Thermal Noise Reduction in Ternary Optical Coatings: From Ti::GeO$_2$-Based Ternary Systems to High Index Materials

Abstract

Minimizing coating thermal noise is crucial for enhancing gravitational wave detector sensitivity, with a target Amplitude Spectral Density Reduction Factor (ASD RF) of $0.5$ relative to standard coatings. This study investigates the design of low-noise dielectric stacks using the 'Double Stack of Doublet' strategy, explored via ad-hoc optimization heuristics specifically developed for efficient parametric analysis of coating performance. We analyze the performance limits of ternary coatings based on SiO$_2$, Ti::SiO$_2$, and Ti::GeO$_2$, considering material property uncertainties and absorption constraints. Optimization results show that this system, even with relaxed absorbance constraint (1 ppm), falls short of the target, achieving a best ASD RF of $\sim 0.69$. Consequently, we explore alternative ternary 'Double Stack of Doublet' designs incorporating higher-refractive-index materials. Simulations demonstrate that incorporating alternative high-index materials offers a promising pathway, potentially enabling the achievement of the project target. We discuss the optimization strategies, performance trade-offs, design robustness, and implications of using high-index, potentially higher-loss materials for next-generation optical coatings.

Figures (20)

• Figure 1: A typical coating made of $N$ layers deposited on a substrate. The Electromagnetic field (Gaussian beam shaped) is incident form above (vacuum half space). In the plane wave approximation the field phasor is $E_{inc} \exp{i \omega t}$ where the angular frequency corresponds to the free space wavelength $\lambda_0=1064$ nm.
• Figure 2: Contour plot of the normalized Braginsky coefficient as a function of Young's Modulus and Poisson's Ratio, for a fixed mechanical loss factor $\phi_{Ti:SiO_2}=1.44 \times 10^{-4}$. Black and cyan dots are data points from LIGO document G2001684 LIGOdoc. The green dots specifically represent measurements from the Glasgow group TiSilicaGraemePri. The red region corresponds to values of $\bar{\eta}_{Ti::SiO_2} < 5.6$, while the blue region corresponds to values of $5.6 < \bar{\eta}_{Ti::SiO_2} < 6.7$.
• Figure 3: This figure presents an uncertainty analysis of the normalized Braginsky coefficient ($\eta_{Ti::SiO_2}$) for Ti::Silica coatings, assuming a mechanical loss value of $\phi_{Ti:SiO_2} = 1.44 \times 10^{-4}$. The pink histogram illustrates the distribution of $\bar{\eta}_{Ti::SiO_2}$ derived from experimental data. The blue distribution is obtained by considering the uncertainties in Young’s Modulus ($Y_{Ti::SiO_2}$) and Poisson’s ratio ($\sigma_{Ti::SiO_2}$), assuming uniform distributions within the ranges of $\sigma_{Ti::SiO_2} = [0.3, 0.4]$ and $Y_{Ti::SiO_2} = [85.6, 110]$ GPa.
• Figure 4: The transmissivity $\tau_c$ (or transmittance $\tau_c=1-|\Gamma|^2$) as a function of the number of doublets for QWL design. Here $\Gamma$ is the reflection coefficient at vacuum interface. Simulation parameters are relative to LMA Silica coating; the configuration does not include a half-wave cap. The labels placed above the horizontal red dashed grid lines explain what each line represents.
• Figure 5: The Carniglia's Limit [ppm] for a binary coating of silica and doped silica, plotted as a function of the extinction coefficient of the doped silica (Ti:SiO$_2$). The two datasets correspond to different assumed real refractive indices for the doped silica layer: $n_r = 1.92$ (black dots) and $n_r = 1.97$ (red dots).