# Optimal Design of Small-Aperture Optical Terminals for Free-Space Links
## 3 Results and Discussion
### Tip-Tilt Bandwidth Optimisation
Hence, to suppress most of the jitter in a FSO link, the bandwidth of the control loop should at least be equal to \(f_{knee}\). Figure 5 also shows the effect of aperture size and turbulence strength on the TT spectrum through the dashed and dot-dashed orange curves, respectively. An increase in aperture size will decrease \(f_{knee}\), reducing the uncompensated jitter, whereas an increase in turbulence strength will uniformly increase the magnitude of the entire TT spectrum. To quantify the actual TT suppression required, the residual jitter should keep the normalised TT-only coupling efficiency above 90%, a choice in line with that presented in [30]. This constraint will bring the total power throughput and scintillation close to the TT-compensated case from Figure 3. Using Equation (11), this requirement constrains the bandwidth like so:  
\[\left(\int df\ |1-H(f)|^{2}S_{TT}(f)\right)^{1/2}<0.22\frac{\lambda}{D}. \tag{15}\]  
Figure 5: Uncompensated (orange) and residual (blue) atmospheric TT spectra for a suppression bandwidth of 10 Hz. The orange dot-dashed and dashed curves show the effect of increasing the aperture size and turbulence strength, respectively. The fast-steering mirror’s residual transfer function is shown as a black, dotted curve. The integrated residual jitter spectra gives the residual jitter variance, \(\sigma_{ jitter,res}^{2}\). Figure 6 shows how \(\sigma_{jit,res}\) varies with the control loop bandwidth. It highlights the minimum bandwidth needed to meet the specified value of \(\tilde{\eta}_{c,TT}(\sigma_{jit,res})\) for a range of practical aperture sizes and fried parameters. The required bandwidth ranges from 1 to 70 Hz across all configurations, and particularly weak turbulence strengths do not require TT compensation at all. In general, a larger aperture size will lead to less residual jitter but also make \(\tilde{\eta}_{c,TT}(\theta)\) more sensitive to jitter. The net result is that the required bandwidth increases with increasing aperture size. This behaviour opposes what Figure 5 suggests and highlights the importance of using the normalised TT-only coupling efficiency to inform the steering control loop bandwidth. Table 2 shows the required TT compensation bandwidth and range for each link when using the optimal aperture diameter, \(D\), identified in Figure 3A. Note that being passively below the \(\sigma_{jit,res}\) requirement does not suggest that TT compensation is not required, but rather that a low (\(\sim\)1 Hz) bandwidth will suffice. In practice, TT compensation is always needed to correct for long-term path-varying effects. As such, any sub-1 Hz bandwidths are entered as \(<\)1 Hz in \[\tilde{\eta}_{c,TT}(\theta)=\exp\left[-0.22\left(\frac{\pi D\theta}{\lambda} \right)^{2}\right]\,. \tag{11}\]0  
For the case of dynamic FSO links, residual jitter arising from imperfect target tracking via a telescope mount or similar is typically much larger than that due to atmospheric turbulence [32]. As a result, the modest improvement in coupling efficiency after TT compensation shown in Figure 1 will be much more substantial for dynamic links but also require a greater fast-steering mirror range. The required bandwidth should not change significantly as tracking errors are typically lower-frequency than atmospheric TT. The TT spectrum of a dynamic link can be obtained using a quadrant photodetector or camera [27; 28]. Once this is carried out, the preceding analysis can be used to constrain