Thermal modulation for suppression of parametric instability in advanced gravitational wave detectors

TL;DR

This study investigates thermal modulation as a method to suppress three-mode parametric instability in advanced gravitational wave detectors, demonstrating effective stabilization through finite element analysis and thermal control techniques.

Contribution

The paper provides a detailed finite element analysis showing thermal modulation can suppress parametric instability without degrading optical performance in gravitational wave detectors.

Findings

Thermal modulation with a 1 Watt, 0.01 Hz CO2 laser stabilizes acoustic modes.

Suppression factor is linearly proportional to peak modulation power.

Rear surface heating compensates for surface deformation effects.

Abstract

Three-mode parametric instability is a threat to attaining design power levels in Advanced gravitational wave detectors. The first observation of three-mode parametric instability in a long optical cavity revealed that instabilities could be suppressed by time variation of the mirror radius of curvature. In this paper, we present three dimensional finite element analysis of this thermo-acousto-optics system to determine whether thermal modulation could provide sufficient instability's suppression without degrading time averaged optical performance. It is shown that deformations due to the time averaged heating profile on the mirror surface can be compensated by rear surface heating of the test mass. Results show that a $CO_2$ laser heating beam with a modulation amplitude of $1$ Watt at 0.01 Hz is sufficient to stabilize acoustic mode with parametric gain up to 3. The parametric gain suppression factor is linearly proportional to the peak modulation power.

Tables

Table 1. Table 1: The main parameters used for the simulation

Parameters	Value
Test mass material

Equations

$$R=\frac{P\Lambda {\omega }_1}{M{\omega }_m{L}^2{\gamma }_m{\gamma }_0{\gamma }_1}\frac{1}{1+({\Delta }_m/{\gamma }_1{)}^2},$$

$$\Delta \omega =\frac{c}{L}(m+n){\cos }^{-1}\left[\pm \sqrt{(1-\frac{L}{R_1})(1-\frac{L}{R_2})}\right],$$

$$\Delta {\omega }^{(1)}\approx \Delta {\omega }^{(0)}+\beta \delta R_2.$$

$$\beta =(m+n)L(L-R_1)/2R_2\sqrt{(L(L-R_1)(L-R_2)(R_1+R_2-L))},$$

$$\delta T\approx P_{abs}/2\pi \kappa w,$$

$$\delta d=\alpha w\delta T/2\approx \alpha P_{abs}/(4\pi \kappa ),$$

$$\delta R_{roc}=(w/d{)}^2\delta d/2,$$

Full text

1 Introduction

Laser interferometer gravitational wave detectors are the most sensitive displacement sensors that human-beings have ever built. They use laser light to sense the test mass motion induced by passing gravitational waves. For reaching high sensitivity, one has to increase the optical power that interacts with test masses to reduce the quantum shot noise. Advanced detectors utilise Fabry-Perot cavities in the interferometer arms, combined with power recycling, to aim to achieve a circulating optical power inside the cavity approaching 1 MW.

However, such a high optical power on the test mass has undesired effects, which include thermal aberrations [Lawrence2004, Hello2006] due to unavoidable optical absorption of the test masses, and instabilities induced by radiation pressure [Braginsky2001, Sidles2006]. One particular effect that we are interested in this paper is the three-mode parametric instability. The coupling of the high power optical field with the internal acoustic vibration of the test mass mirrors can lead to uncontrolled optomechanical oscillations which leads to instability. Parametric instability limits the circulating power, thereby limiting attainable detector sensitivity.

Three-mode parametric instability involves one acoustic mode and two optical modes. It was first predicted in a seminal work by Braginsky et al. in 2001 [Braginsky2001, Braginsky2002] and then investigated in detail using 3D finite element simulation for Advanced LIGO gravitational wave detectors [Zhao2005, Gras2010]. Three-mode parametric instability in free space Fabry-Perot optomechanical cavities was first observed in a desktop optical experiment in 2013 [Chen2015] and later observed in an $80$ m suspended cavity, in which LIGO type conditions were mimicked, at the Gingin High Optical Power Facility in Western Australia [Zhao2015]. In 2014 parametric instability was observed in an Advanced LIGO detector at LIGO Livingston Observatory (LLO) [Evans2015]. An acoustic mode with $Q=12\times 10^{6}$ and resonant frequency of 15.5 kHz was observed to undergo exponential growth with threshold circulating power of 25kW, finally causing the interferometer lose lock. The observation was in a good agreement with predictions [Gras2010].

