by prof. thomas pertsch. - uni-jena.de · by prof. thomas pertsch. the course material in 2019...

This is the 2018 version of the

Computational Photonics script, written

by Prof. Thomas Pertsch.

The course material in 2019 derives

largely from this source, however it may

deviate in scope, notation etc.

You can use this script as an additional

referential resource for the lecture.

Dr. Thomas Kaiser

May 2019

Computational Photonics – old 2018 script version by Prof. Thomas Pertsch 2

Computational Photonics Prof. Thomas Pertsch

Abbe School of Photonics Friedrich-Schiller-Universität Jena

0. Introduction and Motivation ..................................................................... 4

0.1 Why computational photonics? ......................................................................... 4

0.2 Maxwell’s equations .......................................................................................... 5 0.2.1 Maxwell’s equations in time domain ................................................................. 5 0.2.2 Maxwell’s equations in frequency domain......................................................... 6 0.3 Basic numerical operations ............................................................................... 6 0.3.1 Differentiation .................................................................................................... 6

0.3.2 Integration ......................................................................................................... 7

0.3.3 Root Finding & Minimization/ Maximization ...................................................... 8

0.3.4 Linear systems of equations ........................................................................... 11 0.3.5 Eigenvalue problems ...................................................................................... 12 0.3.6 Discrete (Fast) Fourier transform - FFT .......................................................... 12 0.3.7 Ordinary differential equations (ODEs) ........................................................... 13

1. Matrix method for stratified media ......................................................... 16

1.1 Wave equation in volumes of homogeneous media ....................................... 16 1.2 Optical layer systems ...................................................................................... 18

1.3 Derivation of the transfer matrix ...................................................................... 19 1.4 Reflection and transmission problem .............................................................. 21

1.5 Guided modes in layer systems ...................................................................... 26

2. Finite-difference method for waveguide modes ..................................... 29

2.1 Scalar approximation for weakly guiding waveguides with small index differences ...................................................................................................... 29

2.1.1 Stationary solutions of the scalar Helmholtz equation .................................... 29 2.1.2 Matrix notation of the eigenvalue equation ..................................................... 31 2.1.3 Boundary conditions ....................................................................................... 32 2.2 Full vectorial mode solver for waveguides with large index differences .......... 33

3. Beam Propagation Method (BPM) ......................................................... 41

3.1 Categorization of Partial Differential Equation (PDE) problems ...................... 41 3.2 Slowly Varying Envelope Approximation (SVEA) ............................................ 43 3.3 Differential equations of BPM ......................................................................... 44 3.4 Semi-vector BPM ............................................................................................ 46 3.5 Scalar BPM ..................................................................................................... 46 3.6 Crank-Nicolson method .................................................................................. 46

3.7 Alternating Direction Implicit (ADI) .................................................................. 47 3.8 Boundary condition ......................................................................................... 47

3.8.1 Absorbing Boundary Conditions (ABC) ........................................................... 47 3.8.2 Transparent Boundary Condition (TBC).......................................................... 48 3.8.3 Perfectly matched layer boundaries (PML) ..................................................... 50 3.9 Conformal mapping regions ............................................................................ 51 3.10 Wide-angle BPM based on Padé operators .................................................... 53

3.10.1 Fresnel approximation – Padé 0th order .......................................................... 54 3.10.2 Wide angle (WA) approximation – Padé (1,1) ................................................. 54

4. Finite Difference Time Domain Method (FDTD)..................................... 56

4.1 Maxwell’s equations ........................................................................................ 56 4.2 1D problems ................................................................................................... 57 4.2.1 Solution with finite difference method in the time domain for Ez ..................... 58

4.2.2 Yee grid in 1D and Leapfrog time steps .......................................................... 60


4.3 The full 3D problem ........................................................................................ 61 4.3.1 Yee grid in 3D ................................................................................................. 62

4.3.2 Physical interpretation..................................................................................... 63 4.3.3 Divergence-free nature of the Yee discretization ............................................ 64 4.3.4 Computational procedure ................................................................................ 65

4.4 Simplification to 2D problems ......................................................................... 66 4.5 Implementing light sources ............................................................................. 66 4.6 Relation between frequency and time domain ................................................ 67 4.7 Dispersive and nonlinear materials ................................................................. 69 4.8 Boundary conditions ....................................................................................... 71

4.8.1 Perfectly conducting material boundary .......................................................... 71 4.8.2 Floquet-Bloch periodic boundaries ................................................................. 72 4.8.3 Transparent boundary conditions - 2nd order Mur (TBC) ............................... 75 4.8.4 Perfectly matched layer boundaries (PML) ..................................................... 77

5. Fiber waveguides .................................................................................. 81

5.1.1 The general eigenvalue problem for scalar fields ........................................... 81 5.1.2 Properties of guided modes ............................................................................ 81 5.1.3 Cylinder symmetric waveguides ..................................................................... 83 5.1.4 Bessel’s differential equation .......................................................................... 84

5.1.5 Analytical solutions of Bessel’s differential equation ....................................... 85 5.1.6 Specifying the numerical problem ................................................................... 87

5.1.7 Solving the second order singularity ............................................................... 88 5.1.8 Numerical integration methods ....................................................................... 88 5.1.9 Eigenvalue search .......................................................................................... 91

5.1.10 Calculation examples ...................................................................................... 92

6. Fourier Modal Method for periodic systems ........................................... 95

6.1 Formulation of the problem in 2D for TE ......................................................... 96 6.2 Scalar theory for thin elements ....................................................................... 97

6.3 Rigorous grating solver ................................................................................. 100 6.3.1 Calculation of eigenmodes in Fourier space ................................................. 101

6.3.2 Explicit derivation for 2D problems ............................................................... 105

7. Finite Element Method (FEM) .............................................................. 112

7.1 The basic set up ........................................................................................... 112 7.2 Global vs. local labeling ................................................................................ 114 7.3 Weak formulation .......................................................................................... 114 7.4 Example Results ........................................................................................... 117

8. Outlook ................................................................................................ 120


0. Introduction and Motivation

0.1 Why computational photonics? it’s a numerical experiment

provides insides to inaccessible domain

permits to interpret and understand experimental results

simplifies the design of functional elements

explores prospective applications presently not realizable

with available large scale computational resources it became an inevitable tool in the world of micro- and nanooptics

What is the basis of our computational model?

„Light is like an odor an emanation of our body“ Epikur (Greek philosopher, 341-271 BC)

Straight „Light-Ray“ as an abstract imagination Euklid in „Elements“ (365 - ca. 300 BC)

Light is an electromagnetic wave J. C. Maxwell 1873 (Propagation and interaction of light with matter)

Light consists of particles (Photons) A. Einstein 1905 (Creation and absorption)

Light is particle and wave De Broglie 1923 (quantum mechanics)

Formulation of the problem:

For a specific geometry and a particular set of boundary conditions Maxwell‘s equations have to be solved (with or without approximations)

Description of the interaction of electromagnetic waves with matter

Initially published in 1873 by James Clark Maxwell

Experimental proof by Heinrich Rudolf Hertz 1884 (speed of radio waves corresponds to the speed of light)


0.2 Maxwell’s equations

0.2.1 Maxwell’s equations in time domain

makr

ext

( , ) ( , )rot ( , ) , rot ( , ) ( , ) ,

div ( , ) ( , ), div ( , ) 0,

t tt t t

t t

t t t

B r D rE r H r j r

D r r B r

(1)

E(r,t) electric field [V m-1]

H(r,t) magnetic field [A m-1]

D(r,t) dielectric flux density [As m-2]

B(r,t) magnetic flux density [Vs m-2]

ext(r,t) external charge

density [As m-3]

jmakr(r,t) macroscopic current

density [A m-2]

(Usually there are no external charges or currents in optics.)

Matter equations in time domain

0

0

( , ) ( , ) ( , ),

( , ) ( , ) ( , )

t t t

t t t

D r E r P r

B r H r M r (2)

P(r,t) dielectric polarization [As m-2]

M(r,t) magnetic polarization (magnetization) [Vs m-2]

0 permittivity of vacuum 2 1 12

0 0 0( c ) 8.85 10 As/Vm

0 permeability of vacuum 12

0 4 10 Vs/Am

in linear, local, isotropic Media in optics

0

0

( , ) ( , ) ( , ) ,

( , ) 0

t R t t t t

t

P r r E r

M r

(3)

with ( , )R tr being the response function


0.2.2 Maxwell’s equations in frequency domain

Using Fourier transformation to transform into frequency space

1

( , ) ( , )exp( ) , ( , ) ( , )exp( )2

t i t d t i t dt

V r V r V r V r . (4)

Maxwell’s equations in frequency domain by FT it

rot ( , ) ( , ), rot ( , ) ( , ),

div ( , ) ( , ), div ( , ) 0.

i i

E r B r H r D r

D r r B r (5)

Matter equations in frequency domain

( , ) ( , )exp( )R t i t t

r r (6)

with ( , ) r being the material´s susceptibility, which is connected to the

dielectric constant ( , ) r by

( , ) 1 ( , ) r r )

0 0

0

( , ) ( , ) ( , ) ( , ) ( , ) ( , )

( , ) 0 ( , ) ( , )

P r r E r D r r E r

M r B r H r (7)

0.3 Basic numerical operations

0.3.1 Differentiation

Derived from the definition of differentiation, e.g. right-sided/forward difference equation

0

( ) ( ) ( )( ) lim

h

f x f x h f xf x

x h

for finite h

( ) ( )( ) [ ( )]h

f x h f xf x D f x

h

or left-sided/backward

( ) ( )[ ( )]h

f x f x hD f x

h

.

or central operator

( ) ( )[ ( )]

2h

f x h f x hD f x

h

higher order differentiation


2

2

( ) 2 ( ) ( )( ) [ ( )]h

f x h f x f x hf x D f x

h

0.3.2 Integration

11

0

.i

i

xb N

ia x

A dx f x dx f x

with 1[ , ]i i iI x x where

1i ix x h and ( ) /h b a N with 0 , Nx a x b

Decomposition of the full integration interval ,a b into N equivalent partial

intervals 1[ , ]i i iI x x .

Implementation example: rectangle rule

in each interval the mean value of the function is approximated by

1

2

( )2

i ii

hf x f f x

resulting in an integral approximation

11 1 1

1 1 1

0 0 02 2 2

i

i

xN N N

i i ii i ix

b aA dx f h f f

N

Rectangle rule for approximation of integrals.


0.3.3 Root Finding & Minimization/ Maximization

Root finding of a single isolated root in one dimension.

Secant method for 1D

iterative solution by

11

1

( )( ) ( )

i ii i

i i

x xx x f x

f x f x

until

1 10i i if x f x f x

with determined by the desired accuracy

Illustration of the secant method (The individual points are numbered in the order of the iterations.)


Secant method 1 function [x,i,status] = secant_method(x0,x1,tol1,tol2,no)

2

3 %x0,x1: start value

4 %tol1, tol2: tolerance

5 %no: maximum number iterations

6

7 %graphic output

8 ab = -0.4; x_old=1000;fx_old=1000;

9 axis on;

10 plot( [-5 5], [0 0]); % plotting x-axis

11 hold on;

12 set(findobj(gca,'Type','line','Color',[0 0 1]),'Color',

'black','LineWidth',1)

13 p=(-5):0.1:(5);

14 for i=1:1:(101)

15 f(i) = f1(p(i));

16 end

17 plot(p,f,'LineWidth',1);

18 i = 0; fa = f1(x0); fb = f1(x1);

19 if abs(fa) > abs(fb)

20 a = x0; b = x1;

21 else

22 a = x1; b = x0; tmp = fb; fb = fa; fa = tmp;

23 end

24 % iteration of the secant

25 while i < no

26 s=fb/fa; r=1-s;

27 t=s*(a-b); %ascent between two points oft he secant

28 x=b-t/r; % step towards root

29 fx=f1(x); % converging towards root

30 % output of points of the secant

31 if abs(x_old-x)<0.3 && abs(fx_old-fx)<0.3

32 ab=ab-0.2;

33 end

34 x_old=x; fx_old=fx;

35 plot(x,fx,'rO','LineWidth',1);

36 text(x+ab, fx,num2str(i+1),'HorizontalAlignment','left')

37 if t == 0

38 status = 'Method did not converge.'; return

39 end

40 if abs(fx) < tol1

41 status = 'Method converged and calculated solution.';

return

42 end

43 if abs(x-b) < tol2*(1+abs(x))

44 status = 'Method converged and calculated solution.';

return


45 end

46 i = i+1;

47 if abs(fx) > abs(fb)

48 a = x; fa = fx;

49 else

50 a = b; fa = fb; b = x; fb = fx;

51 end

52 end

53 status = 'Number iterations exceeded.';

Minima of higher-dimensional functions by minimization along alternating directions

Approach: solution of multiple one-dimensional minimizations in alternating directions.

Starting from point 1P into direction 1u

1 1,P u

and iteratively minimizing

1 1 1 2 1 1 1

2 2 2 3 2 2 2

1

min

min

min n n n n n n

f P u P P u

f P u P P u

f P u P P u

until no further improvement can be obtained along any direction.

Choosing directions along steepest descent (opposite to gradient)

1i i iiP P f P mit min i iit

f P t f P

Examples of easy and difficult converging surfaces.


0.3.4 Linear systems of equations

System of algebraic equations

11 1 12 2 13 3 1 1

21 1 22 2 23 3 2 2

31 1 32 2 33 3 3 3

1 1 2 2 3 3

...

...

...

...

...

n n

n n

n n

m m m mn n m

a x a x a x a x b

a x a x a x a x b

a x a x a x a x b

a x a x a x a x b

in matrix representation

Ax b

with A being a coefficient matrix and b a column vector

11 12 1

21 22 2

1 2 3 4

...

...A

...

n

n

m m m m

a a a

a a a

a a a a

,

1

2

...

m

b

bb

b

.

alternative formulation of the problem ( ) 0f x with 1

n

i ij j i

j

f x a x b

and

1, , ; 1, ,j n i m

Types of problems:

Ax b

calculating the inverse matrix 1A with

-1ˆ ˆ ˆAA E equivalent to ˆ

j jAx b with 1,...,j N and 1jjb , and all other 0 , then

1

1ˆ ,..., NA x x

Important matrix properties:

1. n m same number of equations and unknowns

2. hermite matrix †ˆ ˆA A (komplex conjugate and transposed)

3. positiv definite ˆ 0vAv v

4. band matrix

5. sparse matrix most matrix coefficients are zero

always use addapted solution schemes


0.3.5 Eigenvalue problems

2

2ˆ ˆ ˆM with M E 0x x x

characteristic polynomial ˆ ˆdet M E ( ) 0P

to have solutions of

ˆ ˆ 0M E x which are not identical zero ( 0)x

0.3.6 Discrete (Fast) Fourier transform - FFT

Starting from a periodic function ( ) ( )f x L f x Periodic original space with set of discrete points with period L and step size a

0, ,2 , ,x a a L a

Periodic frequency space with set of discrete frequencies

2 2 20,1 ,2 , , 1LL L a L

k with L

Na

Definition of discrete Fourier transform (FT)

ikx

x

f k a e f x

with 2, 0La L

N k n n N , x n a

and inverse discrete Fourier transform (FT-1)

1 ikx

L

k

f x e f k

General property: 1( ) FT ( ) ( )f x f k FT f x


Illustration of decomposition of periodic function into harmonic functions.

