$2\cdot {10}^{-13}$ fractional laser frequency stability with a 7-cm unequal-arm Mach-Zehnder interferometer

To achieve sub-picometer sensitivities in the millihertz band, laser interferometric inertial sensors rely on some form of reduction of the laser frequency noise, typically by locking the laser to a stable frequency reference, such as the narrow-linewidth resonance of an ultra-stable optical cavity or an atomic or molecular transition. In this paper we report on a compact laser frequency stabilization technique based on an unequal-arm Mach-Zehnder interferometer that is sub-nanometer stable at $10\mu$Hz, sub-picometer at $0.5$mHz, and reaches a noise floor of $7\mathrm{f}\mathrm{m}\mathrm{/}\text{\hspace{0.17em}⁣}\sqrt{\mathrm{H}\mathrm{z}}$ at 1 Hz. The interferometer is used in conjunction with a DC servo to stabilize the frequency of a laser down to a fractional instability below $4\times {10}^{-13}$ at averaging times from 0.1 to 100 seconds. The technique offers a wide operating range, does not rely on complex lock acquisition procedures, and can be readily integrated as part of the optical bench in future gravity missions.