# Methods of Atmospheric Coherence Length Measurement
## 3 Coherence Length \(r_{0}\) Direct Measurement Method
### Slope Difference Method  
In contrast to the two-observation-point approach in the DIMM, the slope difference method uses an array of slopes measured from the SHWS to capture the turbulence effects in statistics [49; 50; 51; 52; 53]. Since the spatial-temporal slope structure implicates both spatial and temporal fluctuations resulting from the atmospheric turbulence, the structure function is useful for characterizing both statistics. The spatial-temporal slope structure function is defined as the mean squared difference between the slopes measured in any two subapertures. One advantage of the slope structure function is that it is insensitive to system jitter and overall wavefront tilt. The inherent mathematical processing associated with the slope structure function offsets any overall tilt that the subapertures bring. On the other side, the insensitivity to system jitter effectively avoids the systemic errors when measurements are operated with a moving or vibrating platform. The SHWS splits the pupil of the imaging system into a limited number of sub-optical paths. Each sensor element focuses incident light onto the CCD with a subaperture lens. As such, an array of points in the focal plane is created. We assume that the subapertures are squares with the same orientation, but in an arbitrary geometric arrangement. The centroid position of the irradiance points on the detector array is calculated according to occupied pixels in the CCD. The centroid position is related to the average slope or gradient of the wavefront on the subaperture lens. The average slope gradient is written as [49],  
\[\eta_{\chi}(x,t)=\int dr[\
abla Z(r-x)\chi]\theta(r,t), \tag{19}\]  
where \(\eta_{\chi}(x,t)\) is the measured slope in direction \(\chi\) centered at \(x\) and time \(t\); the wavefront phase is specified as \(\theta(r,t)\); and \(\
abla\) is the gradient operator. \(\chi\) is a unit vector in the direction of slope measurement sensitivity. The function \(Z(r)\) is the aperture function with  
\[Z(r)=\left\{ \begin{array}{l}1,r\in R\\ 0,else\end{array} \right., \tag{20}\]  
Figure 5: Schematic sketch of the wavefront phase structure function method. where \(R\) is the aperture diameter. Combining the definition of the structure function and the slope measurements given in Equation (18), the slope structure function can be expressed as  
\[ \begin{array}{l}D_{q}^{X}(x,x^{\prime},t,t^{\prime})=\Big{\langle}[\eta_{X}(x, t)-\eta_{X}(x^{\prime},t^{\prime})]^{2}\Big{\rangle}\\ =\Big{\langle}[\int dr[\
abla Z(r-x)\chi]\theta(r,t)-\int dr[\
abla Z(r-x^{ \prime})\chi]\theta(r,t^{\prime})]^{2}\Big{\rangle},\end{array}  \tag{21}\]  
where \(D_{q}^{X}(x,x^{\prime},t,t^{\prime})\) is the slope structure function of the subaperture slope measurement centered at \(x\) and at moment \(t\) and the phase structure function can be expressed as  
\[D_{\varphi}(x,x^{\prime},t,t^{\prime})=\Big{\langle}[\theta(x,t)-\theta(x^{ \prime},t^{\prime})]^{2}\Big{\rangle}. \tag{22}\]  
Based on the basic variations of the slope structure function, the phase structure function, and the atmospheric coherence length variable, the spatial slope structure function can be expressed by a dimensionless integral as [49]  
\[ \begin{array}{lcl}D_{q}^{X}(\Delta x_{s},\Delta y_{s},\Delta x_{t},\Delta y _{t})&=&\Upsilon_{\beta}d^{-2}(\frac{d}{r_{0}})^{\beta-2}\int du\int\limits_{ 0}^{\infty}dv\mathcal{W}(z)tui(u)\\ &&\times\{[2|\Delta x_{s}+\Delta x_{t}(z)-1,u+\Delta x_{s}+\Delta x_{s}(z)|^{ \beta-2} \tag{23}\] \[-|\Delta x_{s}+\Delta x_{t}(z)+1,u+\Delta y_{s}+\Delta y_{t}(z)|^{ \beta-2}\] \[-2(|0,u|_{\beta-2}-|1,u|_{\beta-2})\},\end{array}