Thermal noise of folding mirrors

Current gravitational-wave detectors rely on the use of Michelson interferometers. One crucial limitation of their sensitivity is the thermal noise of their optical components. Thus, for example, fluctuational deformations of the mirror surface are probed by a laser beam being reflected from the mirrors at normal incidence. Thermal noise models are well evolved for that case but mainly restricted to single reflections. In this work, we present the effect of two consecutive reflections under a non-normal incidence onto mirror thermal noise. This situation is inherent to detectors using a geometrical folding scheme such as GEO 600. We revise in detail the conventional direct noise analysis scheme to the situation of non-normal incidence allowing for a modified weighting function of mirror fluctuations. An application of these results to the GEO 600 folding mirror for Brownian, thermoelastic, and thermorefractive noise yields an increase of displacement noise amplitude by 20% for most noise processes. The amplitude of thermoelastic substrate noise is increased by a factor of 4 due to the modified weighting function. Thus, the consideration of the correct weighting scheme can drastically alter the noise predictions and demands special care in any thermal noise design process.

Figure 1

Core optics of GEO 600 as a prototype of a gravitational-wave detector with folded arms. Here, the folding mirror is probed twice by the laser light.

Figure 2

Geometric path of the laser beams at the folding mirror surface. Here, we present the incoming light beams $E_1$ and $E_2$ as well as the coordinate system used for further calculations.

Figure 3

Photograph of the intensity pattern at the folding mirror surface in GEO 600. The stripe pattern is clearly visible on top of the Gaussian intensity profile. The stripe period agrees with the theoretical prediction of $\mathrm{\Lambda }=2.55\mathrm{mm}$. For a better visibility, the intensity of the image has been inverted.

Figure 4

Sketch of a folding mirror surface deformation to explain its effect on the reflected light using Huygen’s principle. The diagram on top shows the intensity distribution at the folding mirror surface. Along the $y$ axis, the deformation is assumed to be small compared to the intensity period along the folding mirror surface and located at minimum intensity. Also, its height is assumed to be small compared to the wavelength.

Figure 5

Scatter paths of first order due to a surface grating at the folding mirror. In (a), the unperturbed light path is shown with simple reflections at the folding mirror. Case (b) represents the direct backscatter of the incoming light in the negative diffraction order. This light does not enter the folded arm at all. In (c), the light is reflected at the folding mirror and the ETM and scattered back into the folded arm at its second approach to the folding mirror. This scatter path uses the positive diffraction order and shows a double round trip in the folded arm. The sum of all these paths adds up to the total optical field reflected from the IFO arm in first order.

Figure 6

Model system of a coating layer on top of a substrate. In this model, no interaction within the layer and a linear coupling to the substrate is assumed. In scheme (a), the coating is probed by a homogeneous readout, while in (b), a striped readout is illustrated.

Figure 7

Thermoelastic noise amplitude for a striped readout and different stripe periods $\mathrm{\Lambda }$. In analogy to substrate TR noise in a beam splitter [7], a region of constant noise emerges at low frequencies. Its cutoff frequency increases by decreasing the stripe period. The dashed red curve marks the conventional noise contribution. It has to be independently added to the stripe noise level to obtain the total TE noise. For relatively large stripe periods, a significant increase in noise can be found within the typical frequency band of gravitational-wave detectors. At high frequencies, the $1/{\omega }^2$ dependence from Eq. (27) is visible.

Figure 8

Displacement noise of GEO 600’s folding mirror. The diagram shows the noise levels for the striped configuration as opaque lines. For comparison, the results of the conventional case have been added as transparent curves. Further, the substrate contributions are solid lines, while the coating contributions appear dashed.

Figure 9

Thermal strain noise $S_h$ of the folding mirror in GEO 600 compared to the measured sensitivity curve. The latter has been taken from Ref. [26]. Thermal noise of the folding mirror is not to limit the current sensitivity of GEO 600.

Figure 10

Sketch for a qualitative explanation of TE coating noise for the limiting case of small thermal path lengths $r_{\text{th}}$. In this approach, the sample is divided into spheres showing a radius of the thermal path length $r_{\text{th}}$. Within them, the temperature is nearly constant, and temperature fluctuations in two spheres are nearly independent. This scheme allows the correct derivation of the thermal noise level with respect to general dependencies on physical parameters.