Optical design and sensitivity optimization of Cryogenic sub-Hz cROss torsion bar detector with quantum NOn-demolition Speed meter (CHRONOS)

Abstract: We present the optical design and sensitivity modeling of the 2.5 m Cryogenic sub-Hz cROss torsion-bar detector with quantum NOn-demolition Speed meter (CHRONOS), a triangular Sagnac speed-meter interferometer incorporating power- and signal-recycling techniques. Using ABCD-matrix analysis and Finesse3 simulations, we show that stable eigenmodes are obtained with optimized mirror curvatures and focal placements, achieving mode-matching efficiencies above 99.5 %. The resulting configuration reaches a quantum-noise-limited strain sensitivity of $h \simeq 3\times10^{-18},\mathrm{Hz^{-1/2}}$ at 1 Hz, with a ring-cavity finesse $\mathcal{F}\simeq3.1\times10^{4}$ and round-trip Gouy phase $\psi_{\mathrm{rt}}\approx153^{\circ}$. The power-recycling cavity detuning ($\phi_p=-85^{\circ}$) dominates the low-frequency quantum noise, while the signal-recycling cavity detuning ($\phi_s=0^{\circ}$) mainly introduces a uniform quadrature rotation. The optimal homodyne angle ($\zeta_{\mathrm{opt}}\simeq46^{\circ}$) balances shot-noise and radiation-pressure effects to give the best sensitivity near 1 Hz. Assuming end-mirror reflectivity $R_{\mathrm{ETM}}=99.9999\%$ under cryogenic operation at 10 K, CHRONOS can achieve quantum-noise-limited performance on a laboratory scale and serve as a testbed for future long-baseline, cryogenic interferometers probing sub-hertz gravitational waves.