1.3 Black hole perturbation theory
1.3.5 An analytical approximation for the scalar Green function in Kerr spacetime
1.3.5.3 My specific contributions
I performed all the calculations up to this point and wrote an initial draft of the paper, with numerous comments and suggestions from Chen.
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Part I
Topics in gravitational wave
detectors
Chapter 2
Brownian thermal noise in multilayer coated mirrors
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 aris- ing 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 in- terface between the coating and substrate, driven by fluctuating shear stresses of the coating. Although thickness fluctuations of different layers are statistically independent, there exists a finite coherence between the layers and the substrate-coating interface. In addition, photoeleastic coefficients of the thin layers (so far not accurately measured) further influences the thermal noise, although at a relatively low level. Taking into ac- count uncertainties in material parameters, we show that significant uncertainties still exist in estimating coating Brownian noise.
Originally published as T. Hong, H. Yang, E. Gustafson, R. X. Adhikari, and Y. Chen, Phys. Rev. D 87, 082001 (2013). Copyright 2013 by the American Physical Society.
2.1 Introduction
Brownian thermal noise in the dielectric coatings of mirrors limits some high precision experiments which use optical metrology. This thermal noise is currently a limit for fixed spacer Fabry-Perots used in optical clock experiments [1] and is estimated to be the dominant noise source in the most sensitive band of modern gravitational wave detectors (e.g., advanced LIGO, GEO, Advanced VIRGO and KAGRA) [2–6]. Recent work has indicated the possibility of reducing the various kinds of internal thermal noise by redesigning the shape of the optical mode [7, 8] or the structure of the multi-layer
coating [9, 10]. In this paper, we seek a more comprehensive understanding of coating Brownian noise. We first identify all thermally fluctuating physical properties (e.g., different components of the strain tensor) of the coating that can lead to coating Brownian noise, and calculate how each of them contributes (linearly) to the total noise; we then calculate their individual levels of fluctuation, as well as cross correlations between pairs of them, using the fluctuation dissipation theorem [11–
13]. In this way, as we compute the total coating Brownian noise, it will be clear how each factor contributes, and we will be in a better position to take advantage of possible correlations between different components of the noise.
As a starting point, we will assume each coating layer to be isotropic, and hence completely characterized by its complex bulk modulus K and shear modulus µ—each with small imaginary parts related to the energy loss in the bulk and shear motions. The complex arguments of these moduli are often referred to as loss angles. While values of K and µ are generally known, loss angles of thin optical layers vary significantly according to the details of the coating process (i.e., how coating materials are applied onto the substrate and their composition). Since the loss angles are small, we will useK andµto denote the real parts of the bulk and shear moduli, and write the complex bulk and shear moduli, ˜Kand ˜µas
K˜ =K(1 +iφB), µ˜=µ(1 +iφS). (2.1)
Here we have used subscriptsB and S to denote bulk and shear, because these will be symbols for bulk strain and shear strain.
Note that our definition differs from that in previous literature, which usedφk andφ⊥ to denote losses induced by elastic deformations parallel and perpendicular to the coating-substrate inter- face [14]. As we shall argue in Appendix 2.C,φkandφ⊥cannot be consistently used as independent loss angles of a material. Only when assumingφk =φ⊥ =φS = φB will the previous calculation agree with ours — if we ignore light penetration into the coating. There is, a priori, no reason why these loss angles should all be equal, although this assumption has so far been compatible with existing ring-down measurements and direct measurements of coating thermal noise [15].
Brownian thermal fluctuations of a multilayer coating can be divided as follows: (i) thickness fluctuation of the coating layers, (ii) fluctuation of the coating-substrate interface, and (iii) refractive index fluctuations of the coating layers associated with longitudinal (thickness) and transverse (area) elastic deformations—as illustrated in Figure 2.1. Using what is sometimes referred to as Levin’s direct approach [12] (based on the fluctuation dissipation theorem), and writing the coating Brownian noise as a linear combination of the above fluctuations, allows the construction of a corresponding set of forces acting on the coating and calculation of the thermal noise spectrum from the the dissipation associated with the simultaneous application of these forces. This has been carried out by Gurkovsky
substrate nsub
nN-1 nN n2 n1
coating layers
z
(x,y) 1 2
N−1 N
j δlj(x,y)
(δA/A)j(x,y)
δzs(x,y)
layer index
δnj(x,y) incoming power profile I(x,y)
l1 l2 lj
lN-1 lN uin(x,y) uout(x,y)
v1(x,y) v2(x,y) fixed
plane
s1(x,y) s2(x,y)
fixed plane
Figure 2.1: Drawing of a mirror coated with multiple dielectric layers. Shown here are the various fluctuations that contribute to coating noise, i.e., fluctuations in the amplitude and phase of the returning light caused by fluctuations in the geometry [including: layer thickness δlj, layer area stretch (δA/A)j, interface height zs of the coating-substrate configuration] and in the refractive indicesδnj(x, y, z) of the layers.
and Vyatchanin [16], as well as Kondratiev, Gorkovsky and Gorodetsky [17]. However, in order to obtain insights into coating noise that have proven useful we have chosen to calculate the cross spectral densities for each of (i), (ii), and (iii), and provide intuitive interpretations of each. We will show, in Sec. 2.4, that (i) and (ii) above are driven by both bulk and shear fluctuations in the coating, in such a way that thickness fluctuations of the j-th layer δlj, or in transverse locations separated by more than a coating thickness, are mutually statistically independent, yet each δlj is correlated with the fluctuation of the coating-substrate interfacezs—becausezsis driven by thesum of thermal stresses in the coating layers. We will also show that when coating thickness is much less than the beam spot size, the only significant contribution to (iii) arises from longitudinal (thickness) fluctuations, see Appendix 2.A.4 for details.
This paper is organized as follows. In Sec. 2.2, we express the amplitude and phase of the reflected field in terms of fluctuations in the coating structure, thereby identifying the various components of coating thermal noise. In Sec. 2.3, we introduce the loss angles of isotropic coating materials, and use the fluctuation-dissipation theorem to calculate the cross spectral densities of the coating thermal noise ignoring light penetration into the multi-layer coating. In Sec. 2.4, we discuss in detail the cross spectra of all the components of the coating structure fluctuation, thereby obtaining the full formula for coating thermal noise, taking light penetration within the muli-layers into account.
The key formulas summarizing phase and amplitude noise spectrum are given in Eq. (2.94) and Eq. (2.95). In Sec. 2.5, we discuss the effect of light penetration on coating thermal noise, using typical optical coating structures. In Sec. 2.6, we discuss the dependence of thermal noise on the material parameters, and optimize the coating structure in order to lower the thermal noise. In Sec. 2.7, we discuss how only one combination of the two loss angles have been measured in past experiments, and how other different combinations can be measured using a new experimental geometry. Finally, we summarize our main conclusions in Sec. 2.8.