Minimum noise optical coatings for interferometric detectors of gravitational waves

Coating Brownian noise is the dominant noise term, in a frequency band from a few tens to a few hundreds Hz, for all Earth-bound detectors of gravitational waves. Minimizing such noise is mandatory to increase the visibility distance of these instruments, and eventually reach their quantum limit. Several strategies are possible to achieve this goal. Layer thickness optimization is the simplest option, yielding a sensible noise reduction with limited technological challenges. Experimental results confirm the accuracy of the underlying theory, and the robustness of the design.


I. INTRODUCTION
The birth of Gravitational Wave Astronomy will open a new and unique window on the Universe [1].Several gravitational wave (henceforth GW) detectors are being constructed, upgraded or planned, including LIGO [2], GEO [3], VIRGO [4], TAMA [5], ACIGA [6], INDIGO [7], KAGRA (formerly LCGT) [8], and ET [9], in an unprecedented multi-national effort.Gravitational waves, predicted by Einstein relativistic theory of gravitation [10], are ripples in the spacetime fabric produced by massive cosmic objects in accelerated motion, which can be detected using very long baseline optical interferometers [12].The sensitivity of these instruments is limited by noises of different origin (e.g., seismic, thermal and quantum, see Figure 1).A reduction of the noise floor level by a factor p entails a p −3/2 boost of the instrument visibility volume [12].For Earth bound detectors, the power spectral density of the noise floor is minimum in a frequency band between a few tens and a few hundreds Hz, where several cosmic sources of gravitational waves (henceforth GW) are deemed to exist [1], and Brownian noise in the highly reflective coatings of the test masses making up the end-mirrors of the interferometer arms is the dominant noise source [11].Minimizing coating Brownian noise is a must to reach (and eventually beat [13]) the sensitivity quantum limit.
In this paper I review possible approaches to coating noise minimization, with special emphasis on coating design optimization, which was invented at the University of Sannio [14] and successfully implemented in collaboration with the Laboratorie des Matériaux Avancés, Lyon (FR), and the LIGO Lab of the California Institute of Technology, Pasadena CA (USA) [15].

II. COATING THERMAL NOISE
Using the fluctuation-dissipation theorem, the Brownian noise power spectral density in the interferometer test-mass mirror coatings can be cast in the form [11] where k B is Boltzmann's constant, T the absolute temperature, w the half-width of the (Gaussian) laser beam, σ s and Y s are the Poisson's ratio and Young's elastic modulus of the substrate, and φ c is the effective mechanical loss angle of coating .To reduce S B one thus could: i) cool the mirrors (i.e., decrease T ); ii) increase the illuminated area (represented in 1 by w); iii) reduce the coating loss angle φ c .I shall focus here on the third option (the other two will be shortly discussed in Section V).
Coatings are presently made of alternating layers of two dielectric materials (amorphous glassy oxides) with different refractive indexes.In the limit where the materials' Poisson's ratios are vanishingly small, we have a simple formula for the coating loss angle [11]: where d L and d H are the total thicknesses of the low(er) and high(er) index materials, respectively, φ L,H their mechanical loss angles, and Y L,H denoting the materials' Young's moduli.The quantities will be henceforth referred to as the specific loss angles (loss angles per unit thickness).
The quarter-wavelength (or Bragg) coating design, where the thickness of the individual layers is δ L,H = λ 0 /4n L,H , λ 0 being the operating wavelength, yields the minimum number of layers to achieve a prescribed transmittance [14], and is the usual choice for all applications where noise is not an issue.
Careful material downselection led to the choice of SiO 2 (Silica) and T a 2 O 5 (Tantala) as the best available materials for the highly reflective (henceforth HR) coatings of GW detectors [16], yielding the best tradeoff among high dielectric contrast (large ratio n h /n L ), low optical absorption (small extinction coefficients, κ L,H ), and low thermal noise (small specific loss angles γ L,H ).
The specific loss angles of Silica and Tantala, however, are quite different, being γ H ≈ 10γ L .This suggests trying different (non-Bragg) designs, where the coating loss angle (hence the coating noise) is minimized subject to a coating transmittance constraint.

III. COATING THICKNESS OPTIMIZATION
Genetic optimization, where no a-priori assumption is made about the geometric structure of the sought optimal coatings (total number of layers and thicknesses of the individual layers) showed that (except for the coating top and bottom layers), the optimal coating consists of a stack of equal doublets with optical thicknesses z L,H such that z L +z H ≈ 1/2, and z H < 1/4 < z L [17].The number of free design parameters is accordingly reduced to just four: the total number N d of doublets, a quantity ξ ∈ (0, 1/4), such that z L,H = 1/4 ± ξ, and the optical thicknesses of the top and bottom layers, z T and z B .Coating optimization is accordingly most easily implemented sequentally through the following steps [14]: i) start from the quarter wavelength design with (power) transmittance τ 0 closest to the design value, and consisting of N d = N 0 doublets; ii) add one doublet, and adjust the layers thicknesses (by varying the single parameter ξ) to make the coating transmittance equal to τ 0 ; iii) calculate the loss angle φ c , and repeat step ii) until φ c reaches a minimum.This procedure results into a shallow minimum, as shown in Figure 2, suggesting that the design will be robust against possible inaccuraccies in the assumed values of γ L,H , and unavoidable coating deposition tolerances [14].The final steps consists in: iv) adjusting the