Generalization for higher dimensions

with ( x x ; k k ), e.g. for 3 dimensions

3

3

1

ikx

x

ikx

k

f k a e f x

f x e f kL

0.3.7 Ordinary differential equations (ODEs)

General form of ordinary differential equation

( 1), ',..., ,

n

n

n

d f tG f f f t

dt

distinction of initial value problems and boundary value problems

Initial value problems (AWA)

initial values

10 , ' 0 ,..., 0

nf t f t f t


equivalent to system of coupled first order differential equations

11 1 2

22 1 2

33 1 2

1 2

( ), ,..., ,

( ), ,..., ,

( ), ,..., ,

( ), ,..., ,

n

n

n

nn n

df tG f f f t

dt

df tG f f f t

dt

df tG f f f t

dt

df tG f f f t

dt

Simple solution method

Forward Euler scheme

problem:

,f t G f t t

transformed into difference equation:

'( ) ,t

f t t f tf t D f t G f t t

t

with discretization t : :nf f n t

results in difference equation

1 ,n nn

f fG f t

t

and recursion formula for the solution of the ODE

1 ,n n nf f t G f t


Limitation: global error

h

Error

Discretization error

Round off error

Global

Limitation: instability

1

2 3

exakte Lösung

Euler Lösung

p

q

Instability scheme of harmonic oscillator equation.


1. Matrix method for stratified media

1.1 Wave equation in volumes of homogeneous media From Maxwell's equation one can derive wave equations in inhomogeneous media for different fields.

Electric field:

2

0 0( , ) ( , ) ( , ) E r r E r (8)

Dielectric flux density (electric displacement):

2

0

0

1( , ) ( , )

( , )

D r D r

r (9)

Magnetic field:

2

0

0

1( , ) ( , )

( , )

H r H r

r (10)

For homogeneous media these wave equations simplify considerably:

( , ) ( ) r (11)

divergence condition

0 0( ) ( , ) ( ) ( , ) 0 E r E r (12)

curl curl grad div – Laplace

for, e.g. the electric field's wave equation:

0

( , ) ( , ) ( , ) ( , )

E r E r E r E r (13)

Helmholtz equation

2

2( , ) ( ) ( , ) 0

c

E r E r with 2

0 0

1c

(14)

To solve the Helmholtz equation one can choose between different ansatz functions, each having advantages for different investigated problems or coordinate systems. Here we are choosing the plane wave ansatz function:

0( , ) ( )exp(i ( ) ) E r E k r with 2

0 0

1c

(15)

The plane wave ansatz function has the following parameters

complex amplitude vector (polarization & phase):

0

0

0

0

( )

x

y

z

E

E

E

E (16)

wave vector (phase velocity & direction of

energy flow (for isotropic homogeneous media)): ( )

x

y

z

k

k

k

k (17)


These six parameters can't be chosen completely independently. They rather have to fulfill the dispersion relation and must form a divergence-free field:

0 ( ) ( , ) 0 E r (18)

The ansatz must solve the Helmholtz equation which results in an eigenvalue problem, the solution of which determines the dispersion relation.

Inserting the plane wave ansatz into Helmholtz equation:

0( , ) ( )exp(i ( ) ) E r E k r

2

2( , ) ( ) ( , ) 0

c

E r E r (19)

with

0

0

0

( , ) exp( ( )) ( ) ( , )

x

y x y z

z

x E

E i k x k y k z iy

E

z

E r k E r (20)

and

2( , ) ( , ) ( ) ( , ) E r E r k E r (21)

Hence the dispersion relation of free space is

2

2

2( ) ( )

c

k (22)

Without loss of generality we define the z-axis as the axis of propagation.

2

2 2

2k ( ) ( ) k ( ) k ( )z x y

c

(23)

We are looking at the different types of solutions which fulfill the dispersion relation:


Propagating waves

for 2

2 2

2( ) k ( ) k ( )x y

c

22 2

2k ( ) ( ) k ( ) k ( )z x y

c

The sign of the wave vector dictates the direction (forward/backward) of propagation.

source

Evanescent waves

for 2

2 2

2( ) k ( ) k ( )x y

c

22 2

2ik ( ) ( ) k ( ) k ( ) ( )z x y

c

The sign of the wave vector must be chosen to ensure decay.

source

1.2 Optical layer systems Now we are looking for solutions of the wave equation in inhomogeneous media, which are constructed as a stack of homogeneous layers. This results in two categories of physical problems:

Transmission-reflection problem

Bragg mirrors

chirped mirrors for dispersion compensation

z

x

y


interferometers

Guided modes

multi-layer waveguides

Bragg waveguides

The general idea for the solution of these problems is the following:

Separating the domains in regions for which an analytical solution for the wave equation exists = mode expansion (free space plane wave)

Expanding the field into a superposition of these modes = Adjusting the amplitudes of each mode, such that the boundary conditions are fulfilled (exact or approximately)

General comment:

Modes in which the field is expanded should be adopted to the geometry

Fields in the layer system

Prerequisites:

stationary

layers in y-z-plane

incident fields in x-z-plane

Ansatz:

z( , , ) Re ( )expx z t x ik z i t rE E

z( , , ) Re ( )expx z t x ik z i t rH H

Decomposition in TE and TM fields:

TE

0

, 0

0

H

E

H

E H

x

TE y TE

z

TM

0

, 0

0

E

H

E

H E

x

TM y TM

z

1.3 Derivation of the transfer matrix Transition conditions

z

x

y

z

x

given field

to be calculated field

given field

to be calculated field


Fields: Et and Ht continuous TE: E(y) and Hz TM: H(y) and Ez

Calculation of tangential components (normal components can be derived from them)

wave vector component kz conserved throughout the layer system (determined by incidence angle)

wave vector component kx constant in a single homogeneous layer but varies from layer to layer

2

2 2

2ix i zk kc

Field calculation of continuous components (TE)

2

2 22

z2 2( ) 0

xi

i y

k

k E xx c

and z

0

( ) ( )y

iH x E x

x

Solution:

i1 2 x

0 1 2

( ) cos sin

( ) ( ) sin cos

i

i i i

y x

z y x x x

E x C k x C k x

i H x E x k C k x C k xx

Determination of 1 2,C C by

1(0)yE C and 2

0iy xE k C

x

TE:

i 0x

0

1( ) cos (0) sin

sin (0) cos

i i

i i i

y x y x y

y x x y x y

E x k x E k x Ek x

E k k x E k x Ex x

TM:

0

0

1( ) cos (0) sin

1 1sin (0) cos

i i

i

i

i i

iy x y x y

x i

x

y x y x y

i i i

H x k x H k x Hk x

kH k x H k x H

x x

kz

kz

kz

kx

kz

kx

kz

kx

kz

kx

kz

kxkz

kx


Combined TE/TM:

i

i

1( ) cos (0) sin (0)

( ) sin (0) cos (0)

i i

i

i i i

x x

x

x x x

F x k x F k x Gk

G x k k x F k x G

TE: 0 i, , 1y z yF E G i H Ex

TM: 0 i i, , 1/y z i yF H G i E Hx

Summary of matrix method:

1 0 0

ˆm ( ) MN

i i

iD

F F Fd

G G G

with

ix

1cos sin

m

sin cos

i i

i

i i

k d k dkx

k k d k d

ix

ix i

ix i ix ix

TE: i, , 1y yF E G Ex

TM: i i, , 1/y i yF H G Hx

1.4 Reflection and transmission problem

Transmission coefficient (T) und reflection coefficient (R)

in in

T RF FT R

F F

z

x

given: F(0), G(0), kz, i, di

to be calculated fields: F(D), G(D)

2

2 2i z

0

2,

ix zk k k

0

D

z

x

y z

x

yFin FR

FT

Fin FR

FT

substrate, s

cladding, c


Connecting the incident, reflected, and transmitted fields with the transfer matrix by expressing the fields at the interfaces of the layer system based on incident, reflected, and transmitted fields:

At the substrate side:

in S R S

S S in S R S

(0,z) exp( ) exp( ) exp( )

(0,z) exp( ) exp( ) exp( )

z x x

x z x x

F ik z F ik x F ik x

G i k ik z F ik x F ik x

At the cladding side:

T C

C C T C

( ,z) exp( ) exp ( )

( ,z) exp( ) exp ( )

z x

x z x

F D ik z F ik x D

G D i k ik z F ik x D

Connecting the substrate and cladding side by the transfer matrix:

in RT 11 12

S S in RC C T 21 22

( ) ( )

( ) ( ) xx

F FF M D M D

k F Fk F M D M D

ii

Resulting in reflection/transmission coefficients:

22 11 21 12

2

k M k M M k k MR

N

kT

N

s sx c cx s sx c cx

s sx

i

with

22 11 21 12N k M k M M k k M s sx c cx s sx c cx

i

2

2 2

s/c x s/c 0

0

2zk k

Energy flux

defined by the normal component of the Poynting vectors sx

x T x R

x in x in

s s

s s

z

x

ysin sR sxR

sT sxT

sx in

z

x

y z

x

ysin sR sxR

sT sxT

sx in

substrate, s

cladding, c


2

2Re

Re

c cx

s sx

R

kT

k

Example: Bragg mirror

Series of alternating dielectric layers with a chosen thickness, such that reflected light interferes constructively

The thickness of each layer is chosen to match 0 / 4i id n .

Example: 10 layers, 1 1.5n ,

2 2.0n , Design 0.6µm


Introducing a defect will couple light evanescently through the structure


Field distribution

Aim: calculation of the field distribution ( )F x in the entire structure

Starting point: known shape of transmitted vector

Remark: The field at x D is more easily known than at 0x since it only depends on a single wave (transmitted wave). In contrast at 0x the field is a superposition of the incident and reflected field. Hence to fix the field at

0x one would have to solve the reflection problem first, whereas at x D one can determine the field by setting ( )F D to an arbitrary value, e.g.

( ) 1F D and calculating ( )G D from Maxwell´s equations to ( ) c cxG D i k .

1

T

c cxcDD

FF

FFi kG

x

now: 1TF

1. invert the structure (vector transforms into (1, -ickcx))

2. calculate the field vector up to the next layer boundary

3. calculate towards the current x-value, starting from the layer boundary

4. save the first component of the vector

5. turn back the derived field and structure by inverting the order of the calculated fields


Calculating the real fields

z( , , ) Re ( )expx z t x ik z i t rE E

z( , , ) Re ( )expx z t x ik z i t rH H

with

TE: yE ex F x

TM: yH ex F x

1.5 Guided modes in layer systems

no y dependence, phase rotation in z direction

waves propagating without diffraction

allows miniaturization of optics

concept is used in optical communication systems

How can waves be bound by the layer system?

principle mechanism is total internal reflection

Physical reasoning of the field distribution:

zx

y

zx

y FinFR

FT

FinFR

FT

zx

y

zx

y

zx

y

zx

y

Fin FR

FT

Fin FR

FT

Fin FR

FT

FinFR

FT

FinFR

FT

FinFR

FT

z

x

nx

substrate

cladding

layer (system)

cladding

substrate

core


plane wave in propagation direction: exp( )zik z

evanescent wave in substrate and cladding 2

2

,2max ( )k

c

z s c

oscillating solution in core sin( ) cos( )fx fxA k x B k x 2

2

2max ( )i

ik

c

z

general condition for guided waves: 2

2

s,c 2max ( ) max ( )i

ik

c c

z

Modes are resonances of the system

A physical explanation of the correspondence of roots and modes can be seen in the singularities of the reflection and transmission coefficients. Hence there is a field in the layer without an input field.

For comparison this is the reflectivity and transmissivity of the same layer system in the kz-domain corresponding to the reflection/transmission problem addressed in the previous section. Here no singularities are observed.

Dispersion relation of guided waves singularities of R and T

22 11 21 12

22 11 21 12

k M k M M k k MFR

F k M k M M k k M

s sx c cx s sx c cxR

I s sx c cx s sx c cx

i

i

singularity: 22 11 21 12 0D k M k M M k k M s sx c cx s sx c cx

i

0 0.5 1 1.5 20

10

20

30

40

kz/k

0

Inte

nsity

abs(T)

abs(R)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

6000

kz/k

0

Inte

nsity

abs(T)

abs(R)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

kz/k

0

Inte

nsity

abs(T)

abs(R)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

kz/k

0

Energ

y


Roots of the denominator correspond to modes guided along the layer system.

The problem of finding a guided mode is reduced to finding a root.

with: , ,k i k i sx s cx c

2

2

, ,2( ) 0k

c

s c z s c

TE,TM TE,TM TE,TM TE,TM

22 11 21 12 0M M M M s c c ss c c s

Field distributions of guided modes inside a single high-index layer embedded into low-index substrate and cladding.

12 14 16 18 20-100

0

100

200

300

kz [µm

-1]

D

0 0.5 10

2

4

6

8

lateral distance [µm]

abs(E

) [V

2/m

2]

0 1 2 3-2

-1

0

1

2

lateral distance [µm]

abs(E

) [V

2/m

2]


2. Finite-difference method for waveguide modes

2.1 Scalar approximation for weakly guiding waveguides with small index differences

(A similar method is documented in J. Riishede, N. A. Mortensen, and J. Laegsgaard, "A ‘poor man´s approach´ to modeling micro-structured optical fibers," J. Opt. A: Pure Appl. Opt. 5, 534-538 (2003))

Starting from the wave equation

2

2rot rot ( , ) (r, ) ( , )

c

E r E r

Neglecting the divergence of the electric field in weakly inhomogeneous media with grad ( , ) 0 r

div ( , ) 0 D r 0 ( , )div ( , ) 0 r E r

We obtain the Helmholtz equation for inhomogeneous media

2

2( , ) (r, ) ( , ) 0

c

E r E r

Neglecting the vectorial properties of the electric field scalar Helmholtz equation

2(r) ( , ) (r) 0v k v r with 2

2

2( , ) ( , )k

c

r r

2.1.1 Stationary solutions of the scalar Helmholtz equation

Now we search for the stationary states (modes) of the problem with ( , ) ( , , )x y r , by taking the following ansatz function

(r) ( , )expv u x y z i

which results in an eigenvalue equation for the propagation constant

(2) 2 2( , ) ( , , ) ( ) ( , ) 0u x y k x y u x y

This eigenvalue problem is to be solved by a finite difference scheme

2 2

2 2

2 2( , ) ( , , ) ( ) ( , ) 0u x y k x y u x y

x y

with the Laplace operator for two dimensions

2 2

2 2

( , ) ( , )u x y u x y

x y

which must be approximated by a discrete operator in the following way


Discretization of Laplace operator in two dimensions

21 1

2 2

,

( , ) 2 ( , ) ( , )

j k

j k j k j k

x y

u x y u x y u x yu

x h

21 1

2 2

,

( , ) 2 ( , ) ( , )

j k

j k j k j k

x y

u x y u x y u x yu

y h

2 21, 1, , 1 , 1 ,

,2 2 2

,

4( )

j k

j k j k j k j k j k

j k

x y

u u u u uu uu

x y h

Hence the eigenvalue problem in discrete notation reads as

1, 1, , 1 , 1 , 2 2

, ,2

4( ) ( ) 0

j k j k j k j k j k

j k j k

u u u u uk u

h

which is a linear equation for each variable ,j ku coupled to 4 neighboring

equations

( , ) ,( , ) ( )x y j ku x y u

Introducing a computation grid:

quadratic area of size a a

equidistant discretization of the area with N N grid points

-10-5

05

10 -10

-5

0

5

10

0

50

100

150

200

X

y

U(x,y)

xj= yk=h

(xj,k,Uj,k) (xj+1,k,Uj+1,k)

x

y

(xj,k-1,Uj,k-1)

(xj,k+1,Uj,k+1)

(xj-1,k,Uj-1,k)


example at point ( 2, 3)j k :

3,3 1,3 2,4 2,2 2,3

2,3 2

4( )

u u u u uu

h

2.1.2 Matrix notation of the eigenvalue equation

,j ku : originally 2D variable depending on x-direction (j) and y-direction (k)

unfolding of ,j ku into a 1D vector

for each vector component ,j ku results an individual linear equation

matrix dimension: number variables in x-direction times number variables in y-direction

a

N

.

