# Optimal Design of Small-Aperture Optical Terminals for Free-Space Links
## 2 Materials and Methods  
### Atmospheric Turbulence  
The design process begins by quantifying the turbulence of the free-space link. Local temperature and pressure variations in the atmosphere occur down to millisecond timescales. These variations create turbulent cells known as eddies with varying refractive indices [3]. As a laser beam propagates through the atmosphere, it interacts with these eddies and experiences wavefront perturbations. Changes in the gradient of the wavefront are called wavefront TT. Wavefront perturbations are quantified through the Fried parameter, \(r_{0}\), which describes the spatial coherence diameter of the beam after propagation over a given FSO link. The larger a beam is relative to \(r_{0}\), the more perturbed its wavefront will be. The simplest approximation of a propagating beam is a plane wave, which has a Fried parameter given by \(r_{0}=2.1\rho_{0,pl}\), where \(\rho_{0,pl}\) is the plane-wave spatial coherence radius given by  
\[\rho_{0,pl}=\left[1.46\left(\frac{2\pi}{\lambda}\right)^{2}\int_{0}^{L}dz\,C_ {n}^{2}(z)\right]^{-3/5}\,. \tag{1}\]  
Here, \(z\) is the line-of-sight position along the link, \(L\) is the link length, \(\lambda\) is the beam's wavelength, and \(C_{n}^{2}(z)\) is the refractive index structure parameter. For simplicity, this paper assumes a uniform distribution, \(C_{n}^{2}(z)=C_{n}^{2}\). This is a suitable simplification for horizontal links, but for significantly inclined links, \(C_{n}^{2}(z)\) distributions such as the Hufnagel-Valley model [16] should be used.  
Note that Equation (1) is only applicable to approximately plane-wave beams under weak turbulence conditions. However, this paper considers Gaussian beams over a range of turbulence conditions. For general turbulence conditions, the Fried parameter of a Gaussian beam is given by  
\[r_{0}=2.1\rho_{0,pl}\left(\frac{8}{3\left(a_{e}+0.62\Lambda_{e}^{11/6}\right)} \right)^{3/5}\,, \tag{2}\]  
where \(\Lambda_{e}\) is the effective Fresnel ratio beam parameter, \(a_{e}\) is given by  
\[a_{e}=\left\{\begin{array}{cc}\frac{1-\Theta_{e}^{8/3}}{1-\Theta_{e}}&\Theta _{e}\geq 0\\ \frac{1+|\Theta_{e}|^{8/3}}{1-\Theta_{e}}&\Theta_{e}<0\end{array}\right.\,, \tag{3}\]  
and \(\Theta_{e}\) is the effective curvature beam parameter. \(\Lambda_{e}\) and \(\Theta_{e}\) are modifications of the standard diffraction and refraction beam parameters, \(\Lambda\) and \(\Theta\), which are commonly used to describe the beam size and radius of curvature at the receiver plane [17]. \(\Lambda_{e}\) and \(\Theta_{e}\) account for the additional refraction and diffraction caused by turbulence as a beam propagates. They are functions of \(C_{n}^{2}\), \(z\) and \(w_{0}\), the waist size of the transmitted beam, and equations for them can be found in [17].  
Wavefront perturbations may be represented through the Zernike modes. Compensation of a beam's perturbed wavefront up to the \(j\)th Zernike mode, \(Z_{j}\), will increase the beam's spatial coherence length. This compensation can be quantified through the generalised Fried parameter, \(r_{0,j}\)[18]  
\[r_{0,j}=\left(\frac{3.44}{\text{coef}(j)}\right)^{3/5}0.286j^{-0.362}r_{0}\, \tag{4}\]where \(\mathrm{coef}(j)\) is the coefficient of residual wavefront distortion after partial compensation given by [19]. Of interest to this paper are the second term \(Z_{2}\) and third term \(Z_{3}\), which represent wavefront tip and tilt. Equation (4) can be used to give the TT-compensated Fried parameter, \(r_{0,3}=1.347r_{0}\), which is key in determining the improvements offered by TT compensation.