# Comparing Link Budget Requirements for Future Space-Based Interferometers
## 3 Simulations of Phase Tracking Performance
### Simulated Experiments
The tracking error is measured by taking the difference between the weak phasemeter and truth phasemeter phase measurements. Shot-noise-induced phase noise modeled for a 200 femtowatt signal. The 0.03 cycle RMS line (black dashed) marks the maximum tolerable phase error for robust tracking. Dual-stabilized and retro-reflector simulation did not track below a unity gain frequency of 8 Hz due to the acquisition requirements of the gravity signal. profile and shot-noise levels. For each simulated measurement, the phasemeter tracked a simulated optical field with laser frequency noise, shot noise, and a GRACE-FO-like gravity signal present for 4 h. The gravity signal and laser frequency noise for the transponder and dual-stabilized cases were the same as those displayed in Figure 2. Three cases of arm response were explored with satellite separation of 100 km, 200 km, and 300 km. The simulated optical power level was varied between 100 attowatts and 1 picowatt to explore its impact on tracking performance. The key results and simulation parameters are summarized in Table 1.  
The minimum power level for stable tracking differed between the transponder, dual-stabilized, and retro-reflector cases. The transponder tracking error crosses the critical threshold of 0.03 cycles RMS at a carrier noise density ratio of 58.6 dB-Hz--140 femtowatts of optical signal in shot-noise-limited detection. The dual-stabilized case exhibits a more gradual increase in tracking error with a decrease in optical power and crosses the critical threshold for linear operation at a far lower level, 30.0 dB-Hz--200 attowatts of optical signal in shot-noise-limited detection. The cavity-suppressed laser frequency noise enables lower phasemeter bandwidths capable of stably operating at such low power levels. The retro-reflector configuration offers major improvements in tracking uncertainty relative to the transponder configuration and minor improvements over the dual-stabilized configuration. The suppression of the laser frequency noise by the arm response enabled phasemeter operation with lower bandwidths, decreasing the frequency band in which \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Configuration**} & \multicolumn{2}{c}{**Laser Frequency Noise**} & \multirow{2}{*}{**Min Power**} & \multirow{2}{*}{**CNdR (dB-Hz)**} \\ \cline{2-2} \cline{4-5}  & **Satellite A** & & & \\ \hline Transponder & Stabilized & Free running &
\begin{tabular}{c} 140 \\ femtowatts \\ \end{tabular} & 58.6 \\ Dual stabilized & Stabilized & Stabilized & 200 attowatts & 30.0 \\ Retro—100 km to 300 km & Stabilized & Na & 150 attowatts & 29.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key results for minimum trackable optical powers and carrier-to-noise density ratios (CNdR) found from Figure 5. Laser frequency noise column expresses the state of the laser on each satellite for each configuration. The resulting tracked laser frequency information of each case can be found in Equations (4), (5), and (7). Figure 5: Optimal performance RMS as a function of optical power for different mission configurations under optimal phasemeter tuning. For each simulated measurement, the optimal phasemeter unity gain frequency was found by minimizing Equation (1) for the given LFN profile and shot-noise-induced phase noise level. The broken grey line indicates the tracking error value, above which the system becomes susceptible to non-linear error. The key findings are summarized in Table 1.  
shot noise couples into measurement and making the configuration more tolerant to lower received optical power levels. For the laser frequency noise levels explored here, the retro-reflector configuration with arm lengths of 100-300 km decreases the optical power required to achieve a trackable RMS error by three orders of magnitude over the transponder configuration. The minimum trackable power simulated here of \(\sim\)150 attowatts has a carrier-to-noise density ratio of 29.0 dB-Hz, significantly below the previous mission requirement of 70 dB-Hz.