# On-Axis Optical Bench for Laser Ranging Instruments in Future Gravity Missions
## 3 Optical Simulations
The figure shows (in order from top to bottom) the position acquired by the active steering mirror under a given misalignment; the measured in-loop (QPD1) and out-of-loop (QPD2) horizontal and vertical DWS signals; the deviation of the transmitted beam with respect to the inter-satellite line of sight; the RX and LO beam walk at the photodetectors; the TTL coupling experienced at the local S/C; and the TX beam TTL coupling measured at the distant S/C in perfect-transponder mode. Note that the out-of-loop DWS signals are small but orders of magnitude greater than the in-loop signals due to the asymmetry between the reflection and transmission ports of PBS1. The TX beam TTL coupling is limited by the numerical error of double precision. The installation error of M2 directly affects the OB's retro-reflective function by changing the position of the TX RP, thereby introducing extra TTL coupling to the range measurement. This effect is modeled and characterized, and it is determined that with positional tolerances of 0.05 mm and angular tolerances of 50 \(\upmu\)rad, the extra TTL coupling in all three rotation axes is \(\sim\)10 \(\upmu\)m/rad. This performance impact is similar to that of the triple-mirror assembly of the GRACE-FO LRI, where the misalignment of the three mirrors introduced a TTL coupling of less than 20 \(\upmu\)m/rad in all three rotation axes [32]. Finally, a thermal analysis is carried out to ensure that the proposed OB layout is robust against thermoelastic deformation and refractive index variations due to temperature fluctuations. This analysis is of particular importance due to the fact that the RX and LO imaging systems both feature a different set of components on their respective sensitive paths. This means that, e.g., any thermal fluctuations introduced by lenses L3 and L4 are not common-mode between the interfering beams. The expected thermal drift on the optical bench is \(\pm 3\) K/orbit [2], hence, the temperature is swept within this range and both the thermoelastic expansion and the change of refractive index are coherently applied to all optical components. The components are assumed to be attached to a baseplate that expands uniformly, changing the components' positions w.r.t. the baseplate's c.m. ; For the optical components, the thermal coefficients of fused silica are used, \(\alpha=5.5\times 10^{-7}\,1/\)K, and \(\beta=9.6\times 10^{-6}\,1/\)K, where \(\alpha\) measures the fractional length change (\(\Delta L/L\) per Kelvin) and \(\beta\) measures the fractional refractive index change (\(\Delta n/n\) per Kelvin). On the other hand, the baseplate is assumed to be made of titanium with \(\alpha=8.6\times 10^{-6}\,1/\)K. The results are shown in Figure 8. This shows the maximum magnitude of the total TTL coupling of 18.1 \(\upmu\)m/rad, and 1.16 \(\upmu\)m/rad over 300 \(\upmu\)rad in yaw and pitch, respectively. These are both below the GRACE-FO requirement of 80 \(\upmu\)m/rad [13], and thus verify the feasibility of the design in terms of the orbital thermal drift. Figure 6: Depiction of the TX beam TTL coupling simulation. The TX beam (green) propagates to the preset distant OB where it interferes at a single-element photodiode with a Gaussian beam large enough such that the distant system acts as a perfect transponder. Figure 7: Angular motion of the local S/C causes the received beam’s incidence angle at the receive aperture to change (the abscissas). A steering mirror is actuated (**a,b**) via two independent DWS control loops that zero the \(\text{DWS}_{\text{n}}\) and \(\text{DWS}_{\text{v}}\) signals in QPD1 (**c,d**), keeping the RX and LO phase fronts nearly parallel at both detectors. QPD2 is out of loop and measures nearly zero \(\text{DWS}_{\text{n}}\) and \(\text{DWS}_{\text{v}}\) (**e,f**). The TX IS ensures that the TX beam is antiparallel to the RX beam in the inter-S/C path (**g,h**). The RX IS and LO IS ensure that the RX and LO beams experience minimal beam walk at the detectors (**i–l**).