LIGO G050363 00 R OPTIMIZED COATINGS Juri Agresti

LIGO G-050363 -00 -R 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 The Waves Group 12 -17 August 2005, LIGO Hanford Observatory

LIGO G-050363 -00 -R Rationale • Current mirror design: quarter-wavelength (QWL) alternating Si. O 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

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

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

LIGO G-050363 -00 -R • Nice Features Genetic Optimization Multiple, heterogeneous mixed continuous/discrete constraints; Multi-objective and/or best tradeoff optimization; Robust. -structural/rheology-related constraints; Available options include: -multiple-wavelength operation; -several (> 2) materials, etc. … 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)… Status: PIKAIA-based Code-kernel developed. 12 -17 August 2005, LIGO Hanford Observatory

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

Stacked Doublet Optimization LIGO G-050363 -00 -R 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: total (physical) thicknesses C Tantala+ -1 Silica 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. 12 -17 August 2005, LIGO Hanford Observatory

LIGO G-050363 -00 -R Stacked Doublet Optimization z. S = optical length Si. O 2 layer z. T = optical length Ta 2 O 5 layer In units of 0 0. 5 Increasing doublet noise 0. 4 zz. TT 0. 3 0. 2 0. 1 0 z. S+z. T=1/2 0 0. 1 0. 2 0. 3 zz. SS 0. 4 0. 5 12 -17 August 2005, LIGO Hanford Observatory

LIGO G-050363 -00 -R Stacked Doublet Optimization: Approximation # 1 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

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

Constructing Tradeoff Curves LIGO G-050363 -00 -R -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. 12 -17 August 2005, LIGO Hanford Observatory

Stacked Doublet Optimization LIGO G-050363 -00 -R Each point on any curve corresponds to a z. T/z. S value. 8. 3 ppm =10 #d ou ble ts Noise Status: Transmissivity vs. Noise Tradeoff Curves Drawn 12 -17 August 2005, LIGO Hanford Observatory

LIGO G-050363 -00 -R Stacked Doublet Optimization z. T / z. S Noise Noie (arbitrary units) 2. 7 2. 6 2. 5 2. 4 18. 3 ppm |2 | QS QT 10 ± 3%errorbar Current LIGO design z. S +z. T 1 2 circles Unconstrained bullets Tantala noise Silica noise 2. 3 optimum 2. 2 19 20 21 22 23 24 25 26 27 28 29 30 Numberof doublets 12 -17 August 2005, LIGO Hanford Observatory

Quarter Wavelength (yellow bullets) vs. Optimized (grey bullets) Stacked Doublet Design. Transmissivity 8. 3 ppm. Different Si. O 2/Ta 2 O 5 loss ratios. LIGO G-050363 -00 -R 2. 8 1 -| | 8. 3 ppm =30 8. 3 ppm 1 -| | 2. 7 =10 Noise Arbitrary units 2 2. 6 2. 5 2. 4 2. 2 2 2. 3 1. 8 2. 2 40 45 50 55 numberof layers 60 ~14% noise reduction. Nd raised from 19 to 22 -24 (absolute optimum at 23) [Si. O 2]= 249. 583 nm [Ta 2 O 5]= 80. 8843 nm 40 45 50 55 60 numberof layers 65 70 ~24% noise reduction. Nd raised from 19 to 25 (absolute optimum at 29) [Si. O 2]= 280. 216 nm [Ta 2 O 5]= 59. 3664 nm

Quarter Wavelength (yellow bullets) vs. Optimized (grey bullets) Stacked Doublet Design. Transmissivity 1. 12 ppm. Different Si. O 2/Ta 2 O 5 loss ratios. LIGO G-050363 -00 -R 2. 8 2 3 1. 12 ppm 1 -| | Noise Arbitrary units 3. 1 =10 2. 9 2. 8 1 -| | 1. 12 ppm =30 2. 4 2. 2 2 2. 7 2. 6 2 2. 6 1. 8 40 45 50 55 numberof layers 60 40 45 50 55 60 numberof layers 65 70 ~14% noise reduction. Nd raised from 22 to 25 -28 (absolute optimum at 27) ~24% noise reduction. Nd raised from 22 to 28 (absolute optimum at 33) [Si. O 2]= 251. 871 nm [Ta 2 O 5]= 60. 5726 nm [Si. O 2]= 278. 465 nm [Ta 2 O 5]= 60. 5726 nm

LIGO G-050363 -00 -R GA Engineered Prototype 5 (after 10 generations) Goal: 1 -| |2 < 15 ppm. L[Ta 2 O 5] < 2000 nm, =. Ta 2 O 5 layer# 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 thickness [nm] 250 200 150 100 50 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 Si. O 2 layer# 12 -17 August 2005, LIGO Hanford Observatory

GA Prototype, contd. LIGO G-050363 -00 -R vs. nearest-neighbour quarter-wavelengths (QWL) QWL-1 Genetic QWL-2 N (cap included) 36 44 28 1 -| |2 ppm 16. 20 14. 91 235. 46 L(Ta 2 O 5) nm 2359. 43 1815. 61 1835. 11 L(Si. O 2) nm 3479. 98 5217. 4 2747. 35 Ltot 5839. 41 7033. 01 4582. 46 nm 12 -17 August 2005, LIGO Hanford Observatory

Stacked Doublets: Lesson from GA: Tweak End Layers to Improve Reflectivity ! LIGO G-050363 -00 -R 1 -| |2 z. N z 1 (z 1 = 0. 0943, z. N =0. 0437, in units of ) reflectance increased by ~ 10%, noise increased by ~ 1% 12 -17 August 2005, LIGO Hanford Observatory

GA Prototype Characterization LIGO G-050363 -00 -R 18 Mirror frequency response (normal incidence). 16 15 14 1. 75 13 1. 5 12 1030 1040 1050 0 nm 1060 Distribution of 1 -| |2. Random uniform errors, 104 trials. 1. 25 1070 PDF 1 - 2 106 17 1 0. 75 = 14. 94 ppm 3 = 0. 64 ppm 0. 5 0. 25 0 14. 4 14. 6 14. 8 15 15. 2 2 6 1 - 10 12 -17 August 2005, LIGO Hanford Observatory 15. 4 15. 6

Regular Non-Periodic Coatings LIGO G-050363 -00 -R Fractal (e. g. , Cantor); Two main sub-classes: 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] Status: just started 12 -17 August 2005, LIGO Hanford Observatory