.

11 2 3 N

N

.

.

11 . . N

j

k

U|G = UG

.

.

= 0

U j,k

matrix


Matrix equation: schematic picture of the matrix formulation of the 2D Laplace operator

matrix: small number of non-zero values 'sparse' matrix

2.1.3 Boundary conditions

Example: metal boundaries (Metal tube with boundaries i )

U 2,2

=

U 3,2

U N-1,2U 2,3

…

U 3,3 0

U N-2,2

U 4,3

U N-2,3

…

U N-1,3U 2,4U 3,4U 4,4

-4 1 0 0 .. 0 1 0 0 0 0 … 1 -4 1 0 .. 0 0 1 0 0 0 …

1 0 0 0 .. 0 -4 1 0 0 0 … 0 1 0 0 0 … 0 1 0 0 .. 0 1 -4 1 0 0 … 0 0 1 0 0 …0 0 1 0 .. 0 0 1 -4 1 0 … 0 0 0 1 0 …

…

discrete Laplace operator

N

.

k

.

11 j N

x iterates first

accounts for the 1D – problem (tridiagonal matrix)

occures for 2D problems

a

N

.

.

1

- boundaries of the grid ?

- 4 U1,3 + U2,3 + U0,3 + U1,4 + U1,2

h2( U ) 1,3 =

example:

1 20 outside the grid

boundary conditions

(compare with theory of

partial differential equation)


2.2 Full vectorial mode solver for waveguides with large index differences

(according to Zhu et al., Optics Express 10, 853 2002)

starting from directly discretizing Maxwell’s equations using finite differences

t

BE and

t

DH

looking for guided modes propagating in the z-direction, all fields are assumed to have a spatio-temporal dependence like

exp i z t

introducing this ansatz into Maxwell's equation, rescaling the electric field by the free space impedance

0 0 0/Z in order to symmetrize the equations

and performing all derivatives with respect to time t and space z gives

0 0

0 0

0 0

z zx y r x y

z zy x r y x

y yx xz r z

E Hik H i E ik E i H

y y

E Hik H i E ik E i H

x x

E HE Hik H ik E

x y x y

G

U|i = Ui = const

N

.

.

11 . . N

grid with metal boundaries (U|G = 0 )

2 j N-1, 2 k N-1

(N-2) (N-2) – equations

(N-2) (N-2) – unknown

- 4 Uj,k + Uj+1,k + Uj-1,k + Uj,k+1 + Uj,k-1

h2=( U )j,k = 0


now the differential operators are approximated by finite differences which are formulated on a so-called Yee-grid to preserve the inversion symmetry of the original problem also in its discrete mathematical representation

Remarks on the properties of the Yee-grid

Please be aware that electric and magnetic field are evaluated half a unit cell apart! The 'beauty' of the Yee-grid can be seen when illustrating the fundamental laws of electromagnetics. Let's start from Ampere's law by taking the first Maxwell's equation

( , )( , )

tt

t

B rE r

0 ( , ) ( , )i t t H r E r

and integrating over a finite surface F

0 ( , ) ( , )F F

i t t H r f E r f

where f is the surface normal

Applying Stokes law the surface integral on the right hand side is transformed into a boundary integral

0

( )

( , ) ( , )F F

i t t H r f E r s

where s is the tangential element of the boundary

This integral equation can be expressed approximately based on the field components in the Yee-grid as

0 ( , ) x ( 1, ) ( ,( , 1) x ( ,) x )y x y x yz E j l E j l y E j l E jH l lj yi


Resorting the equation gives

0

( 1, ) ( , ) ( , 1) ( , )( , )

y y x xz

E j l E j l E j l E j li H j l

x y

which is the finite difference approximation for the z-component of the H-field in the Maxwell's curl equation.

Now we discretize the complete Maxwell's equations using this ansatz

0

0

0

( , 1) ( , )( , ) ( , )

( 1, ) ( , )( , ) ( , )

( 1, ) ( , ) ( , 1) ( , )( , )

z zx y

z zy x

y y x xz

E j l E j lik H j l i E j l

y

E j l E j lik H j l i E j l

x

E j l E j l E j l E j lik H j l

x y

0

0

0

( , ) ( , 1)( , ) ( , ) ( , )

( , ) ( 1, )( , ) ( , ) ( , )

( , ) ( 1, ) ( , ) ( , 1)( , ) ( , )

z zrx x y

z zry y x

y y x xrz z

H j l H j lik j l E j l i H j l

y

H j l H j lik j l E j l i H j l

x

H j l H j l H j l H j lik j l E j l

x y

with the spatially averaged material parameters


( , ) ( , 1)( , )

2

( , ) ( 1, )( , )

2

( , ) ( 1, 1) ( , 1) ( 1, )( , )

4

r rrx

r rry

r r r rrz

j l j lj l

j l j lj l

j l j l j l j lj l

The spatial averaging of material properties is a technique to increase computational convergence by minimizing the so-called 'staircase approximation error' in finite difference schemes.

As a last step we rewrite the discretized Maxwell's equations in matrix notation and use the perfect electric conductor boundary conditions from above

0

0

0

0

0

0 0 0

0 0 0

0 0 0

x y x

y x y

z y x z

rx x y x

ry y x y

rz z y x z

i

ik i

i

ik i

H I U E

H I U E

H U U E

ε E I V H

ε E I V H

ε E V V H

with

I the identity matrix,

, ,x y zE and , ,x y zH the concatenated vectors of x,y,z-directed field

components , , ( , )x y zE j l and , , ( , )x y zH j l , respectively

rxε the diagonal matrix of spatially averaged material parameters.

Furthermore there are the matrices of differential operators in discrete form on the Yee-grid

1 1

1 11x

x

U ( , 1) ( , )A j l A j l

1 1

1 11y

y

U ( 1, ) ( , )A j l A j l


1

1 11

1x

x

V ( , ) ( , 1)A j l A j l

1

11

1

1

yy

V ( , ) ( 1, )A j l A j l

Please be aware that even though these matrices look asymmetric they are just asymmetric in the notation and not in real space. They represent symmetric finite difference operators since the Yee grid symmetrizes the problem.

For the numerical solution of the eigenvalue problem one can combine the two coupled differential equations of first order for the two fields E and H into a single decoupled differential equation of second order for a single field, i.e. E or H .

For the electric field this results in (quasi TE version):

2xx xy x x

yx yy y y

P P E E

P P E E

with the sub-matrices

2 1 2 1 2

0 0 0

2 1 2 1 2

0 0 0

1 2 2 2 1

0 0 0

1 2 2 2 1

0 0 0

xx x rz y x y x rz x rx y y

yy y rz x y x y rz y ry x x

xy x rz y ry x x x rz x y x

yx y rz x rx y y y rz y x y

k k k

k k k

k k k

k k k

P U ε V V U I U ε V ε V U

P U ε V V U I U ε V ε V U

P U ε V ε V U I U ε V V U

P U ε V ε V U I U ε V V U

These equations might look quite lengthy but finally they consist just of a few matrix operations.

Similarly for the magnetic field this results in (quasi TM version):

2xx xy x x

yx yy y y

Q Q H H

Q Q H H

with the sub-matrices


2 1 2 2 1

0 0 0

2 1 2 2 1

0 0 0

2 1 2 2 1

0 0 0

2 1 2 2 1

0 0 0

xx x y x rz y ry x x y rz y

yy y x y rz x rx y y x rz x

xy ry x x y rz x x y x rz x

yx rx y y x rz y y x y rz y

k k k

k k k

k k k

k k k

Q V U U ε V ε V U I U ε V

Q V U U ε V ε V U I U ε V

Q ε V U U ε V V U I U ε V

Q ε V U U ε V V U I U ε V

Example 1: Photonic crystal fiber

Solving the eigenvalue problem for obtaining the propagation constant and the field distribution of guided modes.

Absolute value of the H-field in the x-y cross section of the first mode.

Example 2: Silver nano wire in vacuum

Ag-cylindrical waveguide (r=λ/3) at λ=630nm


Absolute value of the E-field in the x-y cross section of the first mode.

Intensity of the E-field in the x-y cross section of the second mode.

Dispersion relation of a metallic waveguide with 0 effk n , where

effn is the

so-called effective index of the mode.

Example 3: Silver nano wire on substrate

Ag-cylindrical waveguide (r=λ/3) on a substrate with 1.5n at λ=840nm


Intensity of the E-field in the x-y cross section of the first mode. The light is mainly confined at the interface between the silver nanowire and the substrate.

Dispersion relation of a metallic waveguide on a substrate.


3. Beam Propagation Method (BPM) up to this point we only dealt with eigenmodes

This required invariance of the structure in the third dimension.

What happens if light propagation occurs in a medium where the index distribution changes weakly along the propagation direction?

need for accurate model of very wide range of devices

linear and nonlinear light propagation in axially varying waveguide systems, as e.g.

o curvilinear directional couplers,

o branching and combining waveguides,

o S-shaped bent waveguides,

o tapered waveguides

ultra short light pulse propagation in optical fibers

implementation

finite difference BPM solves Maxwell's equations by using finite differences in place of partial derivatives

computational intensive

entirely in the frequency domain only weak nonlinearities can be modeled

use of a slowly varying envelope approximation in the paraxial direction the device is assumed to have an optical axis, and that most of the light travels approximately in this direction, (paraxial approximation) allows to rely on first order differential equations

3.1 Categorization of Partial Differential Equation (PDE) problems

general second-order Partial Differential Equation

2 2 2

2 20

f f f f fp q r s t u f v

x x y y x y

(23)

The following different types of partial differential equations are distinguished

1. 2 4q pr : elliptic PDE

2. 2 4q pr : parabolic PDE

3. 2 4q pr : hyperbolic PDE


Elliptic PDE

Example: ( , ) ( , )

( , )u x y u x y

x yx y

2 2

2 2+ = Poisson equation

Boundary Value Problem (BVP) limited mainly by computing memory

Hyperbolic PDE

Example: ( , ) ( , )u x y u x y

vt y

2 22

2 2 1D wave equation

Parabolic PDE

Example: ( , ) ( , )u x t u x t

Dt x x

= diffusion equation (IVP)

both are Initial Value Problem (IVP) limited mainly by computing time and stability issues


3.2 Slowly Varying Envelope Approximation (SVEA)

Assumptions

Each component of the optical electromagnetic field is primarily a periodic (harmonic) function of position.

The field changes most rapidly along the optical axis z (with a period on the order of the optical wavelength ).

Slowly Varying Envelope Approximation (SVEA)

replace the quickly varying component, , with a slowly varying one, , as

0( , , ) ( , , )exp( )x y z x y z ikn z with 2 /k

introduction of a reference index 0n

light is travelling mostly parallel to the z axis (paraxial approximation) and is monochromatic with wavelength

SVEA Advantages

requirements on the mesh to represent derivatives by finite differences are relaxed choose fewer mesh points higher speed of the calculation without compromising accuracy

accurate BPM calculations using step sizes z

SVEA Problems

if part of the light strongly deviates from the direction of the axis z improved solution: wide angle BPM


if structure has large index contrast no accurate global choice of

0n finer mesh needed

solution: if variation is in z direction 0( )n z

quality of choice of 0n can be checked by evaluating the speed of phase

evolution of along z in the numerics

3.3 Differential equations of BPM starting from Maxwell’s equations in the frequency domain with an inhomogeneous distribution of the dielectric properties ( , , )x y z

0

0 ( , , )

i

i x y z

E H

H E

+ no charges [ ( , , ) ] 0

0

x y z

E

H

magnetic field can be eliminated by taking curl of curlE -equation

2 ( , , )k x y z E E with 0 0k

using the vector identity 2( ) one gets the wave equation

2 2 ( , , ) ( )k x y z E E E

since BPM is biased for the z -axis, it is natural to treat z -components of the field E and the operator separately from the transverse components x and y

ˆt zE E E z and ˆ

tz

z

the transverse component of the wave equation becomes

2

2 2

2( , , ) ( )t z

t t t t t t

Ek x y z

z z

EE E E (23)

which is an inconsistent problem since it contains transverse and longitudinal components.

Splitting also the divergence equation in the transverse and longitudinal components

( , , )

( , , ) ( , , ) 0zt t z

x y z Ex y z E x y z

z z

E

and neglecting the second term ( is assumed to change only slowly in z )

( , , ) ( , , ) 0zt t

Ex y z x y z

z

E

one can eliminate the longitudinal term in the right hand side of the wave equation (23)


22 2

2

1( , , ) ( ( , , ) )

( , , )

tt t t t t t t tk x y z x y z

z x y z

EE E E E

applying the chain rule on the second divergence term on the right hand side, the first divergence term is canceled out

22 2

2

1( , , ) ( ( , , ))

( , , )

tt t t t t tk x y z x y z

z x y z

EE E E (23)

up to now: Due to the propagation direction, which points mainly along z , the field

tE is varying slowly in x and y , but rapidly in z

now: SVEA is introduced

0( , , ) ( , , )exp( )t tx y z x y z in kz E e

the new field ( , , )t x y ze is slowly varying in all coordinates (compared to )

Inserting SVEA ansatz into wave equation (23) gives

22 2 2

0 02

12 ( ( , , ) ) ( ( , , )) 0

( , , )

t tt t t t t tikn k x y z n x y z

z z x y z

e ee e e

If reference index 0n was chosen correctly, the first term will be much smaller

than the second and hence can be neglected (physics: neglect coupling to backward propagating waves and introducing paraxial approximation)

results in a first-order differential equation for te in the z coordinate and

changes the problem from an elliptic PDE (boundary value problem) to a parabolic PDE (initial value problem)

Next step: collect the transverse 2. order operators in a matrix form and shift them to the right hand side

02x xxx xy

yx yyy y

e eP Pikn

P Pe ez

with the components of the operator matrix being

22 2

02

2

2

22 2

02

1( , , ) ( ( , , ) )

( , , )

1( , , )

( , , )

1( , , )

( , , )

1( , , ) ( ( , , ) )

( , , )

xx

xy

yx

yy

P x y z k x y z nx x y z x y

P x y zx x y z y x y

P x y zy x y z x y x

P x y z k x y z ny x y z y x

Summary of the results

paraxial vector wave equation for the optical electric field


initial value problem: knowledge of the electric field in some transverse plane ( const.z ) is enough to calculate the field in the entire space

reflections of light are neglected

in a finite differencing scheme the operator P is a large sparse matrix

3.4 Semi-vector BPM up to now: full vector character of the electromagnetic field is included

now: if the modeled device operates mainly on a single field component, the other component’s contribution can often be neglected

semi vector TE equation for E-field being mainly transversely polarized

02 xxx x

eikn P e

z

or semi vector TM equation for H-field being mainly transversely polarized

02y

yy y

eikn P e

z

Properties

correctly models differences of TE and TM wave propagation

neglects coupling to other field component

3.5 Scalar BPM if structure has very low index contrast, the x and y term can be

neglected operators commute xxP and yyP reduce to scalar operator

2 2

2 2

02 2( ( , , ) )P k x y z n

x y

3.6 Crank-Nicolson method formal solution of BPM equation can be written as

t 1 t 0

0

( ) exp ( )2

zz z

in k

Pe e with

1 0z z z

however, a rational function is needed to approximate the exponent of the operator

a simple approximation would be

1 (1 )

exp1

xx

x

The case =0.5

called Padé(1,1) approximation

accurate for small x small step size z (higher order Padé approximations can be used, see below as supplementary material)


generates the first three terms of the MacLaurin series expansion for

exp x

leads to Crank-Nicolson scheme

1 0

0 0

( ) ( )4 4

t t

z zz z

in k in k

I P e I P e

since the operator P is applied to the unknown t 1( )ze it is an implicit method

requires solution of set of linear equations

Variable is called scheme parameter, for 0.5 the method is stable and energy conserving, since it corresponds to the discrete operator which preserves the inversion symmetry in z.

