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abstract: 'Braginsky, Gorodetsky, and Vyatchanin have shown that thermo-refractive fluctuations are an important source of noise in interferometric gravitational-wave detectors. In particular, the thermo-refractive noise in the GEO600 beamsplitter is expected to make a substantial contribution to the interferometer’s total noise budget. Here we present a new computation of the GEO600 thermo-refractive noise which takes into account the beam’s elliptical profile and, more importantly, the fact that the laser beam induces a standing electromagnetic wave in the beamsplitter. The use of updated parameters results in the overall reduction of the calculated noise amplitude by a factor of $\sim 5$ in the low-frequency part of the GEO600 band, compared to the previous estimates. We also find, by contrast with previous calculations, that thermo-refractive fluctuations result in white noise between 600 Hz and 39 MHz, at a level of $8.5\cdot 10^{-24}$Hz$^{-1/2}$. Finally, we describe a new type of thermal noise, which we call the thermo-chemical noise. This is caused by a random motion of optically-active chemical impurities or structural defects in the direction along a steep intensity gradient of the standing wave. We discuss the potential relevance of the thermo-chemical noise for GEO600.'
author:
- Bruin Benthem
- Yuri Levin
title: 'Thermo-refractive and thermo-chemical noise in the beamsplitter of GEO600 gravitational-wave interferometer.'
---

Introduction and main results for thermo-refractive noise
=========================================================

The optical layout of GEO600, the German-British gravitational-wave detector, differs in an essential way from that of LIGO and VIRGO, the two other working gravitational-wave interferometers. In both LIGO and VIRGO each of the arms consists of a high-quality Fabry-Perot cavity, where most of the light power is concentrated, while the beamsplitter is located outside of both cavities. By contrast, in GEO600 the light, after passing the beamsplitter, is processed only once through each of the doubly-folded arms [@wilke]. Therefore, GEO600 is much more sensitive to the noise originating at the beamsplitter than its LIGO/VIRGO counterparts.

The beamsplitter in the GEO600 laser interferometer induces a difference in the optical path lengths between the interferometer’s two arms since one of the beams is transmitted through the beamsplitter twice while is the other is being reflected from its surface. This difference in the optical path length can be modified by a change in the refractive index of the beamsplitter due to random fluctuations of temperature in the refracting material. As Braginsky, Gorodetsky, and Vyatchanin have shown [@BGV], the temperature fluctuations result from a random heat flow between different parts of the refractive medium. Since the interrogating beam is not homogeneous but has a Gaussian profile the inhomogeneous temperature fluctuations produce a net (fluctuating) change in the transmitted beam’s phase which is seen as noise in the overall output of the instrument. For a circular Gaussian beam, Braginsky and Vyatchanin [@BV] have computed the following spectral density of this thermo-refractive noise expressed as an equivalent change in the end mirror’s position $\delta z$: $$S^{\delta z}(\omega) = \frac{4k_B\kappa T^2\beta^2 a}{\pi(C\rho r_0^2 \omega)^2}
\label{BVresult}$$ Here, $k_B$ is Boltzmann’s constant, $\kappa$ the thermal conductivity, $T$ the temperature, $\beta = \partial n/\partial T$ where $n$ is the refractive index, $a$ is the beamsplitter’s thickness, $C$ the specific heat, $\rho$ the density and $r_0$ is the beam’s radius defined in terms of the beam intensity, so that $I=I_0\exp(-r^2/r_0^2)$. The expression in Eq. (\[BVresult\]) has been used to estimate the thermo-refractive noise in the GEO600 beamsplitter [@Reid].

