We present the results of an experimental study of Ti:Sapphire pumped Cs laser and theoretical modeling of these results, where we focused on the influence of the pump-to-laser beam overlap, a crucial parameter for optimizing the output laser power. The dependence of the output laser power on the incident pump power was found for varying pump beam cross-section widths and for a constant laser beam. Maximum laser power > 370 mW with an optical-to-optical efficiency of 43% and slope efficiency ∼55% was obtained. Non monotonic dependence of the laser power and threshold power on the pump beam radius (at a given pump power) was observed with a maximum laser power and minimum threshold power achieved at the ratio ∼0.7 between the optimal pump beam and laser beam radius. A simple optical model of the laser, where Gaussian spatial shapes of the pump and laser intensities in any cross section of the beams were assumed, was compared to the experiments. Good agreement was obtained between the measured and calculated dependence of the laser power on the incident pump power at different pump beam radii and of the laser power, threshold power and optimal temperature on the pump beam radius. The model does not use empirical parameters such as mode overlap efficiency but rather the pump and laser beam spatial shapes as input parameters. This model can be applied to different optically pumped alkali lasers with arbitrary spatial distributions of the pump and laser beam widths.