3.7 Alternating Direction Implicit (ADI) Solving the implicit problem in 2D leads to a huge set of linear equations. time consuming numerical solution

However, 1D problems result only in a simple tridiagonal matrix which can be solved very fast.

ADI approximation

splitting the n-dimensional operator into n subsequent 1-dimensional operators

each operation only on a single dimension

fast algorithm: periodic application of 1D operators in x and y direction

3.8 Boundary condition

Easiest boundary condition: reflecting boundaries (electric walls)

assuming that the field is zero at the boundary perfect reflection

Advanced boundary conditions: fields can radiate out of simulation area

fields radiating out, e.g. from waveguide, should not disturb the evolution by being reflected back from the boundaries let the radiation fields out

do not create any additional effects for the fields propagating inside the computation domain (e.g. instability, dissipation etc.)

Typical advanced boundary condition

absorbing boundary conditions (ABC)

transparent boundary conditions (TBC)

perfectly matched layer boundaries (PML)

3.8.1 Absorbing Boundary Conditions (ABC)

problem:


field at the boundary of the computational window is not wanted

solution:

remove field at boundaries by multiplication with factor <1 can be introduced by ( ) 0 at the boundary

however:

All inhomogeneities of the optical properties induce reflections themselves.

This can be shown by computing Fresnel reflection at interface where just the imaginary part of the dielectric function changes.

improvement:

soft onset of the absorbing layer

technical realization:

multiplication of the field in each iteration step with a filter function, e.g. in 1D:

abs

abs abs

abs abs

1 exp expp p

p p Pe e A A

N N

with

abs 3...5N

drawback:

absorption strength absA and absorption width

absN are difficult to adjust for

the smallest reflection

individual optimal parameters for each problem

3.8.2 Transparent Boundary Condition (TBC)

goal:

simulates a nonexistent boundary

Radiation is allowed to freely escape from the simulated area without reflection.

Flux of radiation back into the simulated area is prevented.

method:

assuming that the field in the vicinity of the virtual boundary consists of an outgoing plane wave and does not include any reflected wave from the virtual boundary wave function for the wave traveling to the right with the x -directed wave number

xk is expressed as

( , ) ( )exp(i )xx z A z k x


Derivation

projecting the field onto the following lattice

( )p px

lattice points 1p and p m are outside the computational window

At the left boundary the relation of the field at neighboring mesh points can be expressed as

1

0

exp( )xjk x

with 1 0x x x

from which we can calculate the x -directed wave number xk as

1

0

1lnxk

j x

if ( ) 0xk plane wave is traveling outwards

( ) 0xk plane wave is traveling inwards

since inward waves should not exist

( )xk must be positive ( ) 0xk

Implementation

xk is calculated at the boundary

another mesh point at 1p is artificially added to the mesh

the field at 1p is assumed to be determined by the same plane wave

function

0

1

exp( )xjk x

with

1 0 0 1x x x x x

assuming that the xk -vector is preserved by the wave

1 01 1 0 0

0 1

exp( ) 0xjk x

kx-Vektor boundary

(x)=0exp(ikxx)

x


trick: calculate 1n

xk one step z before applying it to the boundary

takes into account that the wave energy is traveling also forwards

(keep in mind, that we are in the paraxial approximation vor which all

waves propagate mainly forward)

1 1

1 1 0 00 n n n n

3.8.3 Perfectly matched layer boundaries (PML)

Definition

perfectly matched layer = artificial absorbing layer for wave equations used to truncate computational regions in numerical methods to simulate problems with open boundaries

Property

waves incident upon the PML from a non-PML medium do not reflect at the interface strongly absorb outgoing waves from the interior of a computational region without reflecting them back into the interior (impedance matching required)

Different formulations

nice one: stretched-coordinate PML (Chew and Weedon)

coordinate transformation in which one (or more) coordinates are mapped to complex numbers analytic continuation of the wave equation into complex coordinates, transforming propagating (oscillating) waves into exponentially decaying waves

Technical description

to absorb waves propagating in the x direction, the following transformation is applied to the wave equation:

all x derivatives / x are replaced by

1

( )1

i xx x

with being the angular frequency and some positive function of x

Hence, propagating waves in x direction ( 0k ) become attenuated

exp ( ) exp ( ) ( ') 'xk

i kx t i kx t x x

which corresponds to the following coordinate transformation (analytic continuation to complex coordinates)

( ') 'xi

x x x x

or equivalently (1 / )x x i


Properties

for real valued the PML attenuate only propagating waves

purely evanescent waves oscillate in the PML but do not decay more quickly attenuation of evanescent waves can be accelerated by including a real coordinate stretching in the PML corresponds to complex valued σ

PML is reflectionless only for the exact wave equation discretized simulation shows small numerical reflections can be minimized by gradually turning on the absorption coefficient σ from zero over a short distance on the scale of the wavelength of the wave (e.g. with quadratic spatial profile)

3.9 Conformal mapping regions used to simulate curved optical waveguides

solving bends by BPM directly leads to large errors due to the paraxial approximation

can be used to treat losses in curved waveguides

radii of curvature need to be restricted to large values, when first order approximation of the conformal mapping is used (corresponding to linear index gradient)

Method

apply a conformal mapping in the complex plane to transform a curved waveguide ( , )x y in coordinates ( ,x y ) into a straight waveguide described

by a modified epsilon distribution ( )u in new coordinates ( ,u v )

conformal mapping is an angle-preserving transformation in a complex plane

Remarks

This concept led to the emergence of an entire field of optical science, the so-called "transformation optics", where instead of interfaces between

different materials, like lenses, prisms, etc., continuous variations of (x,y)

and (x,y) are exploited for the manipulation of optical fields. However the mathematical apparatus just works in 2D and there is no formulation for 3D.


typical example for local angle preserving transformation: world maps

Demonstration for 2D scalar wave equation

2 2

2 2( , ) ( , ) 0k n x y x z

x z

general transformation

( ) ( )W u iv f x iz f Z

for f being an analytical function in the complex plane

To straighten a bend with radius R in the ( ,u v ) plane, ( )f Z should be taken

as

( ) ln( / )W f Z R Z R

applying the transformation to the wave equation gives

2

2 2

2 2( , ), ( , ) 0

dZk n x u v y u v

dWu v

with the Jacobian of the transformation being

exp( / )dZ

u RdW

a bended waveguide is transformed into straight waveguide with modified refractive index

( , )dZ

n x ydW

limitations imposed by the paraxial approximation are avoided

in first order approximation a linear index gradient is added to account for the bending


3.10 Wide-angle BPM based on Padé operators supplementary material, not covered by the lecture

Principle

expansion via Padé is more accurate than Taylor expansion for the same order of terms larger angles / higher index contrast / more complex mode interference can be analyzed as the Padé order increases

Derivation

starting from scalar wave equation without neglecting second order z derivative, the equation can be formally rewritten as

0 0

0 0

/ 2

1 ( / 2 )( / )

P k nj

z j k n z

(*)

which can be reduced to

N

jz D

with N and D being polynomials determined by the operator P

applying a finite difference scheme we get to iteration equation

1 (1 )l lD j z N

D j z N

with being a control parameter of the finite difference scheme ranging between 0 and 1 ( 0 fully implicit scheme; 1 fully explicit scheme;

0.5 Crank-Nicolson scheme)

the numerator (1 )D j z N can be factorized as

1 2

Nom Nom Nom 2 1( ...) (1 )...(1 )(1 )N N NNA P B P C P c P c P c P

with coefficients 1 2, ,..., Nc c c which can be obtained from the solution of the

algebraic equation

1 2

Nom Nom Nom( (1 ) ) ( ...) 0N N ND j z N A P B P C P

similarly the denominator D j z N can be factorized as

1 2

Den Den Den 2 1( ...) (1 )...(1 )(1 )N N NNA P B P C P d P d P d P

with coefficients 1 2, ,..., Nd d d which can be obtained from the solution of the

algebraic equation

i

0

( ) 0N

i

i

D j z N d P

with 0

0 0 1c d P

Thus, the unknown field 1l at z z is related to the known field

l at z as

1 2

1 Nom Nom Nom1 2

Den Den Den

...

...

N N Nl l

N N N

A P B P C P

A P B P C P


or

1 2 1

2 1

(1 )...(1 )(1 )

(1 )...(1 )(1 )

l lN

N

c P c P c P

d P d P d P

3.10.1 Fresnel approximation – Padé 0th order

starting again from wave equation (*) we account for the z-derivative by the recursion equation

0 0

0 01

/ 2

1 ( / 2 )n

n

P k nj

zj k n

z

(**)

which is used to replace the z-derivative in the denominator of (**)

in paraxial approximation 0

1

/

1

P a Pj j

jz a

a z

with 0 02a k n and 1

0z

comparison to the expansion equations gives

1D and /N P a

and therefore the nominator becomes

Nom(1 ) 1 (1 ) 1P

D j z N j z A Pa

and the denominator analogously

Den1 1P

D j z N j z A Pa

with

Nom

(1 )zA j

a

Den

zA j

a

thus the unknown field 1l at z z is related to the known field

l at z as

1 Nom

Den

1

1

l lA P

A P

or

1Den Nom1 1l lA P A P

which is the standard Fresnel formula

3.10.2 Wide angle (WA) approximation – Padé (1,1)

using again the recursion equation


1

0

/

1

P aj

jz

a z

and inserting the previous result

0

Pj

z a

we get

1

2

/

1

P aj

Pz

a

comparison to the expansion equations gives

2

1P

Da

and /N P a

and therefore the nominator becomes

Nom2 2

1 1(1 ) 1 (1 ) 1 1

P PD j z N j z j z P A P

a aa a

and the denominator analogously

Den2 2

11 1

P PD j z N j z j z P A P

a aa a

with

Nom 2

1 (1 )zA j

aa

Den 2

1 zA j

aa

thus the unknown field 1l at z z is related to the known field

l at z

again as

1Den Nom1 1l lA P A P

which includes now higher order corrections


4. Finite Difference Time Domain Method (FDTD) ab initio, direct solution of Maxwell‘s equations

probably the most often used numerical technique, as its implementation is straight forward (but what is cumbersome: implementation of proper boundary conditions)

requires excessive computational resources for reasonable problems in 3D (implementation requires cluster computers)

implementation is absolutely general but often doesn‘t take explicit advantage of symmetries

in principle all kinds of materials are treatable (dispersive or nonlinear materials) but one has to be careful to preserve the stability of the method

4.1 Maxwell’s equations

makr

ext

( , ) ( , )rot ( , ) , rot ( , ) ( , ),

div ( , ) ( , ), div ( , ) 0,

t tt t t

t t

t t t

B r D rE r H r j r

D r r B r

mater equations

0

0

( , ) ( , ) ( , ),

( , ) ( , ) ( , )

t t t

t t t

D r E r P r

B r H r M r

E(r,t) electric field

H(r,t) magnetic field

D(r,t) dielectric flux density

B(r,t) magnetic flux density

P(r,t) dielectric polarization

M(r,t) magnetic polarization (magnetization)

ext(r,t) external charge density

jmakr(r,t) macroscopic current density

MWEQ in linear isotropic, dispersionless & non-magnetic dielectric Media

0

0

( , ) ( , ),

( , ) ( , ),

t t

t t

D r r E r

B r H r

0 0 makr

0

( , ) ( , )rot ( , ) , ,

div ( , ) 0, di

rot ( ,

,

)

v ( , ) 0

t tt t

t

t

t

t

r E r

H r E rE r H

r

r r

H

j

curl-equations for individual components in Cartesian coordinates

0

( , ) 1(a) rot ( , )

tt

t

H rE r

field sources


0

0

0

1,

1,

1

yx z

y xz

yxz

EH E

t z y

H EE

t x z

EEH

t y x

0

( , ) 1(b) rot ( , )

tt

t

E rH r j

r

0

0

0

1,

1,

1

yx zx

y x zy

y xzz

HE Hj

t y z

E H Hj

t z x

H HEj

t x y

r

r

r

We will assume source-free conditions ( 0j ) for the moment and add the

sources later.

4.2 1D problems Assuming that there is no dependence on y and z all dynamics in x

0 0

1 1( ) 0, ,

y yx z zH EH E H

at t x t x

0 0

1 1( ) 0, ,

y yx z zE HE H E

bt t x t x

r r

grouping together all non-mixing transverse electromagnetic components (TEM)

z-polarized E-field 0 0

1 1,

y yz zH HE E

t x t x

r

y-polarized E-field 0 0

1 1,

y yz zE EH H

t x t x

r

Without loss of generality we concentrate on the first case (zE ).


4.2.1 Solution with finite difference method in the time domain for Ez

Discretization of derivative operators

symmetric discretization is second order accurate

21 1

1 1

1 1

( )n N NN N

N N N

f f fO x x

x x x

while the asymmetric discretization is just first order accurate

11

1

n N NN N

N N N

f f fO x x

x x x

always use symmetric discrete operators if possible / feasible!

introduce discrete space-time index notation: ( , ) ( , )i n i x n t

upper index = time(n )

lower index = space( i )

21 1

1 12

21 1

1 12

( , ) ( )2

( )2

( , ) ( )2

( )2

( )

n n nn i i i

z i

n n n

i i i

n n nn i i i

y i

n n n

i i i

i

E E EE i x n t E O x

x x

E E EO t

t t

H H HH i x n t H O x

x x

H H HO t

t t

i x

Maxwell’s equations

0 0

1 1

1 1

0

1 1

1 1

0

0 0

1 1

1 1

0

1 1

1

0

1 1

1

2 2

1

1 1

1

2 2

1

n ny i iz

i

n n n n

i i i i

i

n n n n

i i i i

i

n ny i iz

n n n n

i i i i

n n n

i i i i

H E HE

t x t x

E E H H

t x

tE E H H

x

H H EE

t x t x

H H E E

t x

tH H E E

x

r

1

n


FDTD Discretization for the H field:

FDTD Discretization for the E field:

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H 5

4

3

2

1

1 2 3 4 5 6 x(i)

t(n)

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

5

4

3

2

1

1 2 3 4 5 6 x(i)

t(n)


4.2.2 Yee grid in 1D and Leapfrog time steps

Equations for a single space-time step

1 0.5 0.5

0.5 0.5

0.5 0.5

0.5 0.5 1

0

0

1

1

n

n n n n

i i i i

i

n n n

i i i i

tE

tH H E

E H

Ex

Hx

Properties

Divergence MWEQs?

The condition of no divergence of the fields is always fulfilled in 1D, since fields are always transverse polarized to the direction of field change.

Resolution of discretization x and t (from physical arguments)?

