# On-Axis Optical Bench for Laser Ranging Instruments in Future Gravity Missions
## 2 On-Axis LRI Optical Bench Design  
Some general design requirements apply to the OB regardless of the topology. One key requirement is that a sufficient amount of light is transferred from the receive aperture (RX aperture) to the photodetectors in order to enable stable phase tracking whilst maintaining the phase fidelity of the light by mitigating disturbances from diffraction due to clipping and from stray light. For a wavelength of approximately 1 \(\upmu\)m, the received light power should exceed 10 pW at the photoreceivers [11], since that corresponds to a carrier-to-noise density (CNR) of \(\sim\)70 dB-Hz considering the current technology of photoreceivers, lasers, and phasemeters. Though phase tracking at lower CNR values, lab experiments have demonstrated [14; 15] that a value of 70 dB-Hz guarantees cycle-slip-free tracking with sufficient margin and robustness against disturbances from thruster firings, sun-blindings, and micro-meteorite impacts, etc. An automatic beam alignment system using a fast steering mirror (FSM) can further enhance the robustness, since it maximizes the interferometric contrast and therefore the CNR with high gain and bandwidth at the local S/C, and ideally also the transmitted power to the remote S/C due to an optimization of the direction of the transmitted beam. Another key requirement of the OB is the feature of a well-defined and stable virtual reference point (RP) that is physically accessible in order for it to be co-located with the S/C center of mass (c.m. ), such that the measured range is invariant under small rotations of the S/C around this point in an arbitrary axis. Considering integration tolerances, thermal variations, launch vibrations, thermoelastic deformations, and settling effects from the transition to space, a reasonable value for the in-flight tilt-to-length (TTL) coupling is of the  
Figure 1: Design concept of an on-axis optical bench for a next generation LRI. The receive and transmit beams (RX and TX, respectively) propagate into and out of the optical bench through the same aperture, where their paths coincide. order of 100 \(\upmu\)m/rad, which has been demonstrated by the GRACE-FO LRI [16], and, to some extent, can be reduced further by in-flight calibrations. The OB also has to ensure that the RX and LO beams remain well aligned, maintaining good interferometric contrast regardless of the attitude of the local S/C. Moreover, the OB has to provide a TX beam that is well aligned to the inter-satellite line-of-sight, thereby optimizing the optimal link. The acceptable pointing stability is typically of the order of the divergence angle. For laser interferometers employing fundamental Gaussian beams, the intensity at the misalignment angle drops by \(e^{-2}=13.5\%\), or by 8.6 dB in terms of CNR. Typical divergence angles are of the order of 100 \(\upmu\)rad (see Table 1 for exact values of the TX beam divergence in various optical bench configurations). Based on the aforementioned requirements, an optical bench design has been carried out as part of the development of a laser ranging instrument for future gravity missions. The design comprises a series of Keplerian telescope imaging systems, in contrast with other realizations employing corner cube retroreflectors, such as that proposed by [PERSON] et al. [10] and by [PERSON] et al. [17]. TTL coupling is the leading source of noise in the range measurement at low frequencies. Careful optimization of the optical bench leads to a design TTL coupling well below 100 \(\upmu\)m/rad. However, it is expected that the final in-orbit TTL coupling is larger, at the level of \(\sim\)100 \(\upmu\)m/rad, due to integration tolerances, settling effects from ground to space, and uncertainties and variations in the c.m. position. These external effects make a prediction of the low-frequency ranging sensitivity difficult, but future missions are expected to exhibit lower satellite pointing errors that drive the TTL error [17] and post-processing correction of the TTL could further reduce the dominant TTL error [16].