Computational Photonics

Computational Photonics, Abbe School of Photonics, FSU Jena, Prof. T. Pertsch, 19.06.2012 1

Computational Photonics Prof. Thomas Pertsch

Abbe School of Photonics Friedrich-Schiller-Universität Jena

0. Introduction and Motivation ..................................................................... 3 0.1 Why computational photonics? ......................................................................... 3 0.2 Maxwell’s equations .......................................................................................... 4 0.2.1 Maxwell’s equations in time domain ................................................................. 4 0.2.2 Maxwell’s equations in frequency domain......................................................... 5 0.3 Basic numerical operations ............................................................................... 5 0.3.1 Differentiation .................................................................................................... 5 0.3.2 Integration ......................................................................................................... 6 0.3.3 Root Finding & Minimization/ Maximization ...................................................... 7 0.3.4 Linear systems of equations ........................................................................... 10 0.3.5 Eigenvalue problems ...................................................................................... 11 0.3.6 Discrete (Fast) Fourier transform - FFT .......................................................... 11 0.3.7 Ordinary differential equations (ODEs) ........................................................... 12

1. Matrix method for stratified media ......................................................... 14 1.1 Optical layer systems ...................................................................................... 14 1.2 Derivation of the transfer matrix ...................................................................... 15 1.3 Guided modes in layer systems ...................................................................... 19

2. Finite-difference method for waveguide modes ..................................... 22 2.1 Stationary solutions of the scalar Helmholtz equation .................................... 22 2.2 Matrix notation of the eigenvalue equation ..................................................... 23 2.3 Boundary conditions ....................................................................................... 25

3. Beam Propagation Method (BPM) ......................................................... 26 3.1 Categorization of Partial Differential Equation (PDE) problems ...................... 26 3.2 Slowly Varying Envelope Approximation (SVEA) ............................................ 28 3.3 Differential equations of BPM ......................................................................... 29 3.4 Semi-vector BPM ............................................................................................ 30 3.5 Scalar BPM ..................................................................................................... 31 3.6 Crank-Nicholson method ................................................................................ 31 3.7 Alternating Direction Implicit (ADI) .................................................................. 32 3.8 Boundary condition ......................................................................................... 32 3.8.1 Absorbing Boundary Conditions (ABC) ........................................................... 32 3.8.2 Transparent Boundary Condition (TBC).......................................................... 33 3.8.3 Perfectly matched layer boundaries (PML) ..................................................... 34 3.9 Conformal mapping regions ............................................................................ 35 3.10 Wide-angle BPM based on Padé operators .................................................... 37 3.10.1 Fresnel approximation – Padé 0th order .......................................................... 38 3.10.2 Wide angle (WA) approximation – Padé (1,1) ................................................. 39

4. Finite Difference Time Domain Method (FDTD)..................................... 40 4.1 Maxwell’s equations ........................................................................................ 40 4.2 1D problems ................................................................................................... 41 4.2.1 Solution with finite difference method in the time domain for Ez ..................... 42 4.2.2 Yee grid in 1D and Leapfrog time steps .......................................................... 44 4.3 3D problems ................................................................................................... 45 4.3.1 Yee grid in 3D ................................................................................................. 46 4.3.2 Physical interpretation..................................................................................... 47 4.3.3 Divergence-free nature of the Yee discretization ............................................ 48 4.3.4 Computational procedure ................................................................................ 49


4.4 Simplification to 2D problems ......................................................................... 50 4.5 Implementing light sources ............................................................................. 50 4.6 Relation between frequency and time domain ................................................ 51 4.7 Boundary conditions ....................................................................................... 53

5. Fiber waveguides .................................................................................. 54 5.1.1 The general eigenvalue problem for scalar fields ........................................... 54 5.1.2 Properties of guided modes ............................................................................ 54 5.1.3 Cylinder symmetric waveguides ..................................................................... 56 5.1.4 Bessel’s differential equation .......................................................................... 57 5.1.5 Analytical solutions of Bessel’s differential equation ....................................... 58 5.1.6 Specifying the numerical problem ................................................................... 59 5.1.7 Solving the second order singularity ............................................................... 60 5.1.8 Numerical integration methods ....................................................................... 60 5.1.9 Eigenvalue search .......................................................................................... 63 5.1.10 Calculation examples ...................................................................................... 63


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 light? • „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 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 tt t

t t t

∂ ∂= − = +

∂ ∂= ρ =

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] − ρext(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 tt t t= ε +

= µ +

D r E r P rB r H r M r

(2)

− P(r,t) dielectric polarization [As m-2] − M(r,t) magnetic polarization (magnetization) [Vs m-2] − 0ε permittivity of vacuum 2 1 12

0 0 0( c ) 8.85 10 As/Vm− −ε = µ = ⋅

− 0µ permeability of vacuum 120 4 10 Vs/Am−µ = π ⋅

in linear, local, isotropic Media in optics

00

( , ) ( , ) ( , ) ,

( , ) 0

t R t t t t

t

∞

′ ′= ε − ∂

=

∫P r r E r

M r (3)

with ( , )R t′r 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∞ ∞

−∞ −∞

= ω − ω ω ω = ωπ∫ ∫V r V r V r V r . (4)

Maxwell’s equations in frequency domain by FT it∂

− ω∂

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

i iω = ω ω ω = − ω ω

ω = ρ ω ω =

E r B r H r D rD r r B r

(5)

Matter equations in frequency domain

( , ) ( , )exp( )R t i t t∞

−∞

χ ω = ω ∂∫r r (6)

with ( , )χ ωr being the material´s susceptibility, which is connected to the dielectric constant ( , )ε ωr by ( , ) 1 ( , )ε ω = + χ ωr r )

0 0

0

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

( , ) 0 ( , ) ( , )

ω = ε χ ω ω → ω = ε ε ω ω

ω = → ω = µ ω

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

( ) ( ) ( )( ) limh

f x f x h f xf xx h→

∂ + −′= =∂

for finite h ( ) ( )( ) [ ( )]h

f x h f xf x D f xh

+ −′ ≈ =

or left-sided/backward ( ) ( )[ ( )]h

f x f x hD f xh

− −= .

or central operator ( ) ( )[ ( )]

2hf x h f x hD f x

h+ − −

=

higher order differentiation


22

( ) 2 ( ) ( )( ) [ ( )]hf 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 intervall the mean value of the function is approximated by

12

( )2i ii

hf x f f x+

≈ = +

resulting in an integral approximation 11 1 1

1 1 10 0 02 2 2

i

i

xN N N

i i ii i ix

b aA dx f h f fN

+− − −

+ + += = =

−≈ = =∑ ∑ ∑∫

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 xf x f x

−+

−

−= −

−

until

( ) ( ) ( )1 10i i if x f x f x−− ≤ ε with ε determined by the desired accuracy

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


Secant method Codeschnipsel 11 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

+ λ = + λ

+ λ = + λ

+ λ = + λ

until no further improvement can be obtained along any direction. Choosing directions along steepest descent (opposite to gradient)

( )1i i iiP P f P+ = − λ ∇

mit ( )( )( )min i ii tf P t f P

∈λ = − ∇

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 ba x a x a x a x ba 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 aa a a

a a a a

=

,

1

2

...

m

bb

b

b

=

.