5

4

3

2

1

1 2 3 4 5 6 x(i)

t(n)

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

E

H

2

1

1 2 3 x(i)

t(n)

E

H

E

H

E

H

E

H

E

H

E

H

1.5 2.5 3.5

2.5

1.5


spatial grid resolution x must be fine enough to display the finest structures of the distribution and the fields rule of thumb (heuristically):

max/ (20 )x n , with maxn being the

highest refractive index in the simulation domain

temporal stepsize t is limited by the speed of light, i.e. the interaction in space can reach only up to the next neighbor in one time-step sets upper limit to the phase velocity /t x c (see notes on stability in section 4.3.4)

Boundaries?

finite size of treatable simulation domain requires truncation of space most simple approach: perfectly conducting boundary ( 0E at the boundary)

Sources?

either initial field distribution or sources in the simulation domain

initial field is difficult in higher dimensions since field must have zero divergence (would require prior knowledge of solution)

sources as e.g. currents:

0.5

0.5

0

1 0.5 0.5

0.5 0.5

0

1n n n n

i i i i

n

i

i i

tE E H

x

tjH

4.3 The full 3D problem space grid

( , , , ) ( , , , )i j k n i x j y k z n t

( , , , )

( , , )

n

x x ijk

ijk

E i x j y k z n t E

i x j y k z

Finite differencing in space and time

1 1 2

1 1

2

( )2

( )2

n nn

x xx i j k i j kijk

n n n

x x xijk ijk ijk

E EEO y

y y

E E EO t

t t


4.3.1 Yee grid in 3D

discretizing MWEQ 0

1x

x yz jt y

H

z

E H

r

1 12 2

1 1 1 12 2 2 2

1 1 1 12 2 2 2

1 12 2

1 12 2

, , , ,

, 1, , , , , 1 , ,

, ,0 , ,

n n

x xi j k i j k

n nn n

y yz zi j k i j k i j k i j k n

x i j k

i j k

E E

H HH Htj

y z

other equations: 1 12 2

1 1 1 12 2 2 2

1 1 1 12 2 2 2

1 12 21 1

2 2

, 1, , 1,

, 1, 1 , 1, , 1, 1, 1,

, 1,0 , 1,

n n

y yi j k i j k

n n n n

x x z z ni j k i j k i j k i j k

y i j ki j k

E E

H H H Htj

z x

1 12 2

1 1 1 12 2 2 2

1 1 1 12 2 2 2

1 12 2

1 12 2

, , 1 , , 1

, , 1 1, , 1 , 1, 1 , , 1

, , 10 , , 1

n n

z zi j k i j k

n n n n

y y x x ni j k i j k i j k i j k

z i j k

i j k

E E

H H H Htj

x y

1 11 12 22 2

3 31 1 1 1 1 12 2 2 2 2 2 2 2

1 12 2

, 1, , 1, , , 1 , , 11

, 1, 1 , 1, 1

0

n n n n

y y z zi j k i j k i j k i j kn n

x xi j k i j k

E E E EtH H

z y

Ez

Ey

Ex

Hy

Hy

Hz

Hz

Hz Hx

Hx

Hy

Hx

x(i)

y(j)

z(k)


1 11 12 22 2

31 1 1 1 1 1 12 2 2 2 2 2 2 2

1 12 2

1 , , 1 , , 1 , , , ,

, , 1 , , 10

n nn n

x xz zn n i j k i j k i j k i j k

y yi j k i j k

E EE EtH H

x z

1 11 1

2 22 2

3 1 1 1 1 1 1 12 2 2 2 2 2 2 2

1 12 2

, , , , , 1, , 1,1

, 1, , 1,

0

n nn n

y yx xi j k i j k i j k i j kn n

z zi j k i j k

E EE EtH H

y x

4.3.2 Physical interpretation

3D Yee grid & Amper’s law

0

( )

( ) (, ) , )(F F

tt

t

r f sH rE r

Hx(i-1/2,j,k)

Hx(i-1/2,j+1,k)

Hy(i,j+1/2,k)

Hy(i-1,j+1/2,k)

Ey(i-1/2,j+1,k-1/2)

Ey(i-1/2,j+1,k+1/2)

Ez(i-1/2,j+3/2,k)

Ez(i-1/2,j+1/2,k)

1 12 2

1 1 1 12 2 2 2

1 12 2

1 11 12 22 2

, , , ,

0 , ,

, , , 1,, , 1, ,

n n

i j k i j k

i j k

n nn n

i j k i jx ki j k i j ky

z

y

z

x

x yt

x y x

E

H H y

E

H H

solving for the unknown component:

1 12 2

1 1 1 12 2 2 2

1 1 1 12 2 2 2

1 12 2

, , , ,

, , 1, , , , , 1,

0 , ,

y y

n n

i j k i j k

n n n n

i j k i j k i j

z z

k i j k

i j k

x xH H H

E E

t

x y

H

3D Yee grid & Faraday’s law

0

( )

( , ) ( , )F F

tt

t

E rf sH r


Hx(i-1/2,j,k)

Hx(i-1/2,j+1,k)

Hy(i,j+1/2,k)

Hy(i-1,j+1/2,k)

Ey(i-1/2,j+1,k-1/2)

Ey(i-1/2,j+1,k+1/2)

Ez(i-1/2,j+3/2,k)

Ez(i-1/2,j+1/2,k)

4.3.3 Divergence-free nature of the Yee discretization

0 0 makr

0

( , ) ( , )rot ( , ) ,

div

r ,

,( , ) 0, div

ot ( , )

( , ) 0

t tt t

t

t

t

t

r E r

H r E rE r H

r

r r

H

j

0div ?t

E 0 0

Yee cell Yee cell

div dV dt t

E E f by Gauss's theorem

1 1 1 12 2 2 2

1 1 1 12 2 2 2

1 1 1 12 2 2 2

0 0 , , 1, ,

Yee cell

Term1

0 , 1, , ,

Term 2

0 , , 1 , ,

Term 3

x xi j k i j k

y yi j k i j k

z zi j k i j k

d E E y zt t

E E x zt

E E x yt

E f

substitute Term 1 with curl equation 0

1 yx zHE H

t y z

r

1 1 1 12 2 2 2

1 1 1 12 2 2 2

, 1, , , , , 1 , ,

1, 1, 1, , 1, , 1 1, ,

Term 1y yz zi j k i j k i j k i j k

y yz zi j k i j k i j k i j k

H HH H

y z

H HH H

y z

collecting the contributions from Term 1, Term 2 and Term 3 results in vanishing of all contributions:


0

Yeecell

Term 1 Term 2 Term 3 0d y z x z x yt

E f

hence, if the field was divergence-free at some time it will conserve this property and stay divergence-free!

0 0 0

Yee cell Yee cell

div ( 0) 0 & div 0 div 0t dV dVt

E E E

However, it is important to ensure that sources do not introduce artificial divergence!

4.3.4 Computational procedure

Using the spatial differences of the E Field that are known for the time step n Δt to calculate the H field at the time step (n+1/2) Δt

Using the spatial differences of the H Field that are known for the time step (n+1/2) Δt to calculate the E field at the time step (n+1) Δt

Using the spatial differences of the E Field that are known for the time step (n+1) Δt to calculate the H field at the time step (n+3/2) Δt

…

Properties of the algorithm

Leapfrog algorithm discretization in all components, explicit formulation in time

Close to the physical world as the spatial and temporal propagation is exactly simulated

Yee grid can ensure inherently divergence-free solution.

Method is second order accurate in all spacetime dimensions (from central differences).

To ensure stability, the Courant-Friedrichs-Lewy condition has to be fulfilled (published very early in [R. Courant, K. Friedrichs, H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik”, Mathematische Annalen 32, (1869)]. This condition links temporal and spatial resolution (as in the 1D case). In three dimensions, the stability condition will be

1/2

2 2 2

1 1 1 1

3

ht

c x y z c

(for a cubic grid).

Note that the temporal and spatial discretization cannot be chosen independently. This means that if you want to double the spatial resolution in a 3D FDTD, your memory consumption rises by a factor of 8 (23), while your computing time rises at least by a factor of 16 (24)!

sou

rce:

urb

an-f

itn

ess.

de


4.4 Simplification to 2D problems Problems are often invariant in one spatial direction, e.g. taking z (as e.g.

for gratings, cylindrical objects)

Derivations of the field along this directions are zero

0

1 yxEH

t z

0

1 yz x zx

HE E Hj

y t y z

r

0

1y z xH E E

t x z

0

1y xy

E Hj

t z

r

0 0

1 1

z

y yz x z xz

H

x

E HH E E Hj

t y x t x y

r

Maxwell can be decoupled into 2 sets of each 3 differential equations

0 0

0

1 1

1

y yz x z xz

yx

H EE H H Ej

t x y t y x

EH

t z

r

0

1 yz x zx

HE E Hj

y t y z

r

0

1y z xH E E

t x z

0

1y xy

E Hj

t z

r

zH

x

TE polarization TM polarization

4.5 Implementing light sources arbitrary light sources can be modeled simply by adding the source field to

the field in the computational domain

a physical model for sources is a macroscopic current density

makr

( , ) ( , )rot ( , ) , rot ( , ) ( , )

t tt t t

t t

B r D rE r H r j r

Simplified implementation by adding a source term to the electric field, which corresponds to an externally driven dipole polarization

Examples for temporal variation of the light source

x-polarized cw-source , , , ,

sin( )n n

x xi j k i j kE E n t

x-polarized impulse ,, , , ,

n n

x x n ni j k i j kE E

Examples for spatial variation of the light source

x-polarized Gaussian wave

22

, , , ,exp sin( )

n n

x xi j k i j kx y

i x j yE E n t


Example calculations (2D-configuration, TM, Hy plotted)

source type: plane wave

homogeneous medium Cylinder with refractive index 2n and diameter 2D

source type: point source

homogeneous medium Cylinder with refractive index 2n and diameter 2D

4.6 Relation between frequency and time domain If one would like to determine the response of an optical system for different excitation wavelengths, one could perform many calculations for different cw excitations with a single frequency. In this case one has to wait for a long computing time until the stationary state of the calculation is reached, since the calculation starts with a step function in the excitation.

Alternatively, on can perform this calculation in the frequency domain:

The frequency spectrum used for the illumination is given by the Fourier-transformation of the time dependent incident field.


With a single calculation with sufficient spectral width, e.g. a delta distribution, we can calculate the entire frequency response of a particular optical structure by detecting the temporal evolution of the field behind the structure.

The contributions of the different frequencies are then separated by taking the Fourier transform of the time series of the field.

For a high resolution in the wavelength domain, we have to record the temporal evolution of the field for a long time (

tN = total number of time

steps) long computing time

Example of a grating waveguide coupler

Periodically structured slab waveguide corresponds to one-dimensional Photonic Crystal waveguide (n1=1.58, n2=1.87, d1=d2=165 nm, TE).

Transmission spectrum - The dips are waveguide resonances that are excited if the transverse k-vector (i.e. the momentum) provided by the grating matches the propagation constant of a waveguide mode.

t

EInc

f

EInc

Tra

nsm

issio

n?

d1

n1

d2

n2

x

z


Field distributions obtained for calculations with cw-excitation at frequencies of the dips in the spectrum, corresponding to guided modes in the grating waveguide.

4.7 Dispersive and nonlinear materials FDTD is not directly applicable for materials with 0 as e.g. metals

Material properties depend strongly on the wavelength (dispersion)

Inclusion of nonlinear response (instantaneous or non-instantaneous) of the material

There exists a great diversity of approaches, but usually all of them require the simulation of the dynamics of an additional quantity

So far we have taken into account E and H

Now we include: j current density, P polarization, and D displacement

Metals

Maxwell's equations

0 0

( , ) ( , )rot ( , ) , rot ( , ) ( , )

t tt µ t t

t t

H r E rE r H r j r

in addition we need an equation relating current and electric field (Drude model to describe mean velocity of electrons driven by an electric field)

2

0 Pt

jj E

with P the plasma frequency and 1 / is governed by the relaxation time

This is usually solved in the frequency domain as

2

P( ) 12 (1 2 )

i

i


Epsilon function of a metal described by the Drude model

Dielectrics

Assuming a 2D geometry (y-z plane) with TM polarization

ˆxE xE

The dielectric material response results in an induced polarization

ˆxP xP

which can be approximated by an Lorentz model similar to Drude model but with non-zero resonance frequency

Lorentz dispersion in time domain: 2

2 2

0 0 0 L2

x xx x

P PP E

t t

Solution can be derived in frequency domain.

Lorentz dispersion in frequency domain: 2

0 0 L, ,2 2

0

x xP Ej

(see also R.W. Ziolkowski et al., JOSA A, Vol. 16, No. 4, 980)

Nonlinear materials

Maxwell's equations

makr

ext

( , ) ( , )rot ( , ) , rot ( , ) ( , ),

div ( , ) ( , ), div ( , ) 0,

t tt t t

t t

t t t

B r D rE r H r j r

D r r B r

Linear material response: D E

Nonlinear material response: 0 D E P

with (1) (2) 2 (3) 3

0( ...) P E E E

Example: instantaneous Kerr (3) nonlinearity

0 D E with intensity dependent refractive index


2 22 2 2

0 2 0 0 2( ) 2n n n E n n n E

Refractive index depends on the square of the E-Field (for 2 0n n )

Problem: implicit equation and solution by Newton iterative procedure necessary

4.8 Boundary conditions Problems appear if the fields at the boundary have to be evaluated.

For keeping the discretized mesh treatable on a computer, we have to limit its size.

But for a proper determination of the field components that are positioned directly at the boundary of the computational domain, we need actually information about field components outside.

Choosing proper boundary conditions

4.8.1 Perfectly conducting material boundary

The easiest boundary condition is again to assume a perfectly conducting material (for E or H ), which surrounds our computational domain. Then the field cannot penetrate the structure.

Setting the field values outside the structure equal to zero

1 12 2

1 1 1 1 1 1 1 12 2 2 2 2 2 2 2

1 1 1 12 2 2 2

; , ; , ; , ; , ; , ; ,

, ,

n n n n n n

x j k x j k y j k y j k z j k z j k

j k j k

t tH H E E E E

z y


Grid size for Perfect Electric Conductor (PEC)

12

12

12

, 1, 1 1, ,

1, , 1 , 1,

1, 1, , , 1

n n

x x y z x x y z

n n

y x y y x y z

n n

z x y z x y z

E N N N H N N N

E N N N H N N N

E N N N H N N N

Index notation for Perfect Electric Conductor boundaries

:,1,: 0, :,:,1 0

:, 1,: 0, :,:, 1 0

x x

x y x z

E E

E N E N

1,:,: 0, :,:,1 0

1,:,: 0, :,:, 1 0

y y

y x y z

E E

E N E N

1,:,: 0, :,1,: 0

1,:,: 0, :, 1,: 0

z z

z x z y

E E

E N E N

4.8.2 Floquet-Bloch periodic boundaries

Floquet-Bloch periodic boundaries have to be applied for periodic objects, like e.g. gratings, photonic crystals.

Floquet-Bloch boundaries in the frequency domain:

( , , ) ( , , )exp(i )exp(i )x y x x y yx m y n x y k m k n

For 1m and 0n this results in


( , , ) (x, , )exp(ik )x xE x y t E y t

Incident plane wave (arbitrary propagation direction):

Incid. exp( )exp( )exp( )exp( )x y zE ik x ik y ik z i t

Example: Floquet-Bloch periodic boundaries applied for calculating the band

structure of a Photonic Crystal (PC)

Left: Periodic geometry of a 2-dimensional photonic crystal slab, consisting of air holes in an otherwise homogeneous material. The red lines indicate the

periodic boundary with a particular xk and yk . The spatial x and y coordi-

nates are measured in numbers of the lattice period a of the photonic crystal. Right: Unit cell of the photonic crystal, which is the size of the computational domain being considered in the numerical simulations.

Lunching an arbitrary field distribution and recording the evolving pattern

on some discrete points in the space

Initial field distribution with random spatial distribution to avoid selective excitation of states with particular symmetry.


Time evolution of the field at some specific spatial position which is recorded during the numerical simulation. The underlying photonic crystal structure was

taken as in the previous picture with air holes of radius 0.3R a in a material

with dielectric constant 13 .

All the frequencies which do not satisfy the periodic boundaries are annihilated and only the modes that are allowed to propagate persist

Spectra obtained by an Fourier Transform of the recorded time series from the previous figure.

