# On-Axis Optical Bench for Laser Ranging Instruments in Future Gravity Missions
## 2 On-Axis LRI Optical Bench Design
The RX/TX RP is imaged into the spacecraft center of mass, where an accelerometer is located. The range measurement is invariant under rotations of the spacecraft around this point. The receive and local oscillator beams interfere at polarizing beamsplitter PBS1, and are captured by the pair of quadrant photodiodes QPD1 and QPD2 in balanced detection configuration. transmit beam propagates antiparallel to the received light, i.e., the local spacecraft \"points\" accurately at the remote spacecraft with its TX beam. In the RX and LO imaging systems, the first lens acts on only one of the two interfering beams, while the second lens acts on both beams. The resulting configuration departs significantly from other interferometers employing imaging systems to suppress TTL coupling, where all lenses are placed after the interferometer's beam combiner [10; 11; 13; 22], and is therefore the subject of scrutiny in this paper. The lenses used in the model are available off-the-shelf from a popular lens manufacturer. The use of free-form lenses for TTL coupling reduction in similar precision interferometers has been studied in [23], and does not offer a significant advantage in terms of raw TTL coupling suppression. Fused silica substrates are chosen due to the excellent transmission properties and low coefficient of thermal expansion. A notable feature of this setup is that a relationship is established between the angular magnifications of the RX, LO, and TX imaging systems,  
\[|m_{\rm RX}|=\frac{|m_{\rm LO}|}{|m_{\rm TX}|}, \tag{1}\]  
where \(m_{\rm RX}\), \(m_{\rm TX}\) and \(m_{\rm LO}\) are the angular magnifications of the respective imaging systems. The signs of the angular magnifications depend on the specific OB layout. The beam angle w.r.t. the optical axis at the entrance pupil of each imaging system (\(\alpha_{i,\rm IN}\)), and the angle at the exit pupil (\(\alpha_{i,\rm OUT}\)) are given by (we work in two-dimensional space for simplicity). \[\alpha_{i,\rm OUT}=m_{i}\cdot\alpha_{i,\rm IN}, \tag{2}\]  
where \(i=\{\rm RX,\,LO,\,TX\}\). The sign of these angles follows the Cartesian sign convention: acute angles are positive when produced by anti-clockwise rotation from the optical axis, and negative when produced by clockwise rotation. In our design \(\alpha_{\rm TX,IN}=\alpha_{\rm LO,IN}\), since the TX and LO imaging systems share lens L4 (see Figure 2). Moreover, by actuating the FSM using DWS, the setup ensures that \(\alpha_{\rm RX,OUT}=\alpha_{\rm LO,OUT}\). Therefore, assuming that \(m_{\rm RX}=m_{\rm LO}/m_{\rm TX}\) can be obtained through a specific OB layout, Equation (2) can be used to show that the setup can ideally produce perfect TX beam pointing to the remote S/C, as follows:  
\[\alpha_{RX,IN}=\frac{m_{LO}}{m_{RX}}\cdot\alpha_{LO,IN}=m_{TX}\cdot\alpha_{ TX,IN}=\alpha_{TX,OUT}. \tag{3}\]  
The sign of the in-plane angular magnification of a Keplerian telescope is determined by the number of reflections in its optical path. An even (odd) number of reflections on the optical path leads to negative (positive) angular magnification. Using beamsplitters and mirrors, the OB can therefore be designed such that \(m_{\rm RX}=m_{\rm LO}/m_{\rm TX}\) is verified. To illustrate the flexibility of the proposed OB design, Table 1 shows the resulting heterodyne efficiency \(\eta\) and carrier-to-noise density CNR for two OB configurations (\"A\" and \"B\") differing in the radius of the injected local beam. Both configurations feature an aperture of radius \(r_{\rm TX/RX\,AP}=8\,\rm mm\), and QPDs with active area radii of 0.5 mm. The effective focal lengths of the selected lenses, measured by \(589\,\rm nm\) light, are: \(f_{\rm L1}=f_{\rm L2}=12.7\,\rm mm\), \(f_{\rm L3}=200\,\rm mm\), and \(f_{\rm L4}=75.6\,\rm mm\). The best performance out of the two proposed configurations is achieved in the \"A\" configuration, yielding a heterodyne efficiency of 0.85 and a CNR of 94 dB.