Based on the solution to the rate equations of interactions between laser and sodium atoms, the excitation probability of sodium atoms is achieved. The return photons at 330 nm are numerically calculated for the 1 W laser power at the sodium layer in the mesosphere. Results show that a long pulse laser with a few microsecond durations and 1000 Hz repetition rates can well excite the polychromatic laser guide star to obtain no less return photons at 330 nm than that by the continuous-wave laser in a given condition. In order to obtain enough return photons to satisfy the requirement of tip-tilt detection, the parameters of linewidth, durations, repetition rates, polarization and launch diameters of the laser should be properly chosen. And effects of the laser launch diameters and the atmospheric turbulence on the return photons are slight. For the wonderful vertical atmospheric transmittance T0=0.5 at the Starfire Optical Range, Mauna Kea and Paranal, considering the effects of geomagnetic field, when the power of launch laser varies from 10 W to 25 W with the zenith angles from 0° to 40°, the return photons at 330 nm reach 10<sup>4</sup> ph/m<sup>2</sup>/s.
The circular-polarized laser can excite Sodium Laser Guide Star (SLGS) to obtain a large number of the return photons. Re-pumping technology can further enhance the return photon flux of SLGS. But laser propagating in the atmosphere suffers the atmospheric turbulence which causes the stochastic distributions of laser intensity in mesosphere. This leads to the fluctuations of return photon flux as the strength of atmospheric turbulence. We study effects of atmospheric turbulence on the return photon flux of SLGS under the Hufnagle-vally5/7(HV5/7), Greenwood and Mod-HV models by numerical simulation. Results show that both the return photon flux and fluctuations under the strong atmospheric turbulence are more than those under the weak one. Comparing re-pumping with no re-pumping, increment of the return photon flux under the three atmospheric turbulence models increase with the decreasing strength of atmospheric turbulence. But the fluctuations of the return photon flux greatly decrease for re-pumping.
Based on the relative intensity distributions of Sodium Laser Beacon (SLB) and analysis of the on-axis imaging of incoherent light, considering the effects of atmospheric turbulence and the changes of telescope receiving diameter on the short-exposure SLB imagings on the focal plane, imagings of an extended source SLB are simulated under the three atmospheric turbulence models. Results indicate that sharpness and peak strehl ratio of SLB imagings increase but sharpness radius decrease with the decrease of atmosphere turbulence strengths. Moreover, the changes of telescope diameter from 3.0m to 1.5m cause the decrease of sharpness and peak strehl ratio but the increase of sharpness radius.