Scanning the k-space and tracing the frequencies that persist as modes

delivers the band structure via FDTD


Left: Calculated temporal frequency spectrum for one particular Floquet-Bloch periodic boundary condition. Right: Band structure computed by tracking the peak positions in the temporal frequency spectrum over a variation of the wave vectors defined by the boundary conditions.

4.8.3 Transparent boundary conditions - 2nd order Mur (TBC)

For simplicity of the explanation we neglect the vectorial aspect of the field equations. Then each field component obeys approximately the scalar wave equation:

2 2 2 2

2 2 2 2 2

10

f f f f

x y z c t

In an operator notation this wave equation can be written as

0Lf

with the operator

2 2 2 22 2 2 2 2

2 2 2 2 2

1x y z tL c

x y z c t

This operator can be decomposed into a product of operators

0Lf L L f

If the two operators describe the waves propagation in the forward ( x ) and backward ( x ) direction they read as

1 21x tL c S wave propagating in the x -direction

1 21x tL c S wave propagating in the x -direction

with

X

M

irreducible Brillouin zone

k


2 2

2

1 1

y z

t t

Sc c

Then the equation

0L f

would describe the backward wave propagation in the x -direction.

Reference: Engquist-Madja exact ABC [Mathematics of computation 31, 629, (1977)]

The direct implementation of the operator is not possible, but the square root

can be expanded as a Taylor-series:

2 1/2 2 412

(1 ) 1 ( )S S O S

First order approximation

2 2

2

1 10

y z

t t

Sc c

nearly plane wave propagating in the x-direction as:

1 0x tL f c f

Second order approximation

2 1/2 212

(1 ) 1S S

resulting in a backward operator

2 212

tx y z

t

cL

c

and eventually in a wave equation in backward (-x) direction

2 2 2 1 212

0xt yy zz ttL f f c f f c f

Writing the differential operators as finite differences [G. Mur, IEEE Trans. Electromagnetic Compatibility 32, 377 (1981)]

Discretizing the operator a half spatial step in front of the boundary (example of the boundary at x=0)

1 1 1 1

1, , 0, , 1, , 0, ,2

1/2, ,

1

2

n n n nn j k j k j k j k

xt j k

f f f ff

t x x

Averaging the second order time derivative at 0x and x x

1 1 1 1

0, , 0, , 0, , 1, , 1, , 1, ,2

2 2

2 21

2

n n n n n n

j k j k j k j k j k j k

t

f f f f f ff

t t

first order approx. second order approximation


Same holds for the second order derivatives in y and z direction

Inserting all those difference schemes leads to

1 1 1 1

0, , 0, , 1 1, , 0, , 2 1, , 0, ,

3 0, 1, 0, , 0, 1, 1, 1, 1, , 1, 1,

3 0, , 1 0, , 0, , 1 1, , 1 1, , 1, , 2

2 2

2 2

n n n n n n

j k j k j k j k j k j k

n n n n n n

y j k j k j k j k j k j k

n n n n n n

z j k j k j k j k j k j k

f f k f f k f f

k f f f f f f

k f f f f f f

with

1

c t xk

c t x

, 2

2 xk

c t x

, and

2

3 2

( )

2 y ( )y

c t xk

x t x

The example above was derived for the boundary at 0x . Similar equations are used for the other boundaries.

Fields have to be stored for 2 different time steps

Simplification by using only the first order Taylor approximation (skipping the derivatives along the y and z directions)

1 1 1 1

0, , 0, , 1 1, , 0, , 2 1, , 0, ,

n n n n n n

j k j k j k j k j k j kf f k f f k f f

Properties

Only the fields components that are evaluated at this (most outer) boundary have to be updated with this equation (the tangential components of the E-field e.g. at the boundary 0x )

Reflection coefficients are in the order of 10-2

Easy to implement

4.8.4 Perfectly matched layer boundaries (PML)

Most efficient currently known boundary conditions

Published by Berenger in 1994 with reflections about 3x103 times less than with 2nd order Mur boundary

Basic idea: constructing a media which absorbs light and at whose interfaces with the region of interest no reflections take place Impedance matching of the medium with the surrounding

Impedance of a medium having both electric and magnetic conductivity

l

i

i

with the electric conductivity

e

and the magnetic conductivity


m

Let the loss-less region being region 1, characterized by and

Impedance:

If the condition e m

holds, the impedances are equal:

l

Plane waves propagating in each of the regions are characterized by their

propagation constants, given as (by choosing and )

kc

(dispersion relation)

resulting in

e m1 1k i ic

And the velocity of propagation is given by

ek ic

the same as in free space but light is additionally absorbed

The reflections coefficients of the interface are given generally by

r2 i 1 t

i

2 i 1 t

cos cos

cos cos

ER

E

and

r

2 t 1 i

i

2 t 1 i

cos cos

cos cos

ER

E

with

1 i 2 tsin sinn n

reflection is zero, only if

angle of incidence = angle of transmittance

The following derivation is done as an example for 2D FDTD in TE polarization [Jean-Pierre Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," Journal of Computational Physics 114, 185-200 (1994) (2D)]

e

e

x zx

y zy

E HE

t y

E HE

t x

m

yz xz

EH EH

t x y

absorb light only in one direction, fields shall propagate parallel to the surface


Berenger’s idea: splitting the H field into a x and y component

z zx zyH H H

(x-derivative of the E field drives the zxH component and vice versa)

Introduction of an anisotropy of all properties

e

e

zx zyxy x

zx zyy

x y

H HEE

t y

H HEE

t x

m

my

yzxx zx

zy xzy

EHH

t x

H EH

t y

Important: waves propagating along the y-axis are not absorbed in the x-boundary

Splitting the field in the boundaries

Adding the electrical and magnetic conductivity to the equations as material parameters

Only the components propagating normal to the boundary are absorbed by adjusting the proper absorption for each component

It can be shown, that the impedance of the Berenger medium equals the impedance of the free space, regardless of the angle of propagation

Problem with sudden change in the material properties: electric and magnetic field are evaluated at different spatial positions (staggered grid)

asymmetric absorption

Choosing an appropriate absorption profile for ex and

mx

Polynomial scaling

e max

m

ex

x

d


Practical considerations: 2 << m << 6 and 10d x 1610R

Electric and magnetic fields are evaluated at half discretization step apart, hence the absorption profile is evaluated likewise at different spatial coordinates

Extensions for TM polarization and 3D is straight forward

[Jean-Pierre Berenger, "Three dimensional perfectly matched layer for the absorption of electromagnetic waves," Journal of Computational Physics 127, 363-379 (1996) (3D)]

e m

e m

e m

e

y xz zx zy y xy zx zy

z xz yx yz z xz yx yz

x yx zx zy x yx zx zy

z yz xy xz

E H H H E Et y t x

E H H H E Et z t z

E H H H E Et x t x

E H Ht z

m

e m

e m

z yz xy xz

x zx yx yz x zx yx yz

y zy xy xz y zy xy xz

H E Et z

E H H H E Et x t x

E H H H E Et y t y


5. Fiber waveguides

5.1.1 The general eigenvalue problem for scalar fields

Restrictions:

weakly guiding structure ( 1n ln ( , ; ) 0x y )

linearly polarized field

Scalar wave equation for a scalar mode field u propagating in the z-direction

2 2

t ( , ) ( , , ) ( ) ( , ) 0u x y k x y u x y

The mode solution ( , )u x y can be assumed to correspond to the power

density u EH and it has two important properties:

u and 'u must be continuous and bound, since the second derivative of u must be finite due to the finite refractive index discontinuities.

u and 'u converge to zero as ,x y , since physically relevant

fields should have finite energy.

These properties determine the eigenvalue problem for the propagation

constant of guided modes.

The discrete solutions are called guided modes.

5.1.2 Properties of guided modes

Orthogonality and normalization

Two solutions au and

bu of the scalar wave equation with the respective

propagation constants a and

b read as

2 2

t ( , , ) ( ) ( , ) 0a ak x y u x y

2 2

t ( , , ) ( ) ( , ) 0b bk x y u x y

By multiplying the first equation with bu and the second with

au and

subtracting the results one obtains

2 2

t ta b a b b a a bu u u u u u

Integration over the infinite cross section of the waveguide A

2 2

t ta b a b b a a bA A

u u dA u u u u dA

Using Green’s law to transform the surface integral on the right side into a line integral, one can show that the right side is equivalent to zero, since the mode field is assumed to tend to zero at infinity.

Hence only the part at the left hand side remains:

2 2 0a b a bA

u u dA


To fulfill the equation for different the integral must be zero

,a b a bA

u u dA

This requires the normalization of the fields according to

2 1

Au dA

For cylinder symmetric problems we will see later that this reads as

,( ) ( )a b a br

Y r Y r r dr

Phase velocity

The propagation velocity of the phase fronts is called phase velocity

p

2 cv

with as the vacuum wavelength and c as the speed of light in vacuum

Group velocity

The propagation velocity of the energy is called group velocity

g 2

2 cv

which can be calculated very accurately by the following approach:

Take two solutions at two different wavelength and

2 2

t ( ) ( ) ( ) ( ) 0k u

2 2

t ( ) ( ) ( ) ( ) 0k u

Do the same as done for deriving the orthogonality relation: multiply the first equation with ( )u and the second with ( )u and subtracting the results one

obtains

2 2 2 2

t t( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )k k u u u u u u

As before we integrate over the infinite waveguide cross section A and transform the right side into a line integral using Green’s law. Similar to the demonstration of the orthogonality of the individual modes above, we can show that the right side is equivalent to zero. Then the remaining left side reads as

2 2 2

2 2

( ) ( )( ) ( ) ( ) ( ) 4 ( ) ( )

A A

u u A u u A

Dividing by ( ) results in

2 2 2

2 2

( ) ( ) 4 ( ) ( )( ) ( ) ( ) ( )

A A

u u A u u A

Now we can introduce the limit or ( ) 0 and obtain


2 2 2

2

1 ( ) ( )( ) 4 ( )

( )A A

u A u A

Using this expression to substitute in our definition for the group

velocity we obtain

22

g 22

2

( )4 ( )

( )( )

A

A

u Acv

u A

Group velocity dispersion

The dependence of the group velocity on the wavelength is expressed by the Group velocity dispersion coefficient

g

1

vD

5.1.3 Cylinder symmetric waveguides

In particular for fibers the refractive index profile of the waveguide is very often cylinder symmetric

2( , ) ( ) ( )x y r n r

For fibers the index profile is then usually characterized by special parameters

Fiber parameter

co cl

1/ 22 22 a

V n n

with

a - core radius

con - maximal refractive index in the core

cln - constant refractive index in the cladding surrounding the core

Normalized index difference

co cl

co

2 2

2

1

2

n n

n

Index profile

co

2 2 1 2 ( )n n f r

with

1

( )1

r af r

r a


Index dispersion

To describe the wavelength dependence of SiO2, which is the standard fiber material, often the Sellmaier equation is used

2

2

21

( ) 1j

i

i i

an

b C

which takes into account electronic transitions in the UV and molecular vibrations in the IR by

2

14

1

10

2

14

1

10

2

1.53714322m

1.968198 10

1.4108754 10

1.790244 10

1.572513 10

C

a

a

b

b

5.1.4 Bessel’s differential equation

Using the Laplace operator in cylindrical coordinates

2

t 2 2

1 1( , )

u uu r r

r r r r

the scalar Helmholtz equation reads as

2

2 2

2 2

1 1( ) 0

u ur k r u

r r r r

Applying the product Ansatz

( , ) ( ) ( )ru r u r u

the scalar cylindrical Helmholtz equation reads as

222 2

2 2 2

1 1( ) 0r r

r r

uu uu u u k r u u

r r r r

multiplying with 2 /( )rr u u the equation can be separate in terms depending

only on r and terms depending only on

22 22 2 2

2 2

1( )r r

r r

ur u r ur k r

u r u r u

Since both sides of the equations are independent, the equation can be

solved non-trivially only when both sides are equivalent to a constant 2m ,

which splits the equation in two independent equations

2 22 2

2 2

2

2

2

1( ) 0

0

r rr

u u mk r u

r r r r

um u


The r -equation is Bessel’s differential equation, the solution of which will be discussed in detail below.

The -equation can be solved analytically very easily as

0cos sin cos( )u A m B m C m .

To obtain a continuous field the so-called azimuthal mode constant m must obey

m

Otherwise unphysical azimuthal discontinuities of the field would result.

Bessel’s differential equation can be simplified further by introducing the dimensionless modal parameters

co

1/ 22 2 2U a k n in the core

cl

1/ 22 2 2W a k n in the cladding

which are connected by

2 2 2V U W

With co

2 2 1 2 ( )n n f r we can express 2 2 2k n as

co co

2 2 2 2 2 2 2 22 ( )k n k n k n f r and hence 2 2

2 2 2

2

( )U V f rk n

a

When introducing additionally the normalized radius /r a we can rewrite

Bessel’s differential equation for ( ) ( )m rY u r in its final form

2 2

2 2

2 2

( ) 1 ( )( ) ( ) 0m m

m

Y Y mU V f Y

5.1.5 Analytical solutions of Bessel’s differential equation

Bessel’s differential equation can be solved analytically for some specific profiles ( )f or in segments where ( )f is constant. Even though we are

interested in the numeric solution of the general problem, we need particular analytic solutions to derive some general properties of the numeric solution, as e.g. the boundary conditions.

Introducing

2 2 2( ) ( )G U V f

the analytic solution in an interval with constant refractive index (here taken as the index in the center of the core ( ) (0) const.f f ) reads as

( ) ( (0))m mY AJ G

with A being an arbitrary scaling constant and ( )mJ being the Bessel

function of the first kind and mth order.

Besides ( )mJ there are other mathematical solutions of Bessel's differential

equation. However these other solutions can be neglected in the


representation of the physical fields in the core area. They can be ruled out by physical reasoning due to their infinite value at the origin or their unboundednes for growing arguments, which would result in infinite fields

( )mY for growing radius .

Similarly the solution in the cladding, where we take ( ) 1f , is based on the

modified Hankel function

2( ) ( )m mY BK W where 1

with B being an arbitrary constant.

Using the requirement that the general solution and its derivative need to be continuous throughout the computed domain, we can derive the following

conditions at the core-cladding interface ( 1 ), which can be written as a set

of linear equations using the arbitrary integration constants A and B

continuity condition for the field: 2( 1) ( ( 1)) 0m mAY BK W

continuity condition for the 1st derivative: 2( 1) ( ( 1)) 0m mAY BWK W

The determinant of this set of linear equations must vanish to obtain unambiguous solutions

2 2 2( 1) ( ( 1)) ( 1) ( ( 1)) ( ) 0m m m mWY K W Y K W F U

This is very elegant, since it transforms the physical boundary condition, which says that the field has to decay to zero at an infinite radius ( ), to

a finite radius at the core cladding interface ( 1 ).

This characteristic equation 2( )F U is only solved for discrete values of the

parameters W or U (W and U are connected) which selects a finite number

of modes. Usually this equation is solved as a function of 2U which can vary

in the interval

2 20 U V .

Example of 2( )F U in the interval (0, )U V for a multimode fiber with large

core and step index profile with azimuthal mode number 0m .


The modes are often categorized in their linearly polarized form as LPmp -

modes, where the index m corresponds to the azimuthal mode number and index p selects the modes according to order of individual solutions of the

above equation. p is therefore often called the radial mode number.

LPmp -modes: m - azimuthal mode number 0,m m

p - radial mode number 1,p p

5.1.6 Specifying the numerical problem

Here we would like to solve Bessel's differential equation for inhomogeneous index profiles using numerical methods

2 2

2 2

2 2

( ) 1 ( )( ) ( ) 0m m

m

Y Y mU V f Y

The numerical problem consists in finding discrete modal solutions for a given azimuthal mode index m with an unknown eigenvalue U .

