# On-Axis Optical Bench for Laser Ranging Instruments in Future Gravity Missions
## 2 On-Axis LRI Optical Bench Design
At higher frequencies (above \(\sim\)40 mHz) the sensitivity is limited by residual frequency fluctuations of the laser, arising from the instability of a reference optical cavity or of an atomic standard. The coupling of laser frequency noise into the range measurement can be \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(r_{\text{TX}}\)/RX AP & 8 mm & 8 mm & 4 mm & 8 mm & 8 mm \\ \(\omega_{0}\) & 2.5 mm & 1 mm & 2.5 mm & 2.5 mm & 6.6 mm \\ \(|m_{\text{X}}^{-1}|\) & 0.064 & 0.064 & 0.125 & 0.064 & 0.064 \\ \(|m_{\text{L}}^{-1}|\) & 0.168 & 0.168 & 0.125 & 0.064 & 0.064 \\ \(|m_{\text{TX}}^{-1}|\) & 2.646 & 2.646 & n/a & n/a & n/a \\ \(\eta\) & 0.85 & 0.23 & 0.67 & 0.2 & 0.85 \\ \(\theta_{\text{TX}}\) & 65 \(\upmu\)rad & 131 \(\upmu\)rad & 149 \(\upmu\)rad & 138 \(\upmu\)rad & 65 \(\upmu\)rad \\ CNR (ideal) & 94.2 dB & 82.9 dB & 81.2 dB & 82.5 dB & 94.9 dB \\ CNR (misaligned) & 88.8 dB & 81.4 dB & 80.0 dB & 81.2 dB & 89.5 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of some optical parameters between two configurations of the proposed LRI OB design, the GRACE-FO LRI OB, and two modified versions of the GRACE-FO LRI OB. \(r_{\text{TX}/\text{RX}}\) AP is the receive/transmit aperture; \(\omega_{0}\) is the waist radius of the local oscillator beam after the fiber injector; \(m_{i}^{-1}\) with \(i=\{\text{RX},\text{LO},\text{TX}\}\) are the linear magnifications of the RX, LO, and TX imaging systems, respectively; \(\eta\) is the heterodyne efficiency of the RX-LO beatnote; \(\theta_{\text{TX}}\) is the half-angle divergence of the TX beam out of the aperture; CNR (ideal) is the resulting carrier-to-noise density for perfectly aligned local and remote satellites; CNR (misaligned) is the resulting carrier-to-noise density considering a 50 \(\upmu\)rad misalignment of the local S/C with respect to the line-of-sight to the remote S/C [11; 13]. estimated by the fractional frequency noise of the laser scaled by the S/C separation [18]. For example, the 1064 nm laser on GRACE-FO was stabilized in-flight to a frequency noise below \(1\,\mathrm{Hz}/\sqrt{\mathrm{Hz}}\) resulting in a ranging sensitivity below \(1\,\mathrm{nm}/\sqrt{\mathrm{Hz}}\) for an S/C separation of approximately 200 km and at a Fourier frequency of 1 Hz [19]. At large inter-satellite distances, the interferometer may become shot-noise-limited due to the decreased received optical power, resulting in a CNR-limited sensitivity. Optical bench designs with intrinsically higher CNR values can therefore lead to a sensitivity enhancement and enable longer baselines. The design of the proposed OB is depicted in Figure 2. The RX beam enters the OB via mirror M1, which images the virtual RP (i.e., the S/C c.m.) into the RX/TX reference point aperture located in the OB. The LO and TX beams both originate from a single beam that is injected into the bench via a fiber injector. After passing a suitably aligned polarizer, this initial beam contains 8% p-polarized and 92% s-polarized light, and through interacting with the polarizing beamsplitter PBS2, the beam is split into the TX beam (p-pol), which is transmitted out of the spacecraft via the same path as the RX beam, and the LO beam (s-pol), which propagates to PBS1 where it interferes with the RX beam. The RX-LO beatnote is then captured by both quadrant photodiodes QPD1 and QPD2 in a balanced detection scheme. A total of three retarder waveplates (WP1, WP2, and WP3) are placed north, west, and east of PBS2 in order to keep all beams in orthogonal polarization states prior to the RX-LO interference at PBS1. The main purpose of introducing WP2 with its slow axis at 22.5 degrees is to allow both the local and the remote S/C optical benches to share identical optical layouts, as they both receive and transmit beams with \(-45\) and \(+45\) degree linear polarizations, respectively.