# Methods of Atmospheric Coherence Length Measurement
## 3 Coherence Length \(r_{0}\) Direct Measurement Method
### Wavefront Phase Variance Method  
The wavefront phase variance method uses the SHWS to measure the variance of the wavefront phase induced by atmospheric turbulence, which is used to obtain \(r_{0}\)[17; 54]. Compared to the wavefront structure function method, it has less exposure time, thus the measurement would be more accurate. Additionally, its operation is tractable.  
According to Equation (17), the phase structure function can be expressed as the time average of the phase difference between two points. To observe the wavefront aberration caused by atmospheric turbulence, a plane wave is introduced as a reference. The instantaneous wavefront phase variance over the aperture can be expressed as  
\[\begin{array}{lcl}\sigma^{2}(R,t)&=&\frac{1}{\pi R^{2}}\int\limits_{\psi=0} ^{\psi=2\pi}\int\limits_{r=0}^{r=R}[(\Phi(r,\psi,t)-\Phi(0,0,t)]^{2}rdrd\psi \\ &=&\frac{1}{\pi R^{2}}\int\limits_{\psi=0}^{r=2\pi}\int\limits_{r=0}^{R}D_{ \varphi}(r)rdrd\psi,\end{array} \tag{24}\]  
where \(R\) is the aperture radius, and the horizontal line indicates the statistical average in time. The relationship between the wavefront error variance and the phase structure function can be obtained by averaging the left and right sides of the above equation in time. According to the Kolmogorov model, the structure function of a plane wave can be expressed as  
\[D_{\Phi}(r)=6.88(\frac{r}{r_{0}})^{5/3}. \tag{25}\]  
If the atmospheric turbulence is assumed to be statistically homogeneous and isotropic, then the wavefront error variance averaged over time is equal to the phase structure function averaged over space. Now Equation (24) becomes  
\[\overline{\sigma^{2}(R,t)}\approx\frac{1}{\pi R^{2}}\int\limits_{\psi=0}^{ \psi-2\pi}\int\limits_{r=0}^{r=R}6.88(r/r_{0})^{5/3}rdrd\psi=3.75(R/r_{0})^{5/ 3}. \tag{26}\]  
It follows that the value of the atmospheric coherence length can be estimated as  
\[r_{0}=(\frac{3.75\times R^{5/3}}{\overline{\sigma^{2}(R,t)}})^{3/5}. \tag{27}\]  
The principle of the wavefront phase variance method is demonstrated in Figure 6.  
The laser launches the optical beam, which is scaled up by a beam expander. Subsequently, the beam travels through an atmospheric chamber mimicking turbulence. On the receiving side, the beam is scaled down to match the aperture size of the SHWS.  
Since the aperture size of the Shack-Hartman sensor is generally smaller than the internal scale \(l_{0}\) of the turbulent cells, when the aperture radius is small, \(r_{0}\) is not a constant. The Kolmogorov model is appropriate within the inertial subrange only. The inner scale \(l_{0}\) is usually between a few millimeters and a few centimeters, decreasing with the turbulence strength. The outer scale \(L_{0}\) is usually in the order of meters. Thus, the Kolmogorov model is inappropriate for the analysis of the FSO link with an optical beam diameter less than \(l_{0}\). That is, the methodology associated with the wavefront phase variance above may fail. In this regard, the optical beam width needs to be extended so that enough turbulence information can be sampled. The beam is scaled down by a beam adjuster after propagating over the atmospheric channel to match the aperture of the SHWS. By doing so, the effective aperture of the sensor is expanded. According to Equation (27), the atmospheric coherence length \(r_{0}\) is evaluated by the use of the measured phase variance and \(R\). It is worth noting that the method depends strongly on the effective aperture of the sensor used in the experimental setup. When the aperture size of the SHWS is close to the inner scale \(l_{0}\), the model that incorporates the inner scale factor, instead of the Kolmogorov model, could be adopted.  
Figure 6: Schematic sketch of wavefront phase variance method.