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

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 $h\simeq 3\times {10}^{-18},\mathrm{H}{\mathrm{z}}^{-1\mathrm{/}2}$ at 1 Hz, with a ring-cavity finesse $\mathcal{F}\simeq 3.1\times {10}^4$ and round-trip Gouy phase ${\psi }_{\mathrm{r}\mathrm{t}}\approx {153}^{\circ }$. 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 (${\zeta }_{\mathrm{o}\mathrm{p}\mathrm{t}}\simeq {46}^{\circ }$) balances shot-noise and radiation-pressure effects to give the best sensitivity near 1 Hz. Assuming end-mirror reflectivity $R_{\mathrm{E}\mathrm{T}\mathrm{M}}=99.9999\mathrm{\%}$ 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.