  We investigate the problem of shaping radially symmetric annular beams into
desired intensity patterns along the optical axis. Within the Fresnel
approximation, we show that this problem can be expressed in a variational form
equivalent to the one arising in phase retrieval. Using the uncertainty
principle we prove rigorous lower bounds on the functional that capture how the
various physical parameters in the problem determine the accuracy of the beam
shaping. We also use the method of stationary phase to construct a natural
ansatz for a minimizer in the short wavelength limit. We illustrate the
implications of our results by applying the method of stationary phase coupled
with the Gerchberg-Saxton algorithm to beam shaping problems arising in remote
delivery of beams and pulses.