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

1

n

i ij j ij

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

ˆj jAx b=

with 1,...,j N= and 1jjb = , and all other 0 , then ( )11

ˆ ,..., 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 xΩ

ω⋅ = λ λ = → −λ =

characteristic polynomial ˆ ˆdet M E ( ) 0P − λ = λ = to have solutions of

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

0.3.6 Discrete (Fast) Fourier transform - FFT Starting from 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 Lk π π π= ⋅ ⋅ − with L N

a=

Definition of discrete Fourier transform (FT)

( ) ( )ikx

xf k a e f x−= ∑

with ( )2, 0La LN k n n Nπ= = ≤ < , x n a= ⋅

and inverse discrete Fourier transform (FT-1)

( ) ( )1 ikxL

kf 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

nn

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 ,..., 0nf 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 t G f f f tdt

df t G f f f tdt

df t G f f f tdt

df t G f f f tdt

=

=

=

=

Simple solution method

Forward Euler scheme problem:

( ) ( )( ),f t G f t t′ = transformed into difference equation:

( ) ( ) ( ) ( )( )'( ) ,tf t t f t

f t D f t G f t tt∆

+ ∆ −= = = ∆

with discretization t∆ : ( ) :nf f n t= ⋅∆ results in difference equation

( )1 ,n nn

f f G f tt

+ −=

∆

and recursion formula for the solution oft he 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 Optical layer systems Previously: transmission problems

− Bragg mirrors − chirped mirrors for dispersion compensation

z

x

y


− interferometers Today: guided modes

− multi layer waveguides − Bragg waveguides

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= − ω rE E

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

Decomposition in TE and TM fields:

TE 0

, 00

HE

H

= =

E Hx

TE y TE

z

TM 0

, 00

EH

E

= =

H Ex

TM y TM

z

1.2 Derivation of the transfer matrix Transition conditions 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

z

x

y


( )2

2 22ix i zk k

cω

= ε ω −

Field calculation of continuous components (TE)

( )2

2 22z2 2 ( ) 0

xi

i y

k

k E xx c

∂ ω

+ ε ω − = ∂

and z0

( ) ( )yiH x E x

x∂

= −ωµ ∂

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

= +

∂ ωµ = = − + ∂

Determination of C1, C2 by 1(0)yE C= and 20

iy xE k Cx∂

=∂

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

xy 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

ε ∂= +

ε ∂

∂ ∂= − +

ε ∂ ε ε ∂

Combined TE/TM:

( ) ( )

( ) ( )i

i

1( ) cos (0) sin (0)

( ) sin (0) cos (0)

i i

i

i i i

x xx

x x x

F x k x F k x Gk

G x k k x F k x G

= +α

= −α +

TE: 0 i, , 1y z yF E G i H Ex∂

= = ωµ = α =∂

TM: 0 i i, , 1/y z i yF H G i E Hx∂

= = − ωε = α α = ε∂

Summary matrix method:

kz

kx

kzkx

kz

kxkz

kx

kzkxkzkx


1 0 0

ˆm ( ) MN

i iiD

F F Fd

G G G=

= =

∏

with ( )( ) ( )

( ) ( )

ix1cos sin

msin cos

i ii

i i

k d k dkx

k k d k d

α= − α

ixix i

ix i ix ix

TE: i, , 1y yF E G Ex∂

= = α =∂

TM: i i, , 1/y i yF H G Hx∂

= = α α = ε∂

Reflection and transmission coefficients for fields

Transmission coefficient (T) und reflection coefficient (R)

in in

T RF FT RF F

= =

( )

( )

22 11 21 12

22 11 21 12

2

k M k M M k k MR

NkT

NN k M k M M k k M

α −α − + α α=

α=

= α + α + −α α

s sx c cx s sx c cx

s sx

s sx c cx s sx c cx

i

i

( )2

2 2s/c x s/c 0

0

2zk k

π= ε λ − λ

Energy flux defined by the normal component of the Poynting vectors sx

z

x

given: F(0), G(0), kz, εi, di

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

( ) ( )2

2 2i z

0

2,ix zk k k

πω = ε ω − λ

0

D

z

x

y z

x

yFin FR

FT

Fin FR

FT

substrate, s

cladding, c


x T x R

x in x in

s ss s

τ = ρ =

( )( )

2

2ReRe

c cx

s sx

R

kT

k

ρ =

ατ =

α

Field distribution Aim: Calculation of F(x) in the entire structure Starting point: known shape of transmitted vector

1

Tc cxcD

D

FFFF i kG

x

= =∂ αα ∂

now: 1TF ≡

1. invert structure (vector transforms into (1, -iαckcx)) 2. calculate 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

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


Calculating the real fields ( )z( , , ) Re ( )expx z t x ik z i t= − ω rE E

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

with TE: ( ) ( ) yE ex F x=

TM: ( ) ( ) yH ex F x=

1.3 Guided modes in layer systems

no y dependence, phase rotation in z direction

− waves propagating without diffraction − miniaturization of optics − optical signal communication

How can waves be bound by the layer system? principle mechanism is total internal reflection

Field distribution:

zx

yz

x

y Fin FR

FT

Fin FR

FT

zx

yz

x

y

zx

yz

x

y

Fin FR

FT

Fin FR

FT

Fin FR

FT

Fin FR

FT

Fin FR

FT

Fin FR

FT

z

x

nx

substrate

cladding

layer (system)


• plane wave in propagation direction: exp( )zik z

• evanescent wave in substrate and cladding 2

2,2 max ( )k

cω

> ε ωz s c

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

22 max ( )ii

kcω

< ε ωz

• general condition for guided waves 2

2s,c 2max ( ) max ( )ii

kc cω ω

ε ω < < ε ωz

Modes are resonances of the system 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 MFRF k M k M M k k M

α −α − + α α= =

α + α + −α αs sx c cx s sx c cxR

I s sx c cx s sx c cx

i

i

singularity: ( ) ( )22 11 21 12 0k M k M M k k Mα + α + −α α =s sx c cx s sx c cxi The problem of finding a guided mode is reduced to finding a root.

with: , ,k i k i= µ = µsx s cx c 2

2, ,2 ( ) 0k

cω

µ = − ε ω >s c z s c

TE,TM TE,TM TE,TM TE,TM22 11 21 12 0M M M Mµ + µ + + µ µα α α =αs c c ss c c s

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

12 14 16 18 20-100

0

100

200

300

kz [µm-1]

D


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.

