  Context: Sodium laser guide stars (LGS) are about to enter a new range of laser powers. Previous theoretical and numerical methods are inadequate for accurate computations of the return flux and hence for the design of the next-generation LGS systems.   Aims: We numerically optimize the cw (continuous wave) laser format, in particular the light polarization and spectrum.   Methods: Using Bloch equations, we simulate the mesospheric sodium atoms, including Doppler broadening, saturation, collisional relaxation, Larmor precession, and recoil, taking into account all 24 sodium hyperfine states and on the order of 100 velocity groups.   Results: LGS return flux is limited by "three evils": Larmor precession due to the geomagnetic field, atomic recoil due to radiation pressure, and transition saturation. We study their impacts and show that the return flux can be boosted by repumping (simultaneous excitation of the sodium D2a and D2b lines with 10-20% of the laser power in the latter).   Conclusions: We strongly recommend the use of circularly polarized lasers and repumping. As a rule of thumb, the bandwidth of laser radiation in MHz (at each line) should approximately equal the launched laser power in Watts divided by six, assuming a diffraction-limited spot size. 
Optimization of cw sodium laser guide star efficiency