Which meets s = xy, and hv stands for photon energy in J. According to the above N-Hexanoyl-L-homoserine lactone site evaluation, we conclude that the recoil effects bring about the red shifts of sodium atoms. Thus, a mass of sodium atoms miss excitation so that the spontaneous emission rate reduces when recoil happens. So as to mitigate these effects, we propose that the laser linewidth ought to be broadened to weaken these recoil effects.3. Solutions and Parameters 3.1. Numerical Simulation Strategies To discover the linewidth broadening mitigating recoil effects of sodium laser guide star, numerical simulations are carried out. A fundamental assumption is the fact that the two-energy level cycle of sodium atoms is capable to become pretty properly maintained resulting from adequate re-pumping. Because the re-pumping power is about ten , even significantly less than 10 , inside the total laser energy [22], this energy is ignored in the numerical simulations. The typical spontaneous emission rates and return photons with respect to this power are attributed to the total values with the cycles between ground states F = 2, m = two and excited states F’ = 3, m’ = 3. As outlined by the theoretical models, Equations (three)ten) are discretized. A numerically simulated system is employed to solve Equation (8). Its discrete formation is written as 1 R= nn iNvD (i )np2 (i )v D v D ,(13)exactly where n = T, = 2, represents the time of decay and when again the excitation of a sodium atom, i is defined because the quantity of velocity groups, NvD (i ) denotes the number of sodium atoms inside the i-th velocity group, and p2 (i ) denotes the excitation probability of sodium atoms in Equation (7). For the goal of getting enough return photons, from Equations (7) and (8), R is necessary to be maximum below the identical other parameters. We set 200001 velocity groups using the adjacent interval v D = 1.0 104 Hz. The array of Doppler shifts is taken from -1.0 GHz to 1.0 GHz. To solve Equation (10), multi-phase screen approach [23] is employed. Furthermore, the atmospheric turbulence model of Greenwood [24] and energy spectrum of Disodium 5′-inosinate Technical Information Kolmogorov [25] are used in simulations of laser atmospheric propagation. Laser intensity distributions are discretized as 512 512 grids. Laser intensity is believed as concentrating on a plane via the entire sodium layer. Then, the return photons are calculated based on Equation (9). Similarly, Equation (11) is discretized as the following kind [21]:Atmosphere 2021, 12,six ofRe f f =1/m,n2 rm,n Ib (m, n)s/m,nIb (m, n)s(14)exactly where Ib (m, n) is intensity of sodium laser guide star in the m-th row and n-th column, and m and n are, respectively, the row and column ordinals of 512 512 grids. Due to the effects of atmospheric turbulence, the distribution of laser intensity is randomized in the mesospheric sodium layer. To simulate laser intensity, the multi-phase screen technique is utilized to resolve Equation (ten) [23]. The power spectrum of Kolmogorov turbulence is taken into account, and its expression is [24]- (k) = 0.033r0 5/3 k-11/(15)3/5 two Cn dwhere r0 is atmospheric coherent length, k is spatial frequency, r0 = 0.2 Cn is refractive index structure continual for atmosphere, and h is the atmospheric vertical height from the ground in m. The atmospheric turbulence model of Greenwood is [25] 2 Cn (h ) = 2.two 10-13 (h + 10)-13 + four.3 10-17 e-h /4000 .h,(16)On the thin layer perpendicular to the laser transmission path, the power spectrum of atmospheric phase is written as [26] n (k ) = 2 (2/)2 0.033k-11/z+z z two Cn d.(17)Then, Equation (17) is filtered by a complex Gaussian.