  The focusing of a propagating untrapped Bose-Einstein condensate is studied
theoretically. We use a scaling solution method comprising a time-dependent
scaling function to analytically examine the dynamics of a falling
Bose-Einstein condensate in different regimes of propagation including the
expansion and compression zones. Our model is based on the Gross-Pitaevskii
equation which involves the interparticle interactions between atoms, and
consequently their influence on the focused structures. We investigate the
focused profile characteristic factors such as the resolution and peak density
for various cases of the focusing optical potential parameters as well as the
factors associated with the moving cloud. Our results are compared with
numerical solutions of the Gross-Pitaevskii equation.
