## 3 Experimental Results and Analysis
### Detection of Atmospheric Turbulence
#### 3.2.2 Non-Kolmogorov Turbulence Detection  
Some research shows that for the atmospheric measurements of the tropopause, stratosphere, and near the ground, the turbulence structure-function or turbulence spectrum obtained in experiments still has power law characteristics, but the power law value deviates from -11/3 of the Kolmogorov turbulence spectrum in most cases [36; 37]. Therefore, non-Kolmogorov turbulence is defined. The Kolmogorov turbulence three-dimensional power spectral density can be expressed as [2]  
\[\Phi_{n}(\kappa)=0.033C_{n}^{2}\kappa^{-\frac{11}{3}},\frac{1}{L_{0}}\leq \kappa\leq\frac{1}{l_{0}} \tag{7}\]  
where \(L_{0}\) is the outer scale of turbulence; \(l_{0}\) is the inner scale of turbulence, and \(\kappa\) is the space wave number. Meanwhile, the non-Kolmogorov turbulence three-dimensional power spectral density has similar expressions with the Kolmogorov turbulence three-dimensional power spectral density [38]:  
\[\Phi_{n}(\kappa,\alpha)=A(\alpha)\widetilde{C_{n}^{2}}\kappa^{-\alpha},\ \ \frac{1}{L_{0}}\leq \kappa\leq\frac{1}{l_{0}} \tag{8}\]  
where \(\alpha\) is the scale index of the three-dimensional power spectrum, also known as the spectral power law, and \(\widetilde{C_{n}^{2}}\) is the atmospheric refractive index structure constant corresponding to the spectral power rate \(\alpha\) within a certain range, which is called the generalized refractive index structure constant, and the unit is \(m^{3-\alpha}\). \(A(\alpha)\) is called the consistency function, and its function is to make the structure function and power spectrum of the scale index \(\alpha\) in a certain range interchangeable, as follows [36]:  
\[A(\alpha)=\frac{\Gamma(\alpha-1)}{4\pi^{2}}\cos\Bigl{(}\frac{\alpha\pi}{2} \Bigr{)},3<\alpha<5 \tag{9}\]  
For non-Kolmogorov turbulence, under weak fluctuation conditions, the relationship between the variance in normalized light intensity fluctuation (scintillation index) and the variance in logarithmic amplitude fluctuation still satisfies Equation (3). However, the variance in the logarithmic amplitude fluctuation in spherical waves under non-Kolmogorov turbulence is no longer as is shown in Equation (4) but is turned into the following form [39]:  
\[\sigma_{\chi}^{2}(\alpha)=0.033\hat{k}^{\frac{7}{6}}\int_{0}^{L}\widetilde{C_{ n}^{2}}(h)B(\alpha)[\sec(\theta)]^{\frac{6}{2}}\Bigl{(}1-\frac{z}{L}\Bigr{)}^{ \frac{6-2}{2}}z^{\frac{6}{2}}dz,3<\alpha<5 \tag{10}\]  
\[B(\alpha)=\frac{-\pi^{3}}{2\Gamma\bigl{(}\frac{\alpha}{2}\bigr{)}\cos\bigl{(} \frac{\pi\alpha}{4}\bigr{)}} \tag{11}\]  
where \(\theta\) is the zenith angle, and the detection path here is the vertical path, so \(\theta=0\). Therefore, the scintillation index of non-Kolmogorov turbulence is written as:  
\[\beta_{I}^{2}(\alpha)=\exp\Bigl{(}4\sigma_{\chi}^{2}(\alpha)\Bigr{)}-1\approx 4 \sigma_{\chi}^{2}(\alpha) \tag{12}\]The detection method and condition for the non-Kolmogorov turbulence profile are the same as that for Kolmogorov turbulence. The laser light source is 532 nm, the emission frequency is 10 Hz, and the iris diaphragm diameter \(d_{0}\) is 0.5 mm. Here, the non-Kolmogorov turbulence profiles at two different time periods are calculated by Newton iteration inversion, and four different values of spectral power law \(\alpha\) are selected as 3.1, 3.5, 3.9 and 4.2, respectively. The result is shown in Figure 12.  
It is clear that the larger the value of the spectral power law, \(\alpha\), the smaller \(\widetilde{C}_{n}^{2}\) is. In the actual measurement, due to the anisotropy of turbulence, the common influence of various spectral power laws should be considered to obtain the final \(\widetilde{C}_{n}^{2}\) profile.