Sensing and control scheme for the inteferometer configuration with an L-shaped resonator

The detection of high-frequency gravitational waves around kHz is critical to understanding the physics of binary neutron star mergers. A new interferometer design has been proposed in [Phys. Rev. X {\bf 13}, 021019 (2023)], featuring an L-shaped optical resonator as the arm cavity, which resonantly enhances kHz gravitational-wave signals. This new configuration has the potential to achieve better high-frequency sensitivity than the dual-recycled Fabry-Perot Michelson. In this article, we propose a sensing and control scheme for this configuration. Despite having the same number of length degrees of freedom as the dual-recycled Fabry-Perot Michelson, the new configuration requires one less degree of freedom to be controlled due to the degeneracy of two length degrees of freedom at low frequencies. We has also shown that introducing the Schnupp asymmetry is ineffective for controlling the signal-recycling cavity length. Therefore, we propose adding control fields from the dark port to control this auxiliary degree of freedom.