Comparative Analysis of Thermal Models for Test Masses in Next-Generation Gravitational Wave Interferometers

TL;DR

This work addresses the challenge of thermal distortions in Terminal Test Masses under high intracavity power by developing a detailed volumetric model that resolves heat deposition in a ternary DSD coating and substrate, and a computationally efficient reduced boundary model that encapsulates the coating’s effect as a surface term. The authors demonstrate that, under realistic high-power scenarios, the temperature rise can localize near the coating interface (up to ~3 K), while the reduced model predicts results within a few millikelvin of the full model and offers substantial computational speed gains. The key contribution is showing that the total coating absorption, rather than the fine-grained internal heat distribution, largely governs the thermal response, enabling rapid parametric studies for thermal management and first-order distortion analysis. This has practical implications for future interferometer design and coating development, guiding where to focus measurement accuracy and computational effort to predict thermal effects on sensitivity.

Abstract

Accurate thermal modeling of Terminal Test Masses (TTMs) is crucial for optimizing the sensitivity of gravitational wave interferometers like Virgo. In fact, in such gravitational wave detectors even minimal laser power absorption can induce performance-limiting thermal effects. This paper presents a detailed investigation into the steady-state thermal behavior of TTMs. In particular, future scenarios of increased intracavity laser beam power and optical coating absorption are considered. We develop and compare two numerical models: a comprehensive model incorporating volumetric heat absorption in both the multilayer coating and the bulk substrate, and a simplified reduced model where the coating's thermal impact is represented as an effective surface boundary condition on the substrate. Our simulations were focused on a ternary coating design, which is a candidate for use in next-generation detectors. Results reveal that higher coating absorption localizes peak temperatures near the coating--vacuum interface. Importantly, the comparative analysis demonstrates that temperature predictions from the reduced model differ from the detailed model by only milli-Kelvins, a discrepancy often within the experimental uncertainties of the system's thermo-physical parameters. This finding suggests that computationally efficient reduced models can provide sufficiently accurate results for thermal management and first-order distortion analyses. Moreover, the critical role of accurately characterizing the total power absorbed by the coating is emphasized.

Figures (9)

• Figure 1: Panel (a) shows a cross-sectional view of the test mass, specifically focusing on the top surface where the incident electromagnetic field, $E_{inc}$, strikes. Different colors corespond to different materials. Panel (b) provides a 3D perspective of the test mass, which is a cylinder. Its dimensions are indicated: radius $R_{cyl}$ and height $L+d$, where $L$ is the substrate thickness. The top face of the cylinder is colored green, representing the surface where the coating (detailed in panel (a)) is deposited. The incident electromagnetic field (i.e., Gaussian beam) is propagating towards the coated top surface.
• Figure 2: An example of a DSD coating structure made of Silica (denoted by "L"), Ti::SiO$_2$, (denoted by "H"), and Ti::GeO$_2$ (denoted by "C"). The sequence of materials and the optical length of each layer are displayed.
• Figure 3: The normalized power absorption per layer (dots) and its continuous exponential fit (line). The model fit is exponential $\sim \exp(-\alpha_0 z)$, and the normalized value of the $\alpha_0$ parameter is displayed in the legend.
• Figure 4: Increase in temperature, with respect to room temperature $T_\infty$, on the radial section of the TTM.
• Figure 5: Increase in temperature ($\Delta T$), with respect to room temperature $T_\infty$, on the $r=0$ axis ($z \in [0,L]$).