# Comparing Link Budget Requirements for Future Space-Based Interferometers
## 2 Tracking Error in Phasemeters  
In inter-satellite interferometric missions, the phasemeter is a digital Phase-Locked Loop (PLL) that tracks the phase of the megahertz interference signal between the incoming weak optical field and a bright local oscillator [13]. The phasemeter tracks the phase in two stages: first, the input signal is compared to an internally generated reference oscillator to determine the phase difference between the two. Then, the measured phase difference is recorded and fed back to update the internally generated oscillator. The phasemeter functions like a lock-in amplifier, except the demodulation frequency is constantly updated to match the input frequency. This enables the phasemeter to track the low-frequency phase information encoded onto dynamic high-frequency carriers over long timescales without needing to record the full rate data of the entire measurement period. To optimize the phasemeter performance, the parameters of the phasemeter must be carefully tuned to minimize tracking uncertainty. In the linear PLL model of a phasemeter, the variance of the tracked phase error (the tracked phase error is calculated by taking the difference between the measured phase signal and the true phase signal (in cycles-squared)) is given by [11]  
\[\sigma_{\phi}^{2}=\int_{0}^{\frac{f_{\mathrm{s}}}{2}}S_{\mathrm{AN}}(f)\bigg{|} \frac{H(f)}{1+H(f)}\bigg{|}^{2}+\Big{(}S_{\mathrm{LFN}}(f)+S_{\mathrm{Sig}}(f) \Big{)}\bigg{|}\frac{1}{1+H(f)}\bigg{|}^{2}df \tag{1}\]  
where \(f_{\mathrm{s}}\) is the sampling frequency, \(H(f)\) is the open-loop transfer function of the phasemeter, \(S_{\mathrm{AN}}(f)\) is the power spectral density (PSD) of the additive noise-induced phase noise (dominated by photon shot noise, which is spectrally white in the phase), and \(S_{\mathrm{LFN}}(f)\) and \(S_{\mathrm{Sig}}(f)\) are the PSDs of the laser frequency noise (LFN) phase fluctuations and tracked signal dynamics, respectively, in units of \(\mathrm{cycles}^{2}/\mathrm{Hz}\). While \(\mathrm{H}(f)\) is dependent on a number of parameters within the phasemeter, under the constraints placed in this work (discussed in Section 3.1), the parameter tuning of the phasemeter can be summarized by one value: the unity gain frequency (UGF). The unity gain frequency of the phasemeter, also here referred to as the bandwidth of the phasemeter, is the frequency band around the tracking frequency in which the phasemeter is sensitive to signal fluctuations. Equation (1) can be intuitively interpreted as follows: the error in phasemeter measurement is proportional to the additive noise present inside the bandwidth of the phasemeter and the signal frequency components present outside the bandwidth of the phasemeter. This motivates the trade-off in phasemeter tuning, as the bandwidth of the phasemeter must be large enough to minimize error contribution from untracked signal components (\(S_{\mathrm{LFN}}(f)+S_{\mathrm{Sig}}(f)\)) but not so large that additional additive noise (\(S_{\mathrm{AN}}(f)\)) is tracked unnecessarily. Figure 2 illustrates the phase amplitude spectral densities (ASD) of the three contributing error terms in Equation (1): the shot-noise-induced phase noise for a 200 femtowatt strength field, a gravity signal taken from GRACE-FO level-1B data at 0.5 Hz [14], and three cases of laser frequency noise that match the three configurations of Figure 1. In the case of shot-noise-limited detection, the shot-noise-induced phase noise is given by the following equation [15]:  
\[A_{\text{AN}}(f)=\sqrt{S_{\text{AN}}(f)}=\sqrt{\frac{\hbar c}{2\pi P_{\text{ Sig}}\lambda\eta_{qe}}}\quad\left(\text{Cycles}/\sqrt{\text{Hz}}\right) \tag{2}\]  
where \(\hbar\) is the reduced Planck's constant, \(c\) is the speed of light in a vacuum, \(\lambda\) is the wavelength of the light, \(P_{\text{Sig}}\) is the optical power in the received weak field and \(\eta_{qe}\) is the detection quantum efficiency.