Optical multilayer coatings of high-reflective mirrors significantly determine the properties of Fabry-Perot resonators. Thermal (Brownian) noise in these coatings produce excess phase noise which can seriously degrade the sensitivity of high-precision measurements with these cavities, in particular in laser gravitational-wave antennas (for example project LIGO), where at the current stage it is one of the main limiting factors. We present a method to calculate this effect accurately and analyze different strategies to diminish it by optimizing the coating. Traditionally this noise is calculated as if the beam is reflected from the surface of the mirror fluctuating due to the sums of the fluctuations of each layer. However the beam in fact penetrates a coating and Brownian expansion of the layers leads to dephasing of interference in the coating and consequently to additional change in reflected phase. Fluctuations in the thickness of a layer change the strain in the medium and hence due to photoelastic effect change the refractive index of this layer. This additional effect should be also considered. It is possible to make the noise smaller preserving the reflectivity by changing the total number of layers and thicknesses of high and low refractive ones. We show how this optimized coating may be constructed analytically rather then numerically as before. We also check the possibility to use internal resonant layers and optimized cap layer to decrease the thermal noise.
The results of measurement of optical mirror coating are presented. These results indicate that Standard Quantum Limit of sensitivity can be reached in the second stage of LIGO project if it is limited by thermoelastic noise in the coating only.
Recent researches show that the fluctuations of the dielectric mirrors coating thickness can introduce a substantial part of the future laser gravitational-wave antennae total noise budget. These fluctuations are especially large in the high-reflectivity end mirrors of the Fabry-Perot cavities which are being used in the laser gravitational-wave antennae. We show here that the influence of these fluctuations can be substantially decreased by using additional short Fabry-Perot cavities, tuned in anti-resonance instead of the end mirrors.
Current interferometric gravitational wave detectors use test masses with mirror coatings formed from multiple layers of dielectric materials, most commonly alternating layers of SiO2 (silica) and Ta2O5 (tantala). However, mechanical loss in the Ta2O5/SiO2 coatings may limit the design sensitivity for advanced detectors. We have investigated sources of mechanical loss in the Ta2O5/SiO2 coatings, including loss associated with the coating-substrate interface, with the coating-layer interfaces, and with the bulk material. Our results indicate that the loss is associated with the bulk coating mat…
We analyze the Brownian thermal noise of a multi-layer dielectric coating, used in high-precision optical measurements including interferometric gravitational-wave detectors. We assume the coating material to be isotropic, and therefore study thermal noises arising from shear and bulk losses of the coating materials. We show that coating noise arises not only from layer thickness fluctuations, but also from fluctuations of the interface between the coating and substrate, driven by internal fluctuating stresses of the coating. In addition, the non-zero photoeleastic coefficients of the thin fil…
Thermal noise of a mirror is one of the limiting noise sources in the high precision measurement such as gravitational-wave detection, and the modeling of thermal noise has been developed and refined over a decade. In this paper, we present a derivation of coating thermal noise of a finite-size cylindrical mirror based on the fluctuation-dissipation theorem. The result agrees to a previous result with an infinite-size mirror in the limit of large thickness, and also agrees to an independent result based on the mode expansion with a thin-mirror approximation. Our study will play an important ro…
We propose a simple way to improve the laser gravitational-wave detectors sensitivity by means of reduction of the number of reflective coating layers of the core optics mirrors. This effects in the proportional decrease of the coating thermal noise, the most notorious among the interferometers technical noise sources. The price for this is the increased quantum noise, as well as high requirements for the pump laser power and power at the beamsplitter. However, as far as these processes depend differently on the coating thickness, we demonstrate that a certain trade-off is possible, yielding a…
The photo-thermal transfer function from absorbed power incident on a dielectric mirror to the effective mirror position is calculated using the coating design as input. The effect is found to change in amplitude and sign for frequencies corresponding to diffusion length comparable to the coating thickness. Transfer functions are calculated for the $Ti$-doped ${\rm Ta_2O_5:SiO_2}$ coating used in Advanced LIGO and for a crystalline ${\rm Al_xGa_{1-x}As}$ coating. The shape of the transfer function at high frequencies is shown to be a sensitive indicator of the effective absorption depth, provi…
We report on thermal noise from the internal friction of dielectric coatings made from alternating layers of Ta2O5 and SiO2 deposited on fused silica substrates. We present calculations of the thermal noise in gravitational wave interferometers due to optical coatings, when the material properties of the coating are different from those of the substrate and the mechanical loss angle in the coating is anisotropic. The loss angle in the coatings for strains parallel to the substrate surface was determined from ringdown experiments. We measured the mechanical quality factor of three fused silica …
In laser gravitational waves detectors optical loss restricts sensitivity. We discuss polarization scattering as one more possible mechanism of optical losses. Circulated inside interferometer light is polarized and after reflection its plane of polarization can turn a little due to reflecting coating of mirror can have slightly different refraction index along axes $x,\, y$ in plane of mirror surface (optical anisotropy). This anisotropy can be produced during manufacture of coating (elasto-optic effect). This orthogonal polarized light, enhanced in cavity, produces polarization optical loss.…
Interferometric gravitational wave detectors use mirrors whose substrates are formed from materials of low intrinsic mechanical dissipation. The two most likely choices for the test masses in future advanced detectors are fused silica or sapphire. These test masses must be coated to form mirrors, highly reflecting at 1064nm. We have measured the excess mechanical losses associated with adding dielectric coatings to substrates of fused silica and calculate the effect of the excess loss on the thermal noise in an advanced interferometer.
