We report on recent progress on our three-dimensional computational fluid dynamics (3D CFD) modeling of supersonic
diode pumped alkali lasers (DPALs), taking into account fluid dynamics and kinetic processes in the lasing medium. For
a supersonic Cs DPAL with laser section geometry and resonator parameters similar to those of the 1-kW flowing-gas
subsonic Cs DPAL [A.V. Bogachev et al., Quantum Electron. 42, 95 (2012)] the maximum achievable output power, ~ 7
kW, is 25% higher than that achievable in the subsonic case. Comparison between semi-analytical and 3D CFD models
for Cs shows that the latter predicts much higher maximum achievable output power than the former. Optimization of the
laser parameters using 3D CFD modeling shows that very high power and optical-to-optical efficiency, 35 kW and 82%,
respectively, can be achieved in a Cs supersonic device pumped by a collimated cylindrical (0.5 cm diameter) beam.
Application of end- or transverse-pumping by collimated rectangular (large cross section ~ 2 - 4 cm2) beam makes it
possible to obtain even higher output power, > 250 kW, for ~ 350 kW pumping power. The main processes limiting the
power of Cs supersonic DPAL are saturation of the D2 transition and large ~ 40% losses of alkali atoms due to
ionization, whereas the influence of gas heating is negligibly small. For supersonic K DPAL both gas heating and
ionization effects are shown to be unimportant and the maximum achievable power, ~ 40 kW and 350 kW, for pumping
by ~ 100 kW cylindrical and ~ 700 kW rectangular beam, respectively, are higher than those achievable in the Cs
supersonic laser. The power achieved in the supersonic K DPAL is two times higher than for the subsonic version with
the same resonator and K density at the gas inlet, the maximum optical-to-optical efficiency being 82%.