# Title-to-Length Coupling Analysis of an Off-Axis Optical Bench Design for NGGM
## 1 Introduction  
With the success of the laser ranging instrument (LRI) in the GRACE Follow-on mission, heterodyne laser interferometry is destined to become a preferred ranging method to measure the miniature range variation between two satellites and map the earth gravity field for future NGGM missions [1]. With the MAGIC mission by NASA/ESA planned to launch a Bender pair of satellites around 2030 [2], satellite gravity will enter a new era of development. The recent success of the first satellite-to-satellite tracking mission in China has also prompted ongoing discussions and debates on what the second generation of gravity missions would be like for China [3]. While the optical design for future MAGIC missions is likely to be based on the heritage of the off-axis triple-mirror assembly (TMA) demonstrated in the GRACE Follow-on mission [4], the mission architecture, and optical design in particular, for future prospective NGGM missions for China is still open. As far as optical design is concerned, an off-axis design is preferred as it minimises the coupling of attitude jitter to the optical path length. However, it is preferable to integrate the function of TMA into the optical bench [5], in contrast to the separation of the optical bench and the TMA in current GFO design. In doing so, we hope to have a more compact and light-weight optical bench. Further, when we look to the long-term future in satellite gravity development, in order to overcome the current outstanding AOD aliasing problemgenerated by the high frequency mass variation signals due to atmosphere and ocean tides, more pairs of satellites are required to enhance the temporal resolution of gravity signals. The desire to realise this multi-pairs scenario and yet keep the financial budget of the mission to a reasonable level naturally leads us to look at the miniaturization of satellites with light-weight payloads (see [6]). Gravity signals with a high sampling rate in time also means that we may turn the atmospheric forcing and ocean tides into signals instead of noises so that the scientific scope of satellite gravity may be broadened. As a modest first step to explore the feasibility of the miniaturization of satellites for satellite gravity, we are currently looking at the option of adopting a more compact optical design for future prospective NGGM missions for China. A new mission architecture is currently under study [7], in which, in place of the triple-mirror assembly adopted in the GRACE Follow-on mission, an optical system comprising lens and rooftop mirrors is employed to generate laser retro-reflection between two satellites. The receiving (RX) beam and transmitting (TX) beam are enforced to be anti-parallel by a control loop with the differential wavefront sensing signal readout from the quadrant photodiodes (QPD) as the controller, and a fast-steering mirror is employed to actuate the pointing of the local beam. An open loop is also designed so that we have the flexibility to adjust on orbit the distance variation between the centre of mass of a satellite and the phase centre of the optical design. This will enable us to minimise the tilt-to-length (TTL) coupling noise in the laser metrology system, which will lead to improvement in the laser ranging accuracy, without the need to subtract the phase centre offset in the post data processing. Due to the presence of the AOD noise in the gravity field recovery, currently the enhancement in sensitivity in the laser ranging will not improve the precision in the measurement of the temporal variation of the gravity field. Still, the precision of static gravity field recovery will benefit, and it is conceivable that, in the long term, when we have more pairs of satellites to improve the temporal resolution of gravity signals, enhancement in the laser ranging precision will help improve the precision in the measurement of the temporal variation of gravity. In a previous work [7], the viability of the new optical design was demonstrated in a preliminary way.