IV. OPTIMIZED COATING PROTOTYPES
The above coating optimization procedure was used to produce a batch of mirrors suited for the Caltech Thermal Noise Interferometer (TNI), an instrument designed for the direct measurement of coating thermal noise [18], shown in Figure 3.The optimized prototypes were designed at the University of Sannio, and manufactured by LMA (Laboratoire Materiaux Avancés of CNRS-In2P3, Lyon, FR), using their large ionbeam sputtering (IBS) chamber, shown in Figure 4.
The optimized coating thermal noise was measured with high accuracy, and compared to that of standard quarterwavelength coatings having the same transmittance (τ = 287 ppm @1064 nm).The optimized and reference (quarter-wavelength) coatings are sketched in Figure 5.The measurement setup, and the data analysis procedure are described in detail in [15].The measured loss angle of the optimized coatings was lower by a factor p = 0.82 ± 0.04 compared to that of the quarter wavelength coatings.This value reproduced, within the estimated uncertainty range of the measurements and the nominal accuracy of the material parameters, our modeling predictions, confirming the validity and effectiveness of our optimization strategy.

A. Optimized Dichroic Coatings
Advanced (2nd generation) interferometers will use the 2nd harmonic of the laser beam for alignment purposes.The test mass coatings must be accordingly dichroic, and besides being highly reflective at the fundamental wavelength λ 0 (with typical transmittances of a few ppm) should provide some reflectance also at λ 1 = λ 0 /2 (with typical values around 0.9).The originally proposed (reference) dichroic design for the AdLIGO coatings consists of a stack of N 1 doublets grown on top of the mirror substrate, with geometrical thicknesses δ L,H such that topped by a second stack of N 0 doublets with geometrical thicknesses ∆ L,H such that Neglecting chromatic dispersion in the materials, i.e., assuming n L,H (λ 0 ) = n L,H (λ 1 ) = n L,H , eq. ( 6) entails Hence, at λ = λ 1 the top stack is transparent, and the bottom stack, which is effectively quarter-wavelength, is designed to provide the prescribed reflectance.Similarly, eq. ( 5) entails Hence, at λ = λ 0 , the bottom stack contributes part of the required reflectance, and the top stack, which is quarterwavelength, is designed to bring the reflectance to the prescribed level [19].
In order to shed light on the structure of minimal noise dichroic coatings, without making any a-priori assumption about the number and thickness of the individual layers, we resorted again to genetic optimization to seek coating configurations which minimize the coating Brownian noise under a dichroic transmittance constraint [19].Genetically optimized coatings were found to consist of a stack of equal doublets (except for the coating top and bottom layers) with thicknesses z L,H such that z H < 1/4 < z L at λ = λ 0 , like for single-wavelenght operation.At variance with this latter, however, in the dichroic case z L +z H = 1/2 .The number of free design parameters is accordingly reduced Coating thickness optimization of the AdLIGO end-test-mass (ETM) dichroic coatings reduces the loss angle by a factor ∼ 0.88 compared to the reference design.The spectral response is also improved, as shown in Figure 7. Modeling predictions have been fully confirmed by TNI measurements on thickness optimized dichroic prototypes [21].

V. MORE COATING NOISE REDUCTION STRATEGIES
In this Section we present a compact overview of different test-mass Brownian noise reduction strategies proposed so far, including those mentioned at the end of Section II .

A. Better Coating Materials
Coating Brownian noise can be reduced by acting on the relevant material properties, represented by the specific loss angles γ L,H in eq. ( 4), and the refraction indexes n L,H .Smaller γ L,H values and larger (contrast) n H /n L ratios (which help reducing the number of layers needed to achieve a prescribed reflectance, and hence the coating ticknesses d L,H ) yield lower Brownian noise.The most successful attempt in this direction is likely represented by the development of T iO 2 :: T a 2 O 5 (Titania-Tantala) [22] and T iO 2 :: SiO 2 (Titania-Silica) [23] mixtures.Mechanical losses in amorphous materials are associated with thermally activated local transitions between the minima of asymmetric bistable potentials, and can be computed from knowledge of (the distributions of) their relevant parameters [24].Modeling efforts to deduce these latter from first principles are ongoing [25], [26].Present knowledge, however is not sufficient for engineering glassy oxide mixtures with prescribed properties, nor even for improving them, and the quest for better coating materials still relies on extensive trialand-error [27].

