Optimizing the longitudinal isolation for LIGO-style test mass suspensions.

We derive the design of a multi-stage mirror suspension which gives optimal isolation performance for upgrades to the Advanced Laser Interferometer Gravitational-wave Observatory (Advanced LIGO). This optimization is only constrained by length, optic mass and total suspension mass. We find that the optimally-isolating suspension with Ν masses, fixed total mass Μ tot , total length L tot , and bottom mass m b , has equal distances between suspended masses, equal ratios between successive suspended payloads, and a highest resonance scaling as (ω N ) 2 ∼ 4g [ L tot ln( M tot / m b )] -1 N(N-1) . This optimization was used to guide the conceptual design for the next planned upgrade, LIGO A # . That conceptual design has several additional constraints, but we show that the isolation performance is within 20% of the theoretical best performance achievable. Additionally, the principles derived from the general optimization are broadly applicable and can be used to inform suspension design for other instruments requiring high-performance vibration isolation, including third-generation gravitational wave observatories such as Cosmic Explorer.