Various methods were proposed for controlling parametric instability [Zhao2005, Vyatchanin2012, Evans2010, Ju2009, Degallaix2007, Miller2011, Gras2009, Gras2015, Evans2008]. Because parametric gain is a function of the difference between the optical cavity mode gap between the fundamental mode and higher order transverse mode, and the acoustic mode frequency, tuning of the optical mode gap through thermal tuning of the mirror radius of curvature proved to be an effective way of suppressing parametric instability. However, due to the high acoustic mode density of the test masses, simple thermal tuning would have limited ranges. At the Gingin 80 m suspended optical cavity, an interesting effect was observed: the amplitude growth of an acoustic mode due to parametric instability was modulated [Zhao2015], leading to an effective suppression of parametric gain. It was shown that this effect was due to radius of curvature modulation. The modulation occurred because the cavity optical transverse mode frequency was modulated due to the low frequency residual fluctuations of the beam spot position. Beam spot position in turn modulated the transverse mode frequencies due to figure errors of the mirror which cause every spot position to ”see” a different radius of curvature. In a system where figure errors of the mirrors are very small, the radius of curvature variation with time could be provided by thermal modulation. This idea was preliminary explored using an ideal thermal modulation model in ref. [Zhao2015].

In this paper, we investigate the feasibility of a practical thermal modulation scheme, using the Advanced LIGO parameters, for suppressing the parametric instability. Using 3D finite modeling analysis, we show that thermal modulation of the mirror, combined with back side static heating, could provide sufficient parametric instability suppression without degrading time averaged optical performance. The paper is organized as follows: in section 2, we give a theoretical background of this work and orders of magnitude estimation; Section 3 presents the simulation results, in which section LABEL:sec3_1 establishes the simulation mode and discusses the static effect that needs to be considered in our scheme. Section LABEL:sec3_2 presents numerical results of the dynamical modulation and the suppression of the instability.

2 The principle and orders of magnitude estimation

In an optical cavity, with suspended test mass mirrors, the thermal fluctuations drive large numbers of acoustic modes. Each acoustic mode scatters the intra-cavity fundamental mode field (frequency $\omega_{0}$) into high order transverse optical modes. There will be beating between the fundamental optical mode and a high order optical mode (frequency $\omega_{1}$), with beat frequency $\Delta\omega=\omega_{0}-\omega_{1}$. If $\Delta\omega$ coincides with the frequency of the acoustic mode of the mirror $\omega_{m}$, and if the mode shape of the high order optical mode and acoustic mode shape have a certain level of overlap, parametric opto-acoustic interaction will occur via radiation pressure associated with the beating. If $\omega_{0}-\omega_{1}>0$, the acoustic mode will be excited, leading to parametric instability. The parametric gain can be expressed as [Ju2006]:

[TABLE]

where $M$ is the mass of the mirror, $L$ is the arm cavity length, $\Lambda$ is the overlapping factor of the optical mode and acoustic mode, $\gamma_{m}$, $\gamma_{0}$, $\gamma_{1}$ are the relaxation rate (half-linewidth) of the acoustic mode and the two optical modes respectively. $\Delta_{m}=(\omega_{0}-\omega_{1})-\omega_{m}$ is the frequency detuning factor which determines the detuning of the cavity mode spacing from the test mass acoustic mode frequency. When $R>1$, the radiation pressure induced ring-up rate $\gamma_{opt}$ is lager than the intrinsic relaxation rate of the acoustic mode $\gamma_{m}$, then the system is unstable.

A critical factor in determine parametric instability is the optical cavity mode gap $\Delta\omega=\omega_{0}-\omega_{1}$, which depends directly on the radii of curvature of the cavity mirrors $R_{1}$ and $R_{2}$. The cavity mode spacing is given by

[TABLE]

where $m$, $n$ are the transverse mode indies, and $R_{1}$ and $R_{2}$ are the radii of curvature of the input and end mirrors of the cavity.

If one of the test mass mirrors (eg. end mirror) is heated up due to absorption of optical power (from the main cavity mode, or from a separate $CO_{2}$ laser beam applied to it), thermal distortion of the mirror will modify its radius of curvature $R_{2}\rightarrow R_{2}+\delta R_{2}$. The mode spacing can be written (expanded to the first order) as:

[TABLE]

Here,

[TABLE]

while $\Delta\omega^{(0)}$ is the original mode spacing before heating.

