Optical Coatings in LIGO

Optical Coatings in LIGO Physics 208, Electro-optics Peter Beyersdorf Document info 1

Optical Coatings in LIGO Stack of alternating λ/4 layers of high index Ta 2 O 5 and low index SiO 2 Materials chosen for low loss and high dielectric contrast (n h -n l ) λ/4 thickness maximizes reflectivity for a given number of layers

Mirror Requirements for LIGO Mirror reflectivity determines finesse of arm cavities and should be kept as high as possible to maximize interferometer response Brownian thermal noise of the mirror coating limits noise floor from 30 Hz-300 Hz Conventional mirror coating design optimizes reflectivity for a given number of layers, but what is needed is optimal thermal noise for a given reflectivity

Significance of Coating Design Thermal noise from optical coatings in gravitational wave detectors Gregory M. Harry, Helena Armandula, Eric Black, D. R. M. Crooks, Gianpietro Cagnoli, Jim Hough, Peter Murray, Stuart Reid, Sheila Rowan, Peter Sneddon, Martin M. Fejer, Roger Route, and Steven D. Penn Gravitational waves are a prediction of Einstein s general theory of relativity. These waves are created by massive objects, like neutron stars or black holes, oscillating at speeds appreciable to the speed of light. The detectable effect on the Earth of these waves is extremely small, however, creating strains of the order of 10 21. There are a number of basic physics experiments around the world designed to detect these waves by using interferometers with very long arms, up to 4 km in length. The next-generation interferometers are currently being designed, and the thermal noise in the mirrors will set the sensitivity over much of the usable bandwidth. Thermal noise arising from mechanical loss in the optical coatings put on the mirrors will be a significant source of noise. Achieving higher sensitivity through lower mechanical loss coatings, while preserving the crucial optical and thermal properties, is an area of active research right now. 2006 Optical Society of America OCIS codes: 310.1620, 310.6870, 350.1270. 1. Introduction Isaac Newton s description of gravity was improved upon in 1915 by Albert Einstein when the latter s general theory of relativity was published. This theory allows for oscillations in space time, caused by motions of masses analogous to electromagnetic waves arising from moving charges in Maxwell s theory. These oscillations, known as gravitational waves, create a strain in space time, so the travel time for a light beam between two inertial masses will change as the wave goes by. The size of this strain is set by the ratio G. M. Harry (gharry@ligo.mit.edu), H. Armandula, and E. Black are with the Laser Interferometer Gravitational-Wave Observatory (LIGO) Laboratory; G. M. Harry, with the Massachusetts Institute of Technology, NW17-161, Cambridge, Massachusetts 01239; H. Armandula and E. Black, with the California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125. D. R. M. Crooks, G. Cagnoli, J. Hough, P. Murray, S. Reid, S. Rowan, and P. Sneddon are with the Department of Physics and Astronomy, The University of Glasgow, Glasgow G12 8QQ, United Kingdom. M. M. Fejer and R. Route are with the Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305. S. D. Penn is with the Department of Physics, Hobart and William Smith Colleges, Geneva, New York 14456. Received 1 March 2005; accepted 7 July 2005. 0003-6935/06/071569-06$15.00/0 2006 Optical Society of America 21 Gmv 2 rc 4 10 m M v 100 Mpc c 2 r, (1) where G is Newton s gravitational constant, m is the mass of the source, v is the velocity of the source, r is the distance from the detector to the source, c is the speed of light, M is a solar mass, and Mpc is a megaparsec. To even approach measurable strains, astronomical-sized masses moving at appreciable fractions of the speed of light are necessary. A typical gravitational wave at Earth from a source at intergalactic distances is expected to have a strain near 10 21 or less. This is roughly a change in length equal to the width of a human hair over the distance between the Sun and the nearest star. There are a number of experiments 1 3 that use interferometry to attempt to detect these waves. A typical Michelson interferometer design with two perpendicular arms is shown in Fig. 1. The tensor field of the gravitational wave is most easily detected by using two perpendicular arms. This is in contrast to the vector electromagnetic field, in which a single linear antenna suffices. The mirrors and other optics of the interferometer hang as pendulums. This gives the best approximation of a freely falling mass; nearly free in the sensitive direction of the interferometer but supported against the static gravitational field of the Earth. To increase the signal, many experiments make each long arm a Fabry Perot cavity to increase the interaction time with the mirrors. 1 March 2006 Vol. 45, No. 7 APPLIED OPTICS 1569 In the most sensitive bandwidth, between 40 Hz and a few hundred hertz, thermal noise is the dominant noise source. This is the thermal motion of the mirror faces themselves and comes primarily from the mechanical loss in the optical coatings. The thermal-noise curve in Fig. 3 assumes the same ion-beam-deposited silica-tantala coating as was used for the initial LIGO. This noise will set the ultimate sensitivity, and thus the astronomical effectiveness, of the Advanced LIGO interferometers. Reducing this noise from the level shown would have big payoffs for gravitational wave detection and astronomy.

