# Comparing Link Budget Requirements for Future Space-Based Interferometers
## 2 Tracking Error in Phasemeters
While this work limits additive noise to optical shot-noise-driven contributions, its results can be generalized to include other additive noise sources, such as laser-relative intensity noise or photodetector electronic noise, through the shot-noise-limited carrier-to-noise density ratio (derived in a similar form in [16]):  
\[\text{CNdR}=10\times\log_{10}\biggl{(}\frac{1}{(2\pi A_{\text{AN}}(f))^{2}} \biggr{)}=10\times\log_{10}\biggl{(}\frac{P_{\text{Sig}}\lambda\eta_{qe}}{2\pi \hbar c}\biggr{)}\quad\left(\text{dB-Hz}\right). \tag{3}\]  
Non-shot-noise-limited systems with spectrally white additive noise across the measurement band, due to sources such as detector noise or laser RIN, can be compared by measuring the carrier-to-noise density ratio and rearranging Equation (3) to compute the effective shot-noise-limited optical power. The three different cases of laser frequency noise presented align with the three mission architectures presented above:  
* **Transponder**--In the transponder case, the phasemeter at satellite B tracks the beatnote between a stabilized incoming signal and its own free-running oscillator. For redundancy, this phase tracking requirement is placed on both satellites. The free-running laser frequency noise is modeled here as \(1/f\) in frequency with a 1 Hz crossing value of 18 kHz. This figure is a conservative estimate for the free-running frequency noise of an Nd:YAG nonplanar ring oscillator (NPRO) laser, as presented in [17]. The resulting phase ASD in Figure 2 is given by the following equation: \[A_{\text{LEN}}(f)=\frac{18\times 10^{3}}{f}\times\frac{1}{2\pi f}\quad\left( \text{Cycles}/\sqrt{\text{Hz}}\right).\] (4)
* **Dual stabilized**--In the dual-stabilized case, both optical sources are locked to independent ultra-stable references. Assuming the two cavities are identical in performance, the signal's phase ASD is \(\sqrt{2}\) greater than either individual signal. The stabilized laser frequency noise of a single laser is modeled here as white in frequency at a level of 0.1 Hz/\(\sqrt{\text{Hz}}\) until crossing with the free-running noise level (at 180 kHz), after which it follows that. This is a conservative model for thermal noise-limited cavity performance above 1 Hz, which follows 0.1 Hz/\(\sqrt{f}\)[18]. The resulting phase ASD in Figure 2 is given by the following equation: \[A_{\text{LEN}}(f)=\frac{\sqrt{2}\times 0.1}{1+\frac{f}{18\times 10^{9}/0.1}} \times\frac{1}{2\pi f}\quad\left(\text{Cycles}/\sqrt{\text{Hz}}\right).\] (5)
* **Retro-reflector**--In the retro-reflector case, a single stabilized optical signal is interfered with a time-delayed version of itself that has traveled to and from satellite B. This means that the stabilized laser frequency noise is further suppressed by the arm response of the interferometer. For time delay \(\tau\), where \(\tau=2L/c\), \(L\) is the inter-space craft separation and \(c\) is the speed of light, the resulting laser frequency noise suppression is given by \[H_{\text{arm}}(f)=1-e^{i2\pi f\tau}\] (6)The resulting phase ASD in Figure 2 is given by the following equation:  
\[A_{\text{LFN}}(f)=\Big{|}1-e^{i2\pi f\tau}\Big{|}\frac{0.1}{1+\frac{f}{18\times 1 0^{7}/0.1}}\times\frac{1}{2\pi f}\quad\Big{(}\text{Cycles}/\sqrt{\text{Hz}} \Big{)}. \tag{7}\]  
The differences in the amplitude and spectral shaping of each of these cases necessitate different phasemeter bandwidths to minimize tracking error and, as such, they can tolerate different levels of additive noise, hence the different optical power levels. As the root mean square (RMS) of the tracking error (the square root of Equation (1)) grows beyond a value of 0.03 cycles (at an RMS of 0.03 cycles, linear trigonometric approximations in the phasemeter begin to become invalid), the phasemeter becomes more susceptible to non-linear measurement errors, such as cycle slips and loss of lock [12]. Cycle slips are probabilistic events where the tracked phase jumps in integer cycles relative to the true phase signal.