March 22 2005 LIGO-G050176-00-R
The Waves Group
Non periodic dielectric mirror coatings
• Giuseppe CASTALDI TWG
• Riccardo DESALVO LIGO
• Vincenzo GALDI TWG
• Vincenzo PIERRO TWG
• Innocenzo PINTO TWG
March 22 2005 LIGO-G050176-00-R
The Waves Group
Non periodic dielectric mirror coatings
• HR mirror coatings have 1/2 wavelength periodicity,
• normally 1/4H, 1/4L refraction index layers• But also 1/8H, 3/8L (CSIRO suggested) work OK• First and last layers are often different to match
other requirements• Fully periodic structures are not optimal• Non periodic structures are difficult to calculate
March 22 2005 LIGO-G050176-00-R
The Waves Group
How to design non periodic coatings
• Genetic Algorithm [J.H. Holland (1975) successfully applied to many constrained design problems]
• Example of one (conservative) optimization: – Minimizing transmittance ( 20 ppm);
– Minimizing Ta2O5 thickness ( 2000 nm).
• Alternatives: Regular non-periodic coatings (e.g. pre-fractal) ?
March 22 2005 LIGO-G050176-00-R
The Waves GroupGenetic synthesis in a nutshell
• Multi-objective optimization: find
• Mimic natural selection, by way of analogy
rx ∈S⊆ RN : fk(
rx) ≤εk, k=1,2,...,M
rx= x1,x2 ,...,xN{ }
chromosome genes
multiple objectives/constraints
One gene assigned to each layer thickness and its compositionMarriages, mutations, hanky-panky,…. AllowedBest partner given first choice of preferred partner (requirement taste)
Darwinian selection based on requirements applied every generation
March 22 2005 LIGO-G050176-00-R
The Waves GroupGenetic synthesis in a nutshell, II
• Evolve initial (random/guess) chromosome population
• according to evolutionary schedule
A(0) =
rx1
(0) ,rx2
(0) ,...,rxP
(0){ }
CrossoverCrossover++
MutationMutation
s
March 22 2005 LIGO-G050176-00-R
The Waves Group
Comparing mirrors (15ppm)
0
50
100
150
200
250
300
1 11 21 31 41
Tantala periodic layer [nm]Silica periodic layer [nm]
layer number (from vacuum)
0
50
100
150
200
250
300
1 11 21 31 41
Tantala a-periodic layer [nm]Silica a-periodic layer [nm]
layer number (from vacuum)
15 ppm 15 ppm
• non-periodic periodic /4+/4• (44 layers, 7033 nm) (38 layers, 6153nm)
1816 nm Ta2O5 2490 nm Ta2O5
5217 nm SiO2 3663 nm SiO2
March 22 2005 LIGO-G050176-00-R
The Waves GroupComparing mirrors (44 layers)
• non-periodic periodic 3/8+/8• (44 layers, 7033 nm) (44 layers, 7766 nm)
1816 nm Ta2O5 1430 nm Ta2O5
5217 nm SiO2 6336 nm SiO2
0
50
100
150
200
250
300
1 11 21 31 41
Tantala a-periodic layer [nm]Silica a-periodic layer [nm]
layer number (from vacuum)
0
50
100
150
200
250
300
1 11 21 31 41
Tantala 1/8 periodic layer [nm]Silica 3/8 periodic layer [nm]
Tantala 1/8 periodic layer [nm]
layer number (from vacuum)
15 ppm 60 ppm
March 22 2005 LIGO-G050176-00-R
The Waves Group
Comparing mirrors (15 ppm)
• non-periodic periodic /8+3/8• (44 layers, 7033 nm) (52 layers, 9178 nm)
1816 nm Ta2O5 1690 nm Ta2O5
5217 nm SiO2 7488 nm SiO2 15 ppm 13 ppm
March 22 2005 LIGO-G050176-00-R
The Waves Group
Non-periodic Coating structure properties
0
2
4
6
8
10
12
14
0.1 0.2 0.3 0.4
non periodic mirror coating
tantala layer thicknesssilica layer thickness
number of layers per 0.1
λ wide bin
[layer thicknessλ]
March 22 2005 LIGO-G050176-00-R
The Waves Group
1.5
2
2.5
4 6 8 10 12 14 16 18
Ta2O5 total thickness [micron]
Coating thickness [micron]
First attemptNon-periodic
March 22 2005 LIGO-G050176-00-R
The Waves Group
Genetically optimized (44 layer, 15ppm)non-periodic coating: Robustness
• Introducing 1 nm r.m.s. error on coating thickness• 10,000 trials
14.25 14.5 14.75 15.25 15.5 15.75
0.25
0.5
0.75
1
1.25
1.5
1.75
14.25 14.50 14.75 15.00 15.25 15.75 transmittance [ppm]
Probability density
0.55 ppm FWHM
March 22 2005 LIGO-G050176-00-R
The Waves Group
Genetically optimized (44 layer, 15ppm)non-periodic coating: Bandwidth
March 22 2005 LIGO-G050176-00-R
The Waves Group
Conclusions
• Non-periodic coatings can be designed by genetic algorithms;
• Multiple heterogeneous (e.g., transmittance, thickness of constituents, etc.) constraints can be introduced;
• Preliminary results suggest better overall performance than non periodic coatings