# Methods of Atmospheric Coherence Length Measurement
## 3 Coherence Length \(r_{0}\) Direct Measurement Method
### Dimm  
The DIMM obtains the atmospheric coherence length \(r_{0}\) by measuring the variance of the differential arrival angle [15; 41; 42]. Since the variance originates from the arrival angles over the two subapertures of the SHWS, DIMM eliminates other optical distortions in the FSO link, such as jitter resulting from the optical terminal platform and inherent distortions of the optics.  
A schematic sketch of the arrival angle deviation is shown in Figure 3. The spot centroid of \((x_{r},y_{r})\) from the aberrant wavefront imaging at the receiving plane deviates from its centroid of \((x_{o},y_{o})\) when there is no aberration, and the relationship between the arrival angle \(\alpha\) and the spot centroid is  
\[x_{r}=\alpha f_{L},y_{r}=x_{r}\cot\alpha, \tag{12}\]  
where \(f_{L}\) denotes the focal length of the imaging lens. The angle-of-arrival undulation variance is  
\[\alpha^{2}=0.340(\frac{\lambda}{D})^{2}(\frac{d}{r_{0}})^{5/3}, \tag{13}\]  
where \(D\) is the diameter of the subaperture of the telescope and \(d\) is the distance between the centers of the two subapertures. The atmospheric coherence length measurement found by using the DIMM is shown in Figure 4.  
In Figure 4, two relatively independent telescope lenses are shown on the left. The upper beam is reflected by the plane mirror to the attenuator and then passes through the filter. The lower light travels through the diaphragm and light wedge, and then reaches the imaging mirror of the CCD together with the upper light. By doing so, the arrival angle deviation can be measured. After several observations, one can obtain the arrival angle In 1990, [PERSON] et al. advanced the method associated with the DIMM measurement of \(r_{0}\), which yields the equations for the lateral, longitudinal and total atmospheric coherence length expressed as [43]  
\[r_{01}=(\frac{2f_{L}^{2}\lambda^{5/3}[0.179(\frac{\lambda}{D})^{1/3}-0.0968( \frac{\lambda}{d})^{1/3}]}{x^{2}})^{3/5}, \tag{14}\]  
\[r_{02}=(\frac{2f_{L}^{2}\lambda^{5/3}[0.179(\frac{\lambda}{D})^{1/3}-0.0968( \frac{\lambda}{d})^{1/3}]}{y^{2}})^{3/5}, \tag{15}\]  
\[r_{0}=(\frac{2f_{L}^{2}\lambda^{5/3}[0.179(\frac{\lambda}{D})^{1/3}-0.0968( \frac{\lambda}{d})^{1/3}]}{x^{2}+y^{2}})^{3/5}. \tag{16}\]  
One is able to measure the variance of the angle of arrival, and then substitute it into the three equations above to obtain the horizontal, vertical and total atmospheric coherence lengths in turn.  
It is worth noting that the above two-point-based differential imaging method exists backward. Experimental studies have shown that the measured atmospheric coherence length \(r_{0}\) at a given time often jumps suddenly by 2-3 cm within a few seconds, and occasionally by nearly 10 cm. The abnormal behavior contradicts the nature of the dynamic disturbance of atmospheric turbulence. It is attributed to the fact that the sample statistics are not sufficient, namely, the wavefront undulation induced by atmospheric turbulence is not identical in each direction, and the differential imaging method only counts the motions of the centroid of two subapertures at a fixed position. Consequently, it requires a long  
Figure 4: Sketch of the DIMM.  
Figure 3: Schematic sketch of the wavefront deviation.  
time to obtain a comprehensive count. However, increasing the number of samples causes a delay in the measurement, while the turbulence has varied. To circumvent this drawback, the authors of [44] designed a multipoint differential imaging method, which increases the number of atmospheric coherence length \(r_{0}\) samples from the spatial perspective. As such, it enhances the central symmetric statistics based on the concept of spatial groups, and significantly reduces the statistical time required to obtain the atmospheric coherence length \(r_{0}\), yielding stable and accurate values.