As a simple example, assume a laser beam with radius $w$ and power $P_{0}$ is applied to a spherical mirror of radius of curvature $R_{roc}$. Due to absorption and finite thermal conductivity of the mirror, a temperature difference between the central part and the edge of the mirror will develop within its substrate [Winkler1991]:

[TABLE]

where $P_{obs}$ is the absorbed power at the centre and $\kappa$ is the thermal conductivity of the mirror material. If the material has a thermal expansion coefficient of $\alpha$, the thermal distortion expressed as a change of sagitta $\delta d$, and the consequent change of mirror RoC are given by [Winkler1991]:

[TABLE]

and

[TABLE]

where $d$ is the initial sagitta with the cord length equal to the heating beam size. For fused silica, $\alpha\sim 5.5\times 10^{-7}$ m/K, $\kappa\sim 1.38$ W/m.K. An absorbed power $\sim 1$ W will lead to $\delta d\sim 30$ nm. For mirrors designed to have figure errors of $\sim 0.5$ nm , this is an enormous effect. The deformation corresponds to a mode-spacing change of the order of $\sim 2000$ Hz. This would be enough to tune $\Delta_{m}$ by several optical mode linewidths and $\sim 10^{3}$ acoustic linewidths; and for LIGO type test masses, to tune across $\sim 10^{2}$ acoustic modes from parametric instability condition.

Thermal tuning based on this method but using radiant heating was used in the advanced LIGO detectors to control parametric instability for an input power of $\sim 1/4$ of the designed power [Evans2015]. However, this simple thermal tuning method may not be sufficient if there are multiple potential unstable acoustic modes present, because tuning the cavity away from one unstable acoustic mode normally leads it to be tuned into resonance with another acoustic mode.

We now consider the application of a periodic $CO_{2}$ laser thermal modulation $P=P_{0}(1+\sin\Omega t)$. The period $2\pi/\Omega$ is chosen to be much larger than the cavity build up time $\tau\sim 4L/(cT_{r})$ ($T_{r}$ being the transmissivity of the input mirror of the cavity), but much smaller than the typical time for exponential growth of parametric instability. The modulation will cause the cavity to sweep through unstable resonant conditions while never remaining within a resonant condition long enough for acoustic unstable modes to build up. The time scale of the modulation $2\pi/\Omega$, typically can have the value from $1-100$ s which is smaller than the thermal conduction time of fused silica, $t_{th}\sim l^{2}c\rho/\kappa\sim 10^{4}s$, where $l$ is the typical length scale (test mass radius of 0.17m), $c$ is the specific heat and $\rho$ is the density.

The modulation power has both a constant (DC) and sinusoidal (AC) component. The corresponding DC temperature of the surface will increase to a certain value when the thermal dissipation balances the power absorption. The thermodynamic process under the periodic heat modulation is non-equilibrium due to the slow thermal response of the material. Because the test masses are suspended by thin wires in the vacuum, the dominant heat dissipation is by radiation. The radiated power can be estimated using the Stefan-Boltzmann law $\sigma l^{2}(T^{4}-T^{4}_{0})$, where $\sigma$ is the Stefan-Boltzmann constant, $T$ and $T_{0}$ are the temperatures of the test mass surface and the environment respectively. For a power absorption of $\sim 20$W, one can estimate the final temperature to be $\delta T\sim P_{abs}/4\sigma l^{2}T^{3}_{0}\sim 8$ K, which implies that the $CO_{2}$ heating does not have a significant effect on the thermal noise level. Use of the Stefan-Boltzmann law is justified because fused silica is effectively a black body at thermal wavelengths. With a modulated heating power, the AC temperature at the heating spot position will vary with the heating power variation. However, the thermal conduction will result in a phase lag of the temperature variation to the heating power variation. The amplitude of the temperature variation decreases as the heating power modulation frequency increases, as does the variation of the radius of curvature and the mode spacing.

3 Simulation results

We used the finite element modeling software package, COMSOL, to perform a thermal elastic analysis of the test mass mirrors. We used a cylindrical fused silica test mass model. Some detailed features of the real test mass for the advanced gravitational wave detectors, such as the flat surfaces on the barrel for suspension, are neglected for simplicity, since they are far from the laser heating center. Fused silica has very high absorption at the $CO_{2}$ laser wavelength of $10\mu m$, the optical power is mostly absorbed at the heating spot surface with a penetration depth of micrometer scale. We treated the heated surface as the heat source which transports energy into the substrate and radiates energy into the environment. Due to the small absorption depth of the fused silica, the highest temperature area will be close to the substrate surface. We used a geometric sequence mesh to ensure enough resolution near the surface. The $CO_{2}$ laser intensity distribution over the light spot is modeled as a two-dimensional Gaussian distribution with a sinusoidal time-dependence: $P(t)=P_{0}(1+\sin\Omega t)$. For studying the cavity mode spacing change and optical mode wavefront deformation of the Fabry-Parot cavity under the in uence of the heating, we used optical Fast Fourier Transformation code, OSCAR [Degallaix2010]. The parameters that used in the simulation is given in Table 3.