## 2 Overview of LRI  
The LRI is a transponder-type heterodyne interferometer: one satellite is treated as the master role where the laser frequency is stabilized by an optical Fabry-Perot (FP) cavity, while the other one acts as a transponder in which the local laser continuously reproduces the phase of the received (RX) light from the master satellite through an optical phase-locked loop (OPLL). Each satellite is equipped with photodiodes and phasemeters to measure the phase of the interference light.  
Figure 1 is a schematic illustration of the LRI optical components. On the master side, by using the Pound-Drever-Hall (PDH) method [19], the laser frequency is required to be stabilized to 30 Hz/Hz\({}^{1/2}\) for Fourier frequencies above 0.01 Hz. It was reported that at frequencies below 0.2 Hz, contributions from gravitational and non-gravitational forces dominate the LRI ranging signal [20; 21]; thus, it is hard to estimate the laser frequency noise for GRACE-FO in flight at low frequencies. However, based on the performance of the LRI cavity measured on the ground and the actual flight data, the Albert Einstein Institution (AEI) provided the following formula to model the laser frequency noise of LRI [10]:  
\[\mathrm{ASD}[\delta f_{\mathrm{cav}}](f)=0.32\,\mathrm{Hz}/\mathrm{Hz}^{1/2} \times\left(\frac{f}{1\,\mathrm{Hz}}\right)^{-0.6} \tag{1}\]  
where ASD is the abbreviation of amplitude spectral density which is defined as the square root of power spectral density (PSD). The frequency-stabilized laser beam is injected to the optical bench (OB) through a fiber collimator and split at a beam splitter (BS) into two parts--the local oscillator (LO) part and the transmitted (TX) part. The local beam interferes with the received beam whose frequency is a bit different due to a frequency offset \(f_{\mathrm{off}}\approx 10\,\mathrm{MHz}\) introduced on the transponder side and a Doppler shift caused by the relative velocity between the satellites. By means of a differential wavefront sensing (DWS) technique [22], the misaligned angle between the two beams is determined and the fast steering mirror (FSM) is controlled to compensate this angle by changing the direction of the local beam. In this situation, the TX beam direction is parallel to the received beam and it points correctly towards the distant satellite after retro-reflection at the TMA as long as the TMA mirrors are pairwise perpendicular. A compensation plate (CP) is needed to compensate part of the TTL coupling effect caused by the BS [3].  
On the transponder side, the optical path is the same while the laser is not in the reference mode and the cavity for frequency stabilization is not used. The laser frequency is locked to the frequency of the incoming light and a constant offset \(f_{\mathrm{off}}\) is added in the frequency-offset digital phase-locked loop (DPLL). This ensures that the beatnote frequency of the interference light at the quadrant photodiodes (QPD) varies around 10 MHz and the phase variations of the incoming light are reproduced by the transponder laser. The frequency-shifted light is then sent back to the master satellite and interferes after the beam splitter mentioned above.  
Ideally, the phase measured on the transponder side is fully contributed by the frequency offset while the full round-trip pathlength measurement is performed on the master side where the contribution of the frequency offset still exists. Therefore, a two-way ranging phase combination is performed by computing the difference of the transponder and master phase, canceling the contributions from the frequency offset and retaining the ranging signals.