $
j!1
N
Gi,jI • k0
2!" jEjVj
# $j!layer #
j%i
Gi,jD • k0
2!" jEjVj # Mi • k02!" iEi
" L • !" i
"#Ei , i ! 1 ,..., N. (5)
The self-term Mi is obtained in a similar manner as for aninfinite homogeneous background:30
Mi ! lim&V!0
!Vi"&V
dr!GD'ri , r!(
!2
3k#2 )'1 " ik# Ri
eff(exp'ik# Rieff( " 1*1, (6)
where Rieff is the effective radius of mesh i:
Rieff ! " 3
4+Vi# 1/3
. (7)
For the integration in Eq. (6) we assumed a spherical ex-clusion volume &V. The corresponding source dyadic is36
L !1
31. (8)
Note in Eq. (6) the effective wave number k# ! k0!"# inlayer #.
The system of Eq. (5) represents the self-consistent in-teraction of N dipoles. Unlike for the coupled dipole ap-proximation in vacuum, each dipole is now a dipole em-bedded in a stratified background, and the interactionincludes all possible reflections and refractions at the L" 1 interfaces.
This system of equations is best solved numericallywith an iterative solver.29,38 Let us mention that, in astratified medium, the Green’s tensor does not have thesame symmetry properties as in an infinite homogeneousbackground. In particular,
G'r, r!( % G'r " r!(. (9)
It is therefore not possible to rewrite Eq. (1) as a convolu-tion and to use a 3D fast Fourier transform to perform theintegration.39 It is, however, possible to use reducedsymmetry properties in the x,y plane to expedite thecomputation.33
One of the advantages of the technique presented inthis paper lies in the fact that only the scatterers must bediscretized, the background being accounted for in the
Green’s tensor. Similarly, the interaction of scattererslocated at large distances from one another does not re-quire the discretization of the stratified background be-tween them. Further, the complex boundary conditionsat the edges of the computational window are automati-cally fulfilled, since they are included in the Green’s ten-sor.
We mentioned that Eq. (1) is an implicit equation forthe field E(r). Actually, this is the case only when r islocated inside a scatterer. When r is located in thestratified background, Eq. (1) gives the field explicitly byintegration on the scatterers’ volume [!"(r!) ! 0 when r!is in the background]. From a physical point of view, thismeans that knowledge of the field inside all the scatterersallows one to compute the field at any point in the strati-fied background. This can be used to expedite the calcu-lation by first computing and storing the solution of Eq.(5) only for the discretized points inside the scatterers andthen using this solution at a later stage to obtain the fieldin the background. Note that the last step does not ne-cessitate the solution of a system of equations but re-quires only simple vector matrix multiplications.
Fig. 3. Solving the scattering problem numerically requires thatonly the scatterers in the structure must be discretized. Thesole constraint on the discretization is that a mesh cannot sitastride a boundary between two layers.
Fig. 4. The incident field must be a solution of the wave equa-tion for the stratified background. It can correspond, for ex-ample, (a) to a plane wave impinging on the system or (b) to awaveguide mode propagating in the stratified background.
856 J. Opt. Soc. Am. A/Vol. 18, No. 4 /April 2001 M. Paulus and O. J. F. Martin
Discretizing the scatterer
Thesis of M. Paulus @ ETHZ
!!i = !!i(r) |r!Vi! !!(ri)
Ei = Ei(r) |r!Vi! E(ri)
Discretizing the equation
Computer Physics Communication, 144, 111 (2002)
Ei = E0i
+N!
j=1, j !=i
Gi,j · k20!!jEjVj
+Mi · k20!!iEi
!L · !!i
!BEi
Calculating the self action terms
one has to solve in principle for
Mi = lim!
Vi!!Vdr"G(ri, r")
difficult to evaluate but detrimental for numerical precision
analytical expressions are available for certain shapes of volumes
A. D. Yaghjian, “Electric dyadic Green’s functions in the source region”,
Proc. IEEE 68, 248 (1980)
for example assuming a sphere
Mi =2
3k2f
!"1! ikfRe!
i
#eikf Reff
i ! 1$1
Re!i =
!34!
Vi
"1/3
25
Calculating the self action terms
one has to solve in principle for
Mi = lim!
Vi!!Vdr"G(ri, r")
difficult to evaluate but detrimental for numerical precision
analytical expressions are available for certain shapes of volumes
A. D. Yaghjian, “Electric dyadic Green’s functions in the source region”,
Proc. IEEE 68, 248 (1980)
for example assuming a sphere
L =131
26
Solving the equation
!
""""""#
Ex1
Ey1
Ez1
Ex2
Ey2
Ez2
$
%%%%%%&=
!
""""""#
E0x1
E0y1
E0z1
E0x2
E0y2
E0z2
$
%%%%%%&
system of linear equations can be solved
by standard matrix inversion techniques
A27
Solving the equation
!
