“propagating ideas”. Rix=2.5 Rix=1 L=0.5 D=0.35 Our Aim.
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Transcript of “propagating ideas”. Rix=2.5 Rix=1 L=0.5 D=0.35 Our Aim.
“propagating ideas”
“propagating ideas”
Rix=2.5
Rix=1
L=0.5 D=0.35 Our Aim
“propagating ideas”
conditions BD. eappropriat
0
0HE
HE
EH
Fields governed by the source free Maxwell equations
The field equations
“propagating ideas”
The Field solver
Robust: Must be capable of dealing with any taper shape
Use mode matching method
Accurate: correctly model high contrast structures
“propagating ideas”
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
“propagating ideas”
MMItapers
Mode convertersPhotonic crystals
Possible applications
“propagating ideas”
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
“propagating ideas”
Field plots in line defect (arbitrary input)
=1.34m
Only 1 mode excited
=1.43m
2 modes excited
“propagating ideas”
Exciting the PC mode
Choose w=0.351
Design an “artificial” waveguide s.t. its fundamental mode has 100% transmission
W W
“propagating ideas”
Optimising the y-junction
The initial structure...
Wavelength response
50% transmission
“propagating ideas”
Setting up the optimisation
D1D2
L
“propagating ideas”
Problem!
Many local minima holes can overlap and vanishdifferent topological configurations
L,D, or W
P
Many local minima
“propagating ideas”
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!
“propagating ideas”
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...
“propagating ideas”
Monitoring interface
Specify your independent variables...Connect them to any structure parameter
define your own objective!
“propagating ideas”
Optimisation results
A
A
B
B
“propagating ideas”
D1= 0.38m , D2 = 0.31m , L= - 0.17m
Optimal point A: transmission=99.8%!
Wavelength response
VERY BAD!
Resonant transmission
“propagating ideas”
Optimal point B: transmission=99.5%!
D1= 0.12m , D2 = 0.47m , L = 0.15m
Wavelength response
MUCH better
steering transmission
“propagating ideas”
Bend optimisation
D
D
LL
“propagating ideas”
Optimisation results
Best point
“propagating ideas”
Best shape : transmission=97%!
L= 0.24m , D = 0.47m
Wavelength response
Resonant transmission
FAIRLY good: variation = 8%
“propagating ideas”
Bend + y junction
transmission=97%!
Input from here
Wavelength scan
Pretty good!
“propagating ideas”
Bend optimisation II
D
D
LL
OFFOFF
Idea: try to find optimal steering transmissions
“propagating ideas”
Optimisation results
2% variation
0.5% variation
“propagating ideas”
The complete crystal
98% transmission, 1% variation!!!
“propagating ideas”
Optimal taper design
Inp ut fie ld
Po we r lo ss
“propagating ideas”
Large losses
…Argh .. Not very good!
56% transmission
“propagating ideas”
Could make it longer ...
Reduced losses
40 m
Too long!
95% transmission
“propagating ideas”
Keep length fixed ...
Inp ut fie ld
Po we r lo ss
Maximise power output
Deform shape ...
“propagating ideas”
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.
“propagating ideas”
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
“propagating ideas”
The first example: length 14um...
Rix = 2.5
Rix = 1.0
P = 84%
Vary ends
|C1
+
|2
“propagating ideas”
Much better ...
P = 91%
|C1
+
|2
“propagating ideas”
Design of optimal taper injector
Replace with
artificial input…and width
Vary taper length
Excite fundamental mode of input waveguide
Optimise offset 5m
“propagating ideas”
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
“propagating ideas”
Field plot at length=9m
99%
“propagating ideas”
The complete result!!!
97% transmission, variation 5%!!!
“propagating ideas”
IMPROVE TAPER FURTHER?
x1
x2
x3 xN),...,,( 21 NxxxP
Optimization problem:
find (x1 , x2 , ... , xN) that maximise P
Could also parametrize shape ...
“propagating ideas”
Here was the original ...
P = 56%
“propagating ideas”
Here is the optimal design ...
15 nodes
P = 88%
“propagating ideas”
P = 97%
39 nodes
Fwd/bwd power “Resonant” region
Using lots of nodes
“propagating ideas”
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
“propagating ideas”
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
“propagating ideas”
3D simulations
Air holes
membrane with refractive index 2.5
Vary height
“propagating ideas”
FDTD 3D
Probes just inside crystal
Input waveguide
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0%
10%
20%
30%
40%
50%
60%
70%
80%
1.1 1.2 1.3 1.4 1.5 1.6
steering
resonant
original
“propagating ideas”
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”