First we need to define the boundary conditions. Starting from the core's

center at 0 we assume that solution ( )Y for 0 behaves like the

Bessel function ( ) ( (0))m mY AJ G of the analytic solution in the previous

section. Since the Bessel function can be expressed as a series like

2

0

( 1)( )

!( 1) 2

n mn

m

n

Jn n m

we can introduce an approximation for small arguments ( 0 ) and use it as

an ansatz function:

( ) ( )m

mY X

where ( )X is an analytic function which is finite valued for finite arguments

. From this ansatz we can also determine the derivative of the solution as

( 1)( ) ( ) ( )m m

mY m X X

Hence the boundary conditions in the center read as

0 : (0) (0)

(0) (0)

1: (0) 0

(0) (0)

1: (0) 0

(0) 0

m Y X

Y X

m Y

Y X

m Y

Y

As was demonstrated in the “Analytic solution” section, the boundary conditions at infinity ( ) can be transformed to the position of the core

cladding interface as

2 2 2( ) ( 1) ( ( 1)) ( 1) ( ( 1)) 0m m m mF U WY K W Y K W


5.1.7 Solving the second order singularity

The governing Bessel’s differential equation

2 2

2 2

2 2

( ) 1 ( )( ) ( ) 0m m

m

Y Y mU V f Y

possesses a second order singularity at 0 which makes it very hard to

integrate numerically.

Using again the ansatz function

( ) ( )m

m mY X

we obtain a new differential equation for ( )X as

2

2 2

2

( ) 2 1 ( )( ) ( ) 0m m

m

X m XU V f X

(24)

which possesses only an ordinary 1st order singularity at 0 .

To use this equation for ( )X instead of the previous equation for ( )Y , for

identifying the modes, the boundary conditions, which we had derived above for ( )mY must be transformed to boundary conditions for ( )X .

Since ( )X is an analytic function it can be expressed as a power series

following the series representation of the Bessel function as

0

( ) n

m n

n

X c

Hence the new boundary conditions in the center of the core, i.e. at 0 , are

independent of m

0(0)X c and (0) 0X

Similarly the boundary condition at the core-cladding interface, i.e. at 1 ,

becomes

2 ( )( ) ( 1) ( 1) 0

( )

mm m

m

K WF U W m X X

K W

with 2( )F U being a characteristic function, which has to be solved for its

roots.

However, we should keep in mind that, once found, the new solution ( )mX

has to be transformed back to ( )mY to derive some physical meaning.

5.1.8 Numerical integration methods

Up to now the problem is a boundary value problem which would require e.g. a relaxation method for its numerical solution in order to find the desired

unknown parameter 2U .

Using the shooting method the problem can be transformed into an initial

value problem starting at 0 which depends on the parameter 2U . Different


2U result in different solutions at 1 and hence also different 2( )F U . The

aim must be to find parameters *U U with

*2( ) 0F U

Then the solutions of the initial value problem would also solve the boundary value problem.

When selecting a specific method for the numerical integration of the differen-tial equation (24) for ( )mX we have to take into account the following

properties of the problem:

At 0 the coefficient in front of function ( ) /mX has a first-order

singularity, which depending on m can lead to instabilities of the numeric solution.

The profile function ( )f can possess discontinuities in the integration

interval, which can reduce the order of the integration methods.

The coefficient in front of ( )mX can go through zero in the integration

interval, which can change the continuity properties of the solution.

Hence to ensure stability of the numerical method one should refer to implicit integration methods, since they are often unconditionally more stable than explicit integration methods.

Furthermore we should reformulate the problem in a notation which is compa-tible to standard numerical schemes. Hence, we transform the second order differential equation into a set of coupled first order differential equations with

( )( )

( )( )( )

I

m

IIm

XZ

XZ

Z

as

2 2

( )( )

( ) 2 1( ) ( ) ( )

III

III II

ZZ

Z mV f U Z Z

In matrix representation the differential equation reads as

2 2

0 1( )

( ) ( , )2 1( ) jm

V f U

ZZ L Z (25)

Now we have to discretize the integration interval [0,1] , e.g. with very

simple equidistant spacing h as:

1

| , 0(1) ,h j j jh j N hN

then the discrete approximation of the solution can be written as


( ),j j j h Z Z

Implicit Runge-Kutta-Midpoint-Method of 2nd order

The midpoint-method, which is a second order accurate Runge-Kutta-method, has the following Runge-Kutta parameter table (see e.g. the explanation of the Butcher tableau in the Wikipedia article on Runge-Kutta methods)

0.5 0.5

1 1

This particular Runge-Kutta method is a single step implicit method. Using this method a single integration step in the solution of an differential equation reads like

1 ( , , )j j j jh h Z Z Φ Z , ( 0,1,..., )j N

where ( , , )j j hΦ Z approximates the ascend of the solution as

1

( , , ) ( , , ),2 2

j j j j j j

hh h h

Φ Z L Z Φ Z

when L is the matrix operator, describing the differential equation.

Using this method to solve our particular problem, where L is defined as in equation (25), gives for the approximation of the ascend of the solution

12 2

2

0 1

22 1( , , )2

22

I I

j

j j jII I

jj

hZ

h mh V f Uhh

Z

Φ Z .

Since we are dealing with a linear problem we can avoid the otherwise necessary iterative numerical solution of the above problem and solve it explicitly

2 2

1 22 2

2 2 2 2

1 22

2 2

2 11

2 4 2

2

2 1

2 2 2

2

2 11

2 4 2

2

I II

j j j

I I

j j

j

j

I II

j j j j

jII II

j j

j

j

h hV f U Z Z

Z Z hh m h h

V f Uh

h h h mV f U Z V f U Z

h

Z Z hh m h h

V fh

2U

The solution accuracy of the method is second order in h , as was already said in the introduction of the midpoint method. The discretization grid

h


should be chosen such that the discontinuity points D of the profile function

( )f coincide with grid points of h

D h

This ensures that the differential equation is only evaluated in continuous intervals at

D 0.5x h . Consequently the second order solution accuracy is

preserved also for discontinuous index profiles ( )f .

Implicit Runge-Kutta-Method of 4th order

To improve the accuracy one could refer to more accurate higher order numerical integration schemes. One such example is the Gaussian type fourth-order method. We are not describing the details here, but give only the respective Butcher scheme

1 11 12

2 21 11

1 2

1 3 1 1 3

2 6 2 2 6

1 3 1 3 1

2 6 2 6 4

1 11

2 2

5.1.9 Eigenvalue search

Since we have transformed the original boundary value problem into an initial

value problem using a shooting method, the search for the eigenvalues 2U is

a simple search for the roots of 2( )F U for parameter m . However, for each

calculation of 2( )F U we have to compute the full differential equation

problem in the interval [0,1] .

The eigenvalue interval for guided modes is restricted to

2 20 U V

containing a finite number of roots. To find all roots the given interval can be

scanned for sign changes of 2( )F U with a fixed step size 2U . The resulting

subintervals can then be refined using e.g. a converging secant method.


5.1.10 Calculation examples

Example of 2( )F U in the interval (0, )U V . This particular example was

calculated for a fiber with a step index profile having a large diameter. This

results in a large fiber parameter V with a value around 52. The calculation was

done for 0m , which restricts to the modes having no azimuthal dependence of the fields.


Radius-dependent field distribution for azimuthal index m=0 (=804nm)



Examples of calculated radial dependence of the mode field for a ring index

fiber for different azimuthal mode indices m , which means different numbers of

oscillation periods of the fields around the azimuthal angle . The particular

ring index fiber which was simulated in these graphs, is similar to a strongly guiding step index fiber with a radius of 30 µm. Additionally, it has a ring-shaped higher index area extending from a radius of 21 µm to 25 µm. The wavelength for the calculation was set to 804 nm.


Examples of calculated intensity distribution of the mode field for a ring index fiber as in the previous figure. Here the full radial and azimuthal dependence of the intensity is shown for two particular modes at two different wavelengths.


6. Fourier Modal Method for periodic systems

Aim

rigorous solution of the diffraction of light at periodic structures (gratings)

Methods

A) thin element approximation

B) formulating the eigenvalue problem in the Fourier space and deriving boundary conditions imposed onto the modes in order to be matched properly to the homogenous space

Examples of diffraction gratings

1D grating 2D grating (biperiodic)

Gratings are characterized by a periodic variation of the dielectric constant in x and y direction with periods x and

y

Illustration of the problem by understanding a grating as a periodic waveguide array embedded between two homogenous media

Dielectric waveguide with rectangular cross section Periodically arranged dielectric waveguide with rectangular cross section


6.1 Formulation of the problem in 2D for TE

Rigorously calculated field distribution around a binary grating for plane wave TE excitation and sketch of the diffraction problem

Incident field

inc, 0 Iexp (sin cos )yE ik n x z with being the angle of incidence

Diffracted field in different diffraction orders i

I, inc, I,exp ( )y y i xi zi

i

E E R i k x k z for 0z

II, II,exp ( ( ))y i xi zi

i

E T i k x k z d for z d

Remaining question: What is the direction and strength of the reflected and transmitted amplitudes?

The wave number of each wave is given by

2 2

0

2x zk n k k

The incident wave is characterized by

I

0

2coszk n

I, I

0

2sinxk n

I

The grating provides a momentum for each diffraction order of

2

xik i

with i being the index of the diffraction order

Hence the diffracted waves are characterized by

0 I

2sinxik i k n

,

2 2 2

I/II, 0 I/IIzi xik k n k

The propagation directions of the diffraction orders are only determined by the grating period , the angle of incidence and the refractive indices

I,IIn of the surrounding media.

The character of the diffraction orders is determined by 2 2 2

0 I/IIxik k n propagating wave

2 2 2

0 I/IIxik k n evanescent field


Illustration of the diffraction orders on the Ewald sphere with:

red xik , green I,zik , blue ,II zik . This illustration is for the simplifying case of

normal incidence.

Illustration of the diffraction orders on the Ewald sphere for oblique incidence.

6.2 Scalar theory for thin elements The field after the structure is given by the incident field multiplied by the transfer function

T 0, , ,U x y T x y U x y with ˆU E y

kx

kz

incident

01

2

-1-2

-3

01

2

3

-1-2

-3

-4

nI

nII

G

kx0

x

z


, , exp ,T x y T x y i x y

For a pure phase grating the amplitude transmission function equals unity and the phase is given by

0 II I( ) ( )x k n n f x

Hence for perpendicular plane wave illumination the field after the structure is given by

exp i x

The amplitudes of the plane waves in the direction of the diffraction orders are given by the Fourier transform of the transmission function

0 II I0

exp ( ) exp expn xnT FT i x ik n n f x ik x dx

Physical interpretation of diffraction efficiency

The diffraction efficiency corresponds to the energy transferred into a diffraction order normalized to the incident energy (given by the Poynting vector) S E H

Introducing the plane waves into the equation leads to

2II,I

T,

II 0

zi

n n

kpT

p k

with

I/II 1p for TE and I/II I/IIp for TM

2I,

R,

0

zi

n n

kR

k

Energy must be conserved for loss-less materials!


Examples for the validity of the thin element approximation

Rigorous simulation of wave propagation through a sinusoidal long period

grating ( 10 ), which is illuminated with a plane wave. Since the grating

period is much larger than and the depth of the grating is much smaller than its period, diffraction effects within the grating have only a negligible influence. This can be verified since no transverse redistribution of the wave's intensity within the volume of the grating is observed in the rigorous simulation. Hence the thin element approach is justified.

Rigorous simulation of wave propagation through a sinusoidal short period

grating ( 2.1 ), which is illuminated with a plane wave. Since the grating

period is in the range of and the depth of the grating is also in the order of its period, diffraction effects within the grating cannot be neglected. This can be verified since a transverse redistribution of the wave's intensity within the volume of the grating is observed in the rigorous simulation. This is a clear signature of diffractive effects inside the grating volume. Hence the thin element approach would fail.

Amplitude Phase

nI=1

nI=2

nI=1

nI=2

nI=1

nI=2

nI=1

nI=2

Amplitude Phase


Limitations of the scalar method

Phase of the field directly after a sinusoidal grating with different grating periods

(grating profile with depth h , I 1n ,

II 2n , TE illumination).

Scalar theory fails if the period gets comparable to the wavelength and the thickness of the grating is significant.

Proper description of the field inside the grating (0 z d ) is necessary.

Maxwell’s equations must be solved also inside the grating!

6.3 Rigorous grating solver

Ansatz

expanding the fields in the different domains in terms of modes (mode = eigensolution of a wave equation)

Modes are just plane waves in the incident and transmitted region.

In grating region the modes are waveguide modes propagating along z , that fulfill the periodic boundary conditions in the transverse direction.

The mode profile must be determined numerically (analytical solutions can be used in special cases).

deriving the proper boundary conditions and solving for the unknown amplitudes of each mode

comparable to the plane interface problem

tangential electric and magnetic field are continuous + transversal wavevector component

Outline of the algorithm

calculate all wave vector components of interest

Fourier transforming the permittivity distribution

calculating the eigenvalues and the eigenvectors of the eigenmodes as supported by the structure in Fourier space


solving the system of linear equations that provide the amplitudes of all relevant field components

calculating derived quantities of interest, such as diffraction efficiency and/or field distributions in the plane of interest

Comment: Note that the algorithm thus far requires invariance of the structure in the propagation direction. However this restriction can be overcome by decomposing a longitudinally varying structure into a system of thin layers, where in each individual layer the structure can be assumed to be longitudinally invariant. But keep in mind that the structures in each layer must be periodic sharing the same period.

6.3.1 Calculation of eigenmodes in Fourier space

starting from Maxwell’s equations for a time harmonic field with the magnetic field scaled by the impedance

0 0( ) ( ) ( ) ( ) ( )ik ik E r H r H r r E r

plugging equations together and eliminating the z-component of the field (unambiguously determined by the divergence equations)

0 0 ( )y z x y x zE E ik H H H ik Ez y x y

r

results in the following dynamic equations

0

0

1 1y y x xE H H ik H

z ik y x y

0

0

1 1x y x yE H H ik H

z ik x x y

0

0

1y y x xH E E ik E

z ik y x y

0

0

1x y x yH E E ik E

z ik x x y

(For details see Li et al., Phys. Rev. E 67, 046607 (2003).)

Fourier expansion of all quantities

To solve the eigenvalue problem derived from the above equations all quantities are transformed into Fourier space: ( ) ( )E r E k , ( ) ( )H r H k ,

( ) ( ) r k , and also 1 1( ) ( ) r k .


, ,

, ,

1 1

( ) exp

( ) exp

( ) exp

( ) exp

ij ij x ij y

ij

ij ij x ij y

ij

ij ij

ij

ij ij

ij

z i k x k y

z i k x k y

i

i

E r E

H r H

r G r

r G r

with , , 0 0 1 2( , ) ( , )ij ij x ij y x yk k k k i j k b b and

1 2ij i j G b b

The integers i , j run from to , which has to be truncated for numerical

calculations.

