# Optimal Design of Small-Aperture Optical Terminals for Free-Space Links
## 3 Results and Discussion
### Power Throughput Optimisation
This analysis does not consider higher-order compensation, focusing on a lower cost and more simple design. Each link length and \(C_{n}^{2}\) value is numerically simulated to obtain the free-space and fibre scintillation indices which are plotted in Figure 3B. The free-space scintillation index is shown by a dashed line. As in the power throughput plots, a band gives the upper and lower limits on the fibre scintillation index without and with TT compensation, respectively. The uncompensated fibre \(\sigma_{ln,l}^{2}\) is calculated from the fibre-coupled power, which is found by convolving the aperture-plane free-space power with the back-propagated fibre mode [29]. The TT-compensated fibre \(\sigma_{ln,l}^{2}\) is found by centering the Fourier transform of the incident beam prior to convolution. This is equivalent to suppressing the G-tilt of the beam, representative of how TT compensation is performed when using a camera or quadrant photodetector [30]. Figure 3: (**A**) \(G_{link}\) (black curve), uncompensated \(\eta_{c}\) (dashed curves) and \(\eta_{net}\) (solid curves) for three point-to-point links of length \(750\,\mathrm{m}\), \(2400\,\mathrm{m}\), and \(10,600\,\mathrm{m}\) between a transmitter and receiver of equal aperture diameter. Three \(C_{n}^{2}\) values of \(1\times 10^{-15}\)\(m^{-2/3}\) (blue), \(5\times 10^{-15}\)\(m^{-2/3}\) (orange) and \(1\times 10^{-14}\)\(m^{-2/3}\) (green) are used. The upper and lower power throughput curves correspond to the TT-compensated and uncompensated cases, and the filled region represents the range for imperfect TT compensation. (**B**) Simulation results of the scintillation indices for free-space (dashed curve), and fibre (solid curves). The lower and upper fibre curves correspond to the TT-compensated and uncompensated cases. Assuming the free-space detector can see the entire aperture of the receiver, TT compensation has no effect on the free-space \(\sigma_{ln,l}^{2}\). Observing Figure 3B, as the turbulence strength increases, all three scintillation indices increase, and when \(D\) increases, aperture averaging effects predictably reduce the free-space scintillation. However, the scintillation in fibre is not as straightforward, since it depends on the scintillation of the incident beam and also on its wavefront perturbations. So, the net trend of fibre scintillation with aperture diameter depends on the balance of these two effects. We see that the scintillation index increases with increasing \(D\) for lower turbulence strengths, and this trend becomes weaker and eventually changes sign as the turbulence strength increases. Compared with shorter links, the fibre and free-space scintillation indices of the 10,600 m link do not change as much when \(C_{n}^{2}\) increases. This is attributed to being close to/within the scintillation saturation regime [22]. Looking at the improvements offered by TT compensation, we see in Figure 3B that TT compensation has the largest improvement on fibre scintillation for lower turbulence strengths. This is because of how Zernike modes are distributed over different turbulence strengths. In weak turbulence, TT is the dominant mode [29], so TT compensation will remove a majority of perturbations from the beam, resulting in large improvements in the scintillation index. In stronger turbulence conditions, wavefront perturbations excite higher-order Zernike modes, so only compensating for TT has less of an impact. Additionally, as the free-space detector is assumed to be insensitive to TT, weak turbulence conditions yield the biggest difference between the fibre and free-space \(\sigma_{ln,l}^{2}\). Note that the \(D\) is already constrained to maximise the power throughput. However, using these \(D\) values in Figure 3B does not lead to a minimised fibre scintillation index. Minimising the fibre scintillation index requires \(D\) in the range of 0.10-0.15 m, which would result in a lower power throughput.