Suppression of parametric oscillatory instability in third generation gravitational wave detectors

•We examine the parametric instability in Einstein Telescope.•We calculate all unstable optical and elastic modes.•We present the method how to suppress the parametric instability.

We discuss one possible method for suppression of the nonlinear effect of a parametric oscillatory instability in a Fabry–Perot cavity of the next generation of gravitational wave detectors by determination of optimal radii of curvature of interferometers' mirrors. Varying the radii of curvature of the mirrors we can change the frequencies of the optical modes of the Fabry–Perot resonators and distributions of optical fields in optical modes, which, in turn, can change the number of unstable modes, leading to parametric oscillatory instability. The determination of the optimal values of the radii of curvature of mirrors to reduce the number of unstable modes by thermal lensing is fulfilled.