# On-Axis Optical Bench for Laser Ranging Instruments in Future Gravity Missions
## 3 Optical Simulations
Figure 3: Amplitude and phase of the “flat-top” RX beam at the receive aperture and at the surface of QPD1, in the horizontal direction. The dashed lines indicate the boundaries of the aperture and the active area of the photodiode, respectively. The RX beam is modeled in IFOCAD using the mode-expansion method. The amplitude of the local oscillator beam at QPD1 is shown in red, showing good spatial overlap with the RX beam, in spite of the much larger peak amplitude. After the OB components are initialized in the model, the imaging systems are optimized by tuning the positions of the photodiodes, of lens L3, of the receiving aperture, and of mirror M2, in the following steps: first, the QPD positions are finetuned along the nominal optical axis in order to minimize the beam walk of the LO beam on the detector planes, under both pitch and yaw motions of the steering mirror; second, the position of L3 is finetuned to minimize the separation between the front focal point of L1 and the effective focal point of the system composed of L3, WP2, PBS2, WP1, and PBS1; third, the position of the receiving aperture is finetuned to minimize the interferometer's TTL coupling, as sensed by both quadrant photodiodes, under pitch and yaw rotations of the S/C with an active steering mirror (i.e., using the DWS signals to actuate on the steering mirror's pitch and yaw degrees of freedom in order to yield optimally overlapped phase fronts between the RX and LO beams at the detectors); lastly, the position of M2 is finetuned, such that the effective focal point of the TX IS is located at the RX/TX RP, and thus near-perfect TX beam pointing to the remote S/C is achieved. In the resulting configuration, the interferometer with closed-loop control of the steering mirror becomes largely insensitive to the angular motion of the S/C. These functionalities are quantitatively evaluated at the local S/C, and TTL coupling is measured both at the local S/C and at the distant S/C in perfect-transponder mode. First, an RX beam of the expected size after having propagated through the inter-satellite path is originated at the local RX RP, i.e., at the center of the receive aperture (Figure 5). This beam is tilted in the pitch and yaw degrees of freedom to simulate angular motion of the S/C around its c.m. ; The RX beam is propagated through the setup, and it is interfered with the LO beam whilst the FSM is being actively controlled using DWS loops to obtain optimal RX-LO overlap at the detectors. The LPS variations are measured, and the local TTL coupling is evaluated by varying the misalignment angles. Then, the position of a distant single-element photodiode (SEPD) representing the remote OB is derived by intersecting the initial RX beam direction with a sphere of 200 km radius centered around the unfolded RX RP (Figure 6). The locally generated TX beam is propagated to this SEPD in the far field, where it is interfered with a large Gaussian beam acting as LO beam for the remote S/C OB. The radius of this LO beam is set large enough such that this distant system acts as a perfect transponder, i.e., such that it measures a range unaffected by the wavefront of the interfering beams. The LPS measured by the distant SEPD is obtained and the TX beam TTL coupling is evaluated. Figure 5: Depiction of local S/C TTL coupling simulation. The RX beam (red) starts at the local RX RP with the expected size after having propagated for the inter-satellite distance, and a certain rotation angle to simulate S/C angular motion. The RX beam interferes with the LO beam (green) and their beatnote is captured by the photodetectors. The results are depicted in Figure 7, which shows the variation of relevant quantities as a function of RX beam angle in both the pitch and yaw degrees of freedom for the interferometer with closed-loop control of the steering mirror.