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

Periodic coatings 3/8 SiO2 +/8 Ta2O5

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