## 2 Principle of Detection  
### Residual Turbulent Scintillation Theory  
The detection principle of Mie-scattering lidar is based on the residual turbulent scintillation theory. When light waves are transmitted in the atmosphere, they will interact with atmospheric molecules and aerosols, and the subsequent scattered signals will be received by Mie-scattering lidar. Due to the existence of turbulence, the received return signal fluctuates. According to the residual turbulent scintillation theory, the signal fluctuation caused by aerosols can be smoothed out by adding a small iris diaphragm to the telescope, retaining only the signal fluctuation information caused by turbulence to measure the atmospheric turbulence intensity. The residual turbulent scintillation theory and its limitations are introduced below. [PERSON] et al. proposed the RTS theory for measuring atmospheric turbulence, expressed as when the light propagates in the atmosphere, if the radius, \(a_{r}\), of the scatterer is smaller than the spatial correlation scale, \(l_{I}\), of the light intensity fluctuation (where \(l_{I}=\sqrt{\lambda L}\), \(\lambda\) is the wavelength of the incident laser and \(L\) is the length of the detection path) for any received aperture, \(D\), the backscattered light intensity signal can weaken the average aperture effect, and the light intensity fluctuation will exist [27]. Under the following four conditions:  
(1) Single scattering approximation and uniform aerosol particle size distribution, average path length \(L\), aerosol particle radius \(\overline{a}\), with the incident laser wavelength \(\lambda\) meeting \(\gg L\gg\overline{a}^{2}/\lambda\);  
(2) Atmospheric optical thickness: \(\tau\ll 1\);  
(3) The relevant scale of radial light intensity \(I_{I\parallel}(l_{I\parallel}\approx L)\) is far more than the scale of the scatterer \(c\tau_{i}/2\), and the scatterer scale is far more than the laser wavelength \(L\gg c\tau_{i}/2\gg\lambda\) (\(c\) is the speed of light and \(\tau_{i}\) is the pulse width);  
(4) Under the condition of weak fluctuations (\(\sigma_{i}^{2}\ll 0.6\)), the telescope receiving aperture \(D\) and the turbulent coherence length of a plane wave \(\rho_{0}\), \(l_{I}\), and \(\rho_{cv}\) meet \(D\gg\rho_{0}\gg l_{l}\gg\rho_{cv}\), where \(\sigma_{i}^{2}\) is the variance of logarithmic light intensity scintillation, \(\rho_{cv}=2L/ka_{ef}\) is given by the van [PERSON] theorem, \(a_{ef}\) is the effective beam size at the distance \(L\) from the light source, and \(k=2\pi/\lambda\) is the wave number. The intensity scintillation of backscattered light has two spot scales: the aerosol spot scale (\(l_{a}=\lambda F/D\)) affected by aerosols; and the turbulence spot scale (\(l_{t}=Fl_{l}/L\)), where \(F\) is the focal length of the telescope. As \(l_{t}/l_{a}=D/\sqrt{\lambda L}\), and, in general, \(D\gg\sqrt{\lambda L}\) in the actual large diameter receiving system, then \(l_{t}\gg l_{a}\) is generally met. Since the scintillation spot has two different spatial scales, the fluctuation in backscattering light intensity can be obtained by changing the size of the aperture \(d_{0}\) of the telescope's field of view:  
\[\left\{ \begin{array}{c}\sigma_{P}^{2}=1+2\sigma_{I}^{2},\;d_{0}\ll l_{a}\\ \sigma_{P}^{2}=\sigma_{I}^{2},\;l_{a}\ll d_{0}\ll l_{t}\\ \sigma_{P}^{2}=O\big{(}\sigma_{I}^{2}/\left(d_{0}/F\sqrt{\lambda/L}\right) \big{)},\;l_{t}\ll d_{0}\end{array} \right. \tag{1}\]  
where \(\sigma_{P}^{2}\) is the variance in the normalized intensity fluctuation in backscattering light passing through an aperture, and \(\sigma_{I}^{2}\) is the variance in the normalized intensity fluctuation in the beam.