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

0 0.5 1 1.5 20

10

20

30

40

kz/k0

Inte

nsity

abs(T)abs(R)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

6000

kz/k0

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/k0

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/k0

Ene

rgy

τρ

0 0.5 10

2

4

6

8

lateral distance [µm]

abs(

E) [

V2 /m2 ]

0 1 2 3-2

-1

0

1

2

lateral distance [µm]

abs(

E) [

V2 /m2 ]


2. Finite-difference method for waveguide modes Starting from the wave equation

2

2rot rot ( , ) (r, ) ( , )cω

ω = ε ω ωE r E r

Neglecting the divergence of the electric field div ( , ) 0ω =D r 0 ( )div ( , ) 0ε ε ω ω ≈E r We obtain the Helmholtz equation

2

2( , ) (r, ) ( , ) 0cω

ω + ε ω ω =E r E r∆

Neglecting the vectorial properties of the electric field scalar Helmholtz equation

2(r) ( , ) (r) 0v k v∆ + ω =r with 2

22( , ) ( , )k

cω

ω = ε ωr r

2.1 Stationary solutions of the scalar Helmholtz equation

Now we search for the stationary states (modes) of the problem with ( , ) ( , , )x yε ω = ε ωr

( )(r) ( , )expv u x y z= βi

which results in an eigenvalue equation for the propagation constant β (2) 2 2( , ) ( , , ) ( ) ( , ) 0u x y k x y u x y ∆ + ω −β ω =

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

2 2

2 22 2 ( , ) ( , , ) ( ) ( , ) 0u x y k x y u x y

x y ∂ ∂ + + ω −β ω = ∂ ∂

with the discrete Laplace operator for two dimensions

2 2

2 2

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

∂ ∂+

∂ ∂

Discretization of Laplace operator in two dimensions

-10-5

05

10 -10-5

05

10

0

50

100

150

200

Xy

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)


linear equation for each variable ,j kU

• quadratic area of size a x a • equidistant discretization of the area with N x N points

example:

2.2 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 times number variables in y

d2Udx2

xj,yk

≈ U(xj+1,yk) - 2 U(xj, yk) + U(xj-1, yk)h2

d2Udy2

xj,yk

≈ U(xj,yk+1) - 2 U(xj, yk) + U(xj, yk-1)h2

d2Udy2

xj,yk

≈ - 4 Uj,k + Uj+1,k + Uj-1,k + Uj,k+1 + U j,k-1

h2d2Udx2 + =( ∆ U )j,k

2 2( , , ) ( ) ( , ) 0k x y u x y + ω −β ω =

∆(x,y) U → (∆ U )j,k

a

N

.

.

11 2 3 N

- 4 U2,3 + U3,3 + U1,3 + U2,4 + U2,2

h2( ∆ U ) 2,3 =


Matrix equation: schematic picture

matrix: small number of non-zero values – ‚sparse matrix‘

N

.

.

11 . . N

j

k

U|∂ΩG = UG

.

.

= 0

U j,k

matrix

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.

1 1 j Nx iterates first

accounts for the 1D – problem (tridiagonal matrix)

occures for 2D problems


2.3 Boundary conditions

Example: Metal boundaries (Metal tube with boundaries i∂Ω

.

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)

∂ΩG

U|∂Ωi = Ui = const

N

.

.

11 . . N

grid with metal boundaries (U|∂ΩG = 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


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

− required invariance of the structure in the third dimension − What happens if light propagation occurs in a medium where the index

distribution weakly changes? • accurately model a 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 it 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 2 0f f f f fp q r s t u f vx x y y x y

∂ ∂ ∂ ∂ ∂+ + + + + ⋅ + =

∂ ∂ ∂ ∂ ∂ ∂ (8)

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

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

∂ ∂ρ

∂ ∂

2 2

2 2+ = Poisson equation

Boundary Value Problem (BVP) limited mainly by computing memory


Hyperbolic PDE

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

∂ ∂=

∂ ∂

2 22

2 2 1D wave equation

Parabolic PDE

( , ) ( , )u x t u x tDt x x

∂ ∂ ∂ ∂ ∂ ∂

= diffusion equation (IVP)

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


3.2 Slowly Varying Envelope Approximation (SVEA) Assumptions • a component of the optical electromagnetic field is a periodic (harmonic)

function of position • it changes most rapidly along the optical axis z (with a period on the order

of the optical wavelength λ )

Slowly Varying Envelope Approximation (SVEA) • replace the quickly varying component, Φ , with a slowly varying one, φ , as 0( , , ) ( , , )exp( )x y z x y z ikn zΦ = φ − with 2 /k = π λ • introduction of a reference index 0n light is travelling mostly parallel to

the z axis (paraxial approximation), and is monochromatic with wavelength λ requirements on the mesh to represent derivatives by finite differences are relaxed choose fewer mesh points higher speed of the calculation without compromising accuracy BPM: accurate calculations using step sizes z∆ > λ

SVEA Problems • if part of the light strongly deviates from the direction of the axis z

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 φ 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 zε

0

0 ( , , )i

i x y z∇× = − ωµ

∇× = ωε ε

E HH E

+ no charges 2( ) 0

0n∇ ⋅ =

∇ ⋅ =E

H

magnetic field can be eliminated by taking curl of curlE -equation 2 ( , , )k x y z∇×∇× = εE E with 0 0k = ω ε µ

using the vector identity 2( )∇×∇× =∇ ∇⋅ −∇ one gets the wave equation

2 2 ( , , ) ( )k x y z∇ + ε =∇ ∇ ⋅E E E since BPM is biased for the z -axis, it is natural to treat z -components of E and ∇ differently separately from the transverse components x and y

ˆt zE= +E E z and ˆt z∂

∇ =∇ +∂

z

the transverse component of the wave equation becomes

2

2 22 ( , , ) ( )t z

t t t t t tEk x y z

z z∂ ∂

∇ + + ε = ∇ ∇ ⋅ +∂ ∂

EE E E (9)

splitting also the divergence equation in the transverse and longitudinal components

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

x y z Ex y z E x y zz z

∂ε ∂∇ ⋅ ε + + ε =

∂ ∂E

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

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

Ex y z x y zz

∂∇ ⋅ ε + ε ≈

∂E

one can eliminate the longitudinal term in the right hand side of the wave equation (9)

2

2 22

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

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

z x y z ∂

∇ + + ε = ∇ ∇ ⋅ − ∇ ⋅ ε ∂ ε

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

2

2 22

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

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

z x y z ∂

∇ + + ε = −∇ ∇ ε ⋅ ∂ ε

EE E E (10)

up to now: 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 λ )


substituting SVEA ansatz into wave equation (10)

2

2 2 20 02

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

t tt t t t t tjkn k x y z n x y z

z z x y z ∂ ∂

− + ε − +∇ +∇ ∇ ε ⋅ = ∂ ∂ ε

e e e e e

if reference index 0n was chosen correctly, the first term will be much smaller than the second and hence can be neglected results in a first order equation in z collect the transverse 2. order operators on the right hand side

02 x xxx xy

yx yyy y

e eP Pjkn

P Pe ez ∂

= ∂

with the components of the operator 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