B. Low Temperatures
Lowering the temperature does not reduce coating Brownian noise as much as one would expect from (1).Most coating materials, including Silica and Tantala (plain as well as T iO 2doped) exhibit a mechanical-loss peak at some low temperature (see Figure 8 and [28]), whose height and width depend on the material composition, and the post-deposition annealing schedule [29].Hafnia (Hf O 2 ) and Titania (T iO 2 ) are notable exceptions [30], [31].Unfortunately, both are prone to the formation of crystallites during the annealing phase, which spoils severely their optical (scattering) and mechanical (loss angle) qualities.This difficulty can be circumvented by doping (co-sputtering) these materials with good glass formers, like Silica [32].This has been demonstrated to suppress crystallite formation during high temperature annealing in Hf O 2 and ZrO 2 [33], as well as T iO 2 [34], at dopant concentrations ∼ 20%.Remarkably, Silica doping does not affect the nice lowtemperature properties of Hafnia [35].Cryogenic measurements on Silica doped Titania are underway [36].Fig. 8. Mechanical losses vs temperature, from [28].Losses increase upon reducing temperature, peaking at a certain temperature.

C. Nanolayered Materials
A possible alternative to co-sputtered glassy mixtures is offered by nm-layered materials.Their macroscopic properties are amenable to simple (effective medium theory based) modeling [37], which makes them easily engineerable.Nanometerscale layered films of Hf O 2 and Al 2 O 3 do not crystallize upon annealing up to very high temperatures [38].The same applies to nm-layered T iO 2 and SiO 2 films [39], [40], as illustrated in Figure 9. Cryogenic measurements of the loss angle of Hafnia/Silica and Titania/Silica nm-layered mixture are underway [36].

D. Wide Beams
Wide beams are effective in reducing coating noise by averaging out thermal fuctuations of the mirror surface over Fig. 9. X-ray diffraction spectra of different Titania/Silica nm-layered films, after 24h annealing @ 300C.All films are designed to have the same refraction index and optical thickness, but differ in the total number and thickness of the individual layers.As the number of layers increases (and the layer thickness decrases) the diffraction peak signaling crystallization (Anatase formation) gradually disappears [40].a larger illuminated area.Different families of "wide beams" have been proposed so far, including "mesa" [41], hyperbolic [42] and Bessel beams [43].See [44] for a broad discussion.Gauss-Laguerre modes, in particular, received considerable attention, since they may fit standard spherical-mirror cavities [45], although imposing much tighter mode-matching and astigmatism requirements [46].

E. Radical Alternatives
A number of radical alternatives to present day mirrors, based on amorphous glassy oxide dielectric coatings, have also been proposed.Among these: replacing the mirrors with anti-resonant cavities obtained by leaving only a few coating layers on the front face of transparent test masses, and placing the remaining ones on the back face (Khalili etalons [47]); adopting non-diffractive, coating-free mirrors, based on total internal reflection and Brewster angle coupling [48]; using diffractive (grating-based) monolithic (e.g., Silicon or Sapphire) mirrors [49]; taking advantage of the extreme low losses of epitaxially grown single-crystalline (e.g., GaP/AlGaP ) coatings [50].All these concepts hold significant potential and are being actively explored, but each of them faces specific technological and/or conceptual problems which hinder, at present, their immediate full-scale applicability to GW detectors.

VI. CONCLUSIONS
Coating design optimization for thermal (Brownian) noise minimization in the test-mass mirrors of interferometric detectors of gravitational waves has been reviewed, with emphasis on geometric (thickness) optimization, which was invented at the University of Sannio, and developed in Collaboration with LMA and Caltech.Among all test-mass Brownian noise reduction techniques proposed so far, coating thickness optimization is by far the simplest, best understood, technologically less demanding, and cheapest option, capable of reducing the coating noise power spectral density level by a factor ∼ 0.8, and correspondingly boosting the instrument's visibility distance by a round ∼ 30%.

Fig. 1 .
Fig. 1.Noise power spectral density budget of the advanced LIGO detector in strain (gravitational wave amplitude) units.

Fig. 2 .
Fig. 2. Loss angle (normalized to that of the reference quarter-wavelengh design) of Silica/Tantala coatings with identical transmittance (here 287ppm) but different number N d of doublets, and different layer thicknesses z L,H .The quarter wavelength and minimum noise (optimized thickness) designs are indicated.thickness z B of the bottom (H)-layer to minimize noise, and v) adjusting the thickness z T of the top (L)-layer to bring back the transmittance to τ 0 .

Fig. 4 .
Fig. 4. The large IBS coater at the Laboratoire des Materiaux Avancés (LMA) of the CNRS-IN2P3, Lyon FR, were the optimized coating prototypes designed for the TNI were manufactured (courtesy LMA).

Fig. 6 .
Fig.6.Constant transmittance/reflectance loci in the (ξ L , ξ H ) plane for a 19-doublets Silica/Tantala coating.The Σ(N d ) region for dichroic response constraints of the interval type (AdLIGO) consists of two disjoint subsets (highlighted by the dashed yellow loops), which collapse into two distinct points in the case of equality constraints.Darker/lighter blue shades indicate higher/lower Brownian noise levels.