We report on 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. Non monotonic dependence of the laser power (optimized over the temperature) on the pump beam radius was observed with a maximum achieved at the ratio ∼ 0.7 between the pump and laser beam radii. The optimal temperature decreased with increasing pump beam radius. Maximum laser power > 370 mW with an optical-to-optical efficiency of 43% and slope efficiency ∼ 55% was obtained. 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 pump power at different pump beam radii and also of the laser power, threshold pump 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. The present results combined with results of the application of the model to K DPAL and Ti:Sapphire pumped Cs laser, indicate that the model can describe the operation of different optically pumped alkali lasers with arbitrary spatial distributions of the pump and laser beam widths.