∂ ∂ ∂= ε ⋅ + + ε − ∂ ε ∂ ∂

∂ ∂ ∂= ε ⋅ + ∂ ε ∂ ∂ ∂

∂ ∂ ∂= ε ⋅ + ∂ ε ∂ ∂ ∂

∂ ∂ ∂= ε ⋅ + + ε − ∂ ε ∂ ∂

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 • reflections of light are neglected • in a finite differencing scheme the operator P is a large sparse matrix

3.4 Semi-vector BPM before: 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

ejkn P ez

∂=

∂

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

02 yyy y

ejkn 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∂ε ∂ and y∂ε ∂ term can be neglected operators commute xxP and yyP reduce to scalar operator

2 2

2 202 2 ( ( , , ) )P k x y z n

x y∂ ∂

= + + ε −∂ ∂

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

t 1 t 00

( ) exp ( )2

zz zjn 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

xxx

+ −α=

−α

The case α=0.5 • called Padé(1,1) approximation • accurate for small x small step size z∆ (higher order Padé

approximations can be used, see below) • generates the first three terms of the MacLaurin series expansion for

[ ]exp x leads to Crank-Nicholson scheme

1 00 0

( ) ( )4 4t t

z zz zjn k jn 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 α is called scheme parameter, for 0.5α = the method is stable and energy conserving


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 (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 ( ) 0nℑ > 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:

absabs abs

abs abs

1 exp expp pp 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 • radiation flux back into the problem region is prevented method: • assuming that the field in the vicinity of the virtual boundary consists of an

outgoing plane wave and does not include the reflected wave from the virtual boundary wave function for the left-traveling wave with the x -directed wave number xk is expressed as

( , ) ( )exp( )xx z A z jk xφ =

Derivation projecting the field onto the following lattice ( )p pxφ = φ

Nodes 1p = − and p m= are outside the computational window

hence the relation of the field at neighboring mesh points can be expressed as

k-Vektor

exp( )np xv v j k x p≈ ∆0

boundary


1

0

exp( )xjk x φ∆ =

φ with 1 0x x x∆ = −

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

1

0

1 lnxkj x

φ= ∆ φ

if ( ) 0xkℜ > plane wave is traveling outwards ( ) 0xkℜ < plane wave is traveling inwards since inward waves should not exist ( )xkℜ must be positive ( ) 0xkℜ ≥

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−

φ∆ =

φ 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 −−

φ φ= ∆ = → = φ φ − φ φ

φ φ

• trick: calculate 1nxk − one step z∆ before applying it to the boundary

takes into account that the wave energy is traveling outwards 1 1

1 1 0 00 n n n n− −−= φ φ − φ φ

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, replacing propagating (oscillating) waves by exponentially decaying waves

Technical description to absorb waves propagating in the x direction, the following transformation is applied to the wave equation: all x derivative / x∂ ∂ are replaced by

1( )1 i xx x

∂ ∂→

σ∂ ∂+ω

with ω being the angular frequency and σ some positive function of x Hence, propagating waves in +x direction (k>0) become attenuated

[ ]exp ( ) exp ( ) ( ') 'xki kx t i kx t x x − ω → −ω − σ ∂ ω ∫

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

( ') 'xix x x x→ + σ ∂

ω ∫ or equivalently (1 / )x x i∂ →∂ + σ ω

Properties • for real valued σ 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 turn on (e.g. with quadratic spatial profile) the absorption coefficient σ from zero over a short distance on the scale of the wavelength of the wave

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 not to be restricted to large values, when first order

approximation of the conformal mapping is used (corresponding to linear index gradient)


Method • conformal mapping in the complex plane to transform a curved waveguide

in coordinates (x,y) into a straight waveguide with a modified refractive index in new coordinates (u,v)

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

plane • typical example for local angle preserving transformation: word maps

Demonstration for 2D scalar wave equation

2 22 2 ( , ) 0k n x y

x z∂ ∂ + + φ = ∂ ∂

general transformation ( ) ( )W u iv f x iz f Z= + = + = for f being an analytical function in the complex plane to straighten a bend radius of R in the (u,v) plane, f should be taken as ln( / )W R Z R= applying the transformation to the wave equation gives

( )2

2 22 2 ( , ), ( , ) 0dZk n x u v y u v

dWu v

∂ ∂+ + φ =

∂ ∂


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 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

/ 21 ( / 2 )( / )

P k njz j k n z∂φ

= − φ∂ + ∂ ∂

(*)

which can be reduced to

Njz D∂φ

= − φ∂

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 ND j z N

+ − ∆ −αφ = φ

+ ∆ α

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

0.5α = Crank-Nicolson scheme) the numerator (1 )D j z N− ∆ −α can be factorized as

1 2Nom Nom Nom 2 1( ...) (1 )...(1 )(1 )N N N

NA 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− −− ∆ −α = + + + = similarly the denominator D j z N+ ∆ α can be factorized as


1 2Den Den Den 2 1( ...) (1 )...(1 )(1 )N N N

NA 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

i0

( ) 0N

i

iD j z N d P

=+ ∆ α = =∑

with 00 0 1c d P= = =

Thus, the unknown field 1l+φ at z z+ ∆ is related to the known field lφ at z as

1 2

1 Nom Nom Nom1 2

Den Den Den

......

N N Nl l

N N NA P B P C PA P B P C P

− −+

− −+ + +

φ = φ+ + +

or

1 2 1

2 1

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

l lN

N

c P c P c Pd P d P d P

+ + + +φ = φ

+ + +

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 njz j k n

z −

∂= −

∂∂ +∂

(**)

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

in paraxial approximation 0

1

/

1

P a Pj jjz aa 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 ) 1PD j z N j z A Pa

− ∆ −α = − ∆ −α = +

and the denominator analogously

Den1 1PD j z N j z A Pa

+ ∆ α = + ∆ α = +

with

Nom(1 )zA ja

∆ −α= −


DenzA ja

∆ α=

thus the unknown field 1l+φ at z z+ ∆ is related to the known field lφ at z as

1 Nom

Den

11

l lA PA P

+ +φ = φ

+

or ( ) ( )1

Den Nom1 1l lA P A P++ φ = + φ 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 ajjza z

∂= −

∂∂ +∂

and inserting the previous result

0

Pjz a∂

= −∂

we get

1

2

/

1

P aj Pza

∂= −

∂ +

comparison to the expansion equations gives

21 PDa

= + and /N P a=

and therefore the nominator becomes

Nom2 21 1(1 ) 1 (1 ) 1 1P PD j z N j z j z P A P

a aa a−α − ∆ −α = + − ∆ −α = + − ∆ = +

and the denominator analogously

Den2 211 1P PD j z N j z j z P A P

a aa aα + ∆ α = + + ∆ α = + ∆ = +

with

Nom 21 (1 )zA j

aa∆ −α

= −

Den 21 zA j

aa∆ α

= +

thus the unknown field 1l+φ at z z+ ∆ is related to the known field lφ at z again as


