# On-Axis Optical Bench for Laser Ranging Instruments in Future Gravity Missions
## 4 Summary and Outlook  
Following on the success of the GRACE-FO LRI, a US-Germany collaboration is conducting preparatory studies for advancing the LRI design and technology under the GRACE-ICARUS project [33]. In parallel, ESA has approved preparatory studies and instrument predevelopments for a Next-Generation Gravity Mission [17; 34]. In this context, we have put forward an optical bench design for a next-generation LRI. The design follows an on-axis topology in which the RX, TX and LO beams' paths coincide. A series of Keplerian telescope imaging systems are configured to: _i._ achieve the retro-reflective function without the need for a retroreflector; _ii._ adapt the size of the received beam and the local oscillator beam to the active area of the quadrant photodiodes; and _iii._ adjust the size of the transmitted beam to match the size of the RX/TX aperture and decrease its divergence to relax the laser power requirements. The imaging systems, in combination with an active steering mirror using differential wavefront sensing loops, ensure optimal overlap between the interfering RX and LO  
Figure 8: Thermal analysis. The temperature is swept in the \(\pm\)3 K range in seven steps, and the TTL coupling simulation is carried out for each step of temperature and RX beam tilt in the pitch and yaw degrees of freedom. The shaded regions shown in the plots are bounded by the maximum and minimum results for TX pointing error (**a,b**), RX beam walk (**c,d**), LO beam walk (**e,f**), local TTL coupling (**g,h**), and TX beam TTL coupling (**i,j**) throughout the temperature range. beams at the detectors, as well as near-perfect TX beam pointing to the remote S/C. The optimal RX/LO overlap maximizes the local carrier-to-noise ratio, and the accurate TX beam pointing maximizes the optical power at the remote S/C, regardless of the attitude of the local S/C. The optical bench functionalities and limitations are demonstrated by means of a computer model using the IFOCAD C++ libraries. The coupling of the attitude of the local S/C to the inter-satellite range measurement is computed, and it is found to be well within the required level of 100 \(\upmu\)m/ rad. A thermal analysis accounting for the effects of thermoelastic expansion of the optical bench components and baseplate, and refractive index fluctuations of the optics, is carried out to verify the robustness of the proposed design against the temperature fluctuations expected in orbit. The proposed layout uses a single telescope in the RX and TX beam path, enhancing the light collecting area and decreasing the TX beam far field divergence. Since this design requires only a single baffle, the footprint of the LRI in the S/C can be made smaller compared with off-axis designs, such as the GRACE-FO LRI, which hosted a retroreflector that was separate from the OB assembly, leading to significantly increased complexity of assembly and thermal control. An experimental realization of the proposed OB design is being conducted at the Albert Einstein Institute, while the primary functions of the OB have been demonstrated via the optical simulations presented in this paper, several questions remain to be investigated experimentally, notably the coupling of polarization imperfections and fluctuations, and stray light, into the length and angular measurements. Conceptualization, Y.Y. and V.M. ; methodology, K.Y. and M.D.A. ; software, K.Y., M.D.A. and D.W.; resources, J.J.E.D. ; writing--original draft preparation, Y.Y., K.Y., M.D.A., D.W., J.J.E.D. and V.M. ; writing--review and editing, M.D.A., J.J.E.D., V.M., J.J. and G.H. ; funding acquisition, G.H. All authors have read and agreed to the published version of the manuscript. This work has been supported by: the Chinese Academy of Sciences (CAS) and the Max Planck Society (MPG) in the framework of the LEGACY cooperation on low-frequency gravitational-wave astronomy (M.IF.A.QOP18098); The Deutsche Forschungsgemeinschaft (DFG, German Research Foundation, Project-ID 434617780, SFB 1464).