Optimizing the longitudinal isolation for LIGO-style test mass suspensions

Abstract 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 (LIGO). This optimization is only constrained by length, optic mass and total suspension mass. We find that the optimally-isolating suspension with N masses, fixed total mass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>tot</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , total length <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>tot</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , and bottom mass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">b</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , has equal distances between suspended masses, equal ratios between successive suspended payloads, and a highest resonance scaling as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>∼</mml:mo> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>g</mml:mi> </mml:mrow> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>tot</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mi>ln</mml:mi> <mml:mo>⁡</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>tot</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">b</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>N</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . This optimization was used to guide the conceptual design for the next planned upgrade, LIGO A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow/> <mml:mtext>#</mml:mtext> </mml:msup> </mml:mrow> </mml:math> . 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.