In this paper, an analytical solution to the problem of optimal dielectric coating design of mirrors for gravitational wave detectors is found. The technique used to solve this problem is based on Herpin's equivalent layers, which provide a simple, constructive, and analytical solution. The performance of the Herpin-type design exceeds that of the periodic design and is almost equal to the performance of the numerical, non-constructive optimized design obtained by brute force. Note that the existence of explicit analytic constructive solutions of a constrained optimization problem is not guara…
Thermal fluctuations in the coatings used to make high-reflectors are becoming significant noise sources in precision optical measurements and are particularly relevant to advanced gravitational wave detectors. There are two recognized sources of coating thermal noise, mechanical loss and thermal dissipation. Thermal dissipation causes thermal fluctuations in the coating which produce noise via the thermo-elastic and thermo-refractive mechanisms. We treat these mechanisms coherently, give a correction for finite coating thickness, and evaluate the implications for Advanced LIGO.
Beyond Bragg-Mirrors for Gravitational Wave Telescopes: A Fabrication Tolerant Hybrid Metasurface-Bragg Mirror Design
Friedrich Schiller University（弗里德里希·席勒大学）；Technische Universität Braunschweig（布伦瑞克工业大学）；Institut für Halbleitertechnik, Technische Universität Braunschweig（布伦瑞克工业大学半导体技术研究所）；Fraunhofer-Institute for Applied Optics and Precision Engineering IOF（弗劳恩霍夫应用光学和精密工程研究所）；Physikalisch-Technische Bundesanstalt（德国联邦物理技术研究院）
We calculate the thermal noise in half-infinite mirrors containing a layer of arbitrary thickness and depth made of excessively lossy material but with the same elastic material properties as the substrate. For the special case of a thin lossy layer on the surface of the mirror, the excess noise scales as the ratio of the coating loss to the substrate loss and as the ratio of the coating thickness to the laser beam spot size. Assuming a silica substrate with a loss function of 3x10-8 the coating loss must be less than 3x10-5 for a 6 cm spot size and a 7 micrometers thick coating to avoid incre…
In this Letter, we investigate the group delay of optical pulses through multilayer dielectric mirrors (MDM) combined with gravitational wave (GW). We find that the delay increases linearly with MDM length for the transmitted wave packet, whereas the Hartman effect disappears. Our study provides insight into the nature of both quantum tunnelling and GW.
Experimental Realization of Optimized Ternary Mirror Coatings
Università del Sannio（桑尼奥大学）；INFN Sezione di Napoli Gruppo Collegato di Salerno（意大利国家核物理研究院那不勒斯分部萨莱诺协作组）；Laboratoire des Matériaux Avancés - IP2I, CNRS, Université Claude Bernard Lyon 1（先进材料实验室-IP2I，法国国家科学研究中心，里昂第一大学）；Massachusetts Institute of Technology（麻省理工学院）；Università di Salerno（萨莱诺大学）；Università di Roma Tre（罗马第三大学）
Cryogenic Mechanical Loss of GaAs/AlGaAs Crystalline Coatings
Syracuse University（雪城大学）；Cardiff University（卡迪夫大学）；Hobart William Smith Colleges（霍巴特威廉史密斯学院）；California Institute of Technology（加州理工学院）；University of Arizona（亚利桑那大学）
Mirror thermal noise is and will remain one of the main limitations to the sensitivity of gravitational wave detectors based on laser interferometers. We report about projected mirror thermal noise due to losses in the mirror coatings and substrates. The evaluation includes all kind of thermal noises presently known. Several of the envisaged substrate and coating materials are considered. The results for mirrors operated at room temperature and at cryogenic temperature are reported.
Increasing requirements in the sensitivity of interferometric measurements is a common feature of several research fields, from gravitational wave detection to quantum optics. This motivates refined studies of high reflectivity mirrors and of noise sources that are tightly related to their structure. In this work we present an experimental characterization of photothermal and thermo-refractive effects in high reflectivity mirrors, i.e., of the variations in the position of their effective reflection plane due to weak residual power absorption. The measurements are performed by modulating the i…
The sensitivity of kilometer-scale terrestrial gravitational wave interferometers is limited by mirror coating thermal noise. We explore the effect of folding the arm cavities of such interferometers. While simple folding alone does not reduce the mirror coating thermal noise, it makes the folding mirror the critical mirror, opening up a variety of design and upgrade options.
