# Title-to-Length Coupling Analysis of an Off-Axis Optical Bench Design for NGGM
## 2 Off-Axis Optical Bench Design
The front focal point of L2 is put on the surface of the FSM, and the back focus of L3 is placed at the transmitter reference point (TX RP). The receiver reference point (RX RP) locates at the centre of the receiving aperture, and the centre of the RX RP and TX RP is designated as the reference point (RP) of the optical bench, which is supposed to coincide with the centre of mass (CoM) of the spacecraft, as well as the accelerometer reference point [8]. The relationship of the angular magnifications among these three lens systems is supposed to be [9]:  
\[\left|\mathrm{m_{a,rx}}\right|=\frac{\left|\mathrm{m_{a,lo}}\right|}{\left| \mathrm{m_{a,tx}}\right|}. \tag{1}\]  
where \(\mathrm{m_{a,rx}},\mathrm{m_{a,lo}},\mathrm{m_{a,tx}}\) denote, respectively, the angular magnifications of the RX, LO, and TX lens systems. The RX and TX beams are maintained to be anti-parallel in the off-axis setup with a combination of lens systems and mirrors, under the control of the DWS feedback loop. Not all optical components are equal in terms of their impacts on the optical pathlength difference (OPD); some are more equal than others. We divide the optical components into critical and uncritical categories. For the off-axis layout, beamsplitter BS1, the rooftop mirrors, and lenses L1, L2, and L3 are critical components, which should be carefully aligned. Those critical components work as beam combiners as well as components directing the beam to a readout target, whose misalignments will cause a significant variation in OPD [10]. The alignment of lenses L4 and L5 are less stringent due to a common mode cancellation of the OPD changes of the incident beams. The effect of the misalignment of the FSM is significant on the local optical bench but can be mostly eliminated when considering the global situation. Beamsplitter BS2 may be categorized as a non-critical component which serves only to split the interfering beams and has a negligible impact on the OPD. Our former work gives a preliminary investigation of the off-axis optical bench by considering a smaller waist radius of 1 mm for the LO beam. To achieve the higher heterodyne efficiency and carrier-to-noise ratio (CNR), the waist radius of the LO beam is enlarged to 2.5 mm, in line with that of the GRACE Follow-on mission, and the parameters of the lenses are chosen to be identical to those of an on-axis optical bench design [9]. The parameters adopted in the simulation are based on the commercial components shown in Table 1. The RX beam clipped by the receiving aperture has an approximately flat wavefront and a constant intensity behind the aperture, known as a flat-top beam [8; 11]. A mode expansion method (MEM) is applied to simulate the propagation of a flat-top beam within the optical bench [9; 11]. An illustration of the simulated optical bench is given in Figure 2. All the parameters and results in the simulation are output in double precision. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \(\mathbf{r_{RX,AP}}\) & \(\mathbf{\omega_{0}}\) & \(\mathbf{f_{L,L,3}}\) & \(\mathbf{f_{L,2}}\) & \(\mathbf{f_{L,4,L5}}\) & \(\left|\mathbf{m_{a,r}^{-1}}\right|\) & \(\left|\mathbf{m_{a,b}^{-1}}\right|\) & \(\mathbf{m_{a,lo}^{-1}}\) & \(\mathbf{\eta}\) & \(\mathbf{\eta_{simulated}}\) & **CNR** \\ \hline Parameters & 8 mm & 2.5 mm & 200 mm & 75.6 mm & 12.7 mm & 0.0635 & 2.6455 & 0.1680 & 0.85 & 0.862 & 94.34 dB-Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the optical components used in the simulation: \(\mathbf{r_{RX,AP}}\) is the radius of the receiving aperture; \(\omega_{0}\) is the waist radius of the LO beam; \(\eta\) is the calculated heterodyne efficiency; \(\eta_{simulated}\) is the simulated heterodyne efficiency; CNR is the carrier-to-noise ratio. The initial setup of the components used in the simulation model is depicted in Figure 2. The off-axis layout is further optimized by adjusting the positions of the lenses, the QPDs, and the receiving aperture [9].