( ) ( )1Den Nom1 1l lA P A P++ φ = + φ

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 an implementation is

straight forward (but cumbersome implementation of proper boundary conditions)

• requires excessive computational resources for reasonable problems in 3D (implementation for clusters)

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

• principally all kinds of materials are treatable (dispersive or nonlinear materials)

4.1 Maxwell’s equations

makr

ext

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

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

t tt t tt t

t t t

∂ ∂= − = +

∂ ∂= ρ =

B r D rE r H r j r

D r r B r

mater equations

0

0

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

t t tt t t= ε += µ +

D r E r P rB 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) − ρext(r,t) external charge density − jmakr(r,t) macroscopic current density

MWEQ in linear isotropic and dispersionless dielectric Media

( )0

0

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

t tt t= ε ε

= µ

D r r E rB r H r

( )

( )

0 0 makr

0

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

div ( , ) 0, di

rot ( ,

,

)

v ( , ) 0

t tt tt

tt

tε ε = =

∂ ∂= −µ = ε ε +

∂ ∂ r E r

H r E rE r H

r

r r

H

j

⇑ field sources


rot-equations for individual components

0

( , ) 1(a) rot ( , )t tt

∂= −

∂ µH r E r

0

0

0

1 ,

1 ,

1

yx zx

y xzy

yxzz

EH E jt z y

H EE jt x z

EEH jt y x

∂ ∂ ∂= − − ∂ µ ∂ ∂

∂ ∂∂ = − − ∂ µ ∂ ∂ ∂ ∂∂

= − − ∂ µ ∂ ∂

( )0

( , ) 1(b) rot ( , )t tt

∂=

∂ ε εE r H r

r

( )

( )

( )

0

0

0

1 ,

1 ,

1

yx z

y x z

y xz

HE Ht y z

E H Ht z x

H HEt x y

∂ ∂ ∂= − ∂ ε ε ∂ ∂

∂ ∂ ∂ = − ∂ ε ε ∂ ∂

∂ ∂∂= − ∂ ε ε ∂ ∂

r

r

r

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 Hat t x t x

∂ ∂∂ ∂ ∂= = =

∂ ∂ µ ∂ ∂ µ ∂

( ) ( )0 0

1 1( ) 0, ,y yx z zE HE H Eb

t t x t x∂ ∂∂ ∂ ∂

= = =∂ ∂ ε ε ∂ ∂ ε ε ∂r r

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

z-polarized E-field ( )0 0

1 1,y yz zH HE Et x t x

∂ ∂∂ ∂= =

∂ ε ε ∂ ∂ µ ∂r

y-polarized E-field ( )0 0

1 1,y yz zE EH Ht x t x

∂ ∂∂ ∂= =

∂ ε ε ∂ ∂ µ ∂r

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


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 11 1

1 1

( )n N NN N

N N N

f f f O x xx x x

+ −+ −

+ −

∂ − ≈ + − ∂ −

while the unsymetric dicretization is just first order accurate

[ ]11

1

n N NN N

N N N

f f f O x xx x x

−−

−

∂ −≈ + −

∂ −

always use symmetric discrete operators

21 1

1 12

21 1

1 12

( , ) ( , ) :

( , ) ( )2

( )2

( , ) ( )2

( )2

( )

n n nn i i i

z i

n n ni i i

n n nn i i i

y i

n n ni i i

i

i n i x n tE E EE i x n t E O xx x

E E E O tt tH H HH i x n t H O xx x

H H H O tt t

i x

+ −

+ −

+ −

+ −

= ∆ ∆

∂ − ∆ ∆ = ⇒ = + ∆ ∂ ∆∂ − = + ∆ ∂ ∆∂ − ∆ ∆ = ⇒ = + ∆ ∂ ∆∂ − = + ∆ ∂ ∆

ε ∆ = ε

Maxwell’s equations

( )0 0

1 11 1

0

1 11 1

0

0 01 1

1 1

0

1 11

0

1 1

12 2

1

1 1

12 2

1

n ny i iz

i

n n n ni i i i

i

n n n ni i i i

i

n ny i iz

n n n ni i i i

n n ni i i i

H E HEt x t x

E E H Ht x

tE E H Hx

H H EEt x t x

H H E Et x

tH H E Ex

+ −+ −

+ −+ −

+ −+ −

+ −+ −

∂ ∂ ∂∂= ⇒ =

∂ ε ε ∂ ∂ ε ε ∂

− −⇒ ≈

∆ ε ε ∆

∆ ⇒ ≈ + − ε ε ∆

∂ ∂ ∂∂= ⇒ =

∂ µ ∂ ∂ µ ∂

− −⇒ ≈

∆ µ ∆

∆⇒ ≈ + −

µ ∆

r

1n


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 0.50.5 0.5

0.5 0.50.5 0.5 1

0

0

1

1

n

n n n ni i i i

i

n n ni i i i

tE

tH H E

E H

Ex

Hx

+ + ++ −

+ −+ + +

∆ ≈ + − ε ε ∆

∆ ≈ + − µ ∆

Properties Divergence?

− That there is no divergence of the fields is always fulfilled in 1D, since fields are always transversally polarized to the direction of field change

Resolution of discretization x∆ and t∆ ? (from physical arguments) − spatial grid resolution x∆ must be fine enough to display the finest

structures of the e distribution and the fields rule of thumb

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


max/ (20 )x n∆ ≤ λ , 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 sets upper limit to the phase velocity /t x c∆ ≤ ∆ in higher dimensions

1/2

2 2 2

1 1 1 1tc x y z

∆ ≤ + + ∆ ∆ ∆

Boundaries? − finite grid size, e.g. with metallic borders ( 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 − sources as e.g. currents:

0.50.5

0

1 0.5 0.50.5 0.5

0

1n n n ni i i i

ni

i i

tE E Hx

t jH+ ++

+ ++ −

∆∆ ≈ + − + ε εε ε∆

4.3 3D problems space grid ( , , , ) ( , , , )i j k n i x j y k z n t= ∆ ∆ ∆ ∆

( , , , )

( , , )

nx x ijk

ijk

E i x j y k z n t E

i x j y k z

⇒ ∆ ∆ ∆ ∆ =

ε ∆ ∆ ∆ = ε

Finite differencing in space and time

1 1 2

1 1

2

( )2

( )2

n nnx xx i j k i j kijk

n n nx x xijk ijk ijk

E EEO y

y y

E E EO t

t t

+ −

+ −

−∂ = + ∆ ∂ ∆

∂ − = + ∆ ∂ ∆


4.3.1 Yee grid in 3D

Properties • no divergence • Leapfrog time steps • central differencing second order explicit method

dicretizing MWEQ ( )0

1x

x yz jt y

Hz

E H ∂ ∂ ∂= − − ∂ ε ε ∂ ∂ 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 nx xi j k i j k

n nn ny yz zi j k i j k i j k i j k n

x i j ki j k

E E

H HH Ht jy z

+ −

+ + + +

+ + + + + +

+ ++ +

=

−−∆ + − − ε ε ∆ ∆

other equaltions: 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 nx 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 Ht jz x

