1. OPTIMIZED COATINGS Juri Agresti, Giuseppe Castaldi, Riccardo de Salvo, Vincenzo Galdi, Vincenzo Pierro, Innocenzo M. Pinto * LIGO Lab / Caltech TWG / University of Sannio at Benevento LIGO G-050363-00-R The Waves Group 12-17 August 2005, LIGO Hanford Observatory

2. Rationale Current mirror design: quarter-wavelength (QWL) alternating SiO 2 and Ta 2 O 5 layers. Yields largest reflectance among all stacked-doublet designs for any fixed no. of layers (or equivalently, smallest no. of layers at any fixed reflectance). Coating (structural) noise dominates thermal-noise budget in key spectral range. QWL coating does not yield mimimum noise for a prescribed reflectivity, hence not optimal. 12-17 August 2005, LIGO Hanford Observatory LIGO G-050363-00-R

3. Coating Design Optimization: Status & Work Plan (2005-2006) 12-17 August 2005, LIGO Hanford Observatory Status/Directions Genetic optimization (running) Stacked-doublet (completed) Regular non-periodic (just started) The choice, for highest design flexibility and insight; Most obvious generalization of stacked quarter wavelength; getting closer to the “perfect mirror”; On top of this: new materials (e.g., JMM TiO 2 -doped Tantala) LIGO G-050363-00-R

4. Funding Proposal 12-17 August 2005, LIGO Hanford Observatory INFN – COAT (2006 - PI Innocenzo M. Pinto) Participants: TWG (algorithm and code), CALTECH LIGO-Lab (substrates & TNI), LMA Lyon, FR (prototyping; bare costs). Goal: prototyping four GA-optimized mirrors to be tested at CALTECH TNI. Time-span: 1 year (2006). Requested budget: 50 KEU (60 K$). Partnerships: ILIAS-Strega (S. Rowan/J. Hough), TAMA (Tsubono K.), VIRGO (F. Vetrano). LIGO G-050363-00-R

5. Genetic Optimization 12-17 August 2005, LIGO Hanford Observatory Educated ignorance attitude (almost no a-priori assumption on structure of sought solution - will shed light on it !); Effective & well established (e.g. microwave antenna and filter design)… -structural/rheology-related constraints; -multiple-wavelength operation; -several (> 2) materials, etc.… Available options include: Nice Features Multiple, heterogeneous mixed continuous/discrete constraints; Multi-objective and/or best tradeoff optimization; Robust. Status: PIKAIA-based Code-kernel developed. LIGO G-050363-00-R

6. Crossover+Mutation s -Problem unknowns  genes; -Point in search space  chromosome; -Set of points in search space  population; -Evolve random initial population according to an evolutionary schedule Genetic Algos in a Nutshell LIGO G-050363-00-R 12-17 August 2005, LIGO Hanford Observatory

7. Stacked Doublet Optimization 12-17 August 2005, LIGO Hanford Observatory Most obvious generalization of current stacked- /4-design. Coating reflectivity is a monotonic (increasing) function of Bloch characteristic exponent (BCE) of transmission matrix of basic doublet (true for any truncated-periodic); Coating noise is closely modeled by a simple (linear) law: LIGO G-050363-00-R  C  Tantala +  -1  Silica  total (physical) thicknesses  related to Young moduli, Poisson ratios & loss angles of both substrate & coating materials. In view of present measurement uncertainties  can be anything between 10 and 30.

8. Stacked Doublet Optimization 00.10.20.30.40.5 0 0.1 0.2 0.3 0.4 0.5 z S +z T =1/2 zTzT zSzS Increasing doublet noise zTzT zSzS z S = optical length SiO 2 layer z T = optical length Ta 2 O 5 layer In units of 0 12-17 August 2005, LIGO Hanford Observatory LIGO G-050363-00-R

9. BCE contour lines very thin: no sensible difference between exact and approximate (z T +z S =1/2) optimization 12-17 August 2005, LIGO Hanford Observatory Stacked Doublet Optimization: Approximation # 1 LIGO G-050363-00-R

10. 12-17 August 2005, LIGO Hanford Observatory Stacked Doublet Optimization: Approximation # 2 LIGO G-050363-00-R …both absorbed by large by uncertainty in 

11. Constructing Tradeoff Curves -Assign number N of doublets; -Assign noise upper-bound noise for whole coating; -Compute corresponding upper-bound for single doublet; -Determine z S and z T so as to maximize BCE; under tha above noise constraint; -Compute terminated N-doublet reflection coefficient. LIGO G-050363-00-R 12-17 August 2005, LIGO Hanford Observatory

12. LIGO G-050363-00-R Stacked Doublet Optimization Status: Transmissivity vs. Noise Tradeoff Curves Drawn Each point on any curve corresponds to a z T /z S value. 8.3 ppm Noise # doublets 12-17 August 2005, LIGO Hanford Observatory  =10

13. Current LIGO design z T +z S =1/2 Tantala noise Silica noise LIGO G-050363-00-R 12-17 August 2005, LIGO Hanford Observatory Stacked Doublet Optimization optimum

16. GA Engineered Prototype 135791113151719212325272931333537394143 SiO 2 layer # 0 50 100 150 200 250 s s e n k c i h t [nm] 2468101214161820222426283032343638404244 Ta 2 O 5 layer# Goal: 1-|  | 2 < 15 ppm. L[Ta 2 O 5 ] < 2000 nm,  = . (after 10 5 generations) LIGO G-050363-00-R 12-17 August 2005, LIGO Hanford Observatory

17. GA Prototype, contd. QWL-1GeneticQWL-2 N (cap included) 364428 1-|  | 2 ppm 16.2014.91235.46 L(Ta 2 O 5 ) nm 2359.431815.611835.11 L(SiO 2 ) nm 3479.985217.42747.35 L tot nm5839.417033.014582.46 vs. nearest-neighbour quarter-wavelengths (QWL) LIGO G-050363-00-R 12-17 August 2005, LIGO Hanford Observatory

18. Stacked Doublets: Lesson from GA: Tweak End Layers to Improve Reflectivity ! z1z1 zNzN 1 -|  | 2 reflectance increased by ~ 10%, noise increased by ~ 1% (z 1 = 0.0943, z N =0.0437, in units of  ) LIGO G-050363-00-R 12-17 August 2005, LIGO Hanford Observatory

19. 10301040105010601070 l 0 nm 12 13 14 15 16 17 18 1 - G 2 0 1 6 Mirror frequency response (normal incidence). Distribution of 1-|  | 2. Random uniform errors, 10 4 trials. GA Prototype Characterization LIGO G-050363-00-R 12-17 August 2005, LIGO Hanford Observatory

20. Regular Non-Periodic Coatings 12-17 August 2005, LIGO Hanford Observatory Status: just started Two main sub-classes: Fractal (e.g., Cantor); Substitutional (e.g. Fibonacci); Goal: large bandwidths (in frequency and wavenumber) [e.g., Optics Lett. 23 (1998) 1573]; Background: applications in antenna array synthesis [e.g., IEEE Trans. AP-53 (2005) 635] LIGO G-050363-00-R