However, inside the GEO600 beamsplitter, the beam is not circular: the $45^\circ$ angle of incidence and the refractive index $n=1.45$ of fused silica introduces ellipticity of the beam. The ratio of the major and minor axes of the beam’s elliptical crossection is given by $\eta=1.23$, while the minor axis equals the width of the original beam. More importantly, the beam is not homogeneous in the longitudinal direction since a standing wave is formed in the beamsplitter as a result of the beam’s reflection from the end mirror. The changes in the material’s refractive index [@footnote1] close to the nodes of the electric-field intensity have less effect on the wave’s phase than changes in the antinodes. Therefore, random thermal fluctuations on the scale of the light wavelength are important, making for a much higher level of noise at frequencies higher than $\sim 600$ Hz than the BV result would suggest.

In this paper we derive the expression for the thermo-refractive noise with these corrections taken into account. We obtain the following result: $$\begin{aligned}
S^{\delta z}(\omega) = &\frac{4k_B\kappa T^2\beta^2a'}{\pi(C\rho r_0^2\omega)^2}\frac{\eta + \eta^{-1}}{2\eta^2}\cdot \nonumber \\ 
&\left[1+\frac{2k^2r_0^2\eta}{(\eta+\eta^{-1})(1+(2kl_{th})^4)}\right] 
\label{result}\end{aligned}$$ Here $a'=a/\cos(i)$ is the optical pathlength through the beamsplitter, where $i=\arcsin(1/\sqrt{2}n)\sim 29^\circ$ is the angle between the beam axis and the normal to the beamsplitter, $k=2\pi n/\lambda$ is the wavevector and $l_{th} = \sqrt{\kappa/(C\rho\omega)}$ is the thermal diffusion length. The numerical values relevant for GEO600 are given in table \[table1\].

    Symbol          (GEO600-)Value
  ---------- ----------------------------
    $k_B$     $1.38 \cdot 10^{-23}$ J/K
   $\kappa$           1.38 W/mK
     $T$               $300$ K
   $\beta$       $8.5 \cdot 10^{-6}$
     $a$         $8.0\cdot 10^{-2}$m
    $\eta$               1.23
     $k$      $8.56 \cdot 10^6$ m$^{-1}$
    $r_0$        $0.71\cdot 10^{-2}$m
     $C$              746 J/kgK
    $\rho$          2200 kg/m$^3$
     $L$                1200 m

  : Parameter values taken from [@Jerome][]{data-label="table1"}

The first term on the right-hand side of Eq. (\[result\]) is the contribution due to the transverse elliptic profile of the beam, and it agrees with the result in [@BV] for $\eta=1$, while the second term is due to the presence of the standing wave. It is common to express the noise amplitude in terms of the dimensionless metric $h$ for easy comparison to other sources. Figure \[plot\] is a plot of $\sqrt{S^h} = \sqrt{S^{\delta z}}/L$ where $L$ is the interferometers [*unfolded*]{} armlength and figure \[plot2\] is a comparison to the measured noise spectrum and the previous estimate for the beamsplitter thermo-refractive noise [@footnote3].

At a typical gravitational wave frequency of 100 Hz the noise amplitude is $\sqrt{S^h} \sim 5.1\cdot 10^{-23}$Hz$^{-1/2}$. For low frequencies the effect of the standing wave is negligible and we have the $1/f$ dependence, consistent with the previous calculations. However, for higher frequencies the standing-wave contribution takes over and the noise spectrum becomes white between $600$Hz and $39$Mhz with an amplitude $\sqrt{S^h} \sim 8.5\cdot 10^{-24}$Hz$^{-1/2}$. At the latter frequency the thermal diffusion length becomes comparable to the wavelength of the beam light, and the $1/f$ dependence is recovered but at much higher value than would be predicted by the BV formula.