Now these Fourier expansions are inserted into the dynamic field equations, here for the following example

0

0

1x y x yH E E ik E

z ik x x y

the individual terms are substituted

, , ,expx mn x m x n y

mn

H H i k x k yz z

, , ,

, , , ,

exp

exp

y mn y m x n y

mn

m x mn y m x n y

mn

E E i k x k yx x

ik E i k x k y

, , , , ,exp expy pq p x q y mn y m x n y

pq mn

E i G x G y E i k x k y

now all terms are multiplied with , ,exp i x j yi k x k y

and integrated over x

and y across one unit cell as

, , , ,exp expi x j y m x n y im jni k x k y i k x k y dxdy

This results in the following example

Real space Reciprocal space


, , , , , ,

,

exp expmn x i x i y m x n y mn x im jn

mn mn

ij x

H i k x k y i k x k y x y Hz z

Hz

which has to be done with all terms

, , , , , ,

, , , , , ,

exp exp exp

exp ( ) ( )

m x n y p x q y i x j y

m x p x i x n y q y j y i m p j n q

i k x k y i G x G y i k x k y dxdy

i G G G x G G G y dxdy

This results in the following set of equations

, 1

, , , , , , 0 ,

0

, 1

, , , , , , 0 ,

0

,

, , , , , 0 , ,

0

,

,

0

ij x

ij x i m j n m x mn y n y mn x ij y

mn

ij y

ij y i m j n m x mn y n y mn x ij x

mn

ij x

ij x im jn m x mn y n y mn x i m j n mn y

mn mn

ij y

ij y

ikE k H k H ik H

z k

ikE k H k H ik H

z k

ikH k E k E ik E

z k

ikH

z k

, , , , 0 , ,im jn m x mn y n y mn x i m j n mn x

mn mn

k E k E ik E

.

These equations can be conveniently written in matrix form as

1 2z z

E TH H T E

where

, ,

, ,

ij x ij x

ij y ij y

E H

E H

E H

and

1 1 2

, ; , , ; , 0 ;;

1 1 2 1

, ; , 0 ; , ; ,0

2 1

, ; , , ; , 0 ;;

2 2 1

, ; , 0 ;0

ij x ij mn mn y ij x ij mn mn x ij mnij mn

ij y ij mn mn y ij mn ij y ij mn mn x

ij x ij mn mn y ij x ij mn mn x ij mnij mn

ij y ij mn mn y ij mn i

k k k k kiT

k k k k kk

k k k k kiT

k k k kk

, ; ,j y ij mn mn xk

This set of two coupled 1st order differential equations for E and H can be combined into a single 2nd order equation for E

2

1 22z

E TT E


For z-invariant geometries this differential equation is transformed into an eigenvalue equation using the ansatz

(z) exp( )i z E E

This results in the eigenvalue equation which has to be solved

2

1 20 TT E E

Eigenvalues

This eigenvalue equation can be solved for its eigenvalues with standard routines.

The infinite Fourier expansions are truncated up to 0 , ,0, ,N N N for

numerical solution.

This results in 02N eigenvalues 2

i with 0i which correspond to

forward or backward propagating solutions.

Eigenvectors

Eigenvectors are associated with the eigenvalues and are given in a matrix

aS with size 0 02 2N N .

They provide the Fourier components of the guided eigenmodes.

Eigenfields are given by the forward and backward propagating eigenmode

a a a

E S E E

with

, ,( ) ( )a a i a a iE z E z

E E

where

, ,( ) exp ( ) expa i i i a i i iE z E i z E z E i z

Here iE and

iE are unknown (free) amplitudes, since the field amplitudes

can be chosen arbitrarily.

(For details see Li et al., Phys. Rev. E 67, 046607 (2003).)

The magnetic fields can be determined from the 1st order differential equations as

1

1

1

1

1

1

a a a

a a a

a a a

z

z

i

H T E

T S E E

T S E E

T E E


with 1

1a ai T T S

Examples of eigenfields

Eigenmodes propagate in the waveguide normal to this plane (periodic boundaries)

6.3.2 Explicit derivation for 2D problems

E.g. for TE polarization

0

0

1y y x xH E E ik E

z ik y x y

2

02

0

10 y x xH E ik E

x z ik y

and

TE

TM


0

0

1 1x y x yE H H ik H

z ik x x y

00 x yE ik Hx z

We substitute the fields by their Fourier series representations.

We start from

0x yE ik Hz

and insert

, ,expx m x m y

m

E E ik yz z

and 0 0 , ,expy m y m y

m

ik H ik H ik y

We get

, , 0 , ,exp expm x m y m y m y

m m

E ik y ik H ik yz

Multiplying with the periodic function ,exp j yik y results in

, , , 0 , , ,exp exp exp expm x m y j y m y m y j y

m m

E ik y ik y ik H ik y ik yz

Integrating over y as

, , , 0 , , ,exp exp exp expm x m y j y m y m y j y

m m

E y ik y ik y ik H y ik y ik yz

gives

, , 0 , ,m x m j m y m j

m m

E ik Hz

, 0 ,j x j yE ik H

z

which are individual equations for the different Fourier components of the fields, as

0, 0 0,

1, 0 1,

, 0 ,

x y

x y

n x n y

E ik Hz

E ik Hz

E ik Hz

0,0, 0

1,1, 0

,, 0

0 0

0 0

0 0

yx

yx

n yn x

HE ik

HE ik

z

HE ik

which in matrix notation reads as

1x yz

E TH with ,

1 0 ,

j n

j nT ik

The other equation we treat equivalently


2

02

0

1y x xH E ik E

z ik y

We do this term by term separately and the recompose the entire equation.

By substituting

, ,expy m y m y

m

H H ik yz z

and, as above, multiplying with the periodic function and integrating over y

we get directly

,i yHz

We apply the same procedure to the second term by substituting

2 2

2

, , , , ,2 2

0 0 0

exp expx m x m y m y m x m y

m m

i i iE E ik y k E ik y

k y k y k

results in

2

, ,

0

i y i x

ik E

k

For the third term

0 0 , , 0 , ,exp exp expx m m n x n y m n x m n y

m n m n

ik E ik iG y E ik y ik E iG y ik y


16


Eigenvalue problem in 2D

(Li et al., Phys. Rev. E, Vol. 67, 046607 2003)

Field expansion inside the grating

The field inside the periodic structure can be written down unambiguously as (here at the example of the x-component of the electric field)

This is the plane wave expansion of the lth eigenmode.

xmnlE are the eigenvectors from the eigenvalue problem.

exp( )li z describes the dynamics of the forward/backward propagating

eigenmodes.

lA and

lB are the amplitudes of the eigenmodes, which are up to now

unknown and which have to be determined from boundary conditions.

The same expansion holds also for the other components of the field

17

TE:

TM:


Example

1D periodic waveguide array

All possible eigenmodes are calculated using the method

shown is the electric field upon propagating through a waveguide in z-direction

Fundamental mode Higher order guided mode

Guided mode with field localization Evanescent mode between the waveguides


6.3.2.1 Solving the reflection/transmission problem

see separate presentation file


7. Finite Element Method (FEM) (Material is derived from "A Tutorial on the Finite Element Method" by Arashi Mafi.)

FEM is a general purpose tool for finding approximate numerical solution to partial differential equations as well as integral equations.

History: Alexander Hrennikoff (1941) and Richard Courant (1942)

(solution of problems on elasticity and structural analysis in aeronautics)

Solution strategy: reduction of partial differential equation to a set of linear equations for steady state problems or to a set of ordinary differential equation for time dependent problems.

Essential ansatz: discretization of a continuous computational domain into discrete sub-domains, which are called elements

Here: demonstration of the basic ideas at the example of a 1D scalar FEM

Professional use of the method: commercial programs are available that employ this computational technique:

Comsol – multiphysics simulations

JCMwave – electromagnetic problems

Strength of the method:

ability to retain various physical models in the simulation simultaneously

takes advantage of a large computational toolbox for the solution of linear equations

applicable to accurate solution of problems involving strongly different scales and where symmetries can't be exploited

Literature:

W. G. Strang and G. J. Fix, "An Analysis of the Finite Element Method," Wellesley Cambridge Press (1973).

O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu, "The Finite Element Method: Its Basis and Fundamentals," Butterworth-Heinemann (2005).

S. Humphries, "Finite-element Methods for Electromagnetics," CRC Press (1997) now freely available at http://www.fieldp.com/femethods.html

7.1 The basic set up The Finite Element Method (FEM) is a weighted residual method that uses compactly-supported basis functions.

Comparison to other methods

Before: The finite difference method approximates an operator (e.g., the derivative) and solves a problem on a set of points (the grid).


Now: Finite Element Method (FEM) uses exact operators but approximates the solution basis functions. Also, FEM solves a problem on the interiors of grid cells (and optionally on the gridpoints as well).

Furthermore one should distinquish between spectral methods, which use global basis functions to approximate a solution across the entire domain and FEM, which uses compact basis functions to approximate a solution on individual elements.

Different formulations of FEM

Strong form

Weak form

Galerkin approximation

Matrix form

Introducing to the FEM method by an example for a 1D problem

Differential equation: 2 2( ) ( ) ( ) 0xR x x x

Global solution interval: / 2, / 2

Neumann boundaries: ,

2 2

( ) 0x xx

Analytical solution: ( ) sin( )x A x with n (integer)

Strict solution: ( )R x is everywhere zero

For numerical solution the global computational domain is partitioned into local segments, the so-called elements:

Position of the element's boundaries: ix (nodes)

Value of the function at the nodes: iU

Solution to the PDE is the (linear/quadratic/…) interpolation between the nodes


7.2 Global vs. local labeling

Local index: The two nodes of the element e are identified as 1

ex and 2

ex .

This labeling is used when discussing a single node.

Global index: The element e is identified in the entire computational domain

as 1,e ex x .

This labeling is used when discussing the entire assemblage of elements.

7.3 Weak formulation Expression of the original problem in a way that the solution has to fulfill less requirements, e.g. smoothness.

The weak form is a variational statement of the problem in which one integrates against a test function. The choice of test function is up to the specific problem. This relaxes the problem since instead of finding an exact solution everywhere, one searches for a solution that satisfies the original problem (strong form) on average over the solution domain.

A solution of the strong form will also satisfy the weak form, but not vice versa.

How is it introduced mathematically?

The global interval / 2, / 2 is partitioned into small local elements.

Introducing a "weight function" ( )i

eQ x for each element such that instead of 2 2( ) ( ) ( ) 0xR x x x in the entire computation domain on requires

2

1

( ) ( ) 0e

e

x

i

e

x

Q x R x x

Inserting 2 2( ) ( ) ( )xR x x x and integration by parts gives

2

1

2( ) ( ) ( ) ( ) 0e

e

x

i i

x e x e

x

Q x x Q x x x (26)

Discretization

Partitioning of the global domain / 2, / 2 into N local elements of equal

size with

Ue-1=U1e

Ue=U2e

xe-1=x1e xe=x2

e


2ex e

N

Linear approximation of the function ( )x on each local element

in element( )

0 elsewhere

e e

e

p x q ex

Globally the solution is expressed as

1

( ) ( )N

e

e

x x

( )e x has to satisfy 1 1( )e e ex U and

2 2( )e e ex U which determines the

parameters of the local element as

ascent 2 1

2 1

e ee

e e

U Up

x x

and starting value

1 2 1 1

2 1

e e e ee

e e

U x U xq

x x

However, please note that the values eU of the final solution are still

unknown.

Now each local element ( )e x can be rearranged to

2

1

( ) U W ( )i i

e e e

i

x x

with the solution independent coefficients

21

2 1( ) e

e

e e

x xW x

x x

and

12

2 1( ) e

e

e e

x xW x

x x

(in element e and otherwise 0 )

Remarks:

This expression would correspond to the Lagrange linear

polynomials for ( )i j

e e ijW x .

The coefficients 1

eW and 2

eW just depend on the discretization.

Consequently the global solution ( )x can be written as

2

1 1

( ) U W ( )N

i i

e e

e i

x x

where we have to solve for the unknown i

eU .

By choosing the weighting function to be ( ) ( )i i

e eQ x W x , inserting everything

into equation (26), and puling the summations out of the integral we get

1

22

1 1

U ( ) ( ) ( ) ( ) 0e

e

xNj i j i j

e x e x m e m

m j x

W x W x W x W x x


which gives for each individual element e an individual equation which has to be fulfilled

1 1

2 22

1 1

U ( ) ( ) U ( ) ( )e e

e e

x x

j i j j i j

e x e x e e e e

j jx x

W x W x x W x W x x

This expression can be reformulated by introducing

1

( ) ( )e

e

x

ij i j

e x e x e

x

A W x W x x

and 1

( ) ( )e

e

x

ij i j

e e e

x

B W x W x x

resulting in

2ij j ij j

e e e eA U B U

which is only related to the single local element e .

Assemblage of global system

The solution is partially redundant since 2 1

1e eU U

The final global assembly of the local elements removes this ambiguity.

Example: 3N (Three elements with three matrix equations):

2ij j ij j

e e e eA U B U with 1,2,3e

switch from local to global labeling scheme:

1 2 1 2 1 2

0 1 1 1 2 2 2 3 3 3, , ,U U U U U U U U U U

11 12 11 120 021 1 1 1

21 22 21 221 11 1 1 1

11 12 11 121 122 2 2 2

21 22 21 222 22 2 2 2

11 12 11 122 223 3 3 3

21 22 21 223 33 3 3 3

U UA A B B

U UA A B B

U UA A B B

U UA A B B

U UA A B B

U UA A B B

These are six equations with only four unknowns.

Remark: In this formulation of the problem the eigenvalue is not

an unknown. It must either be determined by the task or it must be numerically determined from the boundary conditions within the FEM algorithm.

Adding up equations containing the same unknowns:


21 22 2 21 22

1 0 1 1 1 0 1 1

11 12 2 11 12

2 1 2 2 2 1 2 2

21 22 11 12 2 21 11 22 12

1 0 1 2 1 2 2 1 0 2 1 1 2 2

A U A U B U B U

A U A U B U B U

A U A A U A U B U B B U B U

which can be summarized in matrix notation as:

2AU BU

with

11 12

1 1

21 22 11 12

1 1 2 2

21 22 11 12

2 2 3 3

21 22

3 3

0 0

0

0

0 0

A A

A A A A

A A A A

A A

A and

11 12

1 1

21 22 11 12

1 1 2 2

21 22 11 12

2 2 3 3

21 22

3 3

0 0

0

0

0 0

B B

B B B B

B B B B

B B

B

and

0 1 2 3, , ,U U U UU

The construction of the matrices A and B for N elements is straightforward as

2 1 0 0 0

1 4 1 0 0

3

0 1 4 1 0

0 0 1 4 1

0 0 0 1 2

AN

and

1 1 0 0 0

1 2 1 0 0

0 1 2 1 02

0 0 1 2 1

0 0 0 1 1

NB

7.4 Example Results

Example solutions obtained with Mathematica for 10N .


Problem specific mesh generation


8. Outlook

What is missing in this lecture?

Near-field to far-field transformation technique (and vice versa): Richards-Wolf integrals, Stratton-Chu integrals, or Surface integral techniques

Green's function methods

Finite Integration technique by employing the integral form of Maxwell’s equations

Freely available programs

MPB – Powerful tool for calculating photonic band structures http://ab-initio.mit.edu/mpb

MEEP – Parallel FDTD http://ab-initio.mit.edu/meep

Survey of various scattering codes available at http://www.iwt-bremen.de/vt/wriedt

Code for the Discrete Dipole Approximation http://ascl.net/ddscat.html

Code for the T-Matrix approach http://www.giss.nasa.gov/~crmim/t_matrix.html

Codes published under GNU are available, e.g. FDTD http://www.borg.umn.edu/toyfdtd

Commercial programs

Integrated simulation environments incorporating different methods http://www.rsoftdesign.com http://www.optiwave.com http://www.cst.com

Rigorous grating solver http://www.unigit.com http://www.gsolver.com/

Finite Element Method http://www.comsol.com http://www.jcmwave.com http://www.ansoft.com

Finite Difference Time Domain method http://www.lumerical.com

Multiple Multipole Method http://alphard.ethz.ch/hafner/MaX/max1.htm

by prof. thomas pertsch. - uni-jena.de · by prof. thomas pertsch. the course material in 2019...

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