Post on 04-Jan-2016
“propagating ideas”
“propagating ideas”
Rix=2.5
Rix=1
L=0.5 D=0.35 Our Aim
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conditions BD. eappropriat
0
0HE
HE
EH
Fields governed by the source free Maxwell equations
The field equations
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The Field solver
Robust: Must be capable of dealing with any taper shape
Use mode matching method
Accurate: correctly model high contrast structures
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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
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MMItapers
Mode convertersPhotonic crystals
Possible applications
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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
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Field plots in line defect (arbitrary input)
=1.34m
Only 1 mode excited
=1.43m
2 modes excited
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Exciting the PC mode
Choose w=0.351
Design an “artificial” waveguide s.t. its fundamental mode has 100% transmission
W W
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Optimising the y-junction
The initial structure...
Wavelength response
50% transmission
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Setting up the optimisation
D1D2
L
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Problem!
Many local minima holes can overlap and vanishdifferent topological configurations
L,D, or W
P
Many local minima
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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...
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Monitoring interface
Specify your independent variables...Connect them to any structure parameter
define your own objective!
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Optimisation results
A
A
B
B
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D1= 0.38m , D2 = 0.31m , L= - 0.17m
Optimal point A: transmission=99.8%!
Wavelength response
VERY BAD!
Resonant transmission
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Optimal point B: transmission=99.5%!
D1= 0.12m , D2 = 0.47m , L = 0.15m
Wavelength response
MUCH better
steering transmission
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Bend optimisation
D
D
LL
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Optimisation results
Best point
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Best shape : transmission=97%!
L= 0.24m , D = 0.47m
Wavelength response
Resonant transmission
FAIRLY good: variation = 8%
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Bend + y junction
transmission=97%!
Input from here
Wavelength scan
Pretty good!
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Bend optimisation II
D
D
LL
OFFOFF
Idea: try to find optimal steering transmissions
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Optimisation results
2% variation
0.5% variation
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The complete crystal
98% transmission, 1% variation!!!
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Optimal taper design
Inp ut fie ld
Po we r lo ss
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Large losses
…Argh .. Not very good!
56% transmission
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Could make it longer ...
Reduced losses
40 m
Too long!
95% transmission
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Keep length fixed ...
Inp ut fie ld
Po we r lo ss
Maximise power output
Deform shape ...
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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.
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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
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The first example: length 14um...
Rix = 2.5
Rix = 1.0
P = 84%
Vary ends
|C1
+
|2
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Much better ...
P = 91%
|C1
+
|2
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Design of optimal taper injector
Replace with
artificial input…and width
Vary taper length
Excite fundamental mode of input waveguide
Optimise offset 5m
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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
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Field plot at length=9m
99%
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The complete result!!!
97% transmission, variation 5%!!!
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IMPROVE TAPER FURTHER?
x1
x2
x3 xN),...,,( 21 NxxxP
Optimization problem:
find (x1 , x2 , ... , xN) that maximise P
Could also parametrize shape ...
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Here was the original ...
P = 56%
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Here is the optimal design ...
15 nodes
P = 88%
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P = 97%
39 nodes
Fwd/bwd power “Resonant” region
Using lots of nodes
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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
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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
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3D simulations
Air holes
membrane with refractive index 2.5
Vary height
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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
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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”