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: martin@ifh.ee.ethz.ch

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/