Constraining temperature distribution inside LIGO test masses from frequencies of their vibrational modes

Thermal distortion of test masses, as well as thermal drift of their vibrational mode frequencies, present a major challenge for operation of the Advanced LIGO and Advanced VIRGO interferometers, reducing optical efficiency, which limits sensitivity and potentially causing instabilities which reduce duty-cycle. In this paper, we demonstrate that test-mass vibrational mode frequency data can be used to overcome some of these difficulties. First, we derive a general expression for the change in a mode frequency as a function of temperature distribution inside the test mass. Then we show how the mode frequency dependence on temperature distribution can be used to identify the wave function of observed vibrational modes. We then show how monitoring the frequencies of multiple vibrational modes allows the temperature distribution inside the test mass to be strongly constrained. Finally, we demonstrate using simulations, the potential to improve the thermal model of the test mass, providing independent and improved estimates of important parameters such as the coating absorption coefficient and the location of point absorbers.

Figure 1

The geometry used for the COMSOL simulation.

Figure 2

Comparison between the frequency shift predicted by a COMSOL eigenfrequency simulation and the frequency shift predicted by the analytic expression for 225 eigenmodes influenced by an arbitrary thermal disturbance. Excellent agreement is observed.

Figure 3

The frequency shift predicted by the analytic expression for 225 eigenmodes influenced by 1 W of ring heater power plotted against a simulated measurement including 0.1 mHz noise and three modes misidentified (green) compared to the expectation (red).

Figure 4

Comparison of the mode frequency shift between two thermal profiles (inset) that are symmetric left to right. The mode frequencies of the case where the heating is to the left of the optic axis (vertical axis) are indistinguishable from the mode frequencies where the heating is to the right of the optic axis (horizontal axis).

Figure 5

Eigenvalues of the SVD matrix C’.

Figure 6

Profile of estimated temperature distribution inferred from changes in eigenfrequecy (a) Inverting $C_{ij}$ directly, (b) SVD inversion using all 225 eigenfrequencies, (c) SVD inversion using all 225 with 0.1 mHz noise, (d) SVD inversion using 86 components with 0.1 mHz noise.

Figure 7

rms error of estimated temperature as a function of the number of singular value decomposition elements used in the temperature reconstruction. The minimum error occurs with 86 elements for this particular temperature distribution.

Figure 8

Temperature distribution of (a) back surface and (b) front surface and reconstructed temperature distributions from (c) back surface and (d) front surface of the optic. Reconstruction is done using singular value decomposition using 81 elements and negligible (1 nHz) measurement noise.

Figure 9

Comparison of rms error of asymmetric temperature distributions assuming a symmetric (tan) and standard (blue) model. This is compared to the rms of the temperature distribution decomposed into singular value decomposition elements (red), and the rms of the temperature distribution (green dot).

Figure 10

Time evolution of a selection of eigenfrequencies for thermal conductivity of $k=1.37$ (dashed) and $k=1.39$ (solid) and the difference (dash dot).

Figure 11

Log likelihood function example to thermal conductivity. Using the data like that in Fig. 10, including 100 modes and minimal (1 nHz) measurement noise on the simulated measurement of a test mass with $k=1.381\mathrm{W}/(\mathrm{m}.\mathrm{K})$ thermal conductivity.

Figure 12

Likelihood function of point absorber power and thermal conductivity with minimal measurement noise (1 nHz) (colored contour lines) and with 0.1 mHz measurement noise (gray contour lines). It can be seen that with measurement noise the absorbed power may be constrained while the thermal conductivity may only be minimally constrained.

Figure 13

Likelihood function of point absorber location in radial position and depth with 5 nHz measurement noise (in color). With 0.1 mHz measurement noise (gray contour lines) a position bias is introduced.

Figure 14

Exaggerated mode displacements on the left and the form factors on the right. Example of two eigenfrequencies with very similar form factors. They represent mode frequencies 8164 (top) and 8331 Hz (bottom).

Figure 15

Exaggerated mode displacements on the left and form factors for these modes on the right. These modes show a range of spatial scales attainable with existing LIGO measurements. They represent mode frequencies 15016, 15083, 15220, 15534, 23656, 33381 and 33610 Hz (top to bottom).

Figure 16

The seven largest eigenvalued eigenfunctions of the singular value decomposition described in Sec. 4.