+ −

− + + − + +

− + + − + + + − + +

− + +− + +

=

− −∆ + − − ε ε ∆ ∆

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 , ,

, , 10 , , 1

n nz zi j k i j k

n n n ny y x xi j k i j k i j k i j k n

z i j ki j k

E E

H H H Ht jx z

+ −

− + + − + +

+ + − + + − + + −

− + +− + +

=

− −∆ + − − ε ε ∆ ∆

1 1 1 12 2 2 23 31 1 1 1 1 1

2 2 2 2 2 2 2 21 12 2

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

, 1, 1 , 1, 10

n n n ny 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 Hz y

+ + + +

− + + − + + − + + − + ++

− + + − + +

− −∆ = + − µ ∆ ∆

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 nx 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 Hx z

+ ++ ++ + + + − + + + + + +

+ + + +

−−∆ = + − µ ∆ ∆

1 11 1 2 22 23 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 ny yx xi j k i j k i j k i j kn n

z zi j k i j k

E EE EtH Hy x

+ ++ +

+ + + + + + + − + ++

+ + + +

−−∆ = + − µ ∆ ∆

Grid size

( ) ( )( ) ( )( ) ( )

12

12

12

, 1, 1 1, ,

1, , 1 , 1,

1, 1, , , 1

n nx x y z x x y z

n ny x y y x y z

n nz 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

+

+

+

+ + +

+ + +

+ + +

Simple boundary conditions: Perfect Electric Conductor (PEC)

( ) ( )

( ) ( ):,1,: 0, :,:,1 0

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

x y x z

E E

E N E N

= =

+ = + =

( ) ( )

( ) ( )1,:,: 0, :,:,1 0

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

y x y z

E E

E N E N

= =

+ = + =

( ) ( )

( ) ( )1,:,: 0, :,1,: 0

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

z x z y

E E

E N E N

= =

+ = + =

4.3.2 Physical interpretation

3D Yee grid & Amper’s law

0( )

( ) (, ) , )(F F

tt

t∂ε ε ∂ = ∂

∂ ∫ ∫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 ki j k

n nn n

i j k i jx ki jy x yk j

z z

i k

x yt

x y x

E

H H y

E

H H

+ −

− + − +

+ +

− − ++ − +

− ε ε ∆ ∆ ∆

= ∆ + ∆ − ∆ − ∆

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 k i j kx x

z

i k

z

j

H H H

E E

tx y

H

+ −

− + − +

+ − + − − +

+ +

=

− −∆ + + ε ε ∆ ∆

3D Yee grid & Faraday’s law

0( )

( , ) ( , )F F

tt

t∂µ ∂ = − ∂

∂ ∫ ∫ 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 tt

tt

tε ε = =

∂ ∂= −µ = ε ε +

∂ ∂ r E r

H r E rE r H

r

r r

H

j


( )0div ?t∂

ε =∂

E ( )0 0Yee cell Yee cell

div dV dt t∂ ∂

ε = ε∂ ∂∫∫∫ ∫∫E E f

( )

( )

( )

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

+ + − + +

− + + − +

− + + − +

∂ ∂ε = ε − ∆ ∆

∂ ∂

∂+ε − ∆ ∆

∂

∂+ε − ∆ ∆

∂

∫∫ E f

substitute Term 1 with rot equation ( )0

1 yx zHE H

t y z∂ ∂ ∂

= − ∂ ε ε ∂ ∂ 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:

( ) ( ) ( )0Yeecell

Term 1 Term 2 Term 3 0d y z x z x yt∂

ε = ∆ ∆ + ∆ ∆ + ∆ ∆ =∂ ∫∫ E f

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

[ ] ( ) [ ]0 0 0Yee cell Yee cell

div ( 0) 0 & div 0 div 0t dV dVt∂

ε = = ε = ⇒ ε =∂∫∫∫ ∫∫∫E E E

it is important 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 algorihm • Leap-Frog algorithm discretization applies to all components • Close to the physical world as the spatial and temporal propagation is

exactly simulated

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

cylindrical objects) • Derivations of the field along this directions are zero

0

1 yx EHt z

∂∂=

∂ µ ∂ ( )0

1 yz x zx

HE E Hjy t y z

∂∂ ∂ ∂− − = − ∂ ∂ ε ε ∂ ∂ r

0

1y z xH E Et x z

∂ ∂ ∂= −

∂ µ ∂ ∂ ( )0

1y xy

E Hjt z

∂ ∂− = ∂ ε ε ∂ r

( )0 0

1 1

z

y yz x z xz

Hx

E HH E E Hjt y x t x y

∂− ∂

∂ ∂ ∂ ∂ ∂ ∂= − − = − ∂ µ ∂ ∂ ∂ ε ε ∂ ∂ 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 E jt x y t y x

EHt z

∂ ∂ ∂ ∂ ∂ ∂= − = − − ∂ ε ε ∂ ∂ ∂ µ ∂ ∂

∂∂=

∂ µ ∂

r

( )0

1 yz x zx

HE E Hjy t y z

∂∂ ∂ ∂− − = − ∂ ∂ ε ε ∂ ∂ r

0

1y z xH E Et x z

∂ ∂ ∂= −

∂ µ ∂ ∂ ( )0

1y xy

E Hjt z

∂ ∂− = ∂ ε ε ∂ r

zHx

∂− ∂

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 makroskopic current density

makr( , ) ( , )rot ( , ) , rot ( , ) ( , )t tt t tt t

∂ ∂= − = +

∂ ∂B r D rE r H r j r

• Simplyfied implementation by adding a source term to the electric field Examples for temporal variation x-polarized cw-source

, , , ,sin( )n n

x xi j k i j kE E n t= + ∆ ω

x-polarized impulse ,, , , ,

n nx x n ni j k i j k

E E ′= + δ

Examples for spatial variation of the light source


x-polarized Gaussian wave

22

, , , ,sin( )yx

i yi xn n

x xi j k i j kE E e e n t

∆ ∆ − − σσ = + ∆ ω

Example calculations (2D-configuration, TM, Hy) plane wave homogeneous medium Cylinder with n=2 and D=2xlambda

point source homogeneous medium Cylinder with n=2 and D=2xlambda

4.6 Relation between frequency and time domain • The frequency spectrum used for the illumination is given by the Fourier-

transformation of the time dependent incident field


• With a single calculation we can calculate the entire frequency response,

detecting the temporal evolution of the field behind a structure and FT (Nt = total number of time steps)

• For a high resolution in the wavelength domain, we have to record the temporal evolution of the field for an excess in time Kind of disadvantage

Example of a grating wave guide coupler

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

Transmission spectrum - The dips are wave guide resonances that are excited if the momentum provided by the grating matches the propagation constant of a waveguide mode

t

EInc

f

EInc

Transmission?