"""""""#
1!Mxx1 k2
0!"1 + Lxx1
!"1"B
0 0 !Gxx12 k2
0!"2V2 !Gxy12k2
0!"2V2 !Gxz12k2
0!"2V2
0 1!Myy1 k2
0!"1 + Lyy1
!"1"B
0 !Gyx12k2
0!"2V2 !Gyy12k2
0!"2V2 !Gyz12k2
0!"2V2
0 0 1!Mzz1 k2
0!"1 + Lzz1
!"1"B
!Gzx12k2
0!"2V2 !Gzy12k2
0!"2V2 !Gzz12k
20!"2V2
!Gxx21 k2
0!"1V1 !Gxy21k2
0!"1V1 !Gxz21k2
0!"1V1 1!Mxx2 k2
0!"2 + Lxx2
!"2"B
0 0!Gyx
21k20!"1V1 !Gyy
21k20!"1V1 !Gyz
21k20!"1V1 0 1!Myy
2 k20!"2 + Lyy
2!"2"B
0!Gzx
21k20!"1V1 !Gzy
21k20!"1V1 !Gzz
21k20!"1V1 0 0 1!Mzz
2 k20!"2 + Lzz
2!"2"B
$
%%%%%%%&
A =a fraction of the matrix is
28
A proper near-field to far-field transformation technique
(and vice versa)
Richards-Wolf integrals
Stratton-Chu integrals
Surface integral techniques
What is missing?
No matter how long this lecture would last, an
uncountable number of subjects would be always missing
Finite Integration technique by employing the integral form
of Maxwell’s equations
What is missing?
!rx(j, l) =!r(j, l) + !r(j, l ! 1)
2
!ry(j, l) =!r(j, l) + !r(j ! 1, l)
2
!rz(j, l) =!r(j, l) + !r(j ! 1, l ! 1) + !r(j, l ! 1) + +!r(j ! 1, l)
4" (352)
A(j, l + 1)!A(j, l) (353)A(j + 1, l)!A(j, l) (354)A(j, l)!A(j, l ! 1) (355)A(j, l)!A(j ! 1, l) (356)
(357)
A(1, 1) A(1, 2) (358)A(2, 1) A(2, 2) (359)
(360)
D(r, t) = !0E(r, t) + P(r, t) (361)
P(r, t) = !0
! !
0"(r, t")E(r, t! t")dt" (362)
P(r, t) = !0"(r, t)E(r, t) (363)! = 1 + " (364)
P(r, t) = !0"(r, t)E(r, t) + PNL (365)P = !0 (366)
# (367)r (368)
ui = eıkr (369)(370)
(#2 + k2)U = 0 (371)(#2 + k2)U " = 0 (372)
! ! !
V(U#2U " ! U "#2U)dV = !
! !
S
"U
$U "
$n! U " $U
$n
#dS (373)
! !
S
"U
$U "
$n! U " $U
$n
#dS = 0 (374)
E(r) = E0(r) +!
dr"G(r, r") · k20!!E(r) (375)
$
!AE · ds = !
! !
A
$B$t
· dA (376)
#$E = !$B$t
(377)
(378)
17
!rx(j, l) =!r(j, l) + !r(j, l ! 1)
2
!ry(j, l) =!r(j, l) + !r(j ! 1, l)
2
!rz(j, l) =!r(j, l) + !r(j ! 1, l ! 1) + !r(j, l ! 1) + +!r(j ! 1, l)
4" (352)
A(j, l + 1)!A(j, l) (353)A(j + 1, l)!A(j, l) (354)A(j, l)!A(j, l ! 1) (355)A(j, l)!A(j ! 1, l) (356)
(357)
A(1, 1) A(1, 2) (358)A(2, 1) A(2, 2) (359)
(360)
D(r, t) = !0E(r, t) + P(r, t) (361)
P(r, t) = !0
! !
0"(r, t")E(r, t! t")dt" (362)
P(r, t) = !0"(r, t)E(r, t) (363)! = 1 + " (364)
P(r, t) = !0"(r, t)E(r, t) + PNL (365)P = !0 (366)
# (367)r (368)
ui = eıkr (369)(370)
(#2 + k2)U = 0 (371)(#2 + k2)U " = 0 (372)
! ! !
V(U#2U " ! U "#2U)dV = !
! !
S
"U
$U "
$n! U " $U
$n
#dS (373)
! !
S
"U
$U "
$n! U " $U
$n
#dS = 0 (374)
E(r) = E0(r) +!
dr"G(r, r") · k20!!E(r) (375)
$
!AE · ds = !
! !
A
$B$t
· dA (376)
#$E = !$B$t
(377)
(378)
17
Induction law in
Integral form Differential form
Line integral of the E-field around a closed loop equals the negative rate of change of the magnetic
flux through the are enclosed by the loopApplied e.g. in MAFIA
www.cst.com
No matter how long this lecture would last, an
uncountable number of subjects would be always missing
Free available programs
Powerful tool for calculating photonic band structureshttp://ab-initio.mit.edu/mpb/
Survey of various scattering codes available athttp://www.iwt-bremen.de/vt/wriedt
Code for the Discrete Dipole Approximationhttp://ascl.net/ddscat.html
Code for the T-Matrix approachhttp://www.giss.nasa.gov/~crmim/t_matrix.html
University courses offer often free softwarehttp://www.photonik.uni-jena.de/
Codes published under GNU are available, e.g. FDTDhttp://www.borg.umn.edu/toyfdtd/
Commercial programs
Finite Difference Time Domain
http://www.rsoftdesign.com/ or http://www.optiwave.com/
Rigorous grating solvers
http://www.unigit.com/ or http://www.gsolver.com/
Finite Element Methods
http://www.femlab.com/ or http://www.ansoft.com/
Multiple Multipole Methodhttp://alphard.ethz.ch/hafner/MaX/max1.htm/