Optical beam shaping and diffraction free waves: A variational approach

• The problem of shaping radially symmetric beams into desired intensity profiles is studied from a variational perspective.
• Using the uncertainty principle, ansatz free lower bounds on the functional are found.
• The method of stationary phase is used to exactly match the target intensity distribution in the short wavelength limit.
• Potential applications to remote laser ablation are presented.

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 various rigorous lower bounds on the functional; these lower bounds estimate the L2L2 error for the beam shaping problem in terms of the design parameters. 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 the remote delivery of beams and pulses.