# Comparing Link Budget Requirements for Future Space-Based Interferometers
## 3 Simulations of Phase Tracking Performance
### Simulated Experiments  
Figure 4 shows the simulated weak phasemeter tracking error, calculated as the difference between the weak phasemeter measurement (\(\phi_{\text{Weak}}\)) and the truth phasemeter measurement (\(\phi_{\text{Weak}}\)), with change in unity gain frequency for constant optical power. The field amplitude of the weak field was simulated at 200 femtowatts and the truth field to 10 nanowatts. The three cases of laser frequency noise explored were the same as shown in Figure 2. For each case, the unity gain frequency was swept with logarithmic frequency spacing and the signals were continuously tracked at each level. For the stabilized and arm response simulations, the unity gain frequency was swept from 8 Hz to 100 kHz. Figure 3: Block diagram of the hardware-in-loop simulation. The initial frequency input, \(f_{0}\), is combined with numerically generated laser frequency noise before being integrated to produce a phase signal. The output phase signal is combined with an up-sampled phase signal from GRACE-FO data [14] and fed into a numerically controlled oscillator (NCO). The resulting signal is split before gain scaling and additive noise inclusion to simulate a weak, low carrier-to-noise density ratio signal and a strong, high carrier-to-noise density ratio signal. These two signals are tracked with the weak and truth phasemeter, the truth measurement serving as a reference for the weak field measurement. For free-running LFN the unity gain frequency was swept from 1 kHz to 100 kHz, as below 1 kHz the tracking error was above 0.1 cycles RMS, making it almost impossible to track. For each simulation, the gravity signal was seeded such that the phasemeter could acquire a lock of the moving signal successfully. This meant that at lock acquisition, the frequency slew rate of the gravity signal was less than the square of the unity gain frequency, \(f_{\text{slew}}<f_{\text{ugf}}^{2}\), for long enough for the integrators to accrue sufficient information to suppress the low-frequency signal. For the specific gravity signal used, this was not possible below a unity gain frequency of 8 Hz, making this the minimum unity gain frequency explored here. It should be noted, however, that by removing the large amplitude components of the gravity signal that are known in pre-processing, lower unity gain frequencies could be achieved and acquisition requirements could be relaxed.The limits of lock acquisition for weak optical fields and low-bandwidth phasemeters is an area where further research is needed. Figure 4 illustrates that for the three cases of laser frequency noise, the minimum tracking error value and optimal unity gain frequency tuning differ. Lasers with lower noise, i.e., stabilized and arm response cases, have less tracking error at lower unity gain frequencies, while higher noise laser signals, i.e., the free-running case, need higher unity gain frequencies to capture all of their frequency dynamics. The grey dashed line at the 0.03 cycles RMS line represents the point past which the phasemeter begins to exhibit non-linear behavior. As all three configurations have minimum tracking errors below this line, stable tracking is possible for all cases at a received power level of 200 femtowatts. The extra clearance afforded by the stabilized and retro-reflector cases, however, allow these cases to continue to meet this requirement as the optical power decreases. Figure 5 displays the weak phasemeter tracking error as a function of optical power under optimal unity gain frequency tuning. At each power level, the unity gain frequency was tuned using the analytical solutions from Figure 4 given the input laser frequency noise  
Figure 4: Tracking error under change in phasemeter unity gain frequency for transponder (red), dual-stabilized (blue), and retro-reflector with 200 km separation (orange) mission architectures. The model for the residual phase error, with solid colored lines, was calculated from Equation (1).