## 3 Results and Discussion  
### Calculation Parameter Setting  
In order to verify the applicability of the fast calculation model to a wide range of practical applications, we apply \(PSF_{\text{FC}}\) for a fast calculation model to the multiple high-power laser propagation scenarios, which are designed to encounter various effects,including the turbulence effect, the thermal blooming effect, laser beam aberration, and platform jitter. The emitting laser is a flatform beam that is horizontally focused on the target. We modeled HEL propagation of the laser beam through turbulence and thermal blooming as a sequence of 2D thin phase screens, and the wave propagation from one thin phase screen to another using scalar diffraction theory [27; 28]. The fields propagated between the phase-screens were calculated by a sequence of the numerically efficient fast-Fourier-transform (FFT) operations [29; 30]. The random turbulence phase screens were generated with von Karman spectrum distributions by white noise filtering in the Fourier domain. The refractive index structure parameter of atmospheric turbulence near the surface is affected by underlying surface and weather conditions, and its variation range is usually in the range of \(10^{-15}\) m\({}^{-2/3}\) and \(10^{-13}\) m\({}^{-2/3}\)[31; 32]. We selected three typical turbulence refractive index structure parameters to simulate different atmospheric turbulence conditions. The optical turbulence strength of our simulated is given in detail in Table 1. Forty independent statistical turbulence-induced phase screens were laid over the propagation path. Non-adaptive coordinate transformation [29; 33] was adopted to enhance both the speed and the accuracy of the computation. The density changes of thermal blooming was calculated with a time-dependent hydrodynamic model by finite Fourier series method [34]. The system and propagation parameters of interest in the 4D wave optics simulation are described in Table 1, and the scenarios are defined in Table 2, as follows. The strength of turbulence effect is usually measured by atmospheric coherence length (Fried parameter) \(r_{0}\). For a spherical wave and an uplink path, when the zenith angle is 0, \(r_{0}\) is calculated using the formula [26]:  
\[r_{0}=\left[0.423k^{2}\int_{0}^{L}C_{n}^{2}(z)(1-z/L)^{5/3}dz\right]^{-3/5} \tag{13}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value(s)** \\ \hline Grid number, N & 1024 \(\times\) 1024 \\ Grid sampling interval of the emission plane, \(\Delta\)x & 0.0012 m \\ Number of phase screen, N\({}_{\text{PS}}\) & 40 \\ Wavelength, \(\lambda\) & 1 \(\mu\)m \\ Transmitting aperture, D & 0.3 m \\ Propagation distance, L & 3 km \\ Initial aberration of the laser emission system, \(\beta_{0}\) & 1.0 and 5.0 times diffraction limit aberration \\ Turbulence structure constant, \(\text{C}_{n}{}^{2}\) & \(5.0\times 10^{-14}\) m\({}^{-2/3}\), \(1.0\times 10^{-14}\) m\({}^{-2/3}\), \\  & \(5.0\times 10^{-15}\) m\({}^{-2/3}\) \\ Absorption coefficient, \(\alpha\) & \(1.1\times 10^{-5}\) m\({}^{-1}\) \\ Extinction coefficient, \(\xi\), & \(6.1\times 10^{-5}\) m\({}^{-1}\) \\ Wind speed, \(v\) & 2 m/s \\ Angular velocity of light beam scanning, \(\omega\) & 0 rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of interest in the 4D wave optics simulation.