Optical beam shaping and diffraction free waves: a variational approach

TL;DR

This paper presents a variational approach to optical beam shaping, establishing theoretical bounds and practical methods for generating desired intensity patterns in radially symmetric beams using phase retrieval techniques.

Contribution

It introduces a variational formulation for beam shaping, derives lower bounds via the uncertainty principle, and combines stationary phase with algorithms for practical beam design.

Findings

Derived rigorous lower bounds on beam shaping accuracy.

Applied stationary phase and Gerchberg-Saxton algorithm to practical problems.

Demonstrated implications for remote beam delivery and pulse shaping.

Abstract

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.