# Optimal Design of Small-Aperture Optical Terminals for Free-Space Links
## 3 Results and Discussion
### Tip-Tilt Bandwidth Optimisation  
With all the passive optical elements from Figure 2 now chosen, the design of the TT compensation system may be examined. The system must suppress the wavefront TT so the scintillation and total power throughput can match the TT-compensated cases. The two parameters of interest are the mirror's maximum angular range and the frequency response of the overall steering control loop. To constrain these parameters, the effect of beam misalignment on fibre coupling efficiency should be isolated. For the configuration shown in Figure 2, any angular offset in the incoming beam of \(\theta\) will result in a translation of the focused beam along the fibre tip equal to \(f_{\text{eff}}\theta\). By treating the incoming beam and propagation mode of the fibre as Gaussian, the resulting TT-only coupling efficiency is given by [31]  
\[\eta_{c,TT}(\theta)=\left(\frac{2\omega_{1}^{2}\omega_{2}^{2}}{\omega_{1}^{2}+ \omega_{2}^{2}}\right)^{2}\exp\left[-\frac{2(f\theta)^{2}}{\omega_{1}^{2}+ \omega_{2}^{2}}\right]\,, \tag{10}\]  
where \(\omega_{1}\) and \(\omega_{2}\) are the \(1/e^{2}\) spot radius and mode field radius of the focused beam and fibre tip, respectively. Equation (10) can be simplified noting the following: the spot size of the focused Gaussian beam is \(\omega_{1}=2\lambda f/\pi D\) from ray analysis; and the effective focal length \(f_{\text{eff}}\) is constrained via the \(a\) parameter from Equation (6). With this in mind, the isolated effect of TT on coupling efficiency can be expressed through the normalised TT-only coupling efficiency, \(\tilde{\eta}_{c,TT}(\theta)=\eta_{c,TT}(\theta)/\eta_{c,TT}(0)\)  
\[\tilde{\eta}_{c,TT}(\theta)=\exp\left[-0.22\left(\frac{\pi D\theta}{\lambda} \right)^{2}\right]\,. \tag{11}\]  
We now obtain the expected atmospheric TT variation, referred to as jitter, to use with Equation (11) to specify the TT compensation system's required range and bandwidth. Jitter is determined by the TT component of the overall atmospheric noise power spectral density (PSD), \(S_{TT}(f)\). This is approximated by [30] to be  
\[S_{TT}(f)=0.31D^{-2}f^{-8/3}\int_{L}dz\,{C_{n}}^{2}(z)V(z)^{5/3}F_{G}(fD/V(z))\, \tag{12}\]  
Figure 4: Upper and lower quartiles of \(\eta_{net}\) for each link length and turbulence condition from Figure 3A, for three different aperture diameters. Three \(C_{n}^{2}\) values of \(1\times 10^{-15}\), \(5\times 10^{-15}\) and \(1\times 10^{-14}\,m^{-2/3}\) are used, coloured blue, orange, and green, respectively. The light and solid bars correspond to the TT uncompensated and compensated cases, respectively. The white dashes within each bar correspond to \(\eta_{net}\) from Figure 3.  
where \(V(z)\) is the perpendicular wind speed and  
\[F_{G}(y)=\int_{0}^{1}dx\ \frac{x^{5/3}}{\sqrt{1-x^{2}}}J_{1}{}^{2}(\pi y/x). \tag{13}\]  
Equation (12) can be integrated to find the jitter variance, \(\sigma_{ jitter}^{2}\)[30]. The designed maximum optical range of the fast-steering mirror is then set based on the square root of this value. We choose a range of \(3\sigma_{ jitter}\), corresponding to 5 to 75 \(\upmu\)rad for the range of turbulence strengths and optimal aperture sizes being analysed. The bandwidth of the steering control loop dictates its frequency response, \(H(f)\). In the simplest approximation, \(H(f)\) is given by  
\[H(f)=\frac{1}{1+if/f_{3 dB}}\, \tag{14}\]  
where \(f_{3 dB}\) is the bandwidth, imposed by processing delays and actuator dynamics. The residual TT spectrum is then given by \(|1-H(f)|^{2}S_{TT}(f)\), which can be integrated to give the residual jitter variance, \(\sigma_{ jitter,res}^{2}\). Figure 5 shows an example of uncompensated and residual TT spectra for a control loop bandwidth of 10 Hz. The fast-steering mirror is characterised by diminishing suppression as the bandwidth of the control loop is approached. The uncompensated spectrum shifts from an \(f^{-2/3}\) to \(f^{-11/3}\) frequency dependence past the frequency \(f_{knee}=V/D\), due to the effects of aperture averaging [30].