Conventional dielectric mirror coatings HR mirror coatings have λ/2 periodicity normally λ/4, high and low index layers Alternating 3λ/8, λ/8 layers or (n-1)λ/n, λ/n layer geometry is also possible First and last layers are often different to match other requirements sn.

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 (<2 μm). Alternatives: Regular non-periodic coatings (e.g. pre-fractal)? sn.

Genetic Synthesis C={x 1, x 2, x 3, x N } x 1 x 2 Consider the thickness of each layer in an alternating stack of high and low index material to be one gene in a chromosone describing the mirror Create a population of organisms each with one chromosone that has randomized values for each gene Evaluate fitness of each organism as a function of the mirror reflectivity and tantala thickness for the mirror it represents Preferentially select most fit organisms for breeding Mutate random genes Split chromosones of parents and recombine to produce offspring Evaluate fitness of offspring and repeat mating sequence untila solution of acceptable fitness evolves Ch 6, 7

Genetic Synthesis Cut and splice crossover Uniform Crossover Generation of offspring may be allowed to introduce chromosones of new lengths (cut and splice) or may be constrained to a fixed length. Thus the number of dielectric layers can be held constant or allowed to be flexible.

Mirror Reflectivity Detemrine Mirror reflectivity for each organism by solving eigenvalues of field matirx and determining reflection coefficient a 0 a 1 a 2 a 3 a 4 a 6 a 7 a 8 b 0 b 1 b 2 b 3 b 4 a 5 b 6 b 7 b 8 b 5 a 0 1 0 0 0... a 0 b 0 r 01 0 0 t 01 e ikn 1x 1... b 0 a 1 = t 01 e ikn 1x 1 0 0 0... a 1 b 1 0 0 r 12 0... b 1......... Eigenvectors of matrix give steady state field amplitudes. Ratio of b o /a o 2 is mirror reflectivity.

Tantala Thickness Thickness of Tantala is simply the sum of all odd genes (assuming we define the odd genes to be the high index layers) d T a2 O 5 = x=1,3,5,... x i

Fitness Criteria A (largely arbitrary) function defining how well the mirror meets the design goals, i.e. f(r, d T a2 O 5 ) = ( ) 2 ( ) 2 1 R dt a2 O + 5 15 ppm 5000 nm We seek to minimize this function, with an acceptable solution having a value below 1, i.e. a transmission of less than 15 ppm and a total tantala thickness of less than 5000 nm.

Genetic Algorithm Pseudo Code 1.! Choose initial population with each gene of each! organism randomized to a value from 0 to λ/2 2.! Evaluate the fitness f of each organism in the! population 3.! Repeat until organism with f<1 is found 1.! Select best-ranking individuals to reproduce 2.! Breed new generation through crossover and! mutation (genetic operations) and give birth to! offspring 3.! Evaluate the individual fitnesses f of the offspring 4.! Replace worst ranked part of population with! offspring

Comparing mirrors (15ppm loss) non-periodic periodic λ/4+λ/4 (44 layers, 7033 nm) (38 layers, 6153nm) 1816 nm Ta 2 O 5 2490 nm Ta 2 O 5 5217 nm SiO 2 3663 nm SiO 2 15 ppm 15 ppm sn.

Comparing mirrors (44 layers) non-periodic periodic 3λ/8+λ/8 (44 layers, 7033 nm) (44 layers, 7766 nm) 1816 nm Ta 2 O 5, 5217 nm SiO 2!!!!! 1430 nm Ta 2 O 5, 6336 nm SiO 2!!! 15 60 sn.

Periodic coatings 3λ/8 SiO 2 +λ/8 Ta 2 O 5 1-R 2 sn. 15

Comparing mirrors (»15 ppm) non-periodic (44 layers, 7033 nm) 1816 nm Ta 2 O 5 5217 nm SiO 2 15 ppm!!! periodic (3λ/8, λ/8) (52 layers, 9178 nm) 1690 nm Ta 2 O 5 7488 nm SiO 2 13 ppm!!! sn.

Non-periodic Coating structure properties sn.

Number of layers First attempt Non-periodic 1-R 2 (ppm)

Genetically optimized coating robustness 44 layer, 15ppm design Introduce 1 nm r.m.s. error on coating thickness 10,000 trials 1.75 1.5 Probability density 1.25 1 0.75 0.55 ppm FWHM 0.5 0.25 14.25 14.5 14.75 15.25 15.5 14.25 14.50 14.75 15.00 15.25 15.75 transmittance [ppm] sn.

Genetically optimized coating bandwidth 1-R 2 (ppm) sn.

Summary Non-periodic coatings can be designed by genetic algorithms; Multiple heterogeneous (e.g., transmittance, thickness of constituents, etc.) constraints can be introduced; Non λ/4 periodic structures can also be employed sn.

References Juri Agresti, Giuseppe Castaldi, Riccardo DeSalvo, Vincenzo Galdi, Vincenzo Pierro c, and Innocenzo M. Pinto, Optimized multilayer dielectric mirror coatings for gravitational wave interferometers, LIGO-P060027-00-Z