d1

n1

d2

n2


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

4.7 Boundary conditions see separate presentation


5. Fiber waveguides 5.1.1 The general eigenvalue problem for scalar fields Restrictions:

− weakly guiding structure ( 1n∆ ln ( , ; ) 0x y∇ ε ω ≈ ) − linearly polarized field

Scalar wave equation for a scalar mode field u 2 2

t ( , ) ( , , ) ( ) ( , ) 0u x y k x y u x y ∆ + ε ω −β ω =

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 β . The discrete solutions are called 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 aβ and bβ read as

2 2t ( , , ) ( ) ( , ) 0a ak x y u x y ∆ + ε ω −β ω =

2 2t ( , , ) ( ) ( , ) 0b bk x y u x y ∆ + ε ω −β ω =

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 uβ −β = ∆ − ∆

Integration over the infinite cross section of the waveguide A∞

( )2 2t ta b a b b a a bA A

u u dA u u u u dA∞ ∞

β −β = ∆ − ∆∫ ∫

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

( )2 2 0a b a bAu u dA

∞β −β =∫

To fulfill the equation for different β the integral must be zero

,a b a bAu u dA

∞= δ∫


This requires the normalization of the fields according to

2 1A

u dA∞

=∫

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

,( ) ( )a b a brY r Y r r dr

∞= δ∫

Phase velocity The propagation velocity of the phase fronts is called phase velocity

p2 cv ω π

= =β λβ

with λ 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 ∂ω π ∂λ= =∂β λ ∂β

which can be calculated very accurately by the following approach: Take two solutions at two different wavelength λ and ′λ 2 2

t ( ) ( ) ( ) ( ) 0k u ∆ + λ ε λ −β λ λ =

2 2t ( ) ( ) ( ) ( ) 0k u′ ′ ′ ′ ∆ + λ ε λ −β λ λ =

Do the same as done for deriving the orthogonality relation: multiply the first equation with ( )u λ and the second with ( )u ′λ and subtracting the results one obtains ( ) ( )2 2 2 2

t t( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )k k u u u u u u′ ′ ′ ′ ′β λ −β λ − λ ε λ − λ ε λ λ λ = λ ∆ λ − λ ∆ λ

as before integrate over the infinite waveguide cross section A∞ , transform into a line integral using Green’s law, and show that the right side is equivalent to zero

( )2 2

2 2

( ) ( ) 4 ( ) ( )( ) ( ) ( ) ( )A A

u u A u u A∞ ∞

′β λ −β λ ′π ε λ ε λ ′ ′λ λ ∂ = − λ λ ∂ ′ ′ ′λ − λ λ − λ λ λ ∫ ∫

with ′λ → λ one obtains

2 22

1 ( ) ( )( ) 4 ( )( ) A A

u A u A∞ ∞

∂β λ ∂ ε λ λ ∂ = π λ ∂ β λ ∂λ ∂λ λ ∫ ∫

using this expression to substitute ∂λ ∂β in our definition for the group velocity we obtain

2

g 22

2

( )4 ( )( )( )

A

A

u Acvu A

∞

∞

λ ∂− π β λ=

∂ ε λλ λ ∂ ∂λ λ

∫∫


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

g

1v

D∂

=∂λ

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 rε = ε = For fibers the index profile is often characterized by special parameters

Fiber parameter

( )co cl

1/ 22 22 aV n nπ= −

λ

with a - core radius con - maximal refractive index in the core cln - constant refractive index in the cladding surrounding the core

Index difference

co cl

co

2 2

2

12

n nn

−∆ =

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

22

1( ) 1

ji

i i

anb C=

λλ = +

λ −∑

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


2

141

102

141

102

1.53714322m1.968198 10

1.4108754 10

1.790244 10

1.572513 10

Caabb

=

= ×

= ×

= ×

= ×

5.1.4 Bessel’s differential equation Using the Laplace operator in cylindrical coordinates

2

t 2 2

1 1( , ) u uu r rr r r r∂ ∂ ∂ ∆ ϕ = + ∂ ∂ ∂ϕ

the scalar Helmholtz equation reads as

2

2 22 2

1 1 ( ) 0u ur k r ur r r r∂ ∂ ∂ + + ε −β = ∂ ∂ ∂ϕ

Applying the product Ansatz ( , ) ( ) ( )ru r u r uϕϕ = ϕ

the scalar cylindrical Helmholtz equation reads as

22

2 22 2 2

1 1 ( ) 0r rr r

uu uu u u k r u ur r r r

ϕϕ ϕ ϕ

∂∂ ∂ + + + ε −β = ∂ ∂ ∂ϕ

multiplying with 2 /( )rr u uϕ the equation can be separate in terms depending only on r and terms depending only on ϕ

22 2

2 2 22 2

1( )r r

r r

ur u r u r k ru r u r u

ϕ

ϕ

∂∂ ∂ + + ε −β = − ∂ ∂ ∂ϕ

Since both sides of the equations are independent, the equation can be solved non-trivialy only when both sides are equivalent to a constant 2m

2 22 2

2 2

22

2

1 ( ) 0

0

r rr

u u mk r ur r r r

um uϕ

ϕ

∂ ∂+ + ε −β − = ∂ ∂

∂+ =

∂ϕ

The r -equation is the Bessel’s differential equation. The ϕ -equation can be solved analytically as 0cos sin cos( )u A m B m C mϕ = ϕ + ϕ = ϕ+ ϕ .

To obtain a continuous field the so-called azimuthal mode constant must obey m∈ Bessel’s differential equation can be simplified further by introducing the dimensionless modal parameters


( )co

1/ 22 2 2U a k n= −β in the core

( )cl

1/ 22 2 2W a k n= β − 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 −β as

co co2 2 2 2 2 2 2 22 ( )k n k n k n f r−β = −β − ∆ and hence

2 22 2 2

2

( )U V f rk na

−−β =

Now we can rewrite Bessel’s differential equation for ( ) ( )m rY u rρ = with the normalized radius /r aρ = in the final form

2 2

2 22 2

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

Y Y mU V f Y ∂ ρ ∂ ρ

+ + − ρ − ρ = ∂ρ ρ ∂ρ ρ

5.1.5 Analytical solutions of Bessel’s differential equation Bessel’s differential equation can be solved analytically for some specific profiles ( )f ρ or in segments where ( )f ρ 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( ) ( )G U V fρ = − ρ the analytic solution in an interval with constant refractive index ( ( ) const.f ρ = ) reads as ( ) ( (0))m mY AJ Gρ = ρ with A being an arbitrary scaling constant and ( )mJ ρ being the Bessel function of the first kind and mth order. The other solutions of Bessel’s differential equation can be ruled out by physical reasoning due to their infinite value at the origin or their unboundednes for growing arguments. Similarly the solution in the cladding (where ( ) 1f ρ = ) is based on the modified Hankel function 2( ) ( )m mY BK Wρ = ρ where 1ρ > with B being an arbitrary constant. Using the requirement that the general solution and its derivative need to be continuous we can derive the following conditions at the core-cladding interface, which can be written as a set of linear equations using the arbitrary variables A and B

