## 3 Results and Discussion
### Verification of the Fast Calculation Model with PSF Convolution  
#### 3.3.1 Results for Turbulence-Only Effects  
First of all, we consider the turbulence effect to verify the calculation results of the far-field intensity of light propagation by the fast calculation with _PSF_ convolution. The comparison results between the fast calculation results of \(PSF_{\text{FC}}\) and the results of 4D wave optics simulation program are presented. The far-field intensity distribution results of the four propagation scenarios (S1, S2, S3 and S4) under different turbulence, jitter, and systemaberration conditions are shown in Figure 6. The left column is the calculation result of the 4D wave optics simulation program and the right column is the theoretical calculation result of \(PSF_{\text{FC}}\). In the \(PSF_{\text{FC}}\) calculation, Equation (3) is used to convolute the \(PSF\) with different effects, as seen in Table 3. As can be seen from Figure 6, the general shape of light facula generated by \(PSF_{\text{FC}}\) and those by the 4D wave optics simulation program compare relatively well in all scenarios. There is a small deviation in the position of the center of mass for the turbulence effect.  
Figure 6: _Cont._  
In addition to the general qualitative comparison of the relative spot size and shape, we also quantitatively evaluated the far-field irradiance distribution with the normalized encircle-axis energy (\(Enc_{axis}\)) and normalized encircle-peak energy (\(Enc_{peak}\)) as the evaluation metric. The change curve comparison between the fast calculation results and the 4D wave optics simulation results of \(Enc_{axis}\) and \(Enc_{peak}\) with facula expansion multiple \(\beta\) under S1, S2, S3, and S4 scenarios are shown in Figure 7. It can be seen from the figures that, under the turbulence effect, the convolution calculation method proposed in this paper is in good agreement with the results of 4D optic wave simulation for HEL laser propagation. Generally speaking, the error of \(Enc_{axis}\) is a little larger than that of \(Enc_{peak}\).  
Figure 8 shows the relative error between the fast calculation results and 4D wave optics simulation results for \(Enc_{axis}\) and \(Enc_{peak}\) under S1, S2, S3 and S4 scenarios. For all scenarios, the fast calculation results are smaller, which is a slightly more conservative estimate of performance than wave optics simulation. According to the calculation parameters of the scenarios, the size of the Airy spot is 1.22 cm. For \(\beta=5,10,15\) (spot radius represent 6.1 cm, 12.2 cm, and 18.3 cm), the errors are within \(-15\%\), \(-10\%\), and \(-5\%\), respectively.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **Parameters** & **Fast Calculation** \\ \hline S1 & \(\beta_{0}=1,\text{D}/r_{0}=4.0\) & \(PSF_{\text{FC}}=PSF_{\beta=1}\otimes PSF_{D/r_{0}=4.0}\) \\ \hline S2 & \(\beta_{0}=1,\text{D}/r_{0}=16.1\) & \(PSF_{\text{FC}}=PSF_{\beta_{0}=1}\otimes PSF_{D/r_{0}=16.1}\) \\ \hline S3 & \(\beta_{0}\text{= }5, jitter=5\text{\emph{i}}rad,\text{D}/r_{0}=4.0\) & \(PSF_{F_{\beta_{0}=5}}\otimes PSF_{ jitter=5\text{\emph{i}}rad}\otimes PSF_{D/r_{0}=4.0}\) \\ \hline S4 & \(\beta_{0}\text{= }5, jitter=10\text{\emph{i}}rad,\text{D}/r_{0}=16.1\) & \(PSF_{F_{\text{FC}}}=PSF_{ jitter=10\text{\emph{i}}rad}\otimes PSF_{D/r_{0}=16.1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fast calculation with PSF convolution process.  
Figure 6: 4D wave optics simulation results (**a–d**) and the far-field intensity distribution for fast calculation results (**e–h**) with S1, S2, S3, and S4 scenario pairs presented top to bottom, respectively. The center to outermost calibration rings around the central propagation axis in this and subsequent figures have arbitrarily sized diameters to represent the diffraction-limit, 5.0 times diffraction limit, 10.0 times diffraction limit, and 15.0 times diffraction limit, respectively.