We propose thin single-layer grating waveguide structures to be used as high-reflectivity, but low thermal noise, alternative to conventional coatings for gravitational wave detector test mass mirrors. Grating waveguide (GWG) coatings can show a reflectivity of up to 100% with an overall thickness of less than a wavelength. We theoretically investigate GWG coatings for 1064nm based on tantala (Ta2O5) on a Silica substrate focussing on broad spectral response and low thickness.
Thermodynamical fluctuations of temperature in mirrors may produce surface fluctuations not only through thermal expansion in mirror body but also through thermal expansion in mirror coating. We analyze the last "surface" effect which can be larger than the first "volume" one due to larger thermal expansion coefficient of coating material and smaller effective volume. In particular, these fluctuations may be important in laser interferometric gravitational antennae.
We have measured the mechanical loss of a dielectric multilayer reflective coating (ion-beam sputtered SiO$_2$ and Ta$_2$O$_5$) in cooled mirrors. The loss was nearly independent of the temperature (4 K $\sim$ 300 K), frequency, optical loss, and stress caused by the coating, and the details of the manufacturing processes. The loss angle was $(4 \sim 6) \times 10^{-4}$. The temperature independence of this loss implies that the amplitude of the coating thermal noise, which is a severe limit in any precise measurement, is proportional to the square root of the temperature. Sapphire mirrors at 2…
Reduction of coating thermal noise is a key issue in precise measurements with an optical interferometer. A good example of such a measurement device is a gravitational-wave detector, where each mirror is coated by a few tens of quarter-wavelength dielectric layers to achieve high reflectivity while the thermal-noise level increases with the number of layers. One way to realize the reduction of coating thermal noise, recently proposed by Khalili, is the mechanical separation of the first few layers from the rest so that a major part of the fluctuations contributes only little to the phase shif…
In second-generation, ground-based interferometric gravitational-wave detectors such as Advanced LIGO, the dominant noise at frequencies $f \sim 40$ Hz to $\sim 200$ Hz is expected to be due to thermal fluctuations in the mirrors' substrates and coatings which induce random fluctuations in the shape of the mirror face. The laser-light beam averages over these fluctuations; the larger the beam and the flatter its light-power distribution, the better the averaging and the lower the resulting thermal noise. In semi-infinite mirrors, scaling laws for the influence of beam shape on the four dominan…
The sensitivity of current and planned gravitational wave interferometric detectors is limited, in the most critical frequency region around 100 Hz, by a combination of quantum noise and thermal noise. The latter is dominated by Brownian noise: thermal motion originating from the elastic energy dissipation in the dielectric coatings used in the interferometer mirrors. The energy dissipation is a material property characterized by the mechanical loss angle. We have identified mixtures of titanium dioxide (TiO$_2$) and germanium dioxide (GeO$_2$) that show internal dissipations at a level of 1 $…
In interferometric cryogenic gravitational wave detectors, there are plans to cool mirrors and their suspension systems (payloads) in order to reduce thermal noise, that is, one of the fundamental noise sources. Because of the large payload masses (several hundred kg in total) and their thermal isolation, a cooling time of several months is required. Our calculation shows that a high-emissivity coating (e.g. a diamond-like carbon (DLC) coating) can reduce the cooling time effectively by enhancing radiation heat transfer. Here, we have experimentally verified the effect of the DLC coating on th…
Reduction of thermal noise in dielectric mirror coatings is a key issue for the sensitivity improvement in second and third generation interferometric gravitational wave detectors. Replacing an end mirror of the interferometer by an anti-resonant cavity (a so-called Khalili cavity) has been proposed to realize the reduction of the overall thermal noise level. In this article we show that the use of a Khalili etalon, which requires less hardware than a Khalili cavity, yields still a significant reduction of thermal noise. We identify the optimum distribution of coating layers on the front and r…
Thermal noise arising from mechanical dissipation in dielectric reflective coatings is expected to critically limit the sensitivity of precision measurement systems such as high-resolution optical spectroscopy, optical frequency standards and future generations of interferometric gravitational wave detectors. We present measurements of the effect of post-deposition heat treatment on the temperature dependence of the mechanical dissipation in ion-beam sputtered tantalum pentoxide between 11\,K and 300\,K. We find the temperature dependence of the dissipation is strongly dependent on the tempera…
The form of the vacuum stress-tensor for the quantized scalar field at a dielectric to vacuum interface is studied. The dielectric is modeled to have an index of refraction that varies with frequency. We find that the stress-tensor components, derived from the mode function expansion of the Wightman function, are naturally regularized by the reflection and transmission coefficients of the mode at the boundary. Additionally, the divergence of the vacuum energy associated with a perfectly reflecting mirror is found to disappear for the dielectric mirror at the expense of introducing a new energy…
