  We analyze the transfer function of a three-dimensional atomic Bragg
beamsplitter formed by two counterpropagating pulsed Gaussian laser beams. Even
for ultracold atomic ensembles, the transfer efficiency depends significantly
on the residual velocity of the particles as well as on losses into higher
diffraction orders. Additional aberrations are caused by the spatial intensity
variation and wavefront curvature of the Gaussian beam envelope, studied with
(3+1)D numerical simulations. The temporal pulse shape also affects the
transfer efficiency significantly. Thus, we consider the practically important
rectangular-, Gaussian-, Blackman- and hyperbolic secant pulses. For the
latter, we can describe the time-dependent response analytically with the
Demkov-Kunike method. The experimentally observed stretching of the $\pi$-pulse
time is explained from a renormalization of the simple Pendell\"osung
frequency. Finally, we compare the analytical predictions for the
velocity-dependent transfer function with effective (1+1)D numerical
simulations for pulsed Gaussian beams, as well as experimental data and find
very good agreement, considering a mixture of Bose-Einstein condensate and
thermal cloud.
