Which meets s = xy, and hv stands for photon energy in J. Depending on the above evaluation, we conclude that the recoil effects cause the red shifts of sodium atoms. Therefore, a mass of sodium atoms miss excitation in order that the spontaneous emission rate reduces when recoil happens. As a way to mitigate these effects, we propose that the laser linewidth ought to be broadened to weaken these recoil effects.three. Methods and Parameters 3.1. Numerical Simulation Solutions To explore the linewidth broadening mitigating recoil effects of sodium laser guide star, numerical simulations are carried out. A basic assumption is the fact that the two-energy level cycle of sodium atoms is capable to be quite effectively maintained as a consequence of adequate re-pumping. Since the re-pumping energy is about 10 , even much less than ten , in the total laser energy [22], this energy is ignored in the numerical simulations. The typical spontaneous emission prices and return photons with respect to this power are attributed towards the total values with the cycles between ground states F = two, m = two and excited states F’ = 3, m’ = three. In accordance with the theoretical models, Equations (3)ten) are discretized. A numerically simulated method is employed to resolve Equation (eight). Its discrete formation is written as 1 R= nn iNvD (i )np2 (i )v D v D ,(13)where n = T, = two, represents the time of decay and once once again the excitation of a sodium atom, i is defined as the Lufenuron Epigenetics number of velocity groups, NvD (i ) denotes the amount of sodium atoms in the i-th velocity group, and p2 (i ) denotes the excitation probability of sodium atoms in Equation (7). For the objective of acquiring enough return photons, from Equations (7) and (eight), R is expected to be maximum below exactly the same other parameters. We set 200001 velocity groups with all 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 resolve Equation (10), multi-phase screen approach [23] is employed. Moreover, the atmospheric turbulence model of Greenwood [24] and power spectrum of Kolmogorov [25] are utilised in simulations of laser atmospheric propagation. Laser intensity distributions are discretized as 512 512 grids. Laser intensity is believed as concentrating on a plane by means of the whole sodium layer. Then, the return photons are calculated based on Equation (9). Similarly, Equation (11) is discretized as the following form [21]:Atmosphere 2021, 12,6 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 inside 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 inside the mesospheric sodium layer. To simulate laser intensity, the multi-phase screen process is made use of to solve Equation (10) [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 2 Cn dwhere r0 is atmospheric coherent length, k is spatial frequency, r0 = 0.two Cn is refractive index structure continual for atmosphere, and h is definitely the atmospheric vertical height in the ground in m. The atmospheric turbulence model of Greenwood is [25] 2 Cn (h ) = 2.two 10-13 (h + 10)-13 + 4.three 10-17 e-h /4000 .h,(16)On the thin layer perpendicular to the laser transmission direction, the energy spectrum of atmospheric phase is written as [26] n (k ) = two (2/)2 0.033k-11/z+z z two Cn d.(17)Then, Equation (17) is filtered by a complicated Gaussian.