2

2

( 1) ( ( 1)) 0

( 1) ( ( 1)) 0m m

m m

AY BK W

AY BWK W

ρ = − ρ = =

′ ′ρ = − ρ = =

the determinant of which must vanish to obtain unambiguous solutions


2 2 2( 1) ( ( 1)) ( 1) ( ( 1)) ( ) 0m m m mWY K W Y K W F U′ ′ρ = ρ = − ρ = ρ = = =

This characteristic equation 2( )F U is only solved for discrete values of the parameters W or U which selects a finite number of modes. Usually it is solved as a function of 2U which can vary in the interval 2 20 U V< < . The modes are often categorized in their linearly polarized form as LPmp -modes, where the index m corresponds to the azimuthal mode number and p selects the modes according to 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 Repeating the governing equation

2 2

2 22 2

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

Y Y mU V f Y ∂ ρ ∂ ρ

+ + − ρ − ρ = ∂ρ ρ ∂ρ ρ

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 condition. Starting from the core center at 0ρ = we assume that solution ( )Y ρ for 0ρ→ behaves like the Bessel function ( ) ( (0))m mY AJ Gρ = ρ of the analytic solution. Since the Bessel function can be expressed as a series like

2

0

( 1)( )!( 1) 2

n mn

mn

Jn n m

+∞

=

− ρ ρ = + + ∑

we can use an ansatz function like ( ) ( )m

mY Xρ = ρ ρ where ( )X ρ is an analytic finite valued function for finite arguments. From this ansatz we can also take the derivative of the solution as ( 1)( ) ( ) ( )m m

mY m X X−′ ′ρ = ρ ρ + ρ ρ Hence the boundary condition in the center read as

0 : (0) (0)(0) (0)

1: (0) 0(0) (0)

1: (0) 0(0) 0

m Y XY X

m YY X

m YY

= =′ ′=

= =′ =

> =′ =


As was demonstrated in the “Analytic solution” section the boundary conditions at infinity (ρ→∞ ) 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′ ′= ρ = ρ = − ρ = ρ = =

5.1.7 Solving the second order singularity The governing Bessel’s differential equation possesses a second order singularity at 0ρ = which makes it very hard to integrate numerically. Using again the ansatzt function ( ) ( )m

m mY Xρ = ρ ρ we obtain a new differential equation for ( )X ρ as

2

2 22

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

X m X U V f X∂ ρ + ∂ ρ + + − ρ ρ = ∂ρ ρ ∂ρ

which possesses only an ordinary singularity at 0ρ = . At the same time the boundary conditions must be transformed. Since ( )X ρ is an analytic function it can be expressed as a power series following the series representation of the Bessel function as

0

( ) nm n

nX c

∞

=

ρ = ρ∑

Hence the new boundary conditions at 0ρ = are independent of m 0(0)X c= and (0) 0X ′ = Similarly the boundary condition at 1ρ = becomes

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

mm m

m

K WF U W m X XK W

′′= − ρ = − ρ = =

with 2( )F U being a characteristic function. We have to keep in mind that, once found, the new solution ( )mX ρ has to be transformed back to ( )mY ρ 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 and to find the desired parameter 2U . Using the shooting method the problem can be transformed into an initial value problem starting at 0ρ = which depends on the parameter 2U . Different

2U result in different solutions at 1ρ = and hence also different 2( )F U . The aim must be to find a parameter *U U= with *2( ) 0F U =


Then the solutions of the initial value problem would also solve the boundary value problem. For the numerical integration of the differential equation for ( )mX ρ we have to take into account the following properties:

− at 0ρ = the function ( )mX ρ has a first-order singularity, which depending on m can lead to instabilities of the numeric solution

− the profile function ( )f ρ can possess discontinuities in the integration interval, which can reduce the order of the integration methods

− the coefficients of the series representation of ( )mX ρ can go through zero in the integration interval, which can change the continuity properties of the solution

Hence one should refer to implicit integration methods. In the beginning we transform the second order differential equation into a set of coupled first order differential equations with

1

2

( ) ( )( )

( )( )m m

mmm

Z XXZ

ρ ρ ρ = = ′ ρρ

Z

as

12

22 2 1 2

( ) ( )

( ) 2 1( ) ( ) ( )

mm

mm m

Z Z

Z mV f U Z Z

∂ ρ= ρ

∂ρ

∂ ρ + = ρ − ρ − ρ ∂ρ ρ

Now we have to discretize the integration interval [0,1]Ω = ⊂ , e.g. with very simple equidistant spacing h as:

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

Ω = ρ ρ = = =

then the discrete approximation of the solution can be written as ( ),j j j h≈ ρ ρ ∉ΩZ Z

Implicit Runge-Kutta-Midpoint-Method of 2nd order In matrix representation the differential equation reads as

2 2 ( )

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

mm m j jmV f U

∂ ρ = ρ = ρ+ ρ − −∂ρ ρ

Z Z K Z

The midpoint-method has the following Runge-Kutta parameter table

0.5 0.51 1

This is a single step implicit method which results in an iteration step like 1 ( , , )j j j jh h+ = + ρZ Z Φ Z , ( 0,1,..., )j N=


1( , , ) ( , , ),2 2j j j j j j

hh h h ρ = + ρ ρ +

Φ Z K Z Φ Z

applying this method to our particular problem gives

1( ) 1

1 2 2

22( ) 2

0 122 1( , , )

222

j

j j j

jj

hZh mh V f U hh Z

+ ϕ ϕ + ρ = = ρ + − − ϕ + ϕρ +

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 21( ) 2( )

1( 1) 1( ) 22 2

2 2 2 21( ) 2( )

2( 1) 2( )

2 22 11

2 4 22

2 12 2 2

22 11

22

j j j

j j

j

j

j j j j

j

j j

j

h hV f U Z ZZ Z h

h m h hV f Uh

h h h mV f U Z V f U Zh

Z Z hh m h

h

+

+

ρ + − + = ++ + − ρ + − ρ +

+ ρ + − + ρ + − − ρ + = +

++ −

ρ +

22 2

4 2jhV f U ρ + −

The solution accuracy of the method is second order in h . The discretization grid hΩ should be chosen such that the discontinuity points Dρ of the profile function ( )f ρ coincide with grid points of hΩ D hρ ∈Ω 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 profiles ( )f ρ .

Implicit Runge-Kutta-Method of 4th order To improve the accuracy one could also use the following Gaussian type forth order method

1 11 12

2 21 11

1 2

1 3 1 1 32 6 2 2 61 3 1 3 12 6 2 6 4

1 112 2

+ = γ = β + = β

− = γ − = β = β

= β = β


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]Ω = . The eigenvalue interval for guided modes is restricted to 2 20 U V< < containing a finite number of roots. To find all roots the interval can be scanned for sign changes with a fixed step size 2U∆ . The resulting subintervals can 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∈ for a step index profile with 0m =


Examples of calculated mode field for a ring index fiber.

Computational Photonics

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