Method of calculation
=====================

The foundation of this analysis is direct method [@Levin] for calculating thermo-refractive noise. The relevant readout variable is the phase of the output beam, translated into a change in the end mirror’s position $\delta z$. It is a fairly straightforward calculation to show $$\delta z = \int_V d^3\vec{r} \cdot a' \cdot \beta \cdot \delta T(\vec{r}) \cdot q(\vec{r}), 
\label{genvar}$$ where the integral is over the beamsplitter’s volume $V$, $\delta T(\vec{r})$ is the temperature fluctuation at $\vec{r}$, and $q(\vec{r})$ is the form-factor given, taking the elliptic Gaussian beam and standing wave into account, by $$q(\vec{r}) = \frac{2}{\pi r_0^2 a' \eta} \exp\left[-\left(\frac{x^2}{r_0^2}+\frac{y^2}{(\eta r_0)^2}\right)
\right]\sin^2(kz).
\label{formfac}$$

We now calculate the noise of this generalized variable via the hypothetical experiment proceeding in three steps:

1.  Periodically inject entropy into the medium with volume density $$\frac{\delta S(\vec{r},t)}{dV} = F_0\cos(\omega t) a'\beta q(\vec{r})$$

2.  Calculate the dissipated power $W_\text{diss}$ as a result of this entropy injection.

    To do the latter, we solve the heat equation $$C\rho\frac{\partial \delta T}{\partial t} - \kappa \nabla^2\delta T 
    = T\frac{\partial}{\partial t}\frac{\delta S(\vec{r},t)}{dV}$$ while keeping in mind that:

    -   the solution is periodic with frequency $\omega$

    -   the wavelength is much smaller than the beamsplitter thickness so we can ignore boundary effects

    -   the diffusion of the osscilating temperature in the transverse direction is negligible, giving $\nabla^2 = \partial^2/\partial z^2$, since $r_0 >> l_{th}$ for all frequencies of interest.

    The dissipated power is then computed by using the standard expression $$W_\text{diss}=\int_V d^3\vec{r} \frac{\kappa}{T}\langle(\nabla \delta T)^2\rangle$$ taken from e.g. Landau and Lifshitz [@LL], where $\langle ...\rangle$ denotes the time average over one period.

3.  Calculate the spectral density of the noise in $\delta z$ using the fluctuation dissipation theorem [@Levin], [@CW]: $$S^{\delta z}(\omega) = \frac{8k_B T}{\omega^2}\frac{W_\text{diss}}{F_0^2}\label{fdt}$$

These three steps lead directly to the main result in Eq. (\[result\]).

Thermo-chemical noise
=====================

The small scale of the wavelength opens up the possibility for a new type of thermal noise which has not yet been considered. The fused silica used for contemporary beamsplitters contains minute quantities of contaminants such as OH ions, Cl ions and other defects which have an effect on the refractive index depending on their concentration. As these optically-active contaminants diffuse up and down the steep gradient of the standing-wave electric-field intensity, they cause fluctuations of the overall beam’s phaseshift.

Lets compute this thermo-chemical noise. Let $P(\vec{r})$ be the fluctuating volume concentration of the optically active impurities. The optical path change due to these impurities is given by $$\delta z = \int_V d^3\vec{r}\cdot a'\cdot \alpha \cdot \delta P(\vec{r}) \cdot q(\vec{r}),$$ where $\alpha = \partial n/\partial P$ and $q(\vec{r})$ is given by Eq. \[formfac\].

We can follow our earlier treatment of thermal noise [@Levin2] and calculate the dissipated power in the system under the Hamiltonian $$H_{\rm int}=- F_0\cos(\omega t)\delta z$$ This can be easily done by recalling that the formal expression for $P(\vec{r})$ is $$P(\vec{r})=\Sigma_i \delta(\vec{r}-\vec{r}_i),
\label{P}$$ where $\vec{r}_i$ are the positions of individual optical impurities. Then, under the action of the above Hamiltonian, each impurity experiences a force $$f_i=F_0\cos(\omega t) a' \alpha {\partial q(\vec{r_i})\over\partial z_i}.$$ Under the action of this force, the impurity drifts and energy is dissipated. We assume that the impurities achieve their terminal drift velocity $v_i$ on a timescale much shorter than the frequencies of interest, and use Einstein’s relation [@Einstein] $$Df_i = v_ik_BT$$ to compute the drift velocity [@footnote2]. Here $D$ is the diffusion coëfficient. The dissipated power per particle is $<f_iv_i>$; summing over the particles and substituting the result into Eq. (\[fdt\]) yields $$S^{\delta z}(\omega) = \frac{4DP\alpha^2k^2a'}{\pi r_0^2 \eta \omega^2}\label{result2}$$

Taking values of Suprasil 311 SV used in GEO-600 for OH ions [@Takke], $P(OH) = 3.9\cdot 10^{24} \text{m}^{-3}$ and $\alpha = -4.52\cdot 10^{-31} \text{m}^3$ and estimating a value for $D$ from [@Diffusion] to be in the order of $10^{-20}\text{m}^2/\text{s}$ at room temperature we get $$\sqrt{S^h(\omega)} = 4.4 \cdot 10^{-26}\text{Hz}^{-1/2}$$ at a frequency of 100 Hz, suggesting that a simple version of the thermo-chemical noise is outside the realm of relevance for the GEO600 interferometer. However, there may be other optically-active mobile impurities (e.g., small structural defects or localized 2-state systems) in glass that have not yet been considered. Potential presence of such impurities must be thoroughly investigated in future work.

Discussion
==========

The calculations presented here demonstrate that the thermo-refractive fluctuations result in a noise floor which, while substantially below currently measured noise, may become important for future updates of the GEO600 interferometer. The standing-wave contribution should be taken into account in all future calculations of the thermo-refractive noise from transmissive optics.

As a side product, we have identified a new type of thermal noise: the thermo-chemical noise, which is also enabled by the presence of the optical standing wave in the beamsplitter. The naive estimates of this noise, which are based on what is known about the chemical impurities in Suprasil 311, place it beyond the realm of concern. However, one needs to be vigilant about other types of mobile impurities which have not yet been identified, and which could potentially produce a higher thermo-chemical noise.

Acknowledgements
================

We thank Jerome Degallaix for bringing our attention to thermo-refractive noise in GEO600 beamsplitters, and for authoritative updates on the current GEO600 parameters. We thank Stuart Reid for showing us his preprint with the thermo-refracrive noise estimates based on the Braginsky-Vyatchanin formula, and Kip Thorne and Wim van Saarloos for useful discussions. We also thank Ralf Takke of Heraeus for providing information about chemical impurities in the Suprasil 311 fused silica glass.

[99]{} B. Wilke, et al., Classical and Quantum Gravity, 19 (2002) 1377 V.B. Braginsky, M. Gorodetsky, S.P. Vyatchanin, Phys. Lett. A 271 (2000) 303 V.B. Braginsky and S.P. Vyatchanin, Phys. Lett. A 324 (2004) 345-360 Stuart Reid (private communications). In the beamsplitter’s dielectric material the change of the index of refraction is caused by the change in the dielectric constant. Jerome Degallaix (private communications). We remark that there is a significant difference between the updated parameters [@Jerome] and those used for the calculations done previously [@Reid]. Current $\beta$ is $8.5$ ppm compared to the previous value of $15$ ppm, current $r_0$ is $1/\sqrt{2}$cm compared to the previous $0.88/\sqrt{2}$cm, and it is essencial to use the length of the unfolded arm $L=1200$m in the calculation of $S_h$ as opposed to the previously used $L=600$m. All of these corrections, together with the beam ellipticity, reduce the calculated magnitude of the thermo-refractive noise. Y. Levin, Phys. Lett. A 372 (2008) 1941 L.D. Landau and E.M. Lifshitz, Theory of Elasticity, third ed., Pergamon, Oxford, 1986 H.B. Callen, T.A. Welton, Phys. Rev. 83 (1951) 34 Y. Levin, Phys. Rev. D 57 (1998) 659 U.M.B. Marconi et. al., Physics Reports 461 (2008) 111 Another assumption we make is that at frequencies of interest the diffusion scale is much smaller than the light wavelength. This assumption is satisfied with a very high degree of precision. Ralf Takke (private communications). K.M. Davis, M. Tomozawa, J. Non-Cryst. Solids 185 (1995) 203
