Thermorefractive noise of finite-sized cylindrical test masses

We present an analytical solution for the effect of thermorefractive noise considering finite-sized cylindrical test masses. For crystalline materials at low temperatures, the effect of finite dimensions becomes important. The calculations are independently performed using the Fluctuation-Dissipation-Theorem and Langevin’s approach. Our results are applied to the input test mass of the current and future cryogenic gravitational wave detectors CLIO, LCGT, and ET and are compared to the respective standard quantum limit. For a substrate temperature of 10 K, we find that the thermorefractive noise amplitude of the silicon input test mass in ET is only a factor of 2 below the standard quantum limit for frequencies above 500 Hz. Thus, thermorefractive noise of the input test mass could become a severe limitation if one uses techniques to beat the standard quantum limit like, e.g., squeezing. In contrast, the effect of thermorefractive noise of the input test mass is negligible for CLIO and LCGT.

Figure 1

Problem geometry with the substrate (height $H$, diameter $2R$), the laser beam (diameter $2r_0$), and the radial coordinate system ($r$, $z$).

Figure 2

Schematic TR noise spectrum for a finite-sized cylindrical test mass. The solid curve represents the TR noise of the ITM taking standing waves into account as the dashed curve represents the result for a homogeneous readout neglecting standing waves. The diagram is scaled logarithmically on both axes with a factor of 10 between neighbored dashed lines. For silicon at 10 K region III is situated at frequencies above ${10}^9\mathrm{Hz}$. The heat equation loses validity at these high frequencies. Thus, regions III and IV show no physical relevance for silicon. In contrast, for fused silica at room temperature region III shows physical relevance again as it starts at a frequency around 1 kHz [14].

Figure 3

The diagram shows the thermorefractive noise amplitude $\sqrt{S_z(f)}$ for a finite silicon cylinder of ET geometry (solid line) at 10 K compared to an FE calculation of the same geometry (circles). At frequencies above 100 Hz a coincidence with the model of an infinite disc [11] (dashed line) is shown.

Figure 4

Numerical results for the thermorefractive noise amplitude at the output of a simplified interferometer $\sqrt{S_{\mathrm{IF}}(f)}$. Solid lines represent our results considering the finite radial dimension of the ITM [Eq. (31)] for ET, LCGT, and CLIO at 10 K, 20 K, and 30 K. The dashed line indicates the TR noise of the fused silica beam splitter at 290 K. Also the standard quantum limit (SQL) of the simplified interferometers using the corresponding ITM geometry is shown.

Figure 5

Sketch of the ITM geometry underlying the derivation of the ${sin}^2$-term due to standing waves. A small temperature fluctuation at a distance $L$ apart from the end of the ITM is assumed.

Figure 6

Schematic view of an interferometer arm with noise variables inside the arm cavity (mirror positions $x_1$, $x_2$) and outside the cavity ($\phi$, e.g., TR noise of ITM).