“propagating ideas”. rix=2.5 rix=1 l=0.5 d=0.35 our aim

“propagating ideas”


Rix=2.5

Rix=1

L=0.5 D=0.35 Our Aim


conditions BD. eappropriat

0

0HE

HE

EH

Fields governed by the source free Maxwell equations

The field equations


The Field solver

Robust: Must be capable of dealing with any taper shape

Use mode matching method

Accurate: correctly model high contrast structures


Mode Matching Method

x

y z

1c

Nc

S1c

Ncinput

output

Section modes

Propagation constants

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

m

zimsms

zimsms

msms eyxceyxc ψψH

E

1c

1c

Nc

Nc

sc

sc

Continuity at interfaces elimination of intermediate coefficients


MMItapers

Mode convertersPhotonic crystals

Possible applications


Band Gap + line defect

Choose working =1.34m

Vary wavelength...

1,1c

2,1c

3,1c

4,1c

0

2

,m

mNc

Tot. power: Tot. power


Field plots in line defect (arbitrary input)

=1.34m

Only 1 mode excited

=1.43m

2 modes excited


Exciting the PC mode

Choose w=0.351

Design an “artificial” waveguide s.t. its fundamental mode has 100% transmission

W W


Optimising the y-junction

The initial structure...

Wavelength response

50% transmission


Setting up the optimisation

D1D2

L


Problem!

Many local minima holes can overlap and vanishdifferent topological configurations

L,D, or W

P

Many local minima


Search whole function space in intelligent way

Global optimisation

Evolution algorithms (statistical in nature)

• Not guaranteed to find global optimum• Loose a lot of information on the way!


These are algorithms that systematically search the parameter space.

Deterministic global optimisation

Splitting algorithms: • successively subdivide regions in systematic way. • Divide more quickly where optima are “more likely” to exist.

Etc...


Monitoring interface

Specify your independent variables...Connect them to any structure parameter

define your own objective!


Optimisation results

A

A

B

B


D1= 0.38m , D2 = 0.31m , L= - 0.17m

Optimal point A: transmission=99.8%!

Wavelength response

VERY BAD!

Resonant transmission


Optimal point B: transmission=99.5%!

D1= 0.12m , D2 = 0.47m , L = 0.15m

Wavelength response

MUCH better

steering transmission


Bend optimisation

D

D

LL



Best point


Best shape : transmission=97%!

L= 0.24m , D = 0.47m

Wavelength response

Resonant transmission

FAIRLY good: variation = 8%


Bend + y junction

transmission=97%!

Input from here

Wavelength scan

Pretty good!


Bend optimisation II

D

D

LL

OFFOFF

Idea: try to find optimal steering transmissions



2% variation

0.5% variation


The complete crystal

98% transmission, 1% variation!!!


Optimal taper design

Inp ut fie ld

Po we r lo ss


Large losses

…Argh .. Not very good!

56% transmission


Could make it longer ...

Reduced losses

40 m

Too long!

95% transmission


Keep length fixed ...

Inp ut fie ld

Po we r lo ss

Maximise power output

Deform shape ...


The local optimisation algorithm

Use an iterative technique (the quasi-Newton method).

Could approximate these using finite differences:

h

xxxPxhxxP

x

P NkNk

k

,...,,...,,...,,..., 11

…but this requires N field calculations per iteration!

second order convergence, but

Nx

P

x

P

,...,1

requires derivatives per iteration.


Only 2 field calculations per iteration!

GOOD NEWS!

We can derive analytic expressions for Nx

P

x

P

,...,1

dSP

FETaper region

Electric field (solution of wave equations) Adjoint electric field

(solution of adjoint wave equations)

Change in permettivity dueto shape deformation


The first example: length 14um...

Rix = 2.5

Rix = 1.0

P = 84%

Vary ends

|C1

+

|2


Much better ...

P = 91%

|C1

+

|2


Design of optimal taper injector

Replace with

artificial input…and width

Vary taper length

Excite fundamental mode of input waveguide

Optimise offset 5m


OPt transmission vs taper length

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15

Taper langth um

Tra

ns

mis

sio

n F

rac

Choose 9m

Optimal results for length range


Field plot at length=9m

99%


The complete result!!!

97% transmission, variation 5%!!!


IMPROVE TAPER FURTHER?

x1

x2

x3 xN),...,,( 21 NxxxP

Optimization problem:

find (x1 , x2 , ... , xN) that maximise P

Could also parametrize shape ...


Here was the original ...

P = 56%


Here is the optimal design ...

15 nodes

P = 88%


P = 97%

39 nodes

Fwd/bwd power “Resonant” region

Using lots of nodes


Increasing the number of nodes...

Optimisation problem becomes ill posed!

dSP

FE

P

P+P

For “thin enough” :

p 0

E,F are bounded, so


Can improve transmission, but ...

there could be more minima,

Consequences

homing on optimum becomes more difficult:

Power transmission becomes less sensitive to variation of any individual node

Numerical instabilities - inverse problems

Use regularisation techniques.

On Shape Optimisation of Optical Waveguides Using Inverse Problem Techniques

Thomas Felici and Heinz W. Engl, Industrial Mathematics Institute, Johannes Kepler Universität Linz


3D simulations

Air holes

membrane with refractive index 2.5

Vary height


FDTD 3D

Probes just inside crystal

Input waveguide


0%

10%

20%

30%

40%

50%

60%

70%

80%

1.1 1.2 1.3 1.4 1.5 1.6

steering

resonant

original


RAM requirements:

> 1 Gb - you need at least 2Gb of RAM for better performance;

updated version: only 765 Mb !!

Computational performance

Numerical space consists of 290x92x452 grid points ( 12 million points)

we use 8 thousand time steps

Hence we have 96 billion floating point operations per simulation!!

CPU time:

weeks??? - impossible due to the lack of memory (HP station at COM);

days??? Feasible but very slow due to usage of hard disk memory (Pentium 4 PC);

updated version: only3 hours and 55 minutes!!

Speed is even less than in Example1: 142.7 ns per grid point

“propagating ideas”. rix=2.5 rix=1 l=0.5 d=0.35 our aim

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