The linewidth broadening of the circular-polarized continuous wave laser mitigates the recoil effects of the sodium laser guide star very well. By choice of the optimal laser linewidth, the relations between the laser intensity and average spontaneous emission rates are obtained. The numerically simulated results show that the 1–100 MHz linewidth broadening effectively weakens recoil and enhances the average spontaneous emission rates. For laser powers from 10 W to 60 W, considering the intensity distribution with random at the mesospheric sodium layer, when the laser linewidth is broadened to be 1–100 MHz from 0 MHz, increments of the return photons go up to 110% from 50% and do not have an effect on the spot sizes of the sodium laser guide star. Several cases have proven that the linewidth broadening method is correct. Further calculations show that the linewidth broadening method similarly applies to the case of the multi-mode laser. Furthermore, the linewidth broadening of re-pumping should be taken into account.

In the context of normal product, we use the method of the integration within an ordered product (IWOP) of operators to derive three representations of the two-mode Wigner operator: SU(2) symmetric description, SU(1,1) symmetric description and polar coordinate form. We find that two-mode Wigner operator has multiple potential degrees of freedom. As the physical meaning of the selected integral variable changes, Wigner operator shows different symmetries. In particular, in the case of polar coordinates, we reveal the natural connection between the two